ASPECTS OF NONLINEAR

QUANTUM MECHANICS

Marek CzachorCentrum Fizyki Teoretycznej

Polskiej Akademii Nauk

Warszawa

27 October 1993

Contents

1 INTRODUCTION 5

2 QUANTUM MECHANICS

AS A CLASSICAL

HAMILTONIAN SYSTEM 9

3 SOME

GENERALIZATIONS 14

3.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Results of Single Measurements

and Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 COMPOSITE SYSTEMS

AND NONLOCALITY 33

5 A TWO-LEVEL ATOM

IN NONLINEAR QM 42

5.1 A Fully Quantum Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 A Semiclassical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 REMARKS ON THE

MIND-BODY QUESTION

IN NONLINEAR QM 50

7 ENTROPIC FRAMEWORK

FOR NONLINEAR QM 55

7.1 Elements of Information Theory— from Hartley to Renyi . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.2 Poissonian Formulation ofQuantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3 Nambu-like Generalization ofQuantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3.1 The Jacobi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 667.3.2 Composite Systems in the New Framework . . . . . . . . . . . . . 68

1

7.3.3 Density Matrix Interpretation of Solutionsof the Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . 69

7.3.4 Composition Problem for Subsystemswith Different Entropies . . . . . . . . . . . . . . . . . . . . . . . . 73

8 APPENDICES 76

8.1 Elements of Relativistic Formulation— Hamiltonian Formulationof the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.2 Schrodinger Form of the HarmonicOscillator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.3 On Certain Associative Productof Nonlinear Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.4 Polarizing Mach-Zehnder Interferometerand Other Analyzers of Polarizarion . . . . . . . . . . . . . . . . . . . . . 838.4.1 Analyzer of circular polarizations . . . . . . . . . . . . . . . . . . . 848.4.2 Analyzers of linear polarizations . . . . . . . . . . . . . . . . . . . 858.4.3 Polarizing Mach-Zehnder Interferometer . . . . . . . . . . . . . . . 86

2

...Of even greater interest it seems to me is the question of whether quantum mechan-ics is necessarily true. Quantum mechanics has had phenomenal successes in explainingthe properties of particles and atoms and molecules, so we know that it is a very goodapproximation to the truth. The question then is whether there is some other logicallypossible theory whose predictions are very close but not quite the same as those of quan-tum mechanics. It is easy to think of ways of changing most physical theories in smallways. For instance, the Newtonian law of gravitation, that the gravitational force betweentwo particles decreases as the inverse square of the distance, could be changed a little bysupposing that the force decreases with some other power of the distance, close to butnot precisely the same as the inverse square. To test Newton’s theory experimentally wemight compare observations of the solar system with what would be expected for a forcethat falls of as some unknown power of the distance and in that way put a limit on howfar from an inverse square this power of distance can be. Even general relativity couldbe changed a little, for instance by including more complicated small terms in the fieldequations or by introducing weakly interacting new fields into the theory. It is strikingthat it has so far not been possible to find a logically consistent theory that is close toquantum mechanics, other than quantum mechanics itself.

I tried to construct such a theory a few years ago. My purpose was not seriouslyto propose an alternative to quantum mechanics but only to have some theory whosepredictions would be close to but not quite the same as those of quantum mechanics,to serve as a foil that might be tested experimentally. I was trying in this way to giveexperimental physicists an idea of the sort of experiment that might provide interestingquantitative tests of the validity of quantum mechanics. One wants to test quantummechanics itself, and not any particular quantum-mechanical theory like the standardmodel, so to distinguish experimentally between quantum mechanics and its alternativesone must check some general feature of any possible quantum-mechanical theory. Ininventing an alternative to quantum mechanics I fastened on the one general feature ofquantum mechanics that has always seemed somewhat more arbitrary than others, itslinearity...

After some work I came up with a slightly nonlinear alternative to quantum mechanicsthat seemed to make physical sense and could be easily tested to very high acuracy bychecking a general consequence of linearity, that the frequencies of oscillation of any sortof linear system do not depend on how the oscillations are excited. For instance, Galileonoticed that the frequency with which a pendulum goes back and forth does not depend onhow far the pendulum swings. This is because, as long as the magnitude of the oscillationis small enough, the pendulum is a linear system; the rates of change of its displacementand momentum are proportional to its momentum and its displacement, respectively. Allclocks are based on this feature of oscillations of linear systems, whether pendulums orsprings or quartz crystals. A few years ago after a conversation with David Winelandof the National Bureau of Standards, I realized that the spinning nuclei that were usedby the bureau to set time standards provided a wonderful test of the linearity of quantummechanics; in my slightly nonlinear alternative to quantum mechanics the frequency withwhich the spin axis of the nucleus precesses around a magnetic field would depend veryweakly on the angle between the spin axis and he magnetic field. The fact that no sucheffect had been seen at the Bureau of Standards told me immediately that any nonlineareffects in the nucleus studied (an isotope of beryllium) could contribute no more than onepart in a billion billion billion to the energy of the nucleus. Since this work, Wineland

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and several other experimentalists at Harvard, Princeton, and other laboratories haveimproved these measurements, so that we now know that nonlinear effects would have tobe even smaller than this. The linearity of quantum mechanics, if only approximate, isa rather good approximation after all.

None of this was particularly surprising. Even if there are small nonlinear correctionsto quantum mechanics, there was no reason to believe that these corrections should be justlarge enough to show up in the first round of experiments designed to search for them.What I did find disappointing was that this nonlinear alternative to quantum mechan-ics turned out to have purely theoretical internal difficulties. For one thing, I could notfind any way to extend the nonlinear version of quantum mechanics to theories based onEinstein’s special theory of relativity. Then, after my work was published, both N. Gisinin Geneva and my colleague Joseph Polchinski at the University of Texas independentlypointed out that in the Einstein-Podolsky-Rosen thought experiment (...) the nonlinear-ities of the generalized theory could be used to send signals instantaneously over largedistances, a result forbidden by special relativity. At least for the present I have given upthe problem; I simply do not know how to change quantum mechanics by a small amountwithout wrecking it altogether.

This theoretical failure to find a plausible alternative to quantum mechanics, evenmore than the precise experimental verification of linearity, suggests to me that quantummechanics is the way it is because any small change in quantum mechanics would leadto logical absurdities. If this is true, quantum mechanics may be a permanent part ofphysics. Indeed, quantum mechanics may survive not merely as an approximation to adeeper truth, in the way that Newton’s theory of gravitation survives as an approximationto Einstein’s general theory of relativity, but as a precisely valid feature of the final theory.

Steven Weinberg in Dreams of a Final Theory [1]

4

Chapter 1

INTRODUCTION

A reader of this work may feel a bit confused by its title. “Aspects of what?” — one mayask — “And why only aspects?” And, indeed, the name “nonlinear quantum mechanics”suggests that we are dealing with an existing theory, whereas what we have in mind isa collection of various attempts of constructing a generalization of quantum mechanics(QM).

The adjective “nonlinear” corresponds usually to a nonlinear generalization of theSchrodinger equation. Much less often it is associated with the linearity of the Liouville-von Neumann equation, or the linearity of the space of states, or the linearity of thespace of observables. Each of these elements constitutes a specific level of the linearityof ordinary QM, and therefore there are several places where the putative nonlinearitymay be introduced.

The linearity of the Schrodinger (wave) equation results in the quantum mechanicalsuperposition principle. The principle means mathematically that a linear combination ofany number of solutions of the evolution equation is again a solution of this equation; andvice versa — any solution can be decomposed into a linear combination of other solutions.Physically the first part of the principle means that waves of probability can interfere in alinear way. But anybody who observed waves produced by a ship in a harbour knows thatreal waves on water interfere in a nonlinear way: There exist various feedback influencesof the produced waves on their sources. Only in an idealized case of very weak wavesthe linear approximation describes the behaviour of the ship in a correct manner. Thisproperty is shared by practically all oscillations observed at the macroscopic level. Itfollows that the observed phenomena of quantum interferences cannot be considered asan evidence of the fundamental linearity of quantum evolution.

The other part of the superposition principle seems more essential and is related tothe projection postulate. The postulate states that a measurement of a physical quantityis equivalent to a projection of an initial state vector, describing a pure state of the systemin question before the measurement, in some direction in the Hilbert space of pure states.The projected state describes the state of the system after the measurement. Accordingto the Copenhagen interpretation, the onthological status of the projection is the same asin ordinary probability calculus, where a measurement is represented by a projection on asubset of some probability space (the so-called Bayes rule for conditional probabilities). Itis obvious that at the dynamical level the projection postulate can be consistent provided

5

the projection “maps solutions into solutions”, which is typical only of linear evolutions.The projection postulate is useful for calculation of probabilities but for obvious reasonsit does not follow from the Schrodinger dynamics. It raises also interpretational questionswhich shall be discussed in Chapter 6.

The linearity of the Liouville-von Neumann equation reflects another principle: theconvexity of the “figure of states”. Convexity means that a convex combination of twopure states is also a state — a mixed state. One of the properties of QM state figures isthe fact that they are not simplexes, which means that a mixed state can be decomposedin a number of equivalent, physically indistinguishable ways. The pure states (projectorsor propositions) form a boundary of the figure. The projection postulate can be alsoformulated for mixed states: The measurement projects the convex combination into oneof the projectors belonging to the boundary.

The superposition and convexity principles are logically independent. To understandthat, it is sufficient to note that a superposition of two pure states is a pure state, whilea convex combination of two projectors is not a projector. In classical mechanics thetwo aspects are present as well — this can be illustrated by the usual nonlinearity of theHamilton equations of motion and the linearity of the Liouville equation.

The putative nonlinearity can be introduced here in three ways. Either one general-izes the Schrodinger equation and maintains the linearity of the Liouville-von Neumannequation (like in classical mechanics), or generalizes both of them, or keeps the evolutionof pure states linear but modifies the evolution of mixed states. In the approaches ofKibble [2, 3], Bia lynicki-Birula et al. [4, 5, 6, 7], or Weinberg [8, 9, 10] one concentrateson the evolution of pure states. The question how to describe mixed states is left openin those works (one of possibilities — the probability measure approach — is discussedin this context by Bugajski [11], cf. also [12]). In the approach of Mielnik [12, 14, 15],see also [17], a basic principle is the convexity of the figure of states, but pure states canevolve in a nonlinear way, so there is no ordinary superposition principle. The convexityis, in this formulation, an important constituent of the probability interpretation of thetheory. The third possibility, which introduces nonlinear evolution only for mixed states,is discussed here in Chapter 7.

Now, as we know more or less what can be “deformed” in the theory, it remains tofind out how to do it. There are again numerous possibilities.

In the first place, it can be shown that the dynamics of states in QM is a classicalHamiltonian dynamics for pure states (Chapter 2), and Lie-Poissonian dynamics formixed ones (Chapter 7). Accordingly, it turns out that it is sufficient to extend the setof QM observables to make the dynamics nonlinear. Such a modification was proposedby Kibble, Weinberg, Polchinski [18], Jordan [19] and perhaps others. Also the approachof Bia lynicki-Birula and Mycielski can be regarded as belonging to this class of theories.The proposal of Kibble additionally initiated a different direction of investigations, whereone tampers with the manifolds of states. This direction was further explored by theMilano group [21, 22, 23, 24, 25, 26] who discussed dynamical systems on general Kahlermanifolds. Also all attempts of unifying general relativity with QM can be included inthis class of theories [27]. It can be shown that a nonlinear Schrodinger equation appearsnaturally if a gravitational field of an isolated particle is taken into account [28].

All these attitudes to nonlinear generalization of QM can be collected under the name“nonlinear quantum mechanics”. Still, the reader must be hereby warned that none ofthe above proposals is a complete alternative even to the nonrelativistic linear QM.

6

So what aspects of the above formulations will be discussed in this work?First of all, I will concentrate on Hilbert space models. This restriction, significantly

simplifying the discussion, will be sufficiently general to illustrate various difficulties thatshould be met in any nonlinear QM. In majority of papers on nonlinear QM their authorsconcentrated on dynamics of states but left intact the problem of probability interpreta-tion (the important exeptions are the works of Mielnik and Weinberg). Hence, one of themain themes returning in this work in several places will be the probability interpretationof states and observables (alternative definitions of eigenvalues and probabilities).

The other important point which, in my oppinion, is a source of prejudices aboutnonlinear QM, is the problem of composite systems. The question is related to theexistence of “faster-than-light telegraphs” (in separated systems, Chapter 4), “telekineticphenomena” (the role of observers, Chapters 6 and 7), and even description of Rabioscillations in two-level atoms (Chapter 5).

A separate chapter will be devoted to the two-level atom in Weinberg’s nonlinear QM(Chapter 5). The two-level atom will serve as an example helping to understand manyarbitrary elements “smuggled” in usual discussions of possible experimental implicationsof generalized QM.

In Chapters 6 and 7 we shall attempt to develop the suggestion of Wigner whodiscussed a possible relationship between conscious observation and linearity of quantumevolution. We will formulate a hypothesis concerning a role of information for evolutionof quantum states and define “observers” in information theoretic terms. In this way weshall outline a new program for nonlinear QM. The new framework will be free of themain difficulties of the ordinary approaches but, as expected, will lead to open problemsof its own. In the final part we shall formulate the Dirac equation in a Hamiltonian wayespecially suitable for a relativistic extension of our formalism.

A few words have to be said about the topics that have not been included in this work.In particular, the reader will not find here any historical review of previous attemptedgeneralizations. I am aware of the fact that an inclusion of such a review would makethe work much more comprehensive but, simultaneously, much longer. Also a review ofexperiments made in search of putative nonlinearities would be in place here. I hope,however, that after having read the work the readers will understand the reasons foromitting the question. Even the experiments with two- or four-level systems in iontraps include such a number of arbitrary factors (to mention only the infinite number ofpossible descriptions of the “atom+field” composite system) that their serious discussionwould make the work twice bigger. On the other hand, the interferometric experimentsdesigned as tests of the Bia lynicki-Birula–Mycielski equation applied only to quantumsystems in pure states and hence are unrelated to my own proposal.

Finally, I think I should explain that I do not stick to the framework I propose assomething physically “real”. I simply think that I have managed to show a direction forfurther investigations, the direction that has not been fully explored as yet. This may beimportant, since a large number of authors tried to find for nonlinear QM some generalnon-existence theorems, and many claim they have found them. My own viewpoint isalmost the opposite, as the reader will see later.

Most of the results presented in this work appear for the first time in a written form.This concerns first of all the analysis of probabilities and eigenvalues. The examples wereusually invented as counter examples illustrating points that were omitted or presented in

7

an unclear way in other papers. The quantum description of the two level atom, especiallythe questions related to the description of the joint “atom+field” system and the twolevel approximation cannot be found in literature. I have also included my earlier resultson composite systems, but here presented in a more systematic way. The description interms of density matrix is a generalization of the earlier works of Polchinski, Jordan, andBia lynicki-Birula and Morrison. The latter work was an inspiration for my own proposalof nonlinear QM.

8

Chapter 2

QUANTUM MECHANICS

AS A CLASSICAL

HAMILTONIAN SYSTEM

A departure point for the generalizations of ordinary, linear QM discussed below is theobservation that quantum theory can be regarded as a particular classical Hamiltoniantheory. This statement is in apparent conflict with what one usually reads in quantummechanical textbooks. A student who has passed an exam on quantum mechanics willprobably object and formulate two main quantum features of QM:

1. The algebra of quantum observables is non-Abelian as opposed to the Abelianalgebra of functions on a phase space. In particular, the energy is represented inquantum theories by Hamiltonian operators in contrast to Hamiltonian functionsof classical theories.

2. The evolution equation of a pure state of a conservative system is, in quantumtheories, given by the Schrodinger equation in opposition to the Hamilton equationsof classical theories.

It is one of the purposes of the studies on generalized quantum theories to understandwhat is actally the element of the theory that makes quantum theory quantum. Weshall see that the two differences described above are not, in fact, actual. Therefore thereal difference lies somewhere else. It will be argued that the difference is closely relatedto the associativity of the algebra of quantum observables, that is, to the fact that thequantum observables possess a richer algebraic structure than the classical ones.

Let us begin with some Hilbert space H of square integrable functions M → Cn

defined on a finite dimensional manifold M . The manifold M may be, for example,a spacelike hyperplane in the Minkowski space or a mass hyperboloid. Let α ∈ M andψA(α) be a value of an A-th component of the function ψ ∈ H taken at α. Since Cn has anatural structure of a symplectic manifold with the symplectic form being the imaginarypart of the scalar product, it follows that H has a natural structure of a “phase bundle”[29] whose fibers are the finite dimensional Hilbert spaces Hα = Cn with coordinates

9

ϕA(α). In the approach used in this work the Hilbert space and its correspondingprojective space P (H) will be treated as real manifolds. This can be achieved by arealification of the model spaces. (The realification of a complex linear space is obtainedby restricting the field of scalars to real numbers.) Since the manifolds are modelledon complex Hilbert spaces there exist complex valued scalar products in their tangentspaces. The scalar products lead naturally to symplectic and Riemannian structures onthe (realificated) manifolds. Let 〈·|·〉ψ be a scalar product in TψH ≃ H or T[ψ]P (H), forsome ψ [30]. The symplectic form is

ωψ(ψ, φ) = 2νIm〈ψ|φ〉ψ (2.1)

and the Riemannian metric tensor

gψ(ψ, φ) = 2νRe〈ψ|φ〉ψ . (2.2)

The constant ν depends on conventions (say, whether the scalar product is linear in thefirst or in the second argument) or physical applications (for P (H) ν is related to theholomorphic sectional curvature; in the symplectic formulation of quantum mechanics itcan be shown to be h) — here we will put ν = 1 which is equivalent to the units withh = 1 (cf. [31, 32, 30, 21, 22, 23, 24, 25, 26, 2]). Pure states ψ ∈ H are sections of thebundle, in general defined on its dense subsets (manifold domains [32]). The manifold ofstates is an infinite dimensional manifold of the sections; this manifold (a phase space)will be denoted by P .

A basis in a space tangent to a point in Hα (treated as a real manifold) is

∂

∂ϕA(α)and

∂

∂ϕA′(α); (2.3)

here α is fixed and summation can be only over the discrete indices. The correspondingbasis in TϕP is denoted by

δ

δϕA(α)and

δ

δϕA′(α)(2.4)

where one can sum over discrete indices and integrate over the continuous ones (theintegration involves some invariant, or quasi-invariant measure on M). The basis in thedual to TϕP is denoted by

dlϕA(α) and dlϕA′(α) (2.5)

where “dual” means that

dlϕA(α)( δ

δϕB(β)

)

= δBAδ(α− β) (2.6)

dlϕA′(α)( δ

δϕB′(β)

)

= δB′

A′ δ(α− β). (2.7)

The symplectic form and the metric tensor in TϕP are defined as

ω = iδAA′

dlϕA(α) ∧ dlϕA′(α) = iδAA′(

dlϕA(α) ⊗ dlϕA′(α) − dlϕA′(α) ⊗ dlϕA(α))

(2.8)

and

g = δAA′

dlϕA(α) ∨ dlϕA′(α) = δAA′(

dlϕA(α) ⊗ dlϕA′(α) + dlϕA′(α) ⊗ dlϕA(α))

(2.9)

10

where we have introduced the following summation convention: Summation is over re-peated Roman indices and integration is over the repeated Greek ones. This conventionwill be often applied. For two tangent vectors in TϕP

ψ = ψA(α)δ

δϕA(α)+ ψA′(α′)

δ

δϕA′(α′)(2.10)

and

φ = φA(α)δ

δϕA(α)+ φA′(α′)

δ

δϕA′(α′)(2.11)

one finds indeed

ω(ψ, φ) = 2Im 〈ψ|φ〉 (2.12)

g(ψ, φ) = 2Re 〈ψ|φ〉 (2.13)

where the scalar product is anti-linear in the first argument.The two tensors can be combined into one Hermitian form

K =1

2(g + iω) = δAA

′

dlϕA(α) ⊗ dlϕA′(α). (2.14)

The components of K can be derived from the potential

f [ϕ, ϕ] = δAA′

ϕA(α)ϕA′(α) (2.15)

by

KAA′

(α, α′) =δ2f

δϕA(α)δϕA′ (α′). (2.16)

The symplectic form is closed as independent of ϕ, or as derivable from the potential by

ω = i∂∂f (2.17)

(cf. [31, 33])

An analogous language can be formulated for the projective space P (H). Let Ψ = [ψ]be a class of vectors complex proportional to a nonvanishing ψ ∈ H and choose someψK(κ) 6= 0. We define the following local projective coordinates

ΨKκA (α) = ψA(α)/ψK(κ) (2.18)

ΨKκA′ (α′) = ΨKκ

A (α). (2.19)

LetF [Ψ, Ψ] = ln

(1 + δAA

′

ΨKκA (α)ΨKκ

A′ (α)). (2.20)

The symplectic form and the metric tensor for P (H) can be derived from F by

KAA′

Kκ (α, α′) =δ2F

δΨKκA (α)δΨKκ

A′ (α′)

=δAA

′

δ(α− α′)

1 + δBB′ΨKκB (β)ΨKκ

B′ (β)− δAC

′

δCA′

ΨKκC′ (α)ΨKκ

C (α′)(1 + δBB′ΨKκ

B (β)ΨKκB′ (β)

)2 . (2.21)

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In analogy to the Hilbertian formulation

ω = iKAA′

Kκ (α, α′)dlΨKκA (α) ∧ dlΨKκ

A′ (α′), (2.22)

g = KAA′

Kκ (α, α′)dlΨKκA (α) ∨ dlΨKκ

A′ (α′). (2.23)

The metric (2.23) is an infinite dimensional version of the Fubini-Study metric, and playsan important role for geometric phases, such as the Berry phase, (cf. [34] and referencestherein). It it convenient to define also the “inverse” of K defined by

IKκAA′(α, α′) =(

1 + δBB′

ΨKκB (β)ΨKκ

B′ (β))(

δAA′δ(α− α′) + ΨKκA (α)ΨKκ

A′ (α′))

(2.24)

where by the inverse it is meant that

KAA′

Kκ (α, α′)IKκAB′ (α, β′) = δA′

B′δ(α′ − β′) (2.25)

KAA′

Kκ (α, α′)IKκBA′ (β, α′) = δABδ(α− β). (2.26)

Consider now two functionals: The matrix element

H [ψ, ψ] = ψA′(α′)HA′A(α′, α)ψA(α) (2.27)

of a self-adjoint operator H, and its average

H [ψ, ψ] =ψA′(α′)HA′A(α′, α)ψA(α)

ψB′(β)δB′BψB(β). (2.28)

The latter is also a functional defined on P (H):

H [ψ, ψ] = H [Ψ, Ψ] (2.29)

although in the projective coordinates H (also playing a role of a Hamiltonian function)is not a bilinear functional of the projective coordinates.

A straightforward calculation shows [21, 22, 23, 24, 25, 26] that the Hamilton equa-tions

ιXHω = dlH (2.30)

andιXH

ω = dlH (2.31)

are equivalent to the Schrodinger equation

id

dtψ = Hψ (2.32)

and its adjoint expressed either in the Hilbertian or projective coordinates (by the way,the projective form of the Schrodinger equation is nonlinear). The Poisson bracket ofobservables (averages or matrix elements of some self-adjoint operators) written in termsof I and resulting from the Hamilton equations is in one-to-one relationship with thecommutator of the operators. It follows that the Schrodinger equation is equivalent tothe Hamilton equations for an infinite-dimensional Hamiltonian system, and the non-commutability of observables is of the same nature as the classical non-commutability

12

of functions on a phase space with respect to the Poisson bracket (various explicit ex-pressions of the bracket will be discussed in several places in this work). The onlysubtlety that is involved in such calculations is related to the differentiability of aver-ages of unbounded operators. These functions are everywhere discontinuous and thedifferentiability must be understood in a weak sense [32, 21, 22, 23, 24, 25, 26]: we willusually use functional derivatives and all functionals will be assumed to be functionallydifferentiable. The ordinary canonical coordinates “q” and “p” correspond to real andimaginary parts of the state vector’s components.

The above conclusions may suggest that quantum mechanics is just an ordinary clas-sical theory, perhaps more sophisticated from the dynamical viewpoint. On the otherhand we know that the most profound conceptual difficulties of quantum mechanics (theEPR-Bohm and Schrodinger’s cat paradoxes, the Bell theorem, the measurement prob-lem) can be formulated even in two dimensional Hilbert spaces. Such systems, beingdynamically equivalent to a two dimensional harmonic oscillator, are dynamically triv-ial. It follows that there must exist some additional, “non-Hamiltonian”, properties ofquantum mechanics that lead to its physical nontriviality. We will show later that animportant property of linear QM is the additional, associative structure of the algebraof observables. This property is very restrictive and is closely related to the probabilisticnature of the theory.

13

Chapter 3

SOME

GENERALIZATIONS

The observation that quantum dynamics is in fact a classical one immediately openssome possibilities of generalizations. One can either try to modify the Hamiltonian for-mulation itself or simply try to explore general possibilities immanent in the Hamiltonianframework.

In this chapter we shall follow the latter direction in a way analogous to this proposedby Kibble [2] and developed and extended by Weinberg [10], Polchinski [18] and Jordan[19].

3.1 States

We will discuss theories where states are represented in the same way as in ordinaryquantum mechanics (the so-called normal states): by vectors or rays in a Hilbert spaceor by density matrices. We know that even in the ordinary statistical quantum theorythere exist states that are not normal, but we shall not include such subtleties here. Weshould also bear in mind that possible extensions of QM may involve different manifoldsof states [2], for example general Kahler manifolds [21, 22, 23, 24, 25, 26]. A necessity ofsome extension of that kind is suggested by Penrose [35]. We will not include stochasticgeneralizations either, cf. [37].

3.2 Observables

We have shown that the observables in QM belong to a restricted class of functionalsdefined on a phase space. In the Hilbertian formulation they take the form 〈ψ|A|ψ〉(diagonal matrix elements of a self-adjoint operator) where |ψ〉 is not normalized. Itfollows that such observables are not equal to the measurable quantities, and a relationwith these latter is established in two equivalent ways: Either one divides the valueof the observable evaluated at |ψ〉 by the squared norm of |ψ〉 or takes its value inthe normalized |ψ〉. In the projective space formulation the observables are equal tomeasurable quantities (averages).

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The fact that quantum observables are so specifically chosen leads to a special, com-plex Hilbertian type of the quantum mechanical probablity calculus. It seems it wasMielnik who noticed [12, 14, 15, 16] that there exist many different and inequivalentprobability models and tried to relate them to some nonlinear quantum theories. Such aperspective seems quite natural and suggests that one should try to investigate theorieswhere obsevables, like in classical mechanics, belong to a more general set of functionals.

In the Kibble’s approach the set of observables consists of differentiable functionsdefined on a projective space. In the Weinberg formulation [10] one assumes that theobservables are defined on a Hilbert space but satisfy the (1,1)-homogeneity condition

A(λψ, ψ) = A(ψ, λψ) = λA(ψ, ψ). (3.1)

Averages are defined in [10] like in ordinary quantum mechanics. The (1,1)-homogeneityand the definition of averages introduces the projective structure into the set of statesand, effectively, into the dynamics. This is the way the Weinberg approach can beregarded as a variant of the Kibble’s one. (We shall see later that Weinberg’s theorycontains also some elements that are new with respect to the Kibble’s proposal.)

The homogeneity condition (3.1) can be expressed by means of the Euler conditionsas follows

ψn∂A

∂ψn= ψn

∂A

∂ψn= A. (3.2)

(In this section we will often apply finite dimensional conventions; in the full, infinitedimensional framework functional derivatives must be substituted for the partial onesand the Euler homogeneity condition would read, for example,

ψn(α)δA

δψn(α)= ψn(α)

δA

δψn(α)= A

). (3.3)

With the help of (3.2) we can deduce some general properties of the observables. Applying(3.2) twice we find that

A =∂A

∂ψnψn = ψm

∂2A

∂ψm∂ψnψn = ψmAmnψn = 〈ψ|A(ψ, ψ)|ψ〉 (3.4)

and∂A

∂ψm=

∂2A

∂ψm∂ψnψn = (A(ψ, ψ)|ψ〉)m. (3.5)

We can see that the homogeneity condition leads naturally to a nonlinear operator al-gebra. A matrix multiplication of such nonlinear operators can be easily expressed interms of first partial derivatives of observables since

A ∗B :=∂A

∂ψm

∂B

∂ψm= 〈ψ|A(ψ, ψ)B(ψ, ψ)|ψ〉. (3.6)

This ∗-product conserves the (1,1)-homogeneity of observables because the nonlinearmatrices defined in (3.4) are (0,0)-homogeneous. The (0,0)-homogeneity implies, on theother hand, that the matrices are, in certain sense, “almost constant”. Indeed,

( ∂A

∂ψk|ψ〉)

l=

∂

∂ψk

( ∂2A

∂ψl∂ψm

)

ψm = ψm∂

∂ψm

∂2A

∂ψl∂ψk= 0 (3.7)

15

by virtue of the 0-homogeneity Euler condition. Consider now a differentiable functiont→ ψn(t). Eq. (3.7) and its complex conjugate imply that

〈ψ| ddtA(ψ, ψ)|ψ〉 = 0 (3.8)

for any observable A. The algebra of observables equipped with the above ∗-productpossesses a natural left and right unit element n(ψ, ψ) = 〈ψ|ψ〉.

Despite some similarities to the ordinary linear operator algebra our algebra of observ-ables is associative if and only if the operators are linear. This leads to several differenceswith respect to ordinary quantum mechanics. For example, ∗-powers of A will not, ingeneral, ∗-commute with A, a fact that will influence the integrability of the theory.

In linear QM the ∗-powers of an observable are related to higher moments of randomvariables measured in experiments, hence lead to the probabilistic interpretation of thetheory. In nonlinear theory the product (3.6) leads to the first non-unique element of theformalism we shall meet later.

For let us consider an observable A. We know that it can be written in a form of amatrix element 〈ψ|A|ψ〉 where A is the nonlinear operator defined in (3.4) (for simplicitywe shall, from now on, omit in our notation the dependence of A on ψ). The ∗-square ofA can be defined in two ways: Either as ∂A

∂ψm

∂A∂ψm

or as 〈ψ|AA|ψ〉 and the two expressions

are equal. If we want to define the third power we encounter several possibilities. Theapparently most natural choice of (A ∗ A) ∗ A or A ∗ (A ∗ A) is not very reasonable asthey are in general complex and non equal even if A is real. The reality condition leadsuniquelly to 1

2 (A ∗ (A ∗ A) + (A ∗ A) ∗ A) but this uniqueness will be lost for higherpowers. In Section 3.4 we shall see also that ∗ does not lead to a consistent probabilityinterpretation of A. The second definition can be chosen as 〈ψ|AAA|ψ〉 := A∗A∗A. Wesee that once we have defined A, which is unique, all its ∗-powers are unique as well.Notice that since A ∗ B = A∗B both products lead to the same Poisson bracket hencegenerate the same evolution of observables.

Both products can be defined also in terms of density matrices. Quantum observablesare 1-homogeneous in density matrix i.e. A(λρ) = λA(ρ) or ρmn

∂A∂ρmn

= A. The nonlinearoperators are obtained by

Amn =∂A

∂ρnm(3.9)

(notice the reversed order of the indices). The products are

A ∗B = ρmn∂A

∂ρkn

∂B

∂ρmk= A∗B (3.10)

andA∗B∗C = TrρABC. (3.11)

These properties of ∗ hint at the possibility of defining the probability interpretationof nonlinear QM in terms of ∗ instead of ∗. We shall discuss this possibility in a moredetailed way in the context of eigenvalues and dynamics.

Another way of viewing the difficulties with ∗ can be illustrated as follows. An ob-servable A can be defined as a scalar function on a manifold of states. All one-parameter

16

groups of canonical transformations generated by some observables are homogeneity pre-serving. Writing a canonical transformation in terms of components

ψ′n(ψ, ψ) =

∂ψ′n

∂ψmψm +

∂ψ′n

∂ψmψm

︸ ︷︷ ︸

0

, (3.12)

we find the second term vanishing by virtue of the (1,0)-homogeneity Euler conditionfor ψ′

n(ψ, ψ). The canonical transformations (3.12) look holomorphic, just like a changeof a chart on a complex manifold. The “holomorphy” is however only apparent as thetransformation coefficients depend on both ψ and ψ even though the second term in(3.12) vanishes. Consider now the “gradient ”

∂A

∂ψm. (3.13)

We know that, by definition,

ψn∂A

∂ψn= A (3.14)

which may suggest that (3.13) transforms “contravariantly” with respect to the “covari-ant” transformation rule (3.12). This is not the case, however. We find

∂A

∂ψm=∂ψ′

n

∂ψm

∂A

∂ψ′n

+∂ψ′

n

∂ψm

∂A

∂ψ′n

(3.15)

with the last term, in general, nonvanishing. Still, its contraction with ψm does vanishas a result of the (0,1)-homogeneity of ψ′

m(ψ, ψ). The ∗ product is defined as a “scalarproduct” of “gradients” ∂A

∂ψmand ∂B

∂ψmand as such does not represent a scalar. A ∗

B is invariant under a change of coordinates provided the transformation ψ → ψ′ isholomorphic. Such a class is much narrower than the class of canonical transformations.As a particular consequence, ∗-products of two integrals of motion are not, in general,integrals of motion (an integral of motion is a “scalar” with respect to the time evolutionunderstood as a canonical transformation).

One of the most important questions of nonlinear quantum mechanics concerns adefinition of observables that correspond to subsystems of a larger system. Let us beginthe discussion with recalling the ways we do it in ordinary QM.

From a physical viewpoint any system (with the exeption of the Universe itself) is asubsystem of something greater. If we say thatH(ψsubsystem, ψsubsystem) is a Hamilitonianfunction of a subsystem we mean that H(ψsubsystem, ψsubsystem) describes a subsystemwhich is noninteracting and noncorrelated with the rest of the Universe. Quantum me-chanically this means that the state of the Universe is |ψ〉 = |ψsubsystem〉 ⊗ |ψrest〉 (orρ = |ψsubsystem〉〈ψsubsystem| ⊗ ρrest).

Subsystem observables in linear QM can be written as

A(ψ, ψ) = 〈ψ|A⊗ 1rest|ψ〉 = TrρsubsystemA. (3.16)

Let now |r〉 denote a basis in the Hilbert space of the “rest” and |a〉 be the one ofthe subsystem. A general form of the state of the whole system (here we assume for

17

inessential simplicity that the whole system is in a pure state) is

|ψ〉 =∑

a,r

ψa,r|a〉|r〉 =∑

r

|Φ(r)〉|r〉. (3.17)

An average of A in the state |ψ〉 can be expressed by means of the decomposition (3.17)as follows

A(ψ, ψ) =〈ψ|A⊗ 1rest|ψ〉

〈ψ|ψ〉 =∑

r

〈Φ(r)|A|Φ(r)〉〈Φ(r)|Φ(r)〉

〈Φ(r)|Φ(r)〉〈ψ|ψ〉

=∑

r

〈Φ(r)|A|Φ(r)〉〈Φ(r)|Φ(r)〉

〈ψ|1⊗ |r〉〈r|ψ〉〈ψ|ψ〉 . (3.18)

The physical meaning of (3.18) is obvious: The whole ensemble of subsystems is de-composed into subensembles corresponding to different results of measurements of anobservable whose spectral family is given by |r〉〈r|, and the average of A in the sub-system is calculated for each subensemble separately. The form (3.18) is useful as anillustration of the “collapse” of a state vector phenomenon. Indeed, the average is cal-culated as though the ensemble of subsystems consisted of subensembles collapsed by anexternal observer and, in certain sense, the ensemble is treated as the one composed ofindividual objects each of them posessing some “property” measured (via correlations)by the external observer.

The description in terms of the reduced density matrix ρsubsystem suggests a slightlydifferent interpretation of the average. If we believe that density matrices representensembles and not individual systems the form

A(ψ, ψ) = TrρsubsystemA (3.19)

means that we do not care about specific possible decompositions of the ensemble, buttreat it as a whole. The presence of the reduced density matrix in (3.19) does not mean,of course, that we cannot use the state vector’s components — it simply restricts theform of observables to only some functions of ψa,r.

Quantum mechanics when viewed from the first perspective looks as a theory ofindividual systems but with some form of fundamental limitation of knowledge aboutstates of such individuals. The second possibility turns QM into a kind of a “meanfield theory”. Still, since both forms of description are physically indistinguishable suchdiscussions whithin the linear framework seem purely academic.

In nonlinear theories the situation is drastically changed. The nonlinearity is supposedto be fundamental, hence applying to each individual even if the individual is well isolatedfrom other physical systems and no mean-field approximation is justified.

The problem is practically the following. Assuming that we know a form of an ob-servable describing a given subsystem itself (that is, if no interactions or correlations withthe “rest” exist) what is the form of the observable if there do exist some interactions orcorrelations with the “rest”. A reversed problem is simpler: If we know a description ofthe “whole” system and the observable in question is additive then the observable repre-senting the subsystem is this part of the whole one that refers to the subsystem. (Notethat non-additive observables, like higher order correlation functions, can be defined only

18

in terms of the “whole” system, so, by definition, are excluded from our considerationshere.)

The first, explicitly non-mean-field-theoretic, guess for the subsystem’s (additive)observable is suggested by the “collapse-like” form (3.18) of the average, i.e.

Asubsystem(ψ, ψ) =∑

r

A(Φ(r), Φ(r)) (3.20)

where A is a given function defined on a Hilbert space of the subsystem and the otherdefinitions are like in (3.17). It is this definition that was chosen by Weinberg in hisformulation of nonlinear QM. For non-bilinear A(Φ(r), Φ(r)) the form and value of (3.20)depend on the choice of the basis |r〉.

Example 3.1 Consider the following non-bilinear observable

A(ψ, ψ) =〈ψ|σ3|ψ〉2〈ψ|ψ〉 =

(|ψ1|2 − |ψ2|2)2

|ψ1|2 + |ψ2|2(3.21)

where σ3 is the Pauli matrix and |ψ〉 belongs to a two-dimensional Hilbert space. Let

the whole system be described by |ψ〉 =∑2

a=1

∑

r ψa,r|a〉1|r〉2. Definition (3.20) yields

Asubsystem(ψ, ψ) =∑

r

(|ψ1r |2 − |ψ2r|2)2

|ψ1r|2 + |ψ2r|2(3.22)

Let us choose now|ψ〉 = ψ1,1|1〉1

︸ ︷︷ ︸

|Φ(1)〉

|1〉2 + ψ2,2|2〉1︸ ︷︷ ︸

|Φ(2)〉

|2〉2. (3.23)

In this particular stateAsubsystem(ψ, ψ) = |ψ11|2 + |ψ22|2 (3.24)

so that each “collapsed” subensemble contributes to the observable separately. Let uschoose now a different basis |r′〉2 in the Hilbert space of the “rest”, such that |1〉2 =1√2(|1′〉2 + |2′〉2), |2〉2 = 1√

2(|1′〉2 − |2′〉2). Now

|ψ〉 =1√2

(ψ11|1〉1 + ψ22|2〉1

)

︸ ︷︷ ︸

|Φ(1)′ 〉

|1′〉2 +1√2

(ψ11|1〉1 − ψ22|2〉1

)

︸ ︷︷ ︸

|Φ(2)′ 〉

|2′〉2. (3.25)

Substitution of |Φ(1)′〉 and |Φ(2)′〉 into (3.20) gives

Asubsystem(ψ, ψ) =1

2

(|ψ11|2 − |ψ22|2)2

|ψ11|2 + |ψ22|2+

1

2

(|ψ11|2 − |ψ22|2)2

|ψ11|2 + |ψ22|2

=(|ψ11|2 − |ψ22|2)2

|ψ11|2 + |ψ22|2(3.26)

which is also a sum of contributions from two subensembles, but now “collapsed to linearpolarizations”.2

19

The above example reveals two properties of (3.20):

1. This form is basis-dependedent so one should rather write Asubsystem(ψ, ψ) =Asubsystem(ψ, ψ, |r〉);

2. If the “rest” is described by linear QM then no change of |r〉 can influece ob-servables that correspond to “the rest” itself. In particular, if one considers a totalenergy of the whole system then a change of |r〉 will influence its value (we changethe energy of the subsystem and maintain the energy of the “rest”). It follows thatthe form (3.20) can apply only to systems that, as a whole, are open (that is, theremust exist also some another nonlinear system not contained in the “whole” oneand compensating the changes of energy due to the changes of |r〉).

3. The “subsystem+compensating subsystem+rest” system must be described in abasis-independent way.

The last remark indicates that in order to describe systems with “isolated” nonlin-earities, that is with the “rest” linear, one has to describe subsystems in a way differentfrom Weinberg’s. The following modification was proposed by Polchinski [18].

Consider an observable A(ψ, ψ) corresponding to some isolated physical system, andassume that the observable can be expressed as a function of the density matrix ρ =|ψ〉〈ψ|, i.e. A(ψ, ψ) = A(ρ). For example, the observable from the previous examplecould be written in either of the forms

(Tr ρσ3)2

Tr ρ, (3.27)

Tr (ρσ3ρσ3)

Tr ρ, (3.28)

(Tr ρσ3)2Tr (ρ2)

(Tr ρ)3, (3.29)

or(Tr (ρσ3ρσ3))n+1

Tr ρ(Tr ρσ3)2n, (3.30)

and so on. For ρ = |ψ〉〈ψ| all of them, and all their convex combinations, reduce to(3.21). Polchinski’s idea was to start with definitions in terms ρ where ρ is the densitymatrix of the subsystem. For |ψ〉 describing the whole system the observables of thesubsystem would be functions of ρsubsystem = Tr rest|ψ〉〈ψ|.

Example 3.2 Let |ψ〉 = ψ11|1〉1|1〉2 + ψ22|2〉1|2〉2. The reduced density matrix repre-senting the subsystem is

ρ = |ψ11|2|1〉1〈1|1 + |ψ22|2|2〉1〈2|1. (3.31)

The subsystem’s observables are now

(Tr ρσ3)2

Tr ρ=

(|ψ11|2 − |ψ22|2)2

|ψ11|2 + |ψ22|2, (3.32)

20

Tr (ρσ3ρσ3)

Tr ρ=

|ψ11|4 + |ψ22|4|ψ11|2 + |ψ22|2

, (3.33)

(Tr ρσ3)2Tr (ρ2)

(Tr ρ)3=

(|ψ11|2 − |ψ22|2)2(|ψ11|4 + |ψ22|4)

(|ψ11|2 + |ψ22|2)3, (3.34)

and(Tr (ρσ3ρσ3))n+1

Tr ρ(Tr ρσ3)2n=

(|ψ11|4 + |ψ22|4)n+1

(|ψ11|2 + |ψ22|2)(|ψ11|2 − |ψ22|2)2n. (3.35)

As we can see the observables are completely different. If |ψ11| = |ψ22| (like in the spin-1/2 singlet state) the average of the first and third observables vanishes, is non-vanishingbut finite in the second case, and infinite in the last one.

Expression (3.32) seems, at first glance, the most natural one since can be rewrittenas

〈ψ|σ3 ⊗ 1rest|ψ〉2〈ψ|ψ〉 . (3.36)

Kibble in [2] did not explicitly define his way of describing subsystems, but the exam-ples he discussed suggest this type of description. In the next chapter we will discussproblems with nonlocality of nonlinear QM and see that all the descriptions by meansof reduced density matrices are more safe than the proposal of Weinberg. However, ifsome correlations between subsystems actually exist we indeed can “collapse” a state ofa remote individual. We may expect that an average value of this “remote” observableshould be composed of the discussed averages of subensembles. The description by meansof (3.32) does not seem to possess this property: It is the “average spin” that contributesto the observable (3.32). So we first calculate the “average” of σ3 and then square it. Inthe Weinberg description we first square the “spin” and then take the “average”. (Thequotation marks are necessary because we have not defined yet what we mean by eigen-values, or values of single measurements; there exists also another subtle point that has

been ignored in the definition of (3.20) — in nonlinear QM 〈ψ|r〉〈r|ψ〉〈ψ|ψ〉 may not have an

interpretation in terms of probability.)It is interesting that the “collapse”-like property of the average can be seen also in

(3.33) where both values of “spin” are averaged separately. Finally, let us notice thatthe last observable (3.35) is unbounded, so that the values of single measurement cannotbe identified with eigenvalues of σ3.2

3.3 Dynamics

Both Kibble and Weinberg assumed that the dynamics of pure states is given by Hamiltonequations with Hamiltonian functions belonging to the algebra of generalized observablesdiscussed in Section 3.2. More generally, all one parameter flows of canonical transfor-mations are assumed to be integral curves of the Hamilton equations of motion

ιXω = dH (3.37)

which in the Hilbertian formulation can be written as

idψA(α)

dt= δAA′

δH

δψA′(α)(3.38)

21

and

−idψA′(α)

dt= δAA′

δH

δψA(α). (3.39)

Since the evolution is, in general, nonlinear no dynamical separation of states andobservables is possible (the ordinary Heisenberg and Schrodinger pictures can be formu-lated only if the dynamics is linear). Instead, the evolution of observables is governed bythe Poisson bracket resulting from the Hamilton equations (the summation convention)

d

dtF = F,H = iδAA′

( δF

δψA(α)

δH

δψA′(α)− δH

δψA(α)

δF

δψA′(α)

)

. (3.40)

In more general case, where observables are functionals of a density matrix, the evolutioncan be postulated in a form of the “generalized Nambu mechanics” discussed in detail inChapter 7

d

dtF = F,H = [F,H, S] = Ωabc(α, β, γ)

δF

δρa(α)

δH

δρb(β)

δS

δρc(γ)(3.41)

where S = 12Tr (ρ)2 is called, in the Bia lynicki-Birula–Morrison [39] terminology the

entropy. The entropy is not an observable in linear QM (observables are linear inρ); in Weinberg’s nonlinear QM S can be related to the observable S/Tr ρ which is1-homogeneous in ρ.

The formulation of nonlinear QM in terms of the bracket (3.41) has several advan-tages, as shown by Jordan in [19] (see also Chapter 7).

The Hamilton equations (3.38) in Kibble-Weinberg (KW) theory can be written in aSchrodinger-like form

id

dt|ψ〉 = H(ψ, ψ)|ψ〉 (3.42)

where H(ψ, ψ) is Hermitian. All such equations conserve 〈ψ|ψ〉 because

d

dt〈ψ|ψ〉 = i〈ψ|H†(ψ, ψ) − H(ψ, ψ)|ψ〉. (3.43)

(The Hamiltonians arising from the KW equations are (0,0)-homogeneous, but the abovestatement concerns also equations with non-homogeneous nonlinearities.) There existsalso a larger class of nonlinear equations [40] with non-Hermitian Hamiltonians but alsoconserving 〈ψ|ψ〉, namely

id

dt|ψ〉 =

(

H(ψ, ψ) +(

1 − |ψ〉〈ψ|〈ψ|ψ〉

)

U)

|ψ〉 := H ′|ψ〉 (3.44)

where U does not commute with |ψ〉〈ψ|. The term containing U does not occur inaverages:

〈ψ|H(ψ, ψ)|ψ〉 = 〈ψ|H ′(ψ, ψ)|ψ〉. (3.45)

This, by the way, indicates that 〈ψ|H ′(ψ, ψ)|ψ〉 cannot be regarded as a Hamiltonianfunction for (3.44). Another class of nonlinear Schrodinger equations which, for analo-gous reasons, are not equivalent to KW equations although the Hamiltonians are (0,0)-homogeneous is given by equations of the form

id

dt|ψ〉 =

(

H0 +〈ψ|A|ψ〉〈ψ|ψ〉 B − 〈ψ|B|ψ〉

〈ψ|ψ〉 A)

|ψ〉 := H|ψ〉. (3.46)

22

In such a case all the nonlinear terms involving A and B cancel out in 〈ψ|H |ψ〉 =〈ψ|H0|ψ〉.

Example 3.3 Consider a two-level atom irradiated by an external classical light andassume that the Hamiltonian function of the atom takes the Weinberg form

〈ψ|HL + HNL|ψ〉 (3.47)

where the nonlinear 2 × 2 matrix is

HNLmn =∂2HNL

∂ψm∂ψn= (h0 + ~hNL~σ)mn. (3.48)

Let ~s(ψ, ψ) = 〈ψ|~σ|ψ〉. We find

~s = ~s,H = −i〈ψ|[~σ, HL + HNL]|ψ〉= 2(~hL + ~hNL) × ~s = (~ωL + ~ωNL) × ~s. (3.49)

Defining

ω1

ω2

ω3

=

cosωt sinωt 0− sinωt cosωt 0

0 0 1

ωNL1

ωNL2

ωNL3

,

with ω being the frequency of light, we obtain after RWA the following nonlinear Blochequations

u = −∆v − ω3v + ω2w

v = ∆u+ Ωw + ω3u− ω1w (3.50)

w = −Ωv − ω2u+ ω1v.

Choosing (ω1, ω2, ω3) = (−(A/2)v, (A/2)u, 2ǫw) we get

u = −∆v − 2ǫwv +A

2uw

v = ∆u+ Ωw + 2ǫwu− A

2vw (3.51)

w = −Ωv − A

2u2 − A

2v2.

Eqs. (3.51) are identical to Jaynes’s neoclassical Bloch equations [41] where ǫ is theneoclassical Lamb shift and A is equal to Einstein’s coefficient of spontaneous emission.So let us check what kind of the KW-Schrodinger equation can lead to the neoclassicaldescription.

Returning to the non-rotated reference frame we obtain

(ωNL1, ωNL2, ωNL3) = (−(A/2)s2, (A/2)s1, 2ǫs3) (3.52)

and the nonlinear Schrodinger equation we are looking for is

id

dt|ψ〉 =

(

H(ψ, ψ)1− A

4

〈ψ|σ2|ψ〉〈ψ|ψ〉 σ1 +

A

4

〈ψ|σ1|ψ〉〈ψ|ψ〉 σ2 + ǫ

〈ψ|σ3|ψ〉〈ψ|ψ〉 σ3

)

|ψ〉

= H(ψ, ψ)|ψ〉. (3.53)

23

If H is some (0,0)-homogeneous function then the Hamiltonian is also (0,0)-homogeneous.However

〈ψ|H(ψ, ψ)|ψ〉 = 〈ψ|(

H(ψ, ψ)1 + ǫ〈ψ|σ3|ψ〉〈ψ|ψ〉 σ3

)

|ψ〉 (3.54)

cannot play the role of the Hamiltonian function for (3.53) with A 6= 0. For A = 0 therelevant KW Hamiltonian function is

E〈ψ|ψ〉 +1

2ǫ〈ψ|σ3|ψ〉2〈ψ|ψ〉 (3.55)

and

H(ψ, ψ) = E − 1

2ǫ〈ψ|σ3|ψ〉2〈ψ|ψ〉2 . (3.56)

2

3.4 Results of Single Measurements

and Probabilities

Let H be a finite dimensional Hilbert space and H a Hermitian operator. If H = 〈ψ|H |ψ〉is the associated observable, the values of single measurements of H can be defined in atleast three equivalent ways. Since in nonlinear QM the three options will lead to differentresults we will use here different names for each of them.

a) Eigenvalue E of H = 〈ψ|H |ψ〉 is the number satisfying for some eigenstate theequation

H |ψ〉 = E|ψ〉 (3.57)

or, in the “classical style”,∂H

∂ψm= Eψm. (3.58)

b) Diagonal values are solutions of

det(H − E1) = 0 (3.59)

or

det( ∂2H

∂ψm∂ψn− Eδmn

)

= 0. (3.60)

c) Since eigenstates form a complete orthogonal set of vectors in H any solution of theSchrodinger equation i ddt |ψ〉 = H |ψ〉 can be expressed as

ψ1(t)...

ψN (t)

=

ψ1(0)e−iω1t

...ψN (0)e−iωN t

(3.61)

and the frequencies ωn can be termed the eigenfrequencies . The three possibilities canbe used for definitions of the results of single measurements also in nonlinear QM.

In nonlinear QM observables are not represented by linear operators so that we haveto use the partial derivatives forms of the definitions. Let us begin the comparison of thevarious properties of the definitions with the following example.

24

Example 3.4 The so-called “simplest nonlinearity” considered in experiments designedas tests of Weinberg’s nonlinear QM corresponds to the following Hamiltonian function

〈ψ|H0|ψ〉 + ǫ〈ψ|σ3|ψ〉2〈ψ|ψ〉 (3.62)

where H0 =(E1,00,E2

). Denote ψ1 =

√A1e

iα1 , ψ2 =√A2e

iα2 . The eigenvalue conditionreads

E1 + ǫ[

2A1 −A2

A1 +A2−(A1 −A2

A1 +A2

)2]√

A1 = λ√

A1 (3.63)

E2 + ǫ[

−2A1 −A2

A1 +A2−(A1 −A2

A1 +A2

)2]√

A2 = λ√

A2. (3.64)

For A2 = 0 we find λ+ = E1 + ǫ, for A1 = 0 λ− = E2 + ǫ. If A1 6= 0 and A2 6= 0 we get,for ǫ 6= 0,

A1 −A2

A1 +A2=E2 − E1

4ǫ

which implies (A and B are positive)

|E2 − E1| < 4|ǫ| (3.65)

and

λ0 =1

2

(

E1 + E2 −1

8ǫ(E1 − E2)2

)

with the eigenvector

|ψ〉 =

( (1 + E1−E2

4ǫ

)−1/2eiα1

(1 − E1−E2

4ǫ

)−1/2eiα2

)

(3.66)

We find three eigenvalues although the dimension of the Hilbert space is two. The thirdeigenstate exists only for ǫ satisfying the condition (3.65).

Of course, all the eigenstates are stationary. The eigenstate corresponding to λ0 isnot orthogonal to the remaining two eigenstates, which excludes the ordinary probabilityinterpretation in terms of projectors.

Let us now calculate the diagonal values of this Hamiltonian function. The nonlinearHamiltonian is

H =

(E1 + ǫ(8p3 − 20p2 + 16p− 3) 8ǫ|ψ1|2|ψ2|2ψ1ψ2

8ǫ|ψ1|2|ψ2|2ψ1ψ2 E2 + ǫ(−p3 + 4p2 + 1)

)

(3.67)

where the state is assumed normalized and p = |ψ1|2. The matrix is Hermitian and itseigenvalues (=diagonal values of H) are

E± =1

2

(

E1 + E2 − 2ǫ(8p2 − 8p+ 1)

±[

(E1 − E2)2 − 8ǫ

(E1 − E2)(4p3 − 6p2 + 4p− 1)

+2ǫ(20p4 − 40p3 + 28p2 − 8p+ 1)]1/2)

. (3.68)

25

The diagonal values are in the nonlinear case functions as opposed to eigenvalues whichare numbers and the number of them is always equal to the dimension of the suitableHilbert space.

The solution of the respective nonlinear Schrodinger equation is

(ψ1(t)ψ2(t)

)

=

(

ψ1(0)e−i(E1+2ǫ〈σ3〉−ǫ〈σ3〉2)t

ψ2(0)e−i(E2−2ǫ〈σ3〉−ǫ〈σ3〉2)t

)

(3.69)

where the averages are integrals of motion. It follows that the eigenfrequencies are alsostate dependent functions but differ from the diagonal values.

Fig. 3.1 illustrates the mutual relations between the three definitions.

In linear QM mechanics observables correspond to averages and, at experimental level,to random variables. With any random variable one can associate its higher moments.The higher moments, on the other hand, can be used to deduce a probability interpre-tation of the theory. Therefore, one of the essential points of any generalized algebra ofobservables is the question of the representability of higher moments corresponding tosome given observable. In linear QM the problem is solved by the spectral theorem.

The first interesting result concerning the nonlinear eigenvalues , noticed by Weinbergin [10], is the following

Lemma 3.1 Let F and G be two (1,1)-homogeneous observables possessing a commoneigenstate |ψ〉, i.e.

δAA′

δF

δψA′(α)= fψA(α) and c.c., (3.70)

δAA′

δG

δψA′(α)= gψA(α) and c.c., (3.71)

and let

F ∗G = δAA′

δF

δψA(α)

δG

δψA′(α). (3.72)

Then

δAA′

δ(F ∗G)

δψA′(α)= fgψA(α) and c.c. (3.73)

Proof :

δBB′

δ(F ∗G)

δψB′(β)= δBB′

δ

δψB′(β)

(

δAA′

δF

δψA(α)

δG

δψA′(α)

)

= δAA′δBB′

( δ2F

δψB′(β)δψA(α)

δG

δψA′(α)

+δF

δψA(α)

δ2G

δψB′(β)δψA′ (α)

)

= gδBB′

δ2F

δψB′(β)δψA(α)ψA(α)

+δBB′ψA′(α)δ2G

δψB′(β)δψA′ (α)

= fgψB(β)

26

where in the last step we have applied the (1,0)-homogeneity Euler condition for thederivatives over ψ.2

The homogeneity implies that an average of F in an eigenstate is equal to the re-spective eigenvalue. For finite dimensional Hilbert spaces we know also that a number ofeigenvalues is not smaller than the dimension of the space (a result from the Morse the-ory). Moreover, the eigenvalues are critical points of averages. For averages defined onthe whole Hilbert/projective space their maxima and minima must be found at criticalpoints (averages defined on a finite dimensional projective space are smooth functionsdefined on a compact set). These facts suggest that the eigenvalues are correct candi-dates for the results of single measurements of the generalized observables. However, thefollowing examples show that the problem is not that simple.

Example 3.5 The observable

A =〈ψ|ψ〉2〈ψ|σ3|ψ〉

(3.74)

satisfies the KW requirements. Its eigenvalues are ±1 so do not bound the averages. Itfollows that such singular observables must be excluded if we want to have the probabilityinterpretation in terms of the eigenvalues. The algebra of observables would have to berestricted to (1,1)-homogeneous smooth functions defined of the whole Hilbert/projectivespace. This requirement is not very restrictive. Notice that no difficulties will appear ifwe will apply to A the interpretation in terms of diagonal values or eigenfrequencies.2

Example 3.6 In QM we often encounter observables whose eigenstates are degenerate.Consider now

H = E〈ψ|ψ〉 + ǫ〈ψ|σ3|ψ〉3〈ψ|ψ〉2 . (3.75)

Its eigenvalues are E ± ǫ and E. The number of them is hence greater than the di-mension of the Hilbert space no matter how small ǫ is. Again, no problems appear ifwe apply the diagonal values or eigenfrequencies interpretation. Had we substituted thefirst term with some

(E1,00,E2

), where E1 6= E2, we would have obtained the third eigenvalue

provided |E1 − E2| ≤ 6|ǫ| so that the eigenvalue will not appear for sufficiently small ǫ.This phenomenon seems to be related to the Kolmogorov-Arnold-Moser theorem wherea dimension of an invariant torus may not change with nonlinear perturbation if a non-perturbed Hamiltonian system is nondegenerate and the perturbation is not too large[30]. 2

Example 3.7 Let

H = 〈ψ|H0|ψ〉 + ǫ〈ψ|σ3|ψ〉2N〈ψ|ψ〉2N−1

(3.76)

where H0 =(E1,00,E2

). Two eigenvalues corresponding to the eigenstates |±〉 of σ3 are

E+ = E1 + ǫ and E− = E2 + ǫ so are shifted by the same amount. The third eigenstateoccurs if

|ǫ| > |E1 − E2|4N

. (3.77)

The third eigenvalue

E0 =1

2(E1 + E2) + ǫ(1 − 2N)

(E2 − E1

4Nǫ

) 2N2N−1

(3.78)

27

tends to E1 with N → ∞. Although such nonlinearities are not “simplest”, they aresufficiently simple to be borne in mind in experiments whose ambition is to eliminate allnonlinear corrections to QM. 2

Example 3.8 Consider

H = E〈ψ|ψ〉 + ǫ〈ψ|σ3|ψ〉2〈ψ|ψ〉 . (3.79)

The eigenvalues are E and E + ǫ. Probabilities calculated by means of

〈H〉 = (E + ǫ)pE+ǫ + EpE (3.80)

and the normalization of probability are

pE+ǫ = 〈σ3〉2 pE = 1 − 〈σ3〉2. (3.81)

We know that H∗H has eigenvalues (E+ǫ)2 and E2, so we can calculate the probabilitiesby means of 〈H ∗H〉 = (E + ǫ)2pE+ǫ + E2pE and the normalization condition. Since

〈H ∗H〉 = E2 + (4ǫ2 + 2Eǫ)〈σ3〉2 − 3ǫ2〈σ3〉4 (3.82)

we obtain

pE+ǫ =(4ǫ2 + 2Eǫ)〈σ3〉2 − 3ǫ2〈σ3〉4

2Eǫ+ ǫ2(3.83)

which is, of course, inconsistent with the previous result. The troubles arise because of thenonassociativity of ∗. An analogous result was derived by Jordan [20], who showed thatpropositions cannot be defined consistently within the ∗-algebraic approach to Weinberg’sobservables. We conclude that ∗ cannot be applied for a definition of higher momentsand the above lemma is useless from the viewpoint of the probability interpretation. 2

Next candidate for a single result of a measurement of some H is the diagonal value.We have already seen that the diagonal values are state dependent functions. There isno general guarantee that a diagonal value of an integral of motion is itself an integralof motion. In fact, as the KW equations are not necessarily integrable for dimH > 2,whereas the number of the diagonal values is equal to the dimension of H, the diagonalvalues, for N > 2, can be time independent only for integrable systems. In addition, asdiagonal values are roots of algebraic equations, they even do not have to be differentiablefunctions of states. Therefore, we cannot find for them any general equation of motion.

In spite of these disadvantages the diagonal values are in some respects superior toeigenvalues. If for two observables A and B the commutator AB − BA = 0, then A andB can be diagonalized simultaneously and the product AB has eigenvalues values beingproducts of the eigenvalues of A and B. This leads to the following unique probabilityinterpretation. Let U(ψ, ψ) be the unitary transformation diagonalizing H . Then

H =∑

n

En(ψ, ψ)|(Uψ)n|2. (3.84)

The functions pn = |(Uψ)n|2/〈ψ|ψ〉 are probabilities resulting from the higher momentsprocedure.

28

Let us recall that for two observables F ∗ G = F ∗G. The nonassociativity of ∗ isreflected in the ∗-algebra of observables in the following non-uniqueness of ∗. Althoughalways

〈ψ| ˆ(H ∗ . . . ∗H)|ψ〉 = 〈ψ|H . . . H |ψ〉 (3.85)

but for non-bilinear observables

ˆ(H ∗ . . . ∗H) 6= H · . . . · H. (3.86)

The eigenfrequencies approach can be applied only to systems that are close to integrable[30]. For finite dimensional Hamiltonian systems some general KAM theorems are known.It is known that components of state vectors exhibit in such a case, unless some resonantinitial conditions occur, a quasi periodicity described by [10]

ψk(t) =∑

n1...nN

ck(n1 . . . nN)e−i∑

νnνωνt (3.87)

where both the amplitudes and the frequencies are dependent on initial conditions, asshown in our first example in this subsection, and the sums run over all positive andnegative integers. The form (3.87) of the solution is valid only if the frequencies arenonresonant, which means that

∑

ν nνων 6= 0. Now, following Weinberg, we note thatboth

〈ψ|ψ〉 =∑

k

∑

n1...nN

∑

n′

1...n′

N

ck(n1 . . . nN )ck(n′1 . . . n

′N )exp

(

−i∑

ν

(nν − n′ν)ωνt

)

(3.88)

and

H =∑

k

ψk∂H

∂ψk= i∑

k

ψkd

dtψk

=∑

k

∑

n1...nN

∑

n′

1...n′

N

ck(n1 . . . nN )ck(n′1 . . . n

′N )

×(∑

ν

nνων

)

exp(

−i∑

ν

(nν − n′ν)ωνt

)

are, on general grounds, time independent. The condition of nonresonance means thatthe only time independent terms that can contribute to 〈ψ|ψ〉 and H are those withnν = n′

ν . Thus

〈ψ|ψ〉 =∑

k

∑

n1...

|ck(n1 . . .)|2 (3.89)

andH =

∑

k

∑

n1...

|ck(n1 . . .)|2∑

ν

nνων . (3.90)

These formulas lead to the natural interpretation of the eigenfrequencies∑

ν nνων as theresults of single measurements and the respective normalized coefficients as probabilities.The interpretation can be naturally extended to other observables by assuming that allobservables generate one parameter groups of canonical transformations via the Hamiltonequations of motion. The same result would be also obtained by means of the “standard”

29

theory of measurements where we assume that the so-called pre-measurement is describedby an interaction Hamiltonian function which is proportional to the observable measured,and that all the measuring procedures are based on measurements of observables thatare bilinear like in linear QM. To be precise, we must remark that this procedure (chosenfinally by Weinberg) is not unique either because there is no unique description of thecomposite “object+observer” system even if the observer is assumed to be linear (seeremarks in the section devoted to observables).

To better understand the mutual relations between eigenvalues and eigenfrequenciesit is interesting to compare the two notions in situations where a number of eigenvaluesis different from this of eigenfrequencies.

Example 3.9 We know that

H = E〈ψ|ψ〉 + ǫ〈ψ|σ3|ψ〉3〈ψ|ψ〉2 . (3.91)

has three eigenvalues E ± ǫ and E. The eigenfrequencies are

E1(ψ, ψ) = E + ǫ(3〈σ3〉2 − 2〈σ3〉3) (3.92)

E2(ψ, ψ) = E + ǫ(−3〈σ3〉2 − 2〈σ3〉3) (3.93)

The three eigenvalues correspond to eigenvectors satisfying

〈σ3〉 =

0 for ψ0

1 for ψ1

−1 for ψ−1

In this notation

E1(ψ0, ψ0) = E with probability p1(ψ0, ψ0) = 1/2E2(ψ0, ψ0) = E with probability p2(ψ0, ψ0) = 1/2E1(ψ1, ψ1) = E + ǫ with probability p1(ψ1, ψ1) = 1E2(ψ1, ψ1) = E − 5ǫ with probability p2(ψ1, ψ1) = 0

E1(ψ−1, ψ−1) = E + 5ǫ with probability p1(ψ−1, ψ−1) = 0E2(ψ−1, ψ−1) = E − ǫ with probability p2(ψ−1, ψ−1) = 1.

As expected even in an eigenstate there are two eigenfrequencies. The ones which arenot equal to the eigenvalues occur with probabilities 0.2

A reader of the main Weinberg’s paper may be a little bit confused with what hefinally understands as a result of a single measurement. A half of the paper suggeststhat this role will be played by eigenvalues, then he defines probabilities in terms ofeigenfrequencies and, finally, in his analysis of a two-level atom he treats difference ofeigenvalues as the energy difference while the difference of the eigenfrequencies is treatedas the frequency of the emitted photon. It seems that from the viewpoint of the analysisabove the eigenfrequency difference should be treated as the energy difference of atomiclevels.

I think that each of the three possibilities lacks elegance and generality. Therefore inthe last chapter of this work we will attempt to develop a version of nonlinear QM whereobservables will remain like in ordinary QM.

30

Finally, to close the variations on probability interpretation, let us outline some othergeneral possibilities of the relations between averages, probabilities and values of singlemeasurements.

We have stressed several times that in the Hilbertian formulation of QM the Hamil-tonian function is not equal to the energy because the states are not asumed normalized.This freedom may be useful. Notice that observables in physics in general are identifiedby means of their algebraic properties or the way they change under an action of somesymmetry group. It is reasonable to expect that a modification H of some bilinear

HL = 〈ψ|H0|ψ〉

will have the same transformation properties as HL. If we add some nonlinear term toHL we will generally change its transformation properties. Therefore it seems we shouldconsider the generalized observables of the form

H = I(ψ, ψ)〈ψ|H0|ψ〉 (3.94)

where I is (0,0)-homogeneous and depends on ψ and ψ only via averages of Casimirsof the symmetry group and/or, perhaps, superselection operators like charge. It can beproved easily that I is in such a case an integral of motion, and we can define averagevalues by

〈H〉 =H(ψ, ψ)

I(ψ, ψ)〈ψ|ψ〉 . (3.95)

The evolution in each eigensubspace (“superselection sector”) of the Casimirs and thesuperselection operators would be linear, but with different “fundamental constants”.The linearity of the evolution would be violated only for states that are initially superpo-sitions of states belonging to different sectors. It is obvious that the definition of averageswould eliminate any difficulties with probabilities and eigenvalues.

Another general line of extentions would be to consider observables whose nonlin-ear parts vanish on normalized states. For instance the following modification of theBia lynicki-Birula–Mycielski equation

id

dt|ψ〉 = (H0 + b ln〈ψ|ψ〉)|ψ〉 (3.96)

leads to two inequivalent definitions of averages. One is

〈ψ|H0 + bln〈ψ|ψ〉|ψ〉〈ψ|ψ〉 =

H(ψ, ψ)

〈ψ|ψ〉 =〈ψ|H0|ψ〉〈ψ|ψ〉 + b ln〈ψ|ψ〉 (3.97)

and the other is

H( ψ

‖ψ‖ ,ψ

‖ψ‖)

=〈ψ|H0|ψ〉〈ψ|ψ〉 . (3.98)

The distinction between the generator of evolution and the observable energy may be oneof the ways of introducing nonlinearities without affecting the linearity of averages. Sucha distinction between Hamiltonian and energy appears, for example, in some versions ofsymplectic formulation of field theories [29], and the dynamics so derived would have tobe treated as a dynamics with constraints [38].

31

Figure 3.1: Eigenvalues, eigenfreqencies and diagonal values are in the nonlinear casecompletely different concepts.

32

Chapter 4

COMPOSITE SYSTEMS

AND NONLOCALITY

One of the most often quoted “impossibility theorems” about nonlinear QM is that anysuch theory must imply faster than light communications. We will argue in this and lastchapters that this statement is too strong. In fact, we will show that there exists a richclass of theories where the mentioned phenomenon does not occur. As we shall see thetheories must satisfy two conditions.

1. Their interpretation must not be based on the “collapse of a state vector” postulate.It means that any reasoning based on this postulate has to be regarded as unphys-ical. (In linear QM there exist such limitations. For example, all “counterfactual”problems like EPR paradox or Bell theorem can be eliminated trivially by reject-ing reasonings involving alternative measurements.) Interpretations of QM that donot introduce the collapse (projection) postulate exist, to mention the many worldsinterpretation [42].

2. Observables corresponding to subsystems must be functionals depending on densitymatrices of those subsystems.

In the last chapter we shall prove some general, quite strong theorems related to the lattercondition. Here, we shall limit ourselves to several simple examples showing different waysof making quantum mechanical nonlocality “malignant”, to use the marvelous phrase ofBogdan Mielnik.

It seems it was Nicolas Gisin who noted for the first time that a nonlinear evolution canlead to a faster-than-light communication between two separated systems. His argumentwas the following [43]. Let H be a finite dimensional Hilbert space, |ψi〉, |φj〉 ∈ H,i = 1, . . . , n, j = 1, . . . ,m, 〈ψi|ψj〉 = 〈φi|φj〉 = δij . Then the following lemma holds.

Lemma 4.1 If for some nonvanishing probabilities xi, yj

∑

i

xi|ψi〉〈ψi| =∑

j

yj|φj〉〈φj | (4.1)

33

then there exist two orthonormal bases |αi〉, |βj〉 in some Hilbert space H′ and thestate

|χ〉 =∑

i

√xi|ψi〉 ⊗ |αi〉 =

∑

j

√yj |φj〉 ⊗ |βj〉. (4.2)

2

The proof can be found in [43]. The meaning of the lemma is that the two decompositionsof the density matrix can be obtained by means of the EPR correlations. Indeed, we cantake the density matrix |χ〉〈χ| and trace out H′. The resulting density matrices are theseappearing in the lemma. Assume now that we have a nonlinear evolution of pure states

|ψi〉〈ψi| 7→ gt(|ψi〉〈ψi|). (4.3)

Then, in general,∑

i

xigt(|ψi〉〈ψi|) 6=∑

j

yjgt(|φj〉〈φj |) (4.4)

even if initially the two decompositions were equal. The important assumption leading tothe faster-than-light communication is that each of the pure state sub-ensembles evolvesaccording to (4.3), even if the whole state is given by their convex combination. Thisseems reasonable if we assume that the ensemble consists of the collapsed sub-ensembles.If we apply some “no-collapse” interpretation then the argument cannot be consistentlyformulated. However, it has to be stressed that the form of the evolution appearing in(4.4) can be derived from quite general assumptions. It has been shown in a rigorousway in the language of the theory of cathegories by Posiewnik [44] that this form isimplied by Mielnik’s definition of mixed states as probability measures on the set of purestates. This remark has quite nontrivial consequences: The description of mixed statesin terms of probability measures (Mielnik’s “convex approach”) combined with putativenonlinearity of the Schrodinger equation leads to faster that light communication. If onewants to get rid of such difficulties one cannot keep the figure of states convex.

Returning to Gisin, in his second paper [45] he considered an example of a nonlinearevolution taken from the Weinberg paper [10]. Consider an ensemble of pairs of spin-1/2particles in the singlet state and assume that in one arm of the experiment the evolutionis given by the Hamiltonian function

〈ψ|σ3|ψ〉2〈ψ|ψ〉 . (4.5)

An experimenter in the other arm chooses between two settings of his Stern-Gerlachdevice and decomposes, by means of the EPR correlations, the ensemble in the otherarm into subensembles corresponding to, say, spins up or down in the z direction, orspins up or down in some u direction tilted at 45 with respect to the z-axis. Theevolution equation, up to an overall phase, is given by

id

dt|ψ〉 = 2〈σ3〉σ3|ψ〉. (4.6)

If the initial state is either up or down in the z direction the evolution is stationary andthe average of σ2 is always zero. If the state is either up or down in the u direction thespin is precessing around the z axis but the sense of the rotation is opposite for the up

34

and down states. After a time of a quarter of the “Larmor period” the spin will have thesame positive value of 〈σ2〉. And this result can be detected by an observer in the armwith the nonlinearity.

As we can see Gisin in both of his proofs has completely ignored details of the de-scription and the evolution of the “large” system. To be precise he should rather definea Hamiltonian function of the two arms, then solve the nonlinear Schrodinger equationfor the whole system or calculate the evolution of the average of σ2 in one arm. Thisis the crucial point. We know already that the description of composite systems is notunique as long as we know only the evolution of pure states of subsystems. It will beshown below that the Gisin’s telegraph will never work if we describe the whole systemin a correct way. It will be shown also that his proof is valid for the specific choice ofthe description proposed by Weinberg.

But before we shall pass on to the details of the calculations let us consider another“general proof” which was formulated by myself independently and simultaneously withthis of Gisin [46]. Consider the situation depicted in Fig. 4.1 (for a detailed descriptionof the polarizing Mach-Zechnder interferometer (PMZI) see Appendix 8.4.3). Kets rep-resenting, respectively, right- or left-hand polarized photons travelling to right along thehorizontal “x” axis, evolve in the interferometer in the following way (the index y meansthat the photon travels along the vertical “y” axis):

|+, x〉 BS1−→ i|+, y〉 M1−→ i|+, x〉 α−→ ieiα|+, x〉BS2−→ i

1√2eiα(i|+, y〉 + |+, x〉) (4.7)

|−, x〉 BS1−→ |−, x〉 λ/2−→ |+, x〉 M2−→ |+, y〉BS2−→ 1√

2(i|+, x〉 + |+, y〉). (4.8)

Here α denotes a phase shifter which introduces a phase shift α, λ/2 is a half-wave plate,BS1 is a polarizing beam splitter, BS2 a non-polarizing one, and M1 and M2 are mirrors.A transmitted beam is always phase-shifted by π/2 with respect to the reflected one [47].Let us consider now the following two cases:

(1) A source produces a linear polarization state, say

|ψ〉 =1√2

(|+, x〉 + |−, x〉). (4.9)

The M-Z interferometer acts by means of (4.8) thus the whole state transforms into

|ψ′〉 =1

2

(i(eiα + 1)|+, x〉 + (1 − eiα)|+, y〉

). (4.10)

We observe an interference between the two outgoing channels. In particular, for α = 0the whole beam goes through the x channel.

(2) A source produces a singlet state [47]

|φ〉 =1√2

(|+,−x〉|+, x〉 + |−,−x〉|−, x〉), (4.11)

35

where the photons in a pair travel along the x axis but in opposite directions. Now thewhole system consists of two separated subsystems, I and II, where in I the photonsenter the M-Z interferometer. This subsystem is described by a reduced density matrix

ρ1 = Tr2|φ〉〈φ| =1

2|+, x〉〈+, x| +

1

2|−, x〉〈−, x| (4.12)

and is therefore in a mixed state. No interference should be observed, as there is no co-herence between the incoming right- or left-handed photons. Indeed, the transformation(4.8) transforms the density matrix into

ρ′1 =1

2|+, x〉〈+, x| +

1

2|+, y〉〈+, y|, (4.13)

hence the intensities of beams in the outgoing channels are equal and independent of theoptical retarder.

This property of the singlet state was considered by Pykacz and Zukowski [47] as theproof that a superluminal signalisation is impossible in this experiment. This is obviouslytrue in linear quantum mechanics.

Let us, however, recalculate the problem in a different way.We must obtain an identical result if we first transform the whole singlet state ac-

cording to (4.8) then form a density matrix of the whole system, and, finally, take thetrace over II.

The whole state transforms as follows:

|φ′〉 = 12 (ieiα |+,−x〉|+, x〉 −eiα |+,−x〉|+, y〉

+i |−,−x〉|+, x〉 + |−,−x〉|+, y〉). (4.14)

Now we can explicitly see why there cannot appear interference in I: The linear de-pendence of the states which interfered in the linear polarization case is destroyed byorthogonality of their singlet state partners . The lack of the interference in I is thereforea nonlocal phenomenon. The two explanations of the lack of interference are, unfortu-nately, experimentally indistinguishable. As long as ordinary linear devices are appliedto the subsystem II no possibility of a signalisation between I and II is possible, be-cause the kets will always evolve according to some unitary transformation and remainorthogonal.

We know that nonlinear evolutions in a Hilbert space do not conserve scalar products(the “mobility phenomenon” [16]) and states that are initially orthogonal may loose theirorthogonality during the course of the evolution. It is clear that if we will violate in II theorthogonality of |+,−x〉 and |−,−x〉 because of some nonlinearity then the photons in I

will start to interfere and the interference will be the stronger the more violation of theorthogonality has been obtained. Putting it differently, tracing over a space where theevolution does not conserve scalar products may result in some “remains” of the tracedout system in the reduced density matrix describing the remote, separated system.

This argument looks general but a careful reader has probably noticed that one ad-ditional assumption has been smuggled here: It is implicitly assumed that a nonlinearevolution in H can be extended to H⊗H′ in such a way that a solution describing thecomposite system is of the form

|φ〉 ⊗ |+〉 + |φ′〉 ⊗ |−〉 (4.15)

36

where |φ〉 and |φ′〉 are solutions of the subsystem’s nonlinear evolution equation that are“in mobility”, that is whose scalar product is not conserved. This assumption is a stronglimitation. In fact, it never occurs if separated systems are described in a correct way.However, the proposal of Weinberg does allow for such pathological solutions.

In what follows we shall formulate the arguments of Gisin and myself in a precise waywithin the framework of Weinberg’s approach.

Let us assume that we have two separated systems, I and II, described by Hamiltonianfunctions

H1(ϕ, ϕ) = E1〈ϕ|ϕ〉, H2(χ, χ) = E2〈χ|χ〉 + ǫ〈χ|σ3|χ〉2〈χ|χ〉 . (4.16)

According to Weinberg, the Hamiltonian function of the whole I+II system is given by

H(ψ, ψ) =∑

l

H1(ϕ(l), ϕ(l)) +∑

k

H2(χ(k), χ(k)), (4.17)

where ϕk(l) = χl(k) = ψkl.A general solution of

idψkdt

=∂H

∂ψk, (4.18)

with H given by (4.17), in a basis diagonalizing σ3, is

ψ =

(ψ11 ψ12

ψ21 ψ22

)

=

(αx αyαx′ αy′

)

= |ξ1〉 ⊗ |φ〉 + |ξ2〉 ⊗ |ψ〉 (4.19)

where |ξ1〉 =(α0

), |ξ2〉 =

(0α

), α = exp(−iE1t) and |φ〉 =

(xy

), |ψ〉 =

(x′

y′

)are some solutions

|ψ(t)〉 =

(√Ae−iE2+ǫ(A−B)(A+B)−2(A+3B)t−iδ√Be−iE2−ǫ(A−B)(A+B)−2(3A+B)t

)

(4.20)

of

id

dt|ψ〉 = E2ψ + ǫ

(

2〈ψ|σ3|ψ〉σ3

〈ψ|ψ〉 − 〈ψ|σ3|ψ〉2〈ψ|ψ〉2

)

|ψ〉 (4.21)

resulting, via the Hamilton equations, from the Hamiltonian function

H2(ψ, ψ) = E2〈ψ|ψ〉 + ǫ〈ψ|σ3ψ|〉2〈ψ|ψ〉 . (4.22)

Like in ordinary quantum mechanics, the state evolves here by means of separate evolu-tions of the “collapsed states”.

Now, if |ψ〉 is a solution of (4.21), then |φ〉 = iσ2|ψ〉 is another solution correspondingto the same mean value of the energy, and we get

〈ψ|φ〉 = 2i√AB sin

(4ǫ(A−B)(A+B)−1t+ δ

). (4.23)

For ǫ 6= 0, AB 6= 0, δ = 0, and A 6= B the states are orthogonal for t = 0, but later loosetheir orthogonality, although the norm of each of them is conserved. This is an explicitexample of the mobility phenomenon.

37

The Hamiltonian functions

H1(ψ, ψ) =∑

l

H1(ϕ(l), ϕ(l)) and H2(ψ, ψ) =∑

k

H2(χ(k), χ(k)) (4.24)

commute because H1(ψ, ψ) = E1〈ψ, ψ〉. Keeping in mind that there is no interactionpart in H , we conclude that there is no flow of energy between I and II.

Let us now calculate the reduced density matrix of the linear subsystem I. With thenotation of (4.19), we find (we take normalized states)

ρ1 =1

2(|ξ1〉〈ξ1| + |ξ2〉〈ξ2| (4.25)

+〈ψ|φ〉|ξ1〉〈ξ2| + 〈φ|ψ〉|ξ2〉〈ξ1|). (4.26)

The off-diagonal elements vanish in the linear theory (ǫ = 0) if initially the states areorthogonal, and we get a “fully mixed” state. For ǫ 6= 0 these coherences oscillate withthe mobility frequency given by (4.23). There therefore exist observables whose averagevalues oscillate in this way. For example, the components of spin satisfy 〈σ1〉 = 〈σ3〉 = 0,but

〈σ2〉 = Im〈φ|ψ〉 = 2√AB sin

(4ǫ(A−B)(A+B)−1t+ δ

). (4.27)

The result that has been derived is in agreement with our general analysis. Let us returnto the example with the interference and assume that we introduce in II an optical devicethat induces the nonlinear evolution discussed above.

According to the superposition principle, the source emits pairs which are superpo-sitions of the states corresponding to results (+,+) or (−,−). Once the “left” photonenters the non-linearity region, the superposition starts to evolve like the state (4.19).The right- and left-handed states start to rotate around each other because of the mo-bility phenomenon. Their partners do not interfere as long as the states are orthogonal,but the “less orthogonal” they are, the stronger the interference of their partners. Itfollows that the intensities in both of the outgoing channels of the M-Z interferometerin I start to change. In the beginning of the process the intensities were equal. Atthe moment of time in which the orthogonality is broken in the strongest way (let usaccept, for inessential simplicity, that the two states are then proportional to each other;for reversible evolutions, the states can be arbitrarily close, but never proportional) theinterference is strongest and, for α = 0, the intensity in the y channel is 0, while in thex one it is maximal. After this extreme point, the mobility starts to make the states“more and more orthogonal”, and the interference becomes weaker and weaker. At themoment when the states are again orthogonal the intensities are again equal. And so on.The outgoing intensities must oscillate with the mobility frequency which agrees with theresult of the calculation.

Another way of viewing this phenomenon can be the following. We know that theinterference does not take place if one knows a path of a particle (say, a photon) in aninterferometer. This observation inclined many people to propose the so-called delayed-choice experiments. It is sometimes believed that this interpretation means that thephotons in one arm of the interferometer can interfere until a remote observer in theother arm makes an experiment which determines their paths in the interferometer.Such an indirect measurement could be possible if there were a one-to-one relationship

38

between a trajectory in the interferometer and the results of the remote measurements.This, on the other hand, means that the state of the whole system takes the form

ψ = |yes〉|one route〉 + |no〉|another route〉, (4.28)

where the kets |yes〉 and |no〉 must be orthogonal since correspond to different results ofmeasurements made by the remote observer. But our earlier analysis proves that there isno interference in such a case. Therefore the question whether the experiment is delayedor not, is inessential. It is important that in principle one can make such an experiment.

Now, what happens if the photons traverse some non-linear device which intro-duces the mobility of initially orthogonal states |yes〉 and |no〉 before they reach theremote observer? The stronger the violation of their orthogonality, the less informationone gets about the trajectories in question. In the extreme case of the mobility, say|yes〉 → |yes〉, |no〉 → |yes〉, the result “yes” tells us nothing about the trajectories in theother arm, and then one obtains the strongest interference. Again, the “more orthogo-nal” the states are, the greater the probability of a correct prediction of the route of thephotons in the interferometer and the weaker should be the interference. By the way, thisanalysis helps to unify the well known “Feynman rules” for calculation of probabilities inQM: We add amplitudes in case we do not know trajectories, and probabilities in case weknow them. The above analysis shows that we always add amplitudes , but the knowledge(in principle or actual) of trajectories is possible if and only if the states correspondingto different trajectories are correlated with mutually orthogonal states and then thereis no interference terms in the probabilities because of the same reasons as there is nointerference in our Mach-Zehnder interferometer.

Let us complete the analysis of this part with two comments.

1. The telegraph is based on the fact that an average of an observable in the lin-ear system I depends on parameters of the Hamiltonian function of the nonlinearsystem II. Recalling the form of the Poisson bracket equation for observables A1

related to I

A1 = A1, H1 +H2 (4.29)

we can see that such a dependence is possible if and only if

A1, H2 6= 0. (4.30)

It is an easy exercise to check that this condition is met indeed in the case ofA = 〈ψ|σ2|ψ〉 and H2 from (4.17). It follows that the necessary an sufficient con-dition for elimination of telegraphs based on the mobility phenomenon is that anyobservables corresponding to separated systems must be in involution with respectto the Poisson bracket generating the evolution of observables. The observationthat such observables are not necessarily in involution in the Weinberg approach isdue to J. Polchinski [18].

2. If one of the systems is linear, then the telegraph based on the mobility phenomenoncan be used for sending information only from the nonlinear system to the linearone. In Gisin’s telegraph one utilizes the remote preparation of mixtures enteringthe nonlinear system. Accordingly, this kind of telegraph works in the oppositedirection hence cannot be equivalent to this based on the mobility phenomenon.

39

Let us describe now the Gisin’s telegraph [48]. Consider again the same Hamiltonianfunctions of the subsystems and the Weinberg’s description of the whole one (Eqs. (4.16)and (4.17)).

The sender is free to choose a basis in his Hilbert space by a rotation of his Stern-Gerlach device. Let the unitary matrix with unit determinant

(α β−β α

)

(4.31)

describe the freedom in the choice of bases in the linear subsystem and the relationbetween the components of the whole state in the chosen basis and in the spin up-downone is given by

(ψ11 ψ12

ψ21 ψ22

)

=

(α β−β α

)(ψ++ ψ+−ψ−+ ψ−−

)

. (4.32)

The state is initially the singlet, which means that

(ψ11(0) ψ12(0)ψ21(0) ψ22(0)

)

=1√2

(α β−β α

)(0 1−1 0

)

=1√2

(−β α−α −β

)

. (4.33)

Finally the solution for the whole system is

|ψ〉 =1√2e−i(E1+E2−ǫX2)t

(−βe−i2ǫXt αei2ǫXt

−αei2ǫXt −βe−i2ǫXt)

(4.34)

where X = |β|2 − |α|2. The reduced density matrix of the nonlinear system II reads

ρII =1

21 + Re(αβ) sin

(4ǫ(|α|2 − |β|2)t

)σ2 + Im(αβ) sin

(4ǫ(|α|2 − |β|2)t

)σ1. (4.35)

The average of σ2 in the nonlinear system is

〈σ2〉 = 2Re(αβ) sin(4ǫ(|α|2 − |β|2)t

)(4.36)

hence depends on the choice of basis made in the linear one.Notice that since the Hamiltonian function of the linear subsystem is proportional to

〈ψ|ψ〉 it is in involution with any observable. In particular

〈σ2〉, H1 = 0 (4.37)

which means that the commutability of observables corresponding to separated systemsis not a sufficient condition for the nonexistence of faster-than-light telegraphs. Thedependence of 〈σ2〉 on α and β follows from the dependence of H2 on these parameters.We have remarked already in the chapter dealing with observables that the Weinberg’schoice is basis dependent: By a change of basis in I we can change a value of energy inII.

It becomes clear now under what conditions this kind of pathology can be eliminated.Consider a density matrix ρ describing a “large” system. A change of basis in a subsystemI is represented by the unitary transformation

ρ 7→ UI ⊗ 1IIρU−1I ⊗ 1II . (4.38)

40

Any function of ρ that can be written in a form of a series

f(ρ) =∑

k

skρk (4.39)

transforms in the same way. It follows that all expressions like

Tr(f(ρ)1I ⊗ AII

)(4.40)

are UI -independent. Also all functionals depending on reduced density matrices of sub-systems are independent of changes of bases outside of those subsystems. In the lastchapter we shall prove a general theorem stating that if two observables depend on re-duced density matrices of different subsystems then they are in involution with respectto a large class of brackets. In such a case both kinds of telegraphs will be eliminated.

The last point that has to be explained is the question of uniqueness of the descriptionof subsystems. We have shown already that a knowledge of observables on pure states ofsubsystems does not determine their form if the subsystems are correlated with somethingelse. Here we will show that observables that differ only on mixed states will, in general,generate different evolutions of subsystems.

Consider our “canonical” example (4.16) but rewritten in a form including densitymatrices. For the sake of clarity let us also substitute a more general matrix

ǫ =

(ǫ1 00 ǫ2

)

for√ǫσ3.

We have now an infinite number of possibilities. The simplest nontrivial ones are

H1(ρ1) = E1Tr ρ1, H2(ρ2) = E2Tr ρ2 +(Tr ρ2ǫ)

2

Tr ρ2(4.41)

and

H1(ρ1) = E1Tr ρ1, H2(ρ2) = E2Tr ρ2 +(Tr ρ2ǫ)

2

Tr ρ2

Tr (ρ22)

(Tr ρ2)2. (4.42)

Let ρkl =∑

m ψmkψml denote the components of the reduced density matrix ρ2. TheHamiltonian function of the whole system is

H(ρ) = H1(ρ1) +H2(ρ2). (4.43)

The two forms of H2 lead, respectively, to the following two evolution equations

d

dtρkl = −2i

Trρ2ǫ

Tr ρ2ρkl(ǫk − ǫl) (4.44)

andd

dtρkl = −2i

Trρ2ǫ

Tr ρ2

Tr (ρ22)

(Tr ρ2)2ρkl(ǫk − ǫl) (4.45)

where the expressions involving traces are integrals of motion. The equations are differ-ent.

There is only one way out of the above dilemma: The description must from theoutset be given in terms of density matrices. This remark will be the departure point forthe new version of nonlinear QM proposed in the last chapter.

41

Chapter 5

A TWO-LEVEL ATOM

IN NONLINEAR QM

Since the present-day ion trap experimental techniques enable unprecedently precise mea-surements of quantum optical properties of atoms, the question of a correct descriptionof a two-level system in nonlinear QM has accuired particular importance. The aimof the analysis below is to clarify some elementary features of the nonlinear formalisminvolved in calculations of such problems. Therefore, the content of this chapter shouldnot be understood as a complete, unique solution or systematization of all the questionsencountered. Still, I hope that I have managed to point out some elements essential forcorrect calculations in practical, experimental situations.

5.1 A Fully Quantum Approach

In linear QM a two-level system is mathematically equivalent to a spin-1/2 nonrelativisticparticle. All the examples discussed in this work were based on a two-dimensional Hilbertspace and the reader may have a feeling that they could be applied equally well to bothspin-1/2 particles and two-level atoms. This kind of conviction has been shared byall the authors (including Weinberg himself) dealing with theoretical and experimentalaspects of nonlinear QM. The main result of this chapter, as we shall see later, is that,paraphrasing G. Orwell’s words, in nonlinear QM all two-level systems are two-level butsome of them are more two-level than others.

We shall assume that the supposed nonlinearity is of purely atomic origin. This meansthat we shall consider the atom as a nonlinear subsystem of the larger “atom+field”system where both the field and interaction Hamiltonian functions are linear in densitymatrix. (The papers discussing the problem can be divided into two groups: Eitherthe authors do not care about the description of the “atom+field” composite system ortreat it in the way proposed by Weinberg. We know that none of them can be correctunless the atom and the field are in a product state, which is typical for semiclassicaltreatments. It follows that no really quantum description of the problem has been givenas yet.) We shall assume also the dipole and rotating wave approximations. To clarifythe role of the latter we shall treat both ∆m = 0 and ∆m = ±1 cases.

42

We begin with the form of the atomic Hamiltonian function. The simplest one (atlest from the point of view of simplicity of calculations) is

HAT [ρ] = Tr ρHL +(Tr ρǫ)2

Tr ρ(5.1)

where HL is the linear Hamiltonian of the atom and ǫ is an operator commuting withHL. Assuming that we consider the atom in a pure state ρ = |ψ〉〈ψ| we find that ageneral solution of the resulting nonlinear Schrodinger equation is (in this chapter weshall use the ordinary units with h 6= 1)

ψk(t) = ψk(0) exp[

− i

h

(

Ek + 2〈ǫ〉ǫk − 〈ǫ〉2)

t]

. (5.2)

The averages of ǫ are integrals of motion and, of course, depend on all nonvanishingcomponents of |ψ〉. This is an important point. In the analysis of a coupling betweenthe atom and an external electromagnetic field we meet two difficulties. First of allwe have to decide which states will be involved in the absorbtion-emission process. Inlinear QM the situation is simple: We take two stationary states of the noninteractingatom. In nonlinear QM the atomic nonlinearity may lead to stationary states thatare not orthogonal between one another. Atomic creation and annihilation operatorscorresponding to such levels cannot satisfy ordinary anticommutation relations.

To avoid such complications the interaction term we shall choose will be defined inthe ordinary way, that is, in terms of creation and annihilation operators correspondingto the levels of the linear Hamiltonian.

Second, we cannot a priori restrict the atomic Hilbert space to two dimensions be-cause a k-th eigenfrequency depends on amplitudes of all other components of |ψ〉 andthe evolution cannot be naturally “cut into N -dimensional pieces”. Of course, we cannotalso assume that only these ǫk are nonvanishing which we are interested in (a probabilitythat in a given case we will find just those only nonvanishing ones is 0). Therefore, to beperfectly consistent we should resign from the two level approximation in the interactionterm. We shall not, however, consider such complications although this approximationwill further restrict the physical validity of the calculations presented below.

Let bk, b†k be the k-th level atomic annihilation and creation operators satisfying the

fermionic algebra [bk, b†l ]+ = δkl and a, a† be the annihilation and creation operators of a

monochromatic photon field whose frequency is ω. The choice of the creation-annihilationoperators language leads naturally to the following Hamiltonian function of the whole“atom+field” composite system

HA+F (ψ, ψ) = 〈ψ|∑

k

hωkb†kbk + hωa†a+

ihq

2(b†2b1a− a†b†1b2)|ψ〉

+〈ψ|∑k ǫkb

†kbk|ψ〉2

〈ψ|ψ〉 . (5.3)

The state in the Fock basis is |ψ〉 =∑

kn ψkn|k〉|n〉. The nonlinear term is thereforeequivalent to

〈ψ|ǫ⊗ 1F |ψ〉2〈ψ|ψ〉 =

(Tr ρAT ǫ)2

Tr ρAT(5.4)

43

which seems natural but, as we have seen before, is only one out of the whole variety ofinequivalent possibilities. Such a description is consistent with the definition of the atomicHamiltonian function (5.1) (in this sense it is unique) and free from any “malignant”nonlocalities. On the other hand it is in a mean-field style; we have remarked in thesection on observables that

Tr ρAT ǫρAT ǫ

Tr ρAT(5.5)

would look more “fundamental”. Anyway, although the conclusions drawn on a basisof such an analysis are limited in their generality, some choice has to be made andcalculations with (5.5) would be more complicated.

The Hamiltonian resulting from (5.3) is

HA+F =∑

k

hωkb†kbk + hωa†a+

ihq

2(b†2b1a− a†b†1b2)

+2〈ψ|∑k ǫkb

†kbk|ψ〉

〈ψ|ψ〉∑

k

ǫkb†kbk −

〈ψ|∑k ǫkb†kbk|ψ〉2

〈ψ|ψ〉2 (5.6)

and the nonlinear Schrodinger equation is

ihψ1n =

hω1 + 2

∑

lm ǫl|ψlm|2∑

lm |ψlm|2 ǫ1 −(∑

lm ǫl|ψlm|2∑

lm |ψlm|2)2

+ nhω

ψ1n

− i

2hq

√nψ2,n−1

ihψ2n =

hω2 + 2

∑

lm ǫl|ψlm|2∑

lm |ψlm|2 ǫ2 −(∑

lm ǫl|ψlm|2∑

lm |ψlm|2)2

+ nhω

ψ2n

+i

2hq

√n+ 1ψ1,n+1

...

ihψkn =

hωk + 2

∑

lm ǫl|ψlm|2∑

lm |ψlm|2 ǫk −(∑

lm ǫl|ψlm|2∑

lm |ψlm|2)2

+ nhω

ψkn

for k > 2. (5.7)

Writing ψkn = Akn exp(−iαkn/h) we find that for k > 2 Akn = const for all n. Sincealso ‖ψ‖ is time independent (hereafter we put ‖ψ‖ = 1) it follows that for k > 2 theexponents depend on time also via 〈ψ|ǫ12|ψ〉 =

∑

n(ǫ1|ψ1n|2 + ǫ2|ψ2n|2) whose explicitform has to be determined. Decomposing

〈ψ|ǫ12|ψ〉 =1

2〈ψ|(ǫ1 + ǫ2)(b†1b1 + b†2b2)|ψ〉 +

1

2〈ψ|(ǫ1 − ǫ2)(b†1b1 − b†2b2)|ψ〉

:= (ǫ1 + ǫ2)〈ψ|R0|ψ〉 + (ǫ2 − ǫ1)〈ψ|R3|ψ〉, (5.8)

where the first expression is an integral of motion, we see that the problem reduces tocalculating 〈ψ|R3|ψ〉 which is one half the atomic inversion. Denoting ε = ǫ ⊗ 1F , thetotal Hamiltonian can be decomposed now into two parts

H1 = h(ω2 − ω1)R3 + 2(ǫ2 − ǫ1)〈ψ|ε|ψ〉R3 + hωa†a

44

+ihq

2(b†2b1a− a†b†1b2),

H2 = h(ω1 + ω2)R0 +∑

k>2

hωkb†kbk + 2〈ψ|ε|ψ〉

((ǫ1 + ǫ2)R0

+∑

k>2

hǫkb†kbk)− 〈ψ|ε|ψ〉2. (5.9)

Operators, R3, R1 = 12 (b†2b1 + b†1b2), R2 = i

2 (−b†2b1 + b†1b2), a and a† commute with H2

so that the evolution of the atomic operators Rj is generated by the following nonlineargeneralization of the Jaynes-Cummings Hamiltonian

H1 = hω0R3 + 2ǫ0〈ψ|ε|ψ〉R3 + hωa†a+ihq

2(R+a− a†R−), (5.10)

where ω0 = ω2 −ω1 and ǫ0 = ǫ2 − ǫ1. To explicitly distinguish between initial conditionsand dynamical objects we shall decompose 〈ψ|ε|ψ〉 as follows

〈ψ|ε|ψ〉 =∑

k>2,n

ǫk|ψkn|2 + (ǫ1 + ǫ2)〈ψ|R0|ψ〉 + ǫ0〈ψ|R3|ψ〉 := A+ ǫ0〈ψ|R3|ψ〉 (5.11)

where A is a constant depending on initial conditions. Denoting further

hω0 + 2ǫ0A = hω′0, 2ǫ20 = hǫ (5.12)

we finally obtain

H1 = hω′0R3 + hǫ〈ψ|R3|ψ〉R3 + hωa†a+

ihq

2(R+a− a†R−). (5.13)

It is clear now what is the actual meaning of the two-level approximation in nonlinear QM.The evolution of the atomic operators Rj is generated by a two-dimensional Hamiltonian

in analogy to the linear case, but the parameters of H1 depend on components of the wavefunction corresponding to the levels being outside of the two-dimensional Hilbert space.On the other hand, the phases of the remaining components depend on the average ofthe atomic inversion of the two levels.

(It has to be stressed here that one can consider a quantum system whose Hamilto-nian function is a sum of a linear term and some nonlinearity which involves only spinorcomponents of the wave function. Formally, such a case would be equivalent to a com-posite system whose constituents are a scalar particle that evolves according to the lawsof the linear QM and some nonlinear two-level system noninteracting with the parti-cle. We know that various solutions of the nonlinear Schrodinger equation, including theproduct one, will exist and no components other than the spinor ones will be involvedin the nonlinear part of the evolution. Systems with nonlinearities of this kind would be“truely two-level” as opposed to the systems which are two-level in the sense specifiedabove.)

Our next task is to find and solve an equation for the atomic inversion. The Poissonbracket evolution equation reads

ihd

dt〈ψ|R3|ψ〉 = 〈ψ|[R3, H1]|ψ〉 =

q

2〈ψ|R+a+ a†R−|ψ〉 (5.14)

45

so is just like in linear QM [49]. Following the notation of [49] the second derivative isfound equal

d2

dt2〈ψ|R3|ψ〉 =

1

ih

q

2〈ψ|[R+a+ a†R−, H1]|ψ〉

= −q2〈ψ|R3

(N +

1

2

)|ψ〉

+iq

2

(∆′ + ǫ〈ψ|R3|ψ〉

)〈ψ|R+a− a†R−|ψ〉 (5.15)

where N = R3 + a†a and ∆′ = ω′0 − ω. Define

B =∑

k>2,n

hωk|ψkn|2 + h(ω1 + ω2)〈ψ|R0|ψ〉. (5.16)

〈ψ|N |ψ〉, like in the linear case, is constant. In order to get rid of the average in the lastrow of (5.15) we rewrite the whole Hamiltonian function as follows

HA+F = hωA+F = A2 +B + hω〈ψ|N |ψ〉

+h∆′〈ψ|R3|ψ〉 + ǫ20〈ψ|R3|ψ〉2 +ihq

2〈ψ|R+a− a†R−|ψ〉. (5.17)

Denoting w = 2〈ψ|R3|ψ〉, ωA = A2/h, ωB = B/h, ωRWA = ωA+F − ωB we finally get

w = 2∆′(ωRWA − ω〈N〉 − ωA)

+(ǫ(ωRWA − ω〈N〉 − ωA) − ∆′)w − q2〈ψ|R3

(N +

1

2

)|ψ〉

−3

4ǫ∆′w2 − ǫ2

8w3. (5.18)

We have met here the characteristic inconvenience of the Poisson bracket formalism ofnonlinear QM: The nonexistence of the Heisenberg picture. In the linear case we cansolve the Jaynes-Cummings problem completely, independently of any particular initialconditions for states. Here the term 〈ψ|R3

(N + 1

2

)|ψ〉 involves correlations between the

atom and the field and I have not managed to express it solely in terms of constants andw unless the state is an eigenstate of N , or a semiclassical decorelation is assumed. So letinitially the state of the system be a common eigenstate of R3 and a†a with respectiveeigenvalues n′ and n. The atomic inversion satisfies then the general elliptic equation[50]

w = 2∆′(∆′n′ +1

8ǫ) +

(

ǫ(∆′n′ +1

8ǫ) − ∆′ − q2

2

(N +

1

2

))

w − 3

4ǫ∆′w2 − ǫ2

8w3. (5.19)

The general method of solving equations of the form

y = A+By + Cy2 +Dy3 (5.20)

is the following. After multiplying (5.20) by y, integrating over t, and simplifying theresulting expression we obtain

y2 = a+ 2Ay +By2 +2

3Cy3 +

1

2Dy4 (5.21)

46

where a is an arbitrary constant, which has to be determined from initial conditions. Werewrite the equation in the more convenient form

y2 = a+ by + cy2 + dy3 + ey4 (5.22)

and seek a transformation z = z(y) by means of which the equation reduces to thestandard elliptic form

z2 = (1 − z2)(1 − k2z2) := F2(z). (5.23)

Writingy2 = h2(y − α)(y − β)(y − γ)(y − δ) := G2(y) (5.24)

and defining

z2 =(β − δ)(y − α)

(α− δ)(y − β)(5.25)

k2 =(β − γ)(α− δ)

(α− γ)(β − δ)(5.26)

M2 =(β − δ)(α − γ)

4(5.27)

we obtain1

F(z)

dz

dt=

M

G(z)

dy

dt= hM (5.28)

hencez = sn

(hM(t− t0), k

). (5.29)

The applicability of the method is up to problems with vanishing of denominators in theabove fractions. In such a case there exist other transformations analogous to these givenabove [50].

Although we could try to find a general expression for the atomic inversion followingfrom (5.19) it seems more instructive to make here some simplifying assumptions. Firstof all we can take the “two-level initial conditions”, that is, assume that the initial stateof the system is such that the only nonvanishing components of the wave function arethese with k = 1, 2. Then h∆′ = h∆ + ǫ22 − ǫ21 where ∆ = ω0 − ω. (Let me remarkhere that in most of the papers dealing with two-level systems (cf. [41, 8]) their authorsassumed that for the “simplest” nonlinearities ǫ2 = 0 which seemed to suggest that thenonlinearity must shift the resonant frequency. As we can see, the more symmetric “σ3”choice does not change the detuning.) Further, choosing the “detuning” ∆′ = 0 anddenoting ǫ2/8 = 2ς2 we get

w = (2ς2 − Ω2)w − 2ς2w3. (5.30)

This equation can be solved immediately. For example, with the initial condition w(0) =−1 we find

w(t) =

−cn(Ωt, ς/Ω) for Ω > ς−sech(Ωt) for Ω = ς−dn(ςt,Ω/ς) for Ω < ς

(5.31)

The result is analogous to this of Wodkiewicz and Scully [41] who chose the Blochequations approach.

47

It is an appropriate point for a brief comparison of our results with those of Weinbergwho chose his own, basis dependent description of the “atom+field” system. Considerthe same form of the atomic nonlinearity. The Hamiltonian function of the compositesystem in the Fock basis (this basis was chosen by Weinberg) is

HA+F (ψ, ψ) = 〈ψ|∑

k

hωkb†kbk + hωa†a+

ihq

2(b†2b1a− a†b†1b2)|ψ〉

+∑

n

〈ψ|∑

k ǫkb†kbkPn|ψ〉2

〈ψ|Pn|ψ〉. (5.32)

where Pn project on n-photon states. Assume now that initially the state of the wholesystem has only one nonvanishing component ψ11 (one photon and the atom in theground state). It follows that only ψ11 and ψ20 will appear in the nonlinear Schrodingerequation. Our “nonmalignant” choice leads to the Schrodinger equation of the form

ihψ11 =

hω1 + 2ǫ1|ψ11|2 + ǫ2|ψ20|2|ψ11|2 + |ψ20|2

ǫ1 −(ǫ1|ψ11|2 + ǫ2|ψ20|2

|ψ11|2 + |ψ20|2)2

+ hω

ψ11

− i

2hqψ20

ihψ20 =

hω2 + 2ǫ1|ψ11|2 + ǫ2|ψ20|2|ψ11|2 + |ψ20|2

ǫ2 −(ǫ1|ψ11|2 + ǫ2|ψ20|2

|ψ11|2 + |ψ20|2)2

ψ20

+i

2hqψ11, (5.33)

while the Weinberg’s one leads to

ihψ11 =

hω1 + 2ǫ1|ψ11|2|ψ11|2

ǫ1 −(ǫ1|ψ11|2

|ψ11|2)2

+ hω

ψ11

− i

2hqψ20

ihψ20 =

hω2 + 2ǫ2|ψ20|2|ψ20|2

ǫ2 −(ǫ2|ψ20|2

|ψ20|2)2

ψ20

+i

2hqψ11. (5.34)

Simplifying the fractions we obtain

ihψ11 = (hω1 + ǫ21)ψ11 −i

2hqψ20

ihψ20 = (hω2 + ǫ22)ψ20 +i

2hqψ11 (5.35)

which are linear and the only modification with respect to ordinary QM is that theenergy levels of the atom are the nonlinear eigenvalues corresponding to the atom inthe absence of radiation. In addition, the “σ3” nonlinearity satisfies ǫ21 = ǫ22 so that forthis choice of the “simplest” nonlinearity neither the energy difference nor the shape ofthe inversion’s oscillation would be affected, whereas we know already that the correctdescription leads to elliptic oscillations even for w(0) = −1 and N + 1

2 = 1.

48

In the calculations in this section we have not needed any assumption about the“smallness” of the nonlinearity. Moreover, as we have shown before, it is not evident whatis actually meant by a small nonlinearity. This lack of uniqueness is related to the factthat there exist singular nonlinearities that are negligible in the lack of correlations, butcan become dominant if the nonlinear system in question correlates with something else.Anyway, disregarding these subtle points it seems reasonable to expect that a physicalnonlinearity, if any, should be in ordinary situations small in some sense. Therefore, itbecomes interesting to understand in what respect the solutions we have found dependon approximations. In particular the role of RWA should be clarified.

The easiest way of doing that is to consider transitions with the selection rules ∆m =±1 involving circularly polarized light. It can be shown easily that the only differencewith respect to the ∆m = 0 transitions discussed above is the necessity of substitutingqq for q2 in the equations for w, so that no qualitative change in the time dependence ofw will appear.

5.2 A Semiclassical Approach

The derivation of the nonlinear Bloch equations has been given shortly in the examplewith the neoclassical Jaynes theory. We have seen there that Weinberg’s theory cannot,in general, give predictions equivalent to those based on the neoclassical approach. Herewe will briefly follow the Wodkiewicz and Scully paper to compare their semiclassicalresults with these given in the previous subsection.

The nonlinearity chosen by Wodkiewicz and Scully corresponds in our notation toǫ2 = 0 and ǫ1 =

√2hς. We know that this choice leads to the “detuning” ∆′ = ∆ − 2ς.

If no spontaneous emission is introduced phenomenologically into the Bloch equations,they take the form

u = −∆′v − 2ςwv

v = ∆′u+ Ωw + 2ςwu (5.36)

w = −Ωv. (5.37)

Let now ∆′ = 0. Then

u = −2ςwv

v = Ωw + 2ςwu (5.38)

w = −Ωv. (5.39)

These equations have an additional conserved quantity

u2 + v2 +Ω

εu (5.40)

which we shall put equal 0. Differentiating the last Bloch equation we get

w = −2ςΩwu− Ω2w

= (2ς2 − Ω2)w − 2ς2w3 (5.41)

which is identical to our result (here, of course, Ω is the semiclassical Rabi frequency).

49

Chapter 6

REMARKS ON THE

MIND-BODY QUESTION

IN NONLINEAR QM

The present writer has no other qualification to offer

his views than has any other physicist and he believes

that most of his colleagues would present similar op-

pinions on the subject, if pressed.

E. P. Wigner in Remarks on the Mind-Body Question

The problem of measurement seems to be the most profound difficulty of quantum me-chanics. A very clear and concise presentation of the problem was given by Wigner inhis philosophical paper [51], a fragment of which has been chosen as the motto to thischapter. The essential part of his argumentation goes as follows.

Let us consider (...) an initial state of the object which is a linear combi-nation αψ1 +βψ2 of the two states ψ1 and ψ2. It then follows from the linearnature of the quantum mechanical equations of motion that the state of theobject plus observer is, after the interaction, αψ1⊗χ1+βψ2⊗χ2. If I now askthe observer whether he saw a flash, he will with a probability |α|2 say thathe did, and in this case the object will also give me the responses as if it werein the state ψ1. If the observer answers “No” — the probability for this is|β|2 — the object’s responses from then on will correspond to a wave functionψ2. The probability is zero that the observer will say “Yes”, but the objectgives the response which ψ2 would give because the wave function (...) of thejoint system has no ψ2 ⊗ χ1 component. (...) All this is quite satisfactory:the theory of measurement, direct or indirect, is logically consistent so longas I maintain my priviliged position as ultimate observer.

However, if after having completed the whole experiment I ask my friend,“What did you feel about the flash before I asked you?” he will answer, “I told

50

you already, I did [did not] see a flash”, as the case may be. In other words,the question whether he did or did not see the flash was already decidedin his mind, before I asked him. If we accept this, we are driven to theconclusion that the proper wave function immediately after the interactionof friend and object was already either ψ1 ⊗χ1 or ψ2 ⊗χ2 and not the linearcombination αψ1 ⊗χ1 + βψ2 ⊗χ2. This is a contradiction, because the statedescribed by the wave function αψ1 ⊗ χ1 + βψ2 ⊗ χ2 describes a state whichhas properties which neither ψ1 ⊗ χ1 nor ψ2 ⊗ χ2 has. If we substitute for“friend” some simple physical apparatus, such as an atom which may or maynot be excited by the light-flash, this difference has observable effects andthere is no doubt that αψ1 ⊗χ1 + β2 ⊗χ2 describes the properties of the jointsystem correctly, the assumption that the wave function is either ψ1 ⊗ χ1 orψ2 ⊗ χ2 does not. If the atom is replaced by a conscious being, the wavefunction αψ1 ⊗ χ1 + βψ2 ⊗ χ2 (which also follows from the linearity of theequations) appears absurd because it implies that my friend was in a state ofsuspended animation before he answered my question.

It follows that the being with consciousness must have a different rolein quantum mechanics than the inanimate measuring device: the atom con-sidered above. In particular, the quantum mechanical equations of motioncannot be linear if the preceding argument is accepted... ([51], p. 179–180)

The last sentence was the main motivation for the inclusion of the measurement problemin the work on aspects of nonlinear QM.

It seems that an unquestionable element of all the known approaches to the mea-surement problem is the so-called pre-measurement procedure. The pre-measurement isthe stage before the moment the friend of Wigner “starts to realize” that he has [hasnot] seen the flash of light. The pre-measurement prepares the initial conditions for theforthcoming “actualisation” of the result of the measurement and, assuming that thesupposed nonlinearity of the evolution is “localised in the observer”, we return to thetheme tune of our work — the description of composite systems.

I have argued several times that the “collapse-like” description proposed by Weinbergcannot be correct in usual situations, first of all because it is basis dependent. Never-theless, we must bear in mind that so long as we are dealing with observers the linearsuperposition principle is no longer valid for the observers themselves. This difficulty,outlined by Wigner in the above quotation, is known as the pointer basis problem [52].Putting aside the various theoretical proposals of its solution, we remark that once someway of inducing the “pointer superselection rules” is given, no freedom in the choice ofthe observer’s basis is left, and the Weinberg approach may be motivated. The point isthat results of single measurements are evidently, at least operationally, precisely definedfor an observer who notices the flashes of light. A technical difficulty in a nonlineardescription of the collapse (a projection of a solution is not necessarily a solution) is in-telligently passed round by Weinberg who introduces the collapse implicitly at the levelof the Hamiltonian function and not by means of some additional discontinuous evolutionlaw.

A modification of the observed (linear) system can, of course, exert an influence on theobserver; the Gisin’s telegraph belongs to such a class of phenomena, so may not be verypathological in the context of measurements. On the other hand, the telegraph basedon the mobility phenomenon would lead to “telekinetic phenomena”. Indeed, consider a

51

pre-measurement that produces an entangled state of a system and an observer. Let theobserved system be a two-level atom and an electromagnetic field, initially in the state

|1〉|1〉 + |2〉|0〉 = |11〉 + |20〉 (6.1)

where the notation is like in the previous chapter. Assume that a Hilbert space of theobserver is also spanned by two states |±〉. After the pre-measurement the state of thejoint system is

|11〉|+〉 + |20〉|−〉. (6.2)

If now the observer’s state space undergoes a nonlinear evolution with the “σ3” nonlin-earity like in (4.16) then two possibilities occur. First, if only the |±〉 start to rotatewith the mobility frequency then the atomic reduced density matrix is like in linear QM.However, if the nonlinearity violates the orthogonality of the states |1〉|+〉 := |1+〉 and|0〉|−〉 := |0−〉 (i.e. when the observer makes both his own consciousness and the inter-acting photons evolve in a nonlinear way) then the atom starts to “feel” it. It follows that,at least in principle, a suitable form of his “own” nonlinearity can enable the observerinfluence in a statistically observable way (compare (4.27)) a behaviour of a random gen-erator just by watching it! Notice that there is also some limitation on the possibilityof the influence. For consider a situation where there is a number of intermediate statesbetween the random generator and the observer, so that the entangled state takes theform

|1〉|+1〉|+2〉 . . . |+n〉|+〉 + |2〉|−1〉|−2〉 . . . |−n〉|−〉.The random generator will “feel” the mobility only provided the mobility will involvethe underbraced states

|1〉 |+1〉|+2〉 . . . |+n〉|+〉︸ ︷︷ ︸

+|2〉 |−1〉|−2〉 . . . |−n〉|−〉︸ ︷︷ ︸

.

In case the nonlinearity is more localized, say

|1〉|+1〉 |+2〉 . . . |+n〉|+〉︸ ︷︷ ︸

+|2〉|−1〉 |−2〉 . . . |−n〉|−〉︸ ︷︷ ︸

then the generator’s density matrix does not contain the “malignant” terms (one cannotinfluence the generator by watching it on TV; or, it is much easier to influence with mythoughts my own finger than someone else’s).

The above phenomenon is present only if we assume the kind of description a laWeinberg which is neither very elegant nor easy in practical applications. One may hopethat at least in the “correct” description no ways of exerting observer’s influence onexternal world by “thoughts” or “intentions” exist. The following surprising example isdue to J. Polchinski [18] (the version presented here is slightly modified with respect tothe original one).

Consider a process involving four steps.

1. A spin-1/2 ion enters a Stern-Gerlach device, which couples to the linear spin σ3

component and the beam splits into two, “+1” and “−1” sub-beams.

2. The “+1” beam evolves freely; in the path of the “−1” beam a macroscopic observer(or a random generator) takes one of two actions: (a) nothing (say, with probabilityλ1), or (b) rotates the spin into the “1” direction with a magnetic field coupled toσ2 (with probability λ2).

52

3. The two beams are rejoined and the ion enters a region of field coupled to thenonlinear observable

f(Tr ρσ1)2

Tr ρ. (6.3)

4. The observer again measures the spin with a Stern-Gerlach device coupled to σ3.

The steps 1 and 2 prepare the initial condition for the nonlinear evolution, and thereduced density matrix of the ion after step 2 is

ρ0 = λ1

(1/2 00 1/2

)

+ λ2

(3/4 1/41/4 1/4

)

. (6.4)

The ion’s nonlinear Hamiltonian for the step 3 is

H = 2fTr ρσ1

Tr ρσ1 (6.5)

where we have assumed the “correct” form of the evolution, i.e. that it is generated bythe Hamiltonian function depending only on the reduced density matrix of the ion. Thereduced density matrix satisfies

iρ = 2fTr ρσ1

Tr ρ[σ1, ρ] (6.6)

whose solution is

ρ(t) = e−2if〈σ1〉σ1tρ(0)e2if〈σ1〉σ1t

=λ1

21 +

λ2

4

(

21 + σ1 + σ3 cos 2λ2ft+ σ2 sin 2λ2ft)

. (6.7)

In the analogous manner we can calculate the evolution of the projector P± = 12 (1±σ3).

We find in the “Heisenberg picture” (i.e. we solve the Heisenberg equations of motionwith the nonlinear Hamiltonian)

P±(t) =1

2

(

1± σ3 cos 2λ2ft∓ σ3 sin 2λ2ft)

. (6.8)

The linear case, where in step three the ion couples to the linear σ1, would yield

P±(t) =1

2

(

1± σ3 cos 2ft∓ σ3 sin 2ft)

. (6.9)

Assuming that the time of the interaction during the third step satisfies 2ft = π we get,in the linear case,

P±(t) =1

2

(

1∓ σ3

)

= P∓(0), (6.10)

which means that the spin changes its sign during the evolution. In the nonlinear case,however,

P±(t) =1

2

(

1 ± σ3 cosλ2π ∓ σ3 sinλ2π)

(6.11)

hence, in particular, the evolution of P+ depends on λ2 — the probability of one of thetwo actions taken by the observer or the random generator in case the spin turned out

53

to be −1. The result means that the evolution of the ion depends on “intentions” of theobserver concerning his possible actions he would have undertaken had the spin of the ionturned out to be −1 — even in case the spin is +1 and the observer is passive! The pointis that we assume that the ion evolves linearly after having left the nonlinearity region,so that we can apply the ordinary interpretation of results of single measurements. Forexample, for λ2 = 0, that is when the observer is decided not to take any actions, thespin state of the ion would remain unchanged. In the opposite case, λ2 = 1, the evolutionwould be like in linear QM.

I cannot find any mistake in this reasoning. Notice also that since the interpreta-tional background must be some “no collapse” interpretation of QM, the effect meansthat different (Everett) branches of the Universe can somehow influence results of sin-gle measurements, which is not so bizarre if one accepts the spirit of the many-worldsinterpretation.

It seems that the essential point of this argumentation is the assumption that a singlemember of a beam of ions is described by the same density matrix as the whole ensemble.This is exactly opposite to the reasoning leading to Gisin’s telegraph. And I think thisexample indicates one of the most fundamental conceptual, or practical, difficulties ofnonlinear QM: The fact that we do not really know how to treat beams of sigle objects.Intuitively, weak beams should be “linear” while strong ones could be, perhaps, “non-linear” in some mean-field sense. The second hint for futher generalizations of linearQM is a possibility, suggested by Wigner, that the only domain of fundamental nonlin-earities could be the consciousness of observer. Then the density matrix representingthe consciousness could evolve nonlinearly and no decompositions of the observer intosub-ensembles would make any sense. Still, the Polchinski’s phenomenon can describesomething like intuition: A perception of a single event depends on the overall propertyof an ensemble of such events since the whole density matrix is involved.

A question that arises immediately is how can we distinguish between systems thatare “conscious” (observers) and “non-conscious”. The only natural characterisation thatcomes to my mind is the way they behave from the point of view of information gain —the notion defined in theory of information.

This viewpoint will become the starting point for my own proposal of nonlinear QMpresented in the next chapter. It seems it is the first version of the theory free of all themain difficulties presented until now.

54

Chapter 7

ENTROPIC FRAMEWORK

FOR NONLINEAR QM

This chapter is devoted to a new proposal of a general framework for nonlinear QM.A fundamental drawback of the Hamiltonian generalizations a la Kibble or Weinbergis, as we have seen, the fact that one has to modify observables in order to modifydynamics. A modification of observables will, in general, lead to the known problemswith results of single measurements and probabilities. These, on the other hand, formthe fundamental physical and mathematical body of the theory. Physically, averages areinherently linear concepts and higher order moments of observable quantities (randomvariables) are related to the associativity of the algebra of observables. Mathematically,the spectral theory of self-adjoint linear operators leads to the required structures in avery natural way and, last but not least, is a beautiful and elaborated theory.

The Hamiltonian form of the quantum mechanical equations of motion, with theaverages playing effectively the role of generators of canonical transformations, indicatesthat quantum dynamics is deeply rooted in probability. In fact, not only the probabilities— via average values of energy — generate the evolution of quantum systems, but theevolving being is again... the probability, since the canonical coordinates for a quantumsystem are the real and imaginary parts of probability amplitudes. Let me stress it again:The evolution of states is generated by objects that describe statistical regularities ofquantum ensembles.

Another important, or even essential, physical characterization of a statistical physicaltheory is provided by the notion of entropy. Now, is the entropy of quantum systems anobservable quantity or not? Can we take a single member of an ensemble and measureits entropy in analogy to the measurement of energy? No, we cannot. Why? Because,logically, the entropy is not a linear sum of contributions from the constituents of theensemble, but — rather — describes an overall property of the ensemble. It follows thatthere does not exist a linear operator of entropy whose average value could describe the(average) entropy of the ensemble while its eigenvalues would be the elementary entropiesof single systems. This property of entropy can be seen in its standard definition (thevon Neumann entropy)

S[ρ] = −Tr (ρ ln ρ) (7.1)

55

or, in a form of averages,S[ρ] = −〈ln ρ〉. (7.2)

Sometimes another, similar expression is considered [55, 56]

S[ρ] = − ln〈ρ〉 = − ln Tr (ρ2) (7.3)

which we shall term the Stueckelberg entropy , as it seems it was he who discussed it forthe first time in his nowadays almost forgotten papers [57].

Now, what is the role of entropy for quantum evolution? A first look at generalstructures of quantum mechanics suggests that there is no role whatever. What we cansay generally is that both entropies are conserved by unitary evolutions, because unitarityimplies conservation of Tr (ρn) for any n, a property apparently typical only for linearevolutions.

Although it seems there is no physical principle relating evolution and entropy (withthe notable exeption of the second law of thermodynamics, but this concerns a differentlevel of discussion), one of the main purposes of this chapter is an attempt to proposesuch a principle. A dynamics resulting from the principle is rather the Nambu [39] thanthe Hamilton or Poisson dynamics, although it contains the Hamiltonian and Poissonianframeworks as particular cases. It is surprising, as we shall see, that most of the propertiesof QM that are usually attributed to the linearity are typical for a large class of theories(which are not necessarily linear!) resulting from the generalized formalism.

A departure point of the new construction will be a generalization of the ordinaryPoisson bracket to a triple bracket involving an additional functional S which, followingBia lynicki-Birula and Morrison (BBM) [39], will be interpreted as a measure of entropy.Since the generalized framework will be based on different measures of entropy (or in-formation), it is best to begin the discussion with an introduction of some elementaryconcepts from information theory.

7.1 Elements of Information Theory

— from Hartley to Renyi

A logarithmic measure of information was introduced by R. V. Hartley in 1928 [58].According to him, to characterize an element of a set of size N we need log2N units ofinformation. It follows that a unit of information (1 bit) is the amount of informationnecessary for a characterization of a pair. Of course, one can choose also other unitssuch that the unit is the amount of information necessary for a characterization of a setwith 0 < k ∈ N elements, or even with 0 < r ∈ R elements in average. The respectivemeasures of information in arbitrary units a are logaN . The most important featureof the logarithmic information measure is its additivity: If a set E is a disjoint unionof M N -tuples E1, . . . , EM , then we can specify an element of this MN -element set Ein two steps: First we need logaM units of information to describe which Ek of thesets E1, . . . , EM contains the element, then we need logaN further units to tell whichelement of this Ek is the considered one. The information necessary for a characterizationof an element of E is the sum of the partial informations: logaMN = logaM + logaN .Next step in the developement of the measures of information was done independentlyby C. E. Shannon [59] and N. Wiener [60] in 1948 who derived a formula analogous

56

to Boltzman’s entropy. Their formula has the following heuristic motivation. Let Ebe the disjoint union of the sets E1, . . . , En having N1, . . . , Nn elements respectively(∑n

k=1Nk = N). Let us suppose that we are interested only in knowing the subset

Ek. (This is typical for classical statistical problems in physics: Statistical quantitiesdepend on classes of microscopic conditions and not on single microscopic properties.)The information characterizing an element of E consists of two parts: The first specifiesthe subset Ek containing this particular element and the second locates it within Ek. Theamount of the second piece of information is, by Hartley formula, logaNk thus dependson the index k. On the other hand, to specify an element of E we need logaN units ofinformation. The amount necessary for the specification of the set Ek is therefore

Ik = logaN − logaNk = logaN

Nk= loga

1

pk. (7.4)

It follows that the amount of information received by learning that a single event ofprobability p took place equals

I(p) = loga1

p. (7.5)

In statistical situations measured quantities correspond to averages of random variables.Therefore the average information is

I =∑

k

pk loga1

pk. (7.6)

This is the Shannon’s formula and I is called the entropy of the probability distributionp1, . . . , pn. If all the probabilities are equal 1/N then the Shannon’s formula is equal tothe Hartley’s one. The mean we have applied is the so-called linear mean. Renyi observedthat there exist information theoretic problems where the measures of information arethose obtained by more general ways of averaging — the Kolmogorov–Nagumo functionapproach [61]. Let ϕ be a monotonic function on real numbers. The Kolmogorov–Nagumo average information can be defined by means of ϕ as

I = ϕ−1

(∑

k

pkϕ(

loga1

pk

))

. (7.7)

If the generalized information measure is to satisfy the postulate of additivity, the func-tion ϕ cannot be arbitrary. Indeed, let a chance experiment be a union of two independentexperiments. Let us suppose that we obtain Ik units of information with probability pkin the first experiment and Jl units of information with probability ql in the second one.Thus we receive Ik +Jl units of information with probability pkql. If the average amountof information obtained from the joint experiment is the sum of the average amounts ofinformation obtained from both experiments, then

ϕ−1(∑

k,l

pkqlϕ(Ik + Jl))

= ϕ−1(∑

k

pkϕ(Ik))

+ ϕ−1(∑

l

qlϕ(Jl))

(7.8)

must hold for any probability distributions pk and ql. If Jl = J for all l then

ϕ−1(∑

k,l

pkqlϕ(Ik + Jl))

= ϕ−1(∑

k

pkϕ(Ik))

+ J (7.9)

57

which can hold if and only if ϕ is linear or exponential function. The linear functioncorresponds to Shannon’s information. The exponential functions provide a large classof new measures of information. Consider a function ϕ(x) = a(1−α)x. We can alwayschoose the units of information in such a way that

I = ϕ−1

(∑

k

pkϕ(

loga1

pk

))

=1

1 − αloga

(∑

k

pαk

)

= loga

((∑

k

pαk

)1/(1−α))

. (7.10)

For pk = 1/N we obtain again the Hartley formula. Formula (7.10) describes Renyi’sα-entropy which, from now on, will be denoted Iα(P), where P denotes the probabilitydistribution. We see that the essential part of the definition is played by

I∗α(P) = aIα(P) =(∑

k

pαk

)1/(1−α)

(7.11)

which is independent of the choice of the unit a. (The power 1/(1 − α) could be alsoreplaced by K/(1 − α), where K is a positive constant. The entropy would be thenmultiplied by K; we shall need this freedom later.) To distinguish between α-entropyand I∗α(P) we shall call the latter α∗-entropy (∗ will remind us that this quantity ismultiplicative in opposition to the additivity of Iα(P)). (The observation that whatis in fact informationally fundamental in Iα(P) is I∗α(P) is strenghtened by Daroczy’sdefinition of entropy of order α [62] defined as

(21−α − 1)−1(∑

k

pαk − 1)

. (7.12)

This expression possesses many ordinary properties of the entropy and in the limit α→ 1becomes, the so-called Shannon’s information function.)

The limit α → 1 is interesting also for α-entropies. It can be shown that I1 =limα→1 Iα equals Shannon’s entropy.

Iα(P) is a monotonic, decreasing function of α. For negative α Iα(P) tends to in-finity if one of pk tends to zero. This property excludes α < 0 because adding a newevent of probability 0 to a probability distribution, what does not change the probabilitydistribution, turns Iα(P) into infinity.

A fundamental notion in information theory is the gain of information. Consideran experiment whose results are A1, . . . , An having probabilities pk = P (A = Ak). Weobserve an event B related to the experiment and obtain a result B = Bl. Now theconditional probabilities are pkl = P (A = Ak|B = Bl). Consider now a system (an“observer”) whose information is measured by some α-entropy. How much informationabout the random variable A has he received by observation of B = Bl? The amount ofinformation he would have obtained by observing A = Ak would be equal to

loga1

pk(7.13)

if he had not measured B. After having observed B = Bl the amount of information hewould have obtained by observing A = Ak would be

loga1

pkl. (7.14)

58

It follows that the measurement of B = Bl has given him already

loga1

pk− loga

1

pkl= loga

pkpkl

(7.15)

units of information about A. The expression (7.15) is called the decrease of uncertaintyabout A = Ak by observing B = Bl. We define the gain of information about A, obtainedwhen the probability distribution pk is replaced by pkl, by

ϕ−1

(∑

k

pklϕ(

logapkpkl

))

=1

1 − αloga

(∑

k

p2−αkl

p1−αk

)

. (7.16)

If we define the increase of the uncertainty by minus decrease of uncertainty we cancalculate the average “loss of information” defined by

ϕ−1

(∑

k

pklϕ(

logapklpk

))

=1

1 − αloga

(∑

k

pαklpα−1k

)

. (7.17)

For Shannon’s entropy the gain is minus the loss. For α-entropies the two concepts areinequivalent.

The gain of information defined by (7.16) for α > 2 has the same pathological prop-erties as Iα for α < 0 so, it seems, cannot be consistently applied. This is the reasonwhy Renyi defined the gain of information as minus the loss. From the viewpoint of ourquantum mechanical applications the situation is not so clear, however.

It should be emphasized that when we speak about information, what we have inmind is not the subjective “information” possessed by a particular, animate observer.In reality the information contained in an observation is a quantity independent of thefact whether it does or does not reach the perception of the observer (be it a man, someregistering device, a computer, or some other physical system). On the other hand,different kinds of entropies introduced above may be characteristic for different systems.The entropy (information) is objective in the same sense as probability, and in the samesense it is reasonable to expect that there are classical and quantum informations, asthere are classical and quantum probabilities.

A fundamental fact about physical interpretation of the statistical nature of QM isthat the probabilities do not result from our lack of knowledge about some deeper levelof quantum systems but are, in certain sense, fundamental. It means that there doesnot exist (in principle!) any deeper level of description. When we think about a classicalstatistical system we can imagine that its internal order, or disorder, can change due tointeractions with the rest of the world. Putting it more formally, we can say that aninformation characterizing a classical system should allow different gains of informationin different situations. A contemplation of the quantum case suggests that a quantuminformation might be of such a kind that its corresponding gain of information is zerounder all circumstances. It is tempting to develop this hypothesis a little and find whethera measure of information possessing this property exists.

The Shannon’s information gain is given by

−∑

k

pkl logapklpk

(7.18)

59

and vanishes only if A and B are independent. So this case can be excluded because wewant the gain of information to be 0 for all probability distributions. For α-entropies wefind that the vanishing of (7.16) implies

∑

k

p2−αkl

p1−αk

=∑

k

pk(p2−αkl pα−2

k

)= 1 (7.19)

which can hold for all pk and pkl if and only if α = 2. It follows that the correct candidatefor the quantum entropy is the Renyi’s 2-entropy which reads

− loga

(∑

k

p2k

)

. (7.20)

Expressing the probabilities by means of a density matrix and choosing the unit ofinformation with a = e we obtain

I2[ρ] = − ln Tr (ρ2) (7.21)

hence the entropy of Stueckelberg! A possible interpretation of this result is that α-entropies for 1 ≤ α ≤ 2 represent a continuous transition between classical and quantumentropies.

7.2 Poissonian Formulation of

Quantum Mechanics

Let H be a Hilbert space. Consider the Hamilton equations

ωAA′

(α, α′)dψA(α)

dt=

δH

δψA′(α′)(7.22)

and

ωAA′

(α, α′)dψA′(α′)

dt=

δH

δψA(α)(7.23)

where the bars denote complex conjugations and the conventions concerning primedand unprimed indices are assumed like in the spinor abstract index calculus [36]. Thesummation convention is like in Chapter 2: We sum over repeated Roman indices andintegrate over repeated Greek ones. In QM the “symplectic form” is given by the deltadistribution

ωAA′

(α, α′) = iδAA′

δ(α− α′) = ωAA′

δ(α− α′) (7.24)

δAB′

= δAB′ = 1 if A = B′ and 0 for A 6= B′. The inverse of ωAA′

(α, α′) is

IAA′(α, α′) = −iδAA′δ(α − α′) = IAA′δ(α− α′) (7.25)

where by the inverse we understand that

ωAA′

(α, α′)IBA′(β, α′) = δABδ(α− β) (7.26)

ωAA′

(α, α′)IAB′(α, β′) = δA′

B′δ(α′ − β′). (7.27)

60

Accordingly, the form of the Hamilton equations we shall use is

dψA(α)

dt= IAA′

δH

δψA′(α)(7.28)

anddψA′(α)

dt= −IAA′

δH

δψA(α). (7.29)

(7.28) and (7.29) describe a quantum evolution of pure states. Assume now that allobservables of the theory depend on |ψ〉 and 〈ψ| by the density matrix ρ = |ψ〉〈ψ|. Let Fand G be two such observables, that is F [ψ, ψ] = F [ρ] and G[ψ, ψ] = G[ρ]. The Poissonbracket resulting from the Hamilton equations is

F,G = IAA′

( δF

δψA(α)

δG

δψA′(α)− δG

δψA(α)

δF

δψA′(α)

)

. (7.30)

Applying the chain rule to the components of the pure state density matrix

ρAA′(α, α′) = ψA(α)ψA′(α′) (7.31)

we find that

F,G = IAA′

( δF

δρAB′(α, β′)ρCB′(γ, β′)

δG

δρCA′(γ, α)−(F ↔ G

))

. (7.32)

So long as the density matrix in (7.32) is given by (7.31) the bracket is equivalent to thePoisson bracket (7.30). Jordan, in a context of the Weinberg’s theory [19] and for a finitedimensional Hilbert space, investigated properties of the bracket (7.32) with ρ being anarbitrary density matrix. For reasons that will be explained below I will term such ageneral bracket the Bia lynicki-Birula–Morrison–Jordan (BBMJ) bracket.

We will now show that (7.32), for a general ρ, can be written in a form of a generalizedNambu bracket. Let ρ be arbitrary. The BBMJ bracket can be rewritten as

F,G = ρAA′(α, α′)ΩAA′

BB′CC′(α, α′, β, β′, γ, γ′)δF

δρBB′(β, β′)

δG

δρCC′(γ, γ′)(7.33)

with

ΩAA′

BB′CC′(α, α′, β, β′, γ, γ′) = δACδA′

B′IBC′δ(α− γ)δ(α′ − β′)δ(β − γ′)

−δABδA′

C′ICB′δ(α − β)δ(α′ − γ′)δ(γ − β′)

= Ωabc(α, β, γ) (7.34)

where, in analogy to the spinor calculus, we have clumped together the respective pairsof indices into composite indices (a = AA′, (α, α′) = α, etc.).

The “structure kernels” Ωabc(α, β, γ) satisfy conditions characteristic for Lie-algebraicstructure constants:

Ωacb(α, γ, β) = −Ωabc(α, β, γ) (7.35)

and

Ωabc(α, β, γ)Ωcde(γ, δ, ǫ)+Ωaec(α, δ, γ)Ωcbd(γ, β, δ)+Ωadc(α, δ, γ)Ωceb(γ, ǫβ) = 0. (7.36)

61

These two conditions imply the Jacobi identity. The composite index form of the BBMJbracket

F,G = ρa(α)Ωabc(α, β, γ)δF

δρb(β)

δG

δρc(γ)(7.37)

shows that it takes the same form as the generalized BBM-Nambu bracket written interms of the Wigner function for a scalar field [39]. As a matter of fact, the BBMJbracket is simply a different representation of the BBM bracket. The formula (7.37)looks much the same as the Poisson bracket related to the Kiryllow form on coadjointrepresentations of Lie groups [30]. In fact, such brackets for general structure constantsare called the Lie-Poisson brackets (cf. [63]). As shown in [39], the structure kernelsappearing in the BBMJ bracket correspond to the Weyl-Heisenberg Lie algebra.

It remains to find out how to formulate the explicit triple bracket equivalent to (7.37).In order to do this we first have to define a “metric tensor” to lower the upper index

in the structure kernels. The apparently natural guess

gab(α, β) = Ωcad(γ, α, δ)Ωdbc(δ, β, γ) (7.38)

is incorrect as (7.38) involves expressions like δ(0) which are not distributions in theSchwartz sense.

The correct definitions are

gab(α, β) = −IAB′IBA′δ(α − β′)δ(β − α′) (7.39)

gab(α, β) = −ωAB′

ωBA′

δ(α− β′)δ(β − α′). (7.40)

The metric tensor is symmetric

gab(α, β) = gba(β, α) (7.41)

and satisfies the invertibility conditions

gab(α, β)gbc(β, γ) = gcb(α, β)gba(β, γ) (7.42)

= δACδA′

C′δ(α − γ)δ(α′ − γ′) (7.43)

= δac δ(α− γ). (7.44)

The metric tensor is a useful tool. Consider for example a ρ-independent Fb(β) =FBB′(β, β′). Then

F [ρ] = gab(α, β)ρa(α)Fb(β)

= δAB′

δBA′

δ(α− β′)δ(β − α′)ρAA′(α, α′)FBB′(β, β′)

= ρB′

A′(β′, α′)FA′

B′(α′, β′) = Tr ρF (7.45)

and we see that bilinear observables can be naturally expressed with the help of (7.40).This example is important also as an illustration of the convention concerning loweringand raising of indices (following from the use of the metric tensor). For notice that

δF

δρB′

A′(β′, α′)= FA

′

B′(α′, β′) (7.46)

although the staggering of indices like F A′

B′ (β′, α′) might seem more natural.

62

Example 7.1 To show explicitly how the “kernel formalism” works consider the non-relativistic kinetic Hamiltonian function

H [ρ] = Tr Hρ =

∫

d3xd3yδ(~x− ~y)−∆x

2mρ(~x, ~y)

=

∫

d3xd3yδ(~x− ~y)

∫

d3p/(2π)3p2

2me−i~p·~xρ(~p, ~y)

=

∫

d3x

∫

d3p/(2π)3p2

2me−i~p·~xρ(~p, ~x)

=

∫

d3x

∫

d3p/(2π)3p2

2me−i~p·~x

∫

d3yei~p·~yρ(~y, ~x)

=

∫

d3xd3y

∫

d3p/(2π)3p2

2mei~p·(~y−~x)ρ(~y, ~x)

=

∫

d3xd3y

∫

d3p/(2π)3p2

2mei~p·(~x−~y)ρ(~x, ~y)

=

∫

d3xd3yδH [ρ]

δρ(~x, ~y)ρ(~x, ~y)

=

∫

d3xd3yH(~y, ~x)ρ(~x, ~y). (7.47)

The latter two equations are consistent with the definition of the functional derivative interms of the directional derivative:

∇fH [ρ] = limλ→0

H [ρ+ λf ] −H [ρ]

λ=:

∫

d3xd3yδH [ρ]

δρ(~x, ~y)f(~x, ~y). (7.48)

Eq. (7.47) can be also written as

∫

d3xd3yH(~y, ~x)ρ(~x, ~y)

=

∫

d3xd3yd3x′d3y′δ(~x − ~y′)δ(~y − ~x′)H(~x, ~x′)ρ(~y, ~y′)

= g(x, y)H(x)ρ(y) = gxyHxρy.2 (7.49)

The fully covariant form of the structure kernels is

Ωabc(α, β, γ) = gad(α, δ)Ωdbc(δ, β, γ)

= −IAD′IDA′δ(α− δ′)δ(δ − α′)

×(δDC δ

D′

B′ IBC′δ(δ − γ)δ(δ′ − β′)δ(β − γ′)

−δDB δD′

C′ ICB′δ(δ − β)δ(δ′ − γ′)δ(γ − β′))

= −(IAB′ICA′IBC′δ(α − β′)δ(γ − α′)δ(β − γ′)

−IAC′IBA′ICB′δ(α − γ′)δ(β − α′)δ(γ − β′)). (7.50)

One easily verifies that Ωabc(α, β, γ) is totally antisymmetric (where the permutation ofindices is understood in the sense of (7.35)).

63

Following Bia lynicki-Birula and Morrison let us introduce the functional

S2 =1

2gab(α, β)ρa(α)ρb(β) =

1

2δAB

′

δBA′

ρAA′(α, β)ρBB′ (β, α) =1

2Tr (ρ2). (7.51)

We can see that S2 is one half of the inverse of Renyi’s 2∗-entropy and for this reason itwill be termed just entropy.

The BBMJ bracket is now equal to the following triple bracket

F,G = [F,G, S2] = Ωabc(α, β, γ)δF

δρa(α)

δG

δρb(β)

δS2

δρc(γ). (7.52)

The antisymmetry of the triple bracket means that S2 is the Casimir for the BBMJbracket Lie algebra of observables. Another Casimir is

Tr ρ = δAA′

δ(α− α′)ρAA′(α, α′) (7.53)

because

Tr ρ,G = −iδAA′

(

δAB′

δ(α− β′)ρCB′(γ, β′)δG

δρCA′(γ, α)

− δG

δρAB′(α, β′)ρCB′(γ, β′)δCA

′

δ(γ − α))

= −i(

ρCA′(γ, α)δG

δρCA′(γ, α)

− δG

δρAB′(γ, β′)ρAB′(γ, β′)

)

= 0. (7.54)

Notice that until now we have not made any specific assumptions about the form of theobservables (including the Hamiltonian function in (7.28), (7.29)) like, say, linearity or1-homogeneity in ρ. The results are therefore general. The wave functions have beeneliminated from the dynamical equations, but the Hilbert space background is implicitlypresent in the structure kernels and the metric tensor which are defined in terms of ωand I, and in the very notion of the density matrix which acts in the Hilbert space.

Components of the pure state density matrix satisfy

d

dtρAA′(α, α′) = ρAA′(α, α′), H. (7.55)

which holds also for general density matrices as can be seen from the familiar, operatorversion of the Liouville–von Neumann equation. It follows that the density matrices forma Poisson manifold, as opposed to state vectors that form a phase space.

7.3 Nambu-like Generalization of

Quantum Mechanics

At a first glance it is rather surprising that in linear QM the triple bracket [F,G, S2],whose antisymmetric form does not favour any of the three functions, involves observablesF and G which are necessarily linear in ρ, and S2 which is not linear in ρ. A closer look

64

shows, however, that the bracket with S2 makes [F,G, S2] linear in ρ, so that a timederivative of an observable is also an observable. It is clear that this property holds onlyfor entropies quadratic in ρ, which suggests that although S2 itself is not an observable,generalizations of S2 to some more general S make no sense for physical reasons. Ifwe replace the linearity with the weaker homogeneity requirement, like in the Kibble-Weinberg approach, the triple bracket will map the set of observables into itself providedthe entropy will have the same homogeneity as S2.

From the viewpoint of nonlinear generalizations of QM it is important to understandwhether it is indeed physically necessary to have a bracket which maps observables intothemselves. To fix our attention let us consider the nonrelativistic position operator. Anaverage velocity of an ensemble of particles can be calculated either by first calculatingan average position and then taking its time derivative, or by first measuring the velocityof each single particle and then taking the average. We can say that the first procedure isa calculation of the time derivative of an average, whereas the latter is taking the averageof the time derivative. The situation can be described symbolically by the equation

d

dt〈~q〉 = 〈 d

dt~q〉. (7.56)

Still, it is an important property of the QM formalism that we cannot realize the twoprocedures simultaneously, as ~v = ~p/m and ~q are complementary. It follows that in aconcrete experiment we have to decide which way of measuring to choose. In this meaningif we can measure ~q, we cannot measure d

dt~q, and vice versa. To express it differently, if

~q is observable (not an observable!) then ddt~q is not. This restriction is very important

in QM because it can be used to eliminate the paradoxes of the Einstein-Podolsky-Rosenvariety.

In the version of nonlinear QM discussed below we will meet the following problem.The observables will be defined as quantities that are in one-to-one relationship to someexperimentally measured random variables. Two observables will be said to be comple-mentary if there does not exist a physical situation where the two respective randomvariables can be measured simultaneously, that is in a single run of an experiment. Wewill therefore define observables as functionals linear in ρ but the nonlinearity will beintroduced through the entropy by means of the triple bracket evolution equation. Wewill see that in this formulation a derivative of an observable will not be linear in ρ hencewill not be considered as an observable, unless the derivative is 0, of course. For example,if a position of some system is given by ~q[ρ] = Tr ~qρ, where ~q is a linear operator, then

there will not exist, in general, any linear ddt ~q such that

d

dtTr ~qρ = Tr

d

dt~qρ. (7.57)

It will be assumed, however, that there does exist an operator ddt ~q with the interpreta-

tion of velocity, which can be measured in some (complementary) experiment and whose

average will correspond to Tr ddt ~qρ but which will not have to satisfy (7.57). To put it

differently, each observable can be measured in a standard frequency-like way but deriva-tives of observables cannot be considered simultaneously with the observables themselves.And vice versa, if we measure an observable which is the time derivative of some otherobservable then the latter cannot be considered simultaneously with its derivative. This

65

is a somewhat stronger version of the complementarity principle which has to be acceptedin the nonlinear framework.

After these remarks I can begin with my own proposal.Let us replace S2 with some arbitrary S and introduce the “S-bracket”

F,GS = [F,G, S]. (7.58)

where now for different S we will have different brackets.The main postulate of the generalized framework is that the evolution equation for

density matrices is the triple bracket analogue of the Liouville-von Neumann equation,i.e.

d

dtρAA′(α, α′) = ρAA′(α, α′), HS (7.59)

where S has the same homogeneity as S2. This implies that, like in ordinary QM, thescaling by a constant ρ → λρ is a symmetry of the dynamics. We will assume thatobservables are linear in ρ, so can maintain the associative structure of random variablesjust like in the linear theory.

The first step in this new program is to understand to what degree the structure ofordinary QM depends on the form of S2.

We have explained already that only for S = S2 the set of linear observables isclosed with respect to ·, ·S. Next question concerns the Jacobi identity, or under whatconditions the manifold of states can be regarded as a Poisson manifold. We shall seethat the identity holds for all S that are differentiable functions of f2[ρ] = Tr (ρ2), i.e.S[ρ] = S(f2[ρ]). It seems that in more general cases the Jacobi identity will not have tohold (I am working on this point presently but there still exist some unclear elements inthe proof of a more general theorem). However, even if the manifold of states will nolonger be a Poisson manifold and the dynamics will not be Poissonian, the dynamicalsystem will have many properties characteristic for ordinary QM. We will find a largeclass of conserved quantities (including energy) and show that, at least for a large classof initial conditions, the solution of the evolution equation is a density matrix with timeindependent eigenvalues.

7.3.1 The Jacobi Identity

Let F , G, H and S be arbitrary twice functionally differentiable functionals. Let

F,GS = Ωabc(α, β, γ)δF

δρa(α)

δG

δρb(β)

δS

δρc(γ)

= ΩabcδF

δρa

δG

δρb

δS

δρc(7.60)

where in the second line the discrete and continuous indices have been clumped into singleones. (We could do this in majority of calculations. The fact that I have decided to usenormally the more complicated form involving both Roman and Greek letters followsfrom fear of loosing control over the convention; here the mathematical operations arereduced mainly to permutations of indices.) We consider the expression

J =F,GS , H

S+H,FS, G

S+G,HS, F

S

66

= Ωabcδ

δρa

(

ΩdefδF

δρd

δG

δρe

δS

δρf

)δH

δρb

δS

δρc

+ Ωabcδ

δρa

(

ΩdefδH

δρd

δF

δρe

δS

δρf

) δG

δρb

δS

δρc

+ Ωabcδ

δρa

(

ΩdefδG

δρd

δH

δρe

δS

δρf

) δF

δρb

δS

δρc(7.61)

=δ2F

δρaδρd

δG

δρe

δS

δρf

δH

δρb

δS

δρc

(ΩabcΩdef + ΩdefΩbac

)

+δF

δρd

δ2G

δρaδρe

δS

δρf

δH

δρb

δS

δρc

(ΩabcΩdef + ΩedfΩabc

)

+δ2H

δρaδρd

δF

δρe

δS

δρf

δG

δρb

δS

δρc

(ΩabcΩdef + ΩdefΩbac

)

+δF

δρd

δG

δρe

δ2S

δρaδρf

δH

δρb

δS

δρc

(ΩabcΩdef + ΩfecΩbda + ΩadcΩebf

)(7.62)

=δF

δρd

δG

δρe

δ2S

δρaδρf

δH

δρb

δS

δρc

(ΩdefΩabc + ΩbdfΩaec + ΩebfΩadc

)(7.63)

δ2Sδρaδρf

= gaf for S = S2 and (7.63) vanishes in virtue of (7.36). For more general

S = S(f2[ρ]) we find

δS

δρc= 2

∂S

∂f2ρc (7.64)

δ2S

δρaδρf= 4

∂2S

∂f22

ρaρf + 2∂S

∂f2gaf . (7.65)

Inserting these expressions into (7.63) we obtain

J = 8δF

δρd

δG

δρe

∂2S

∂f22

ρaρfδH

δρb

∂S

∂f2ρc(ΩdefΩabc + ΩbdfΩaec + ΩebfΩadc

)= 0 (7.66)

since Ωabcρaρc = 0. With this choice of S we obtain the dynamics given by

d

dtρAA′(α, α′) = ρAA′(α, α′), HS2C[ρ] (7.67)

where C[ρ] = 2 ∂S∂f2

= C(f2[ρ]) is an integral of motion, as we shall see later. The onlydifference with respect to ordinary QM would be in a ρ-dependent rescaling of time.

For general S the work on Jacobi identity is in progress but, as of now, we haveto accept the possibility that the (mixed) states in the generalized QM do not form aPoisson manifold. Accordingly, the observables will not form a Lie algebra which maysuggest that the generalized framework cannot be formulated in a Poincare covariantway. In Appendix 8.1 I present a formulation of the Dirac equation a la Tomonaga-Schwinger [53, 54] which can be used for a generalization of the Dirac equation along thelines proposed in this work.

67

7.3.2 Composite Systems in the New Framework

Let the Hilbert space in question and the density matrix of some composite system beH = H1 ⊗H2 and

ρa(α) = ρAA′(α, α′) = ρA1A2A′

1A′

2(α1, α2, α

′1, α

′2). (7.68)

The same doubling of indices concerns

IAA′(α, α′) = −iδA1A′

1δA2A′

2δ(α1 − α′

1)δ(α2 − α′2). (7.69)

Reduced density matrices of the two subsystems are

ρIA1A′

1(α1, α

′1) = δA2A

′

2δ(α2 − α′2)ρA1A2A′

1A′

2(α1, α2, α

′1, α

′2) (7.70)

ρIIA2A′

2(α2, α

′2) = δA1A

′

1δ(α1 − α′1)ρA1A2A′

1A′

2(α1, α2, α

′1, α

′2) (7.71)

and satisfy

δρIA1A′

1(α1, α

′1)

δρB1B2B′

1B′

2(β1, β2, β′

1, β′2)

= δB1

A1δB2B

′

2δB′

1

A′

1δ(β1 − α1)δ(β2 − β′

2)δ(β′1 − α′

1) (7.72)

and

δρIIA2A′

2(α2, α

′2)

δρB1B2B′

1B′

2(β1, β2, β′

1, β′2)

= δB2

A2δB1B

′

1δB′

2

A′

2δ(β2 − α2)δ(β1 − β′

1)δ(β′2 − α′

2). (7.73)

The structure kernels for the composite system are

Ωabc(α, β, γ) = Ωa1a2b1b2c1c2(α1, α2, β1, β2, γ1, γ2)

= −i(

δA1B′

1δC1A′

1δB1C′

1δA2B′

2δC2A′

2δB2C′

2×

δ(α1 − β′1)δ(γ1 − α′

1)δ(β1 − γ′1)δ(α2 − β′2)δ(γ2 − α′

2)δ(β2 − γ′2)

− δA1C′

1δB1A′

1δC1B′

1δA2C′

2δB2A′

2δC2B′

2×

δ(α1 − γ′1)δ(β1 − α′1)δ(γ1 − β′

1)δ(α2 − γ′2)δ(β2 − α′2)δ(γ2 − β′

2))

.

(7.74)

The following two results solve generally the question of faster-than-light telegraphs inboth Hamiltonian and triple bracket frameworks.

Lemma 7.1

Ωabc(α, β, γ)δρId1(δ1)

δρa(α)

δρIId2(δ2)

δρb(β)= 0. (7.75)

Proof : It is sufficient to contract (7.74) with (7.72) and (7.73).2

Theorem 7.2 Let F = F [ρI ] and G = G[ρII ], that is depend on ρ via (7.70) and (7.71),then for any S

F,GS = 0. (7.76)

68

Proof : By virtue of the lemma one has

0 = Ωabc(α, β, γ)δρId1(δ1)

δρa(α)

δρIId2(δ2)

δρb(β)

δF

δρId1(δ1)

δG

δρIId2(δ2)

δS

δρc(γ)= F,GS . (7.77)

2

Notice that we have not assumed anything but differentiability not only about S butalso about F and G. So, in particular, for arbitrary (nonlinear) observables and S = S2

we obtain the Polchinski-Jordan result for Weinberg’s nonlinear QM.

7.3.3 Density Matrix Interpretation of Solutions

of the Evolution Equation

One of the essential questions we have to clarify concerns the density matrix interpreta-tion of the solutions of (7.59). In fact, it seems there is no general a priori guaranteethat the generalized dynamics will conserve the positivity of ρ. The next theorems willgive a partial answer to this problem.

In order to attack the question we have to make the language of the S-brackets morereadable. Consider the triple bracket [F,G,H ] of arbitrary functionals F , G and H . Wefind that

i[F,G,H ] =δF

δρB′

A′(β′, α′)

δG

δρC′

B′(γ′, β′)

δH

δρA′

C′(α′, γ′)

− δF

δρC′

A′(γ′, α′)

δG

δρA′

B′(α′, β′)

δH

δρB′

C′(β′, γ′). (7.78)

Applying the notation of (7.46) (where now the “operator” kernels are in general ρ-dependent) we transform (7.78) into

FA′

B′(α′, β′)GB′

C′(β′, γ′)HC′

A′(γ′, α′)

−FA′

C′(α′, γ′)GB′

A′(β′, α′)HC′

B′(γ′, β′) = Tr([F , G]H

). (7.79)

In the last line we have introduced an abbreviated convention based on the assignmentto any functional F an operator

F =δF

δρ(7.80)

which is defined by the kernel form used in (7.79). For example

ρ =δS2

δρ, (7.81)

andδTr

(ρn)

δρ= nρn−1, (7.82)

the latter being the shortened form of

δTr(ρn)

δρAA′(α, α′)= nδB

′

nAδA′B2δB

′

2B3 . . . δB′

n−1Bn

× δ(β′n − α)δ(α′ − β2)δ(β′

2 − β3) . . . δ(β′n−1 − βn)

× ρB2B′

2(β2, β

′2) . . . ρBnB′

n(βn, β

′n). (7.83)

69

The first of these implies the known result

[F,G, S2] = −iTr(ρ[F , G]

). (7.84)

Consider now a functional S (differentiable in fk)

S[ρ] = S(f1[ρ], . . . , fn[ρ], . . .

)(7.85)

where fk[ρ] = Tr (ρk).

Theorem 7.3 For any m ∈ N, and any G, if S satisfies (7.85) then

[fm, G, S] = 0. (7.86)

Proof :

[Tr (ρm), G, S] =∑

n

[Tr (ρm), G, fn]∂S

∂fn= −im

∑

n

nTr([ρm−1, G]ρn−1

) ∂S

∂fn

= −im∑

n

nTr(G[ρm−1, ρn−1]

) ∂S

∂fn= 0. (7.87)

2

This remarkable result covers many interesting and nontrivial generalizations of S2. As aby-product it shows also that the same property holds for the Polchinski-Jordan version ofWeinberg’s nonlinear QM because we have not assumed that G is linear in ρ (moreover, itincludes other theories where observables do not satisfy any homogeneity condition). Theparticular case m = 1 implies that Tr ρ is conserved by all evolutions, a fact importantfor a definition of averages. For pure states Tr (ρm) = (Tr ρ)m so that the integralsfm are not necessarily independent, but for all m,n fm and fn are in involution withrespect to ·, ·S. Jordan proved in [19] by an explicit calculation that in his formulationof Weinberg’s nonlinear QM Tr ρ and Tr ρ2 are conserved — our theorem considerablygeneralizes this result.

Theorem 7.4 Let S satisfy (7.85) and ρt be a self-adjoint solution of (7.59). If ρ0 ispositive and has a finite number of nonvanishing eigenvalues pk(0), 0 < pk(0) ≤ 1, thenthe eigenvalues of ρt are integrals of motion, and the evolution conserves positivity of ρt.

Proof : Since the nonvanishing eigenvalues of ρ0 satisfy 0 < pk(0) ≤ 1 < 2, it followsthat for any α pk(0)α can be written in a form of a convergent Taylor series. By virtueof the spectral theorem the same holds for ρα0 and Tr (ρα0 ). Each element of the Taylorexpansion of Tr (ρα0 ) is proportional to fn[ρ0], for some n. But fn[ρ0] = fn[ρt] hence

Tr (ρα0 ) = Tr (ραt ) =∑

k

pk(0)α =∑

k

pk(t)α (7.88)

for all real α. Since all pk(0) are known (the initial condition), we know also∑

k pk(0)α =∑

k pk(t)α for any α. We can now use the known result used in the information theory[61] stating that the knowledge of

∑

k pk(t)α for all α uniquely determines pk(t). Thecontinuity in t implies that pk(t) = pk(0). 2

70

The spectral decomposition of the density matrix

ρt =∑

k

pk|k, t〉〈k, t|, (7.89)

where t 7→ |k, t〉 defines a one-parameter continuous family of orthonormal vectors, leadsto the unitary (although ρ-dependent) transformation |k, t〉 = U(ρt, ρ0)|k, 0〉. The densitymatrix evolves then as follows

ρt = U(ρt, ρ0)ρ0U(ρt, ρ0)−1. (7.90)

Example 7.2 Let us consider the simplest nontrivial case of a two dimensional Hilbertspace. For S = Sn+1 discussed below (cf. (7.97)) the evolution equation (up to constantfactors) is given by

id

dtρ = [H, ρn]. (7.91)

Take, for simplicity, a natural even n = 2m. Let

ρ = µ(1 + ~ν~σ), (7.92)

then

ρn = µn( n∑

2k=0

ν2k1 +n−1∑

2k+1=1

ν2k+1~n~σ)

. (7.93)

Finally, for H = H01 + ~H~σ,

µ = 0,

ν = 0,

~ν = 2µnn−1∑

2k+1=1

ν2k ~H × ~ν, (7.94)

hence we get Bloch equations with the frequencies depending on constant functions ofinitial conditions. Traces of all powers of ρ depend only on µ and ν so are integrals ofmotion. It follows that in such a case the nonlinearity of the equation of motion does notlead us too far from the linear evolution of ordinary QM. The Bloch vector is rotatingbut the angles of rotation depend nonlinearly on initial conditions. 2

The question whether the same holds good for ρ0 having an infinite number of nonva-nishing eigenvalues will be left open here. In any case, it seems that the above theoremis sufficient at least “for all practical purposes”.

The fact that for any S the evolution conserves Tr (ρα) suggests that the generalizationof S2 to (7.85) may be in a sense trivial. To see that this is not the case we have to makeour proposal a little bit more concrete — we have to choose some explicit “physical”class of S — and here the information theoretic introduction may be helpful.

We have noted that the functional Tr (ρ2) is a natural measure of quantum entropy —it describes a system that cannot “learn” or, more formally, whose gain of informationis 0 under all circumstances. Let us assume that the appearance of S2 in the BBMJbracket is not just a coincidence but reflects the fact that the linear theory is describedby the 2-entropy of Renyi. We formulate therefore the following

71

Hypothesis 7.1 The fundamental law of quantum physics is that the evolution of den-sity matrices is governed by

d

dtρAA′(α, α′) = ρAA′(α, α′), HS (7.95)

where S is a measure of entropy and H = Tr ρH is a measure of average energy. Averagesof observables are defined by

Tr ρF

Tr ρ(7.96)

which implies that ρ→ λρ, for any constant λ ∈ C, must be a symmetry of (7.95).

The suggestion of Wigner that a natural area for nonlinear generalizations of the linearformalism of QM is the domain of observations leads to investigation of systems thatcan gain information hence are described by α 6= 2 entropies. A homogeneity preservinggeneralization of S2 for other α-entropies can be, for instance,

Sα[ρ] =(

1 − 1

α

)(Tr (ρα)

)1/(α−1)

(Tr ρ)1/(α−1)−1. (7.97)

The choice of the denominator is important only from the point of view of the homogene-ity of the evolution equation. The multiplier 1 − 1/α guarantees that the evolution ofpure states is the same, hence linear , for all α ( this is reasonable as pure states have thesame, vanishing α-entropies). The generalized Liouville-von Neumann equation followingfrom (7.97) is

id

dtρ =

(Tr (ρα)

)1/(α−1)−1

(Tr ρ)1/(α−1)−1[H, ρα−1]. (7.98)

For pure states and Tr ρ = 1, ρn = ρ and the equation reduces to the ordinary, linearone; for mixed states the evolution is nonlinear unless the states are “so mixed” that ρis proportional to the unit operator (which makes sense in finite dimensional cases, ofcourse) and all α-entropies reduce to the Hartley formula.

The evolution of (now linear) observables is governed by

id

dtF =

(Tr (ρα)

)1/(α−1)−1

(Tr ρ)1/(α−1)−1Tr(ρα−1[F , H ]

)(7.99)

which shows that for the generalized S the time derivative of an observable is not linearin the density matrix. For α = 2 the equations reduce again to the ordinary linearequations.

It seems that the following choice of Sα is also interesting:

Sα[ρ] =1

2

(Tr (ρα)

)1/(α−1)

(Tr ρ)1/(α−1)−1. (7.100)

For pure states the expression reduces to the linear form 12 〈ψ|ψ〉2 = 1

2Tr (ρ2). The densitymatrix would satisfy then the equation

id

dtρ =

1

2

α

α− 1

(Tr (ρα)

)1/(α−1)−1

(Tr ρ)1/(α−1)−1[H, ρα−1] (7.101)

72

which for pure states and normalized ρ would become

2α− 1

αid

dtρ = [H, ρ] (7.102)

and the “Boltzmann-Shannon classical limit” α → 1 of the Renyi entropy is analogousto the h→ 0 classical limit of QM.

7.3.4 Composition Problem for Subsystems

with Different Entropies

Of course, even assuming that the ideology advocated here is meaningful we shouldbear in mind that the regions of our real world where some nonlinearities reside may berather few and far between. So one of the first things we have to understand is againthe composition problem: What is the form of the equations of motion of a compositesystem whose parts are described by different entropies?

I think it is best to approach the question again in an information theoretic way. Tobegin with, let us consider a system whose entropy is Iα, and whose subsystems haveentropies of the same kind: for all k a k-th system’s entropy satisfies Iαk

= Iα. Let thek-th subsystem be described by a reduced density matrix ρk. The entropy of the “large”system should be defined, as usual in information theory, as the average entropy of thesubsystems. The overall entropy of the large systems should not depend on the way wedecompose it into subsystems. Therefore the average cannot have the apparently naturalform

I1...n[ρ] =

n∑

k

pkIα[ρk], (7.103)

where pk are some weights, because the LHS is sensitive to correlations between thesubsystems whereas the RHS is not, so that the entropy would be sensitive to the de-compositions which are arbitrary. It seems we have to assume that in such a case thecomposition takes the trivial form

I1...n[ρ] =∑

k

pkIαk[ρ] =

∑

k

pkIα[ρ] = Iα[ρ]. (7.104)

Consider now a situation where the different subsystems have different entropies, say,Iαk

. The average entropy of the composite system is now defined in analogy to (7.104)as

Iα1...αn[ρ] =

∑

αk

pkIαk[ρ] (7.105)

where the probabilities pk are weights describing the “percentage” of each of the entropiesin the overall entropy of the system. We do not know how to determine the weights —they can play a role of parameters characterizing the system.

The above definitions imply that the α∗-entropy of the large system is

I∗α1...αn[ρ] =

∏

αk

I∗αk[ρ]pk , (7.106)

so it is natural to defineSα1...αn

[ρ] =∏

αk

Sαk[ρ]pk . (7.107)

73

Denoting the latter expression by S, we obtain the evolution of the density matrix of thewhole system governed by

d

dtρb(β) = [ρb(β), H, S]

=∑

αk

[ρb(β), H, Spkαk

]Sp1α1. . . Spk−1

αk−1Spk+1αk+1

. . . Spnαn

=∑

αk

pk[ρb(β), H, Sαk]Sp1α1

. . . Spk−1αk−1

Spk−1αk

Spk+1αk+1

. . . Spnαn

=∑

αk

pk[ρb(β), H, Sαk]S

Sαk

. (7.108)

Consider again a system which consists of subsystems equipped with the entropy of thesame kind. Then Sαk

= Sαl= S for all k and l and the system evolves according to

d

dtρb(β) = [ρb(β), H, S] (7.109)

as expected.Next possibility is that the entropies that sum to the overall entropy are again sums

of some other entropies. The description of the whole system should not depend on theorder in which the partial entropies are summed up. So consider two entropies SI andSII , with appropriate weights λI and λII , and let the entropies SI and SII consist of someother entropies SIk and SIIl appearing with weights pIkNk=1 and pIIl Ml=1, respectively.Then

d

dtρb(β) = ρb(β), HS = λIρb(β), HSI

S

SI+ λIIρb(β), HSII

S

SII

=∑

k

λIpIkρb(β), HSIk

SI

SIk

S

SI+∑

l

λIIpIIl ρb(β), HSIIl

SII

SIIl

S

SII

=∑

k

λIpIkρb(β), HSIk

S

SIk+∑

l

λIIpIIl ρb(β), HSIIl

S

SIIl(7.110)

which shows that the evolution can be indeed consistently composed of “sub-entropies”.If all Sαk

[ρ] depend on ρ only via fm[ρ], like in our definitions (7.97) and (7.101),we know that on general grounds they are integrals of motion. Consider a subsystemdescribed by ρk and which is noninteracting with the subsystem “where the nonlinearityresides”. In such a case the overall Hamiltonian function is

H [ρ] =∑

k

Hk[ρk] (7.111)

andd

dtρk b(β) =

∑

l

pl[ρk b(β), Hk[ρk], Sαl]S

Sαl

. (7.112)

A system described by ρk and Sαkcan be totally isolated from the “rest of the Universe”,

if for k 6= l,[ρk b(β), Hk[ρk], Sαl

] = 0 (7.113)

74

and

pk =Sαk

S. (7.114)

However, even in such a case the global properties of the large system leave their markon the local properties of all the subsystems as

pkS

Sαk

(7.115)

is at most an integral of motion hence depends on initial conditions. Let, for example,

S[ρ] = Sα1...αn[ρ] =

∏

αk

Sαk[ρk]pk . (7.116)

where the different subsystems have different entropies. Each of the sub-entropies satisfies

d

dtSαk

[ρk] = [Sαk[ρk], H, S]

=∑

αl

pl[Sαk[ρk], H, Sαl

(ρl)]S

Sαl

= 0 (7.117)

in virtue of the theorem 7.2 (we have used here the fact that F,HS = −F, SH), andthe antisymmetry of the triple bracket. Therefore not only the overall entropy, but alsothe sub-entropies are integrals of motion even if the Hamiltonian function H containsinteraction terms .

Consider now the reduced density matrix ρk of one of the subsystems. Using thesame theorem we find that

d

dtρk b(β) = pk[ρk b(β), H, Sαk

]S

Sαk

. (7.118)

where S/Sαkis an integral of motion but its value depends on initial conditions. If the

change of the initial conditions does not affect the reduced density matrices in (7.116),the integral of motion is also unchanged. Therefore in order to change this quantity wehave to change correlations between the subsystems. In particular, a time dependence ofa linear system which is noninteracting with the nonlinear one is insensitive to changesof initial conditions within the linear system if the particular form (7.116) holds. Forglobal entropies different from (7.116) some kind of sensitivity appears but the influ-ences between the subsystems cannot propagate faster than light unless we introduce theprojection postulate.

Such a trace of nonlinearity observed in some linear system might be used to detectthe nonlinearity. Following Santilli [64] we can expect that an evolution of an inside partof a hadron may be nonliner (like in hadronic mechanics). In such a case correlationsbetween a hadron (say, a proton) and some linear system (say, an electron) could beobserved in a form of a ρ-dependent rescalig of time in the electron’s evolution.

75

Chapter 8

APPENDICES

8.1 Elements of Relativistic Formulation

— Hamiltonian Formulation

of the Dirac Equation

Consider the Dirac equation(iγa∇a −m)ψ = 0 (8.1)

where ∇a = ∂a+ ieΦa and Φa is an electromagnetic potential world-vector. We are goingto rewrite the equation in a form of the “proper-time” covariant Hamilton equations ofmotion. The “proper time” will be defined in terms of spacelike hyperplanes constructedas follows. Let στ (x(τ)) = 0 be an equation defining a family of spacelike hyperplanes.The field of timelike, future-pointing, normalized vectors naτ (x) ∝ ∂aστ (x), satisfying thecontinuity equation ∂an

aτ (x) = 0, defines the field of “proper time” directions. Integral

curves τ 7→ xa(τ) of naτ (x), where τ is the parameter of the family στ, play the roleof the world-lines. We shall need the continuity equation to guarantee the reality of theHamiltonian function. Notice that this condition eliminates some physically meaningfulhyperplanes, like the proper-time hyperboloid στ (x) = xaxa − τ2 = 0, but admits si-multaneity hyperplanes στ (x) = naxa − τ = 0. The “proper time” following from theconstruction should not, for this reason, be identified with the ordinary proper time ofthe electron. The “proper time” derivative at x is defined as

d

dτ= naτ (x)∂a. (8.2)

Let φA(x) and χA′

(x) be the spinor components of a bispinor. Our aim is to rewrite theDirac equation in the Hamiltonian form

ωAA′ d

dτφA(x) =

δH

δφA′(x), (8.3)

ωAA′

d

dτχA

′

(x) =δH

δχA(x), (8.4)

ωAA′ d

dτφA′(x) =

δH

δφA(x), (8.5)

76

ωAA′

d

dτχA(x) =

δH

δχA′(x). (8.6)

We cannot do this directly in terms of (8.1) because the derivatives are already contractedover the world-vector idices. So let us transform (8.1) into new, rather unusual form.Multiplying (8.1) from left by the Dirac matrices we obtain

0 =(iγaγb∇b −mγa

)ψ

=( i

2[γa, γb]+∇b +

i

2[γa, γb]∇b −mγa

)ψ

=(i∇a + σab∇b −mγa

)ψ. (8.7)

Writing the four-potential explicitly in

i∂aψ =(−σab∂b + eΦa + ieσabΦb +mγa

)ψ (8.8)

and contracting with naτ (x) we get

id

dτψ(x) =

(

−σabnaτ (x)∂b + enaτ (x)Φa(x) + ieσabnaτ (x)Φb(x)

+ mnaτ (x)γa

)

ψ(x) (8.9)

= Hψ(x). (8.10)

The Hamiltonian operator is here a four-scalar, as opposed to the ordinary Dirac Hamil-tonian which transforms as p0. The Dirac equation (8.8) can be written in a more familiarform if, for Φa = 0, we introduce the Pauli-Lubanski vector

i

2σabpb = −1

2σab∂b = γ5Sa. (8.11)

The equation1

2pa = γ5Sa +

m

2γa (8.12)

expresses the relationship between the four-velocity operator γa and the four-momentumpa. For m = 0 it shows that spin is helicity × momentum. It can be shown also thatspinor equations for any spin and m = 0 (including the source-free Maxwell equations)always take this form.

In what follows we will need the spinor version of (8.9). The Dirac equation will bewritten as

i∇AA′

φA = µχA′

= ig AA′

a ∇aφA (8.13)

i∇AA′χA′

= −µφA = igaAA′∇aχA′

(8.14)

where µ = m/√

2 and g AA′

a are the Infeld-van der Waerden symbols [36]. We will needthe identities [65]

gaXA′gbY A′

+ gb XA′gaY A′

= gabε YX (8.15)

gaXA′gbY A′ − gb XA′gaY A

′

= 4σab YX (8.16)

gaAX′gbAY′ − gb AX′gaAY

′

= 4σab Y ′

X′ (8.17)

77

where σab YX and σab Y ′

X′ are generators of (12 , 0) and (0, 1

2 ) representations of SL(2,C).The generators written in a purely spinorial way are

σAA′BB′XY =1

4εA′B′

(εAXεBY + εBXεAY

)(8.18)

σAA′BB′X′Y ′ =1

4εAB

(εA′X′εB′Y ′ + εB′X′εA′Y ′

). (8.19)

Repeating essentially the same calculations as those leading from (8.1) to (8.8) we obtain

i∇aφX = −4iσ YabX ∇bφY + 2µgaXX′χX

′

(8.20)

i∇aχX′

= 4iσ X′

abY ′ ∇bχY′ − 2µg XX′

a φX (8.21)

and the equations obtained by their complex conjugation. These equations are especiallysimple if we express generators and Infeld-van der Waerden symbols in purely spinorialterms. Remembering that naτ naτ = 1 implies nAA′τ n

BA′

τ = 12ε

BA we get after some

calculations

id

dτφX(x) = i nY Y

′

τ (x)∇XY ′φY (x) + µnτXX′(x)χX′

(x)

+ e naτ (x)Φa(x)φX(x) (8.22)

id

dτχX

′

(x) = i nτY Y ′(x)∇Y X′

χY′

(x) − µnXX′

τ (x)φX (x)

+ e naτ (x)Φa(x)χX′

(x) (8.23)

−i ddτφX′(x) = −i nY Y ′

τ (x)∇Y X′ φY ′(x) + µnτXX′(x)χX(x)

+ e naτ (x)Φa(x)φX′ (x) (8.24)

−i ddτχX(x) = −i nτY Y ′(x)∇XY ′

χY (x) − µnXX′

τ (x)φX′ (x)

+ e naτ (x)Φa(x)χX(x) (8.25)

Let dστ (x) be some invariant measure on the hyperplane στ . The equations can bederived from the Hamiltonian function

H [ψ, ψ] = 〈ψ|H |ψ〉

=

∫

στ

i φX′(x)nXX′

τ (x)nY Y′

τ (x)∇XY ′φY (x)

− i χX(x)nτXX′(x)nτY Y ′(x)∇Y X′

χY′

(x) (8.26)

+1

2µ(

φX′(x)χX′

(x) + χX(x)φX (x))

+ e naτ (x)Φa(x)nXX′

τ (x)(

φX(x)φX′ (x) + χX(x)χX′(x))

dστ (x)

provided∂YX

′

nτXX′(x) = ∂XX′

nτXY ′(x) = 0 (8.27)

78

and the wave functions vanish at boundaries of the hyperplane στ . Reality of H isguaranteed by the same conditions. The Hamiltonian function is not positive definite,which is correct since we are working here in first quantized formalism. Contraction of(8.27) over the remaining indices implies the continuity equation discussed above.

The explicit form of the Hamilton equations is

i nXX′

τ (x)d

dτφX(x) =

δH

δφX′(x), (8.28)

i nτXX′(x)d

dτχX

′

(x) =δH

δχX(x), (8.29)

−i nXX′

τ (x)d

dτφX′(x) =

δH

δφX(x), (8.30)

−i nτXX′(x)d

dτχX(x) =

δH

δχX′(x). (8.31)

or, in the Poissonian way,

id

dτφX(x) = 2nτXX′(x)

δH

δφX′ (x), (8.32)

id

dτχX

′

(x) = 2nXX′

τ (x)δH

δχX(x), (8.33)

−i ddτφX′(x) = 2nτXX′(x)

δH

δφX(x), (8.34)

−i ddτχX(x) = 2nXX

′

τ (x)δH

δχX′(x). (8.35)

We can see that i nτXX′(x) = ωτXX′(x) are the components of the symplectic (sincederivable from a Kahler potential) form on στ at point x ∈ στ , and the Poissonian formIτXX′(x) = −2i nτXX′(x).

Now, we can repeat the whole reasoning that has led us to the BBMJ bracket, itstriple bracket generalization and the entropic framework. It seems there is no generalobstacle for such a generalization. Note that now we can do that in a manifestly covariantway, so that the possible non-Poissonian structure of the manifold of states is unrelatedto the relativistic covariance of the theory.

It is clear, on the other hand, that the representation of the covering of the Poincaregroup we are dealing with acts in the Hilbert space of the bispinor (1

2 , 0) ⊕ (0, 12 ) repre-

sentation of SL(2,C). The operators of the “space-time translations” which determinethe origin of a Minkowski space coordinate system are, of course, linear operators in op-position to the generator of evolution which will not be a linear operator. This is a veryinteresting problem. A hint for its clarification is, perhaps, our observation that an evo-lution of the desity matrix may be determined by a unitary ρ-dependent transformationacting in the Hilbert space.

79

8.2 Schrodinger Form of the Harmonic

Oscillator Equations

Consider an arbitrary number of independent harmonic oscillators. Their Hamiltonianfunction is

H(~q, ~p) =∑

k

( p2k

2mk+mkω

2kq

2k

2

)

. (8.36)

Let us now introduce the complex coordinates

zk =pk√2mk

− i

√mk

2ωkqk (8.37)

whose dimension is [energy]12 , and let

√ε0 be a constant with the same dimension. It

is natural to introduce the dimensionless coordinates ψk = zk/√ε0. The Hamiltonian

function isH(~ψ, ~ψ) =

∑

k

zkzk = ε0∑

k

ψkψk. (8.38)

Let ω0 be a constant with the dimension of frequency. The Hamilton equations of motionread

iε0ω0

d

dtψk =

ωkω0

∂H

∂ψk(8.39)

−i ε0ω0

d

dtψk =

ωkω0

∂H

∂ψk. (8.40)

The same equations can be obtained from the Hamiltonian function

H ′(~ψ, ~ψ) =∑

k

ε0ω0ωkψkψk. (8.41)

Introducing another constant ε0ω0

:= h we get the Schrodinger equation

ihd

dtψk =

∂H ′

∂ψk= hωkψk (8.42)

−ih ddtψk =

∂H ′

∂ψk= hωkψk (8.43)

which is nothing but a different representation of the ordinary harmonic oscillator equa-tion for independent oscillators. It is surprising, however, that the Hamiltonian H ′ isnot equal to energy. The energy H is, on the other hand, proportional to the “squarednorm”

∑

k ψkψk.Consider now a three-dimensional harmonic oscillator oscillating with the frequency

ω. There exists a conserved quantity ~J = ~q × ~p. In the dimensionless coordinates

~J = ihω0

ω~ψ × ~ψ. (8.44)

A k-th component of ~J is

Jk(~ψ, ~ψ) = −ihω0

ωeklmψlψm (8.45)

80

hence an observable bilinear like in QM. Analogously to H ′ we can introduce the con-served quantity ~J ′ = (ω/ω0) ~J . Both ~J ′ and ~J can be represented in terms of matrices

(J ′k

)

lm= −ih eklm (8.46)

satisfying[J ′k, J

′l ] = ih eklmJ

′m. (8.47)

The same Lie-algebraic property follows from the Poisson bracket resulting from the H ′

version of the Hamilton equation. The bracket following from the H version is analogousbut involves a modified, ω dependent “h”.

8.3 On Certain Associative Product

of Nonlinear Observables

What is the role of Weinberg’s multiplication rule?...

Can it be used like that for algebraic relations of ob-

servable quantities? Or is its role only to express the

Lie bracket as a commutator? Is there another mul-

tiplication rule to be used for algebraic relations of

physical quantities? None has been proposed so far.

T. F. Jordan in [20]

In linear QM the associativity of observables follows from their bilinearity in ψ and ψ.We know that this property makes possible two equivalent formulations of the algebra ofobservables: The operator and Weinberg’s ∗-product ones. The ∗-product appears nat-urally because the Poisson bracket (which is equivalent to the commutator of operators)can be written as

F,G = F ∗G−G ∗ F. (8.48)

In what follows I will present another product having this property, but in additionbeing associative for any infinite differentiable functions defined on a Hilbert space (Imust admit that initially I hoped that such products will play an important role for theprobability interpretation of Weinberg’s nonlinear QM, but this program has finally lednowhere). The product is analogous (although not equivalent) to the Moyal product, andthe proof of its associativity is identical to this for general ∗λ-products investigated byFlato et al. [66]. The amount of typographical errors in [66] makes the proof practicallyunreadable, so let me repeat it. I am using the finite dimensional conventions but itsextension to functional derivatives is straightforward.

The Weinberg product is denoted here by

F ∗G = δAA′

∂F

∂ψA

∂G

∂ψA′

:= ∂AF ∂AG. (8.49)

We shall consider the product

F ⋆ G =

∞∑

k=0

λk

k!δA1A′

1. . . δAkA′

k

∂kF

∂ψA1 . . . ∂ψAk

∂kG

∂ψA′

1. . . ∂ψA′

k

81

=∞∑

k=0

λk

k!

∂kF

∂ψA1 . . . ∂ψAk

∂kG

∂ψA1 . . . ∂ψAk

=

∞∑

k=0

λk

k!∂A1...AkF ∂A1...Ak

G

=∞∑

k=0

λk

k!P k(F,G) (8.50)

which, for bilinear observables, reduces only to the first two terms

F ⋆ G = F G+ λF ∗G, (8.51)

and the commutator

F ∗G−G ∗ F = λ−1(F ⋆ G−G ⋆ F

). (8.52)

For λ = −1, and normalized states (8.51) is a correlation. Accordingly, ⋆ can be regardedas an associative generalization of correlations and standard deviations for generalizedobservables.

Theorem 8.1 ⋆-product is formally associative.

Proof : Since

(F ⋆ G) ⋆ H =

∞∑

t=0

λt∑

r + s = tr, s ≥ 0

1

r!s!P r(P s(F,G), H

)(8.53)

and

F ⋆ (G ⋆ H) =∞∑

t=0

λt∑

r + s = tr, s ≥ 0

1

r!s!P r(F, P s(G,H)

), (8.54)

the two expressions will be equal if

Tt(F,G,H) = Ut(F,G,H) (8.55)

where

Tt(F,G,H) =∑

r + s = tr, s ≥ 0

1

r!s!P r(P s(F,G), H

)(8.56)

Ut(F,G,H) =∑

r + s = tr, s ≥ 0

1

r!s!P r(F, P s(G,H)

). (8.57)

82

Taking into account that

P r(F, P s(G,H)

)= ∂A1...ArF

r∑

r′=0

(rr′

)

∂A1...Ar′∂B1...BsG ∂Ar′+1...ArB1...Bs

H (8.58)

we obtain

Ut(F,G,H) =∑

r′+s≤t

1

r′!s!(t− r′ − s)!∂ A1...Ar′Ar′+1...At−s

︸ ︷︷ ︸

t−s

F (8.59)

× ∂ A1...Ar′∂B1...Bs

︸ ︷︷ ︸

r′+s

G ∂ Ar′+1...At−sB1...Bs

︸ ︷︷ ︸

t−r′

H

Tt(F,G,H) =∑

r′+s≤t

1

r′!s!(t− r′ − s)!∂ Ar′+1...At−sB1...Bs

︸ ︷︷ ︸

t−r′F (8.60)

× ∂ B1...Bs∂A1...Ar′

︸ ︷︷ ︸

r′+s

G ∂ A1...Ar′Ar′+1...At−s

︸ ︷︷ ︸

t−s

H. (8.61)

The proof is completed by the exchange between r′ an s.2

8.4 Polarizing Mach-Zehnder Interferometer

and Other Analyzers of Polarizarion

In Chapter 4 we have ilustrated the physical meaning of the “malignant” nonlocal behav-ior of separated systems in Weinberg’s nonlinear QM with the help of a specific polarizingMach-Zehnder interferometer. A detailed description of the interferometer and other an-alyzers of polarizations is given below. The results presented in this section are a part of[67].

From a quantum field theoretic viewpoint an optical device is a scattering area and,as such, has to be treated in a quantum manner. Any scattering process is describedby means of a unitary transformation, the S-matrix, which maps a system’s “in” Fockspace at time t = −∞ into the “out” Fock space at t = ∞. The limits t = ±∞ play arole of a mathematical idealisation expressing the fact that at such times the “in” and“out” fields can be assumed free. It must be stressed that, even in the most elaboratedaxiomatic approaches, like the Haag-Ruelle scattering theory [68], one does not developany physical interpretation of what is going on at the very scattering region and, instead,concentrates on relations between the asymptotic fields.

In practical applications in quantum optics the situation considerably simplifies withrespect to the general scattering theory. All the scattering areas are localized withinsmall regions of space and the free field “±∞” asymptotics is achieved in a trivial wayby simply assuming that the evolution is free outside of the device. Like in the moregeneral case, one does not attempt to describe all the physical phenomena occuringduring, say, a photon’s transmission through a half-wave plate.

In the Heisenberg picture the S-matrix transforms the creation operators accordingto

S†aoutS = ain (8.62)

83

The S-matrix at the LHS of Eq. (8.62) maps an input state |Ψin〉 into |Ψout〉 = S|Ψin〉.In the majority of practical applications the linear mapping (8.62) can be written in aform of a unitary, finite dimensional matrix U acting in some subspace af the whole “in”CCR algebra. Such a situation occurs, for example, if the whole linear transformationdecomposes into irreducible parts corresponding to some given frequency, direction ofpropagation or polarization. The form of this matrix depends on the choice of the “in”and “out” bases in the Fock space. Therefore, in order to describe a given optical deviceit is often easier to find the form of U than this of S, and once such a matrix is giventhe optical device is defined.

One of the simplest optical systems, a lossless, symmetric half-silvered mirror, isdescribed by a 2 × 2 unitary matrix corresponding to two “in” and two “out” modes ofthe electromagnetic field.

Inclusion of polarizations generally increases the number of the required modes tofour.

8.4.1 Analyzer of circular polarizations

A lossless analyzer of circular polarizations (ACP) can be described by a 4 × 4 unitarymatrix corresponding to two “in” ports with two polarization modes for each of them;operators with subscripts “+” and “−” correspond to, respectively, the right- and left-handed modes. The “in” and “out” annihilation operators must, by definition of ACP,satisfy the following relations

〈Ψ|S†Ca

out†1+ aout

1+ SC |Ψ〉 = 〈Ψ|ain†1+a

in1+|Ψ〉

〈Ψ|S†Ca

out†2− aout

2−SC |Ψ〉 = 〈Ψ|ain†1−a

in1−|Ψ〉

aout1−SC |Ψ〉 = aout

2+ SC |Ψ〉 = 0 (8.63)

for any |Ψ〉 satisfying ain2+|Ψ〉 = ain

2−|Ψ〉 = 0.Conditions (8.63) are minimal in the sense that we assume the analyzer to analyze at

least one input channel’s circular polarizations. We do not make here any assumptionsabout the possible output state of light had some photons been introduced also throughthe second input port.

In what follows we shall use the annihilation operators ordered in vectors of thefollowing form (the superscript T denotes a transpose of a vector)

aT = (a1, a2, a3, a4) = (a1+, a1−, a2+, a2−). (8.64)

Let us fix now the circular polarization “in” and “out” annihilation operators accordingto the ordering given by (8.64). The action of ACP can be characterized by the matrices

SC and UC defined by S†Ca

outk SC = UC kla

inl . The relations (4) supplemented with the

unitarity condition lead to

UC =

eiϕ 0 0 00 0 U23 U24

0 0 U33 U34

0 eiχ 0 0

(8.65)

where the matrix U ′ =(U23 U24

U33 U34

)is unitary. The nonsymmetric form of (8.65) follows only

from our conventions concerning the numbering of the annihilation operators; (8.65) isevidently unitarily equivalent to a block diagonal unitary matrix.

84

The difference ϕ − χ corresponds to a phase shift between the reflected and thetransmitted beams and the freedom in the choice of U ′ reflects the freedom of the choiceof the possible behavior of the ACP with respect to the second input port. One canconsider, for example, either symmetric or non-symmetric ACP. For the purposes of thispaper the choice of ϕ and χ is irrelevant and remains a question of convention as we willuse the ACP only in the context of the polarizing Mach-Zehnder interferometer and, ofcourse, all the phase shifts can be collected into an overall phase of the interferometer’sphase shifter. Therefore, in no way restricting the generality of the solution, we will laterput χ = 0. We shall also put eiϕ = i in order to obtain an agreement with conventionsappearing in other papers (cf. [46]).

One can check now that (8.65) represents indeed an analyzer of circular polarizationsfor the “1-in” port by explicitly writing the “out” circular polarization modes annihilationoperators in terms of the “in” ones:

aout1+ = SCe

iϕain1+S

†C

aout1− = SC(U23a

in2+ + U24a

in2−)S†

C

aout2+ = SC(U33a

in2+ + U34a

in2−)S†

C

aout2− = SCe

iχain1−S

†C . (8.66)

(8.66) means that the lefthanded photons in the “1-out” port can arrive only from the“2-in” port. The same concerns the righthanded photons in “2-out”. On the otherhand, all the righthanded photons coming from “1-in” are reflected into “1-out” and allthe lefthanded ones entering through “1-in” are transmitted into “2-out”. By the way,we see from (8.66) that the device acts also as an analyzer of the orthogonal modesU23a

in2+ + U24a

in2− and U33a

in2+ + U34a

in2− of the second input port.

To complete this part of the discussion let us note that although the two inputsevolve independently of each other, we cannot use 2×2 matrices to describe the analyzer.This follows from the fact that the outputs do not actually correspond to 1-dimensionalsubspaces of the Fock space. We can always place another analyzer in one of the outputports of our analyzer and this device will analyze again some two orthogonal polarizations.Therefore the “vacuum input port” “2” must be introduced in order to make us capableof considering problems of the Malus law variety.

8.4.2 Analyzers of linear polarizations

The Wollaston quartz or calcite prisms, nicol, or Glan-Thomson and Glan-Foucault po-larizers are analyzers of linear polarizations (ALP).

Let us introduce the following linear polarization “in” annihilation operators

ain1‖α =

1√2

(−eiαain1+ + ain

1−)

ain1⊥α =

1√2

(eiαain1+ + ain

1−)

ain2‖α =

1√2

(−eiαain2+ + ain

2−)

ain2⊥α =

1√2

(eiαain2+ + ain

2−). (8.67)

85

The linear polarizations are parametrized by the relative phase between some fixed cir-cular polarizations. Such a parametrization is “spinorial” in the sense that a change ofα by π rotates the polarization plane by π/2. Let now the “out” linear polarizationmodes be constructed from aout

1± and aout2± in the same way. The ALP is defined in the

way analogous to (8.63), i.e. by

〈Ψ|S†Lαa

out†1⊥ aout

1⊥SLα|Ψ〉 = 〈Ψ|ain†1⊥a

in1⊥|Ψ〉

〈Ψ|S†Lαa

out†2‖ aout

2‖ SLα|Ψ〉 = 〈Ψ|ain†1‖ a

in1‖|Ψ〉

aout1‖αSLα|Ψ〉 = aout

2⊥αSLα|Ψ〉 = 0 (8.68)

for any |Ψ〉 satisfying ain2⊥α|Ψ〉 = ain

2‖α|Ψ〉 = 0.

Repeating the calculations leading to (8.65) we will obtain the following general for-mula for the ALP

aout1⊥α = SLαe

iϕain1⊥αS

†Lα

aout1‖α = SLα(U23a

in2⊥α + U24a

in2‖α)S†

Lα

aout2⊥α = SLα(U33a

in2⊥α + U34a

in2‖α)S†

Lα

aout2‖α = SLαe

iχain1‖αS

†Lα (8.69)

8.4.3 Polarizing Mach-Zehnder Interferometer

Let us consider PMZI depicted in Fig. 4.1 S is a source of light, BS1 analyzer of circularpolarizations (one may think about some other orthogonal polarizations — the algebraicrelations will in general remain the same), M are fully reflecting mirrors, α an opticalretarder, λ/2 a half-wave plate and BS2 an ordinary symmetric beam splitter.

The description of BS1 has been given in the subsection 8.4.1. In the rest of thispaper we will always assume that a state of the electromagnetic input field is annihilatedby the annihilation operators of the “2-in” port of BS1. Therefore we can further simplifyour calculations by putting U ′ = I and this will not affect the physical results and theirinterpretations.

The remaining parts of PMZI are given by

Uλ/2 =

1 0 0 00 1 0 00 0 0 10 0 1 0

(8.70)

Uα =

eiα 0 0 00 eiα 0 00 0 1 00 0 0 1

(8.71)

U2 =1√2

i 0 1 00 i 0 11 0 i 00 1 0 i

. (8.72)

86

The whole interferometer acts therefore as

UIα = U2UαUλ/2U1 =1√2

−eiα 1 0 00 0 ieiα 1ieiα i 0 0

0 0 eiα i

(8.73)

and the annihilation operators at the detector positions are given by

aout1+ = SIα

1√2

(−eiαain1+ + ain

1−)S†Iα

aout1− = SIα

1√2

(ieiαain2+ + ain

2−)S†Iα

aout2+ = SIα

1√2

(ieiαain1+ + iain

1−)S†Iα

aout2− = SIα

1√2

(eiαain2+ + iain

2−)S†Iα. (8.74)

Recalling our conclusions from Sec. 8.4.2 we understand that the interferometer acts asan analyzer of two orthogonal linear polarizations incoming through “1-in” and also ofsome two linear polarizations arriving from “2-in”. Indeed, rewriting (8.74) with the helpof (8.67) we obtain

aout1+ = SIαa

in1‖αS

†Iα

aout1− = SIαa

in2⊥(α+π/2)S

†Iα

aout2+ = SIαia

in1⊥αS

†Iα

aout2− = SIαia

in2‖(α+π/2)S

†Iα. (8.75)

Let us note that this ALP belongs to the class of analyzers mentioned in the secondremark after Eq. (8.68). We have here the required 1–1 relationship between the inputpolarization state and the relevant output channel, but no matter what the polarizationof the input beam is, any light arriving from “1-in” leaves the PMZI in the righthandedstate (and also any light arriving from “2-in” will leave the PMZI in the lefthanded state).

This property of the PMZI is easy to understand. Any state arriving from “1-in” issplit into circular polarization components at BS1. The righthanded part goes throughthe “upper” path, is scattered on BS2 and does not change the polarization, whereas thelefthhanded component changes its polarization in the half-wave plate and also reachesBS2 in the righthanded polarization state.

The unitarity of the transformation implies that any lefthanded “out” photon mustarrive from “2-in”.

The linear polarization annihilation operators (8.67) satisfy the following formulas

ain1‖α =

1

2

(

1 + ei(α−β))

ain1‖β +

1

2

(

1 − ei(α−β))

ain1⊥β

ain1⊥α =

1

2

(

1 − ei(α−β))

ain1‖β +

1

2

(

1 + ei(α−β))

ain1⊥β (8.76)

87

(the Malus law). With the help of (14) one finds that

SIβain1‖αS

†Iβ =

1

2

(

1 + ei(α−β))

aout1+ − i

2

(

1 − ei(α−β))

aout2+

SIβain1⊥αS

†Iβ =

1

2

(

1 − ei(α−β))

aout1+ − i

2

(

1 + ei(α−β))

aout2+ . (8.77)

The particular case β = α+ π reads

SI(α+π)ain1‖αS

†I(α+π) = −iaout

2+

SI(α+π)ain1⊥αS

†I(α+π) = aout

1+ . (8.78)

A comparison of (8.78) with (8.75) shows that a change of the phase shift from α to α+πis analogous to a rotation of a polarizer (say, the Wollaston prism) by π/2, or a rotationof the Stern-Gerlach device by π.

88

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