Quantum interest in (3þ 1)-dimensional Minkowski space

Gabriel Abreu* and Matt Visser†

School of Mathematics, Statistics, and Computer Science, Victoria University of Wellington, PO Box 600, Wellington, New Zealand(Received 2 September 2008; published 5 March 2009)

The so-called ‘‘quantum inequalities,’’ and the ‘‘quantum interest conjecture,’’ use quantum field theory

to impose significant restrictions on the temporal distribution of the energy density measured by a timelike

observer, potentially preventing the existence of exotic phenomena such as ‘‘Alcubierre warp drives’’ or

‘‘traversable wormholes.’’ Both the quantum inequalities and the quantum interest conjecture can be

reduced to statements concerning the existence or nonexistence of bound states for a certain one-

dimensional quantum mechanical pseudo-Hamiltonian. Using this approach, we shall provide a simple

variational proof of one version of the quantum interest conjecture in (3þ 1)-dimensional Minkowski

space.

DOI: 10.1103/PhysRevD.79.065004 PACS numbers: 03.70.+k, 04.60.�m, 04.62.+v, 11.10.Kk

I. INTRODUCTION

It is well known that quantum physics permits arbitrarilylarge negative energy densities at individual points [1,2],though averages and total energies over volumes or linesare much more tightly constrained. This is of criticalimportance when developing singularity theorems, andother theorems based on global analysis, in that this simpleobservation is enough to guarantee that the so-called ‘‘clas-sical energy conditions’’ are not fundamental physics; theyare at best classical approximations to a more subtle quan-tum universe [3].

Note that without something similar to the energy con-ditions to constrain the spacetimes one wishes to consideras ‘‘physical,’’ one can construct arbitrarily weird space-times containing such exotic objects as warp drives [4,5],traversable wormholes [6–9], singularity-free ‘‘blackholes’’ [10], GNACHOs [11], violations of the generalizedsecond law [12], violations of cosmic censorship [13], andeven time machines [14–16]. Because of the need to keepsuch oddities somewhat constrained, and avoid a completefree-for-all, much work has gone into developing ‘‘aver-aged energy conditions’’ of various types [17–19], thougheven here there are issues associated with quantum fieldtheory anomalies [20].

In particular, within the framework of semiclassicalgeneral relativity (GR), the quantum inequalities (QIs)are an extremely important tool for constraining bothnegative energies, and more generally exotic phenomenawhich violate the classical energy conditions. Originallyintroduced by Ford [21], and subsequently extensivelyinvestigated by Ford and Roman [22,23], and their collab-orators [24,25], the QIs impose a lower bound for theintegral of the renormalized stress-energy tensor along asuitable time interval, weighted by a suitable test function.Initially the test function was chosen to be a Lorentzian

function [23], but subsequently Flanagan found a resultindependent of the specific choice of the test function inflat (1þ 1) spacetime [26]. With a distinct approach,Fewster and Eveson also obtained a bound independentof the weight function in 4- and 2-dimensional Minkowskispace [27], though their result in 2 dimensions was some-what weaker than that of Flanagan [26].A related but distinct restriction for negative energies is

imposed by the quantum interest conjecture (QIC), intro-duced by Ford and Roman [28]. Informally speaking, theQIC states that ‘‘overall’’ the energy density must bepositive, and any negative energy density in one regionmust be overcompensated by positive energy in anotherregion. When restricted to isolated pulses of energy, theQIC not only constrains the amount of negative energy in apulse, but also the time interval between any negativeenergy pulse and the larger positive energy pulse thatmust also be present. The (positive) net energy of the twopulses is called the quantum interest, and grows monotoni-cally with the time separation. Initially Ford and Romanused �-function energy pulses to prove the conjecture, andit had not been generalized to arbitrarily shaped energypulses in 4-dimensional flat spacetime, nor was it initiallyformulated for curved spacetime.A much more general approach to the QIC that is not

restricted to isolated pulses was developed by Fewster andTeo in [29], wherein they created a Sobolev-space-basedtechnical apparatus to reinterpret the QIs and the QIC aseigenvalue problems for a one-dimensional quantum me-chanical pseudo-Hamiltonian. This technique relates the(1þ 1)-dimensional QIs and QIC to the nonexistencebound states for the Schrodinger equation (SDE) in onedimension, while the (3þ 1)-dimensional QIs and QIC arerelated to the nonexistence bound states for the biharmonicSchrodinger equation (bSDE). Within this framework,Fewster and Teo [29], and later Teo and Wong [30], wereable to prove the QIC in (1þ 1)-dimensional Minkowskispace; however they could not extend their proof to the(3þ 1)-dimensional case. In Teo and Wong’s approach

*gabriel.abreu@mcs.vuw.ac.nz†matt.visser@mcs.vuw.ac.nz

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[30] critical use is made of a theorem by Simon [31],regarding the existence of bound states for the 1-dimensional SDE. That is, Simon’s theorem is used toprove a version of the QIC in (1þ 1) dimensions, but thegeneralization to (3þ 1) dimensions is troublesome.

Furthermore, in [29] Fewster and Teo proved that is notpossible for a positive �-function function pulse to com-pensate a negative �-function pulse in 4 dimensions, re-gardless of how small the time separation between themcould be. Additionally, in [32] Pretorius found an upperbound for the separation between two general energypulses by applying a scaling argument to the test function.With this strategy, Pretorius also showed that the quantuminterest grows almost linearly as the pulse separation in-creases, which is the same result obtained in [28].

Turning to the current article: Using the approach ex-plained in [29], we shall provide a simple variational prooffor a variant of the QIC in 4-dimensional flat spacetime,(and in fact in any even-dimensional spacetime), by prov-ing the equivalent of Simon’s theorem for the biharmonicSchrodinger equation (and more generally for the multi-harmonic Schrodinger equation). We shall do this via avariational argument coupled with a power series expan-sion of the test function used in the related one-dimensional pseudo-Hamiltonian problem.

The layout is as follows: In Sec. II we explore the QIsand the QIC from the point of view of eigenvalue prob-lems, to obtain the related Schrodinger equations (SDE andbSDE). In Sec. III we will use an appropriate class of testfunctions to prove Simon’s theorem via variational tech-niques applied to a power series expansion, and then inSec. IV repeat this strategy for the (3þ 1)-dimensionalrelated problem to prove the equivalent of Simon’s theo-rem for the bSDE. (Furthermore, as shown in theAppendix, this naturally leads to a version of Simon’stheorem for multiharmonic Schrodinger equations.)When translated back to the underlying even-dimensionalMinkowski spacetime this guarantees the existence ofquantum interest, though this particular calculation ismute as to the amount of interest and ‘‘interest rate.’’ Weconclude with a brief discussion in Sec. V.

II. QUANTUM INEQUALITIES AND THEQUANTUM INTEREST CONJECTURE

A. The quantum inequalities

The quantum inequalities give a lower bound for theexpectation value of the renormalized stress-energy tensor(in a quantum state c ) when evaluated along a completetimelike geodesic,

Ic ;w �Z 1

�1hTren

00 ðt; 0ÞicwðtÞdt (1)

�IhTren

00 ðt; 0ÞicwðtÞdt: (2)

Here wðtÞ is a more-or-less freely specifiable test function(weighting function), which is non-negative and integratesto unity:

Rþ1�1 wðxÞdx ¼ 1. Initially, the stress tensor is

treated in the test-field limit, so the background is simplytaken to be Minkowski space, and the timelike geodesic is

specified by the 4-velocity Va ¼ ð1; ~0Þ. Since our integralsalmost always run from �1 to þ1 we shall for brevityoften use the shorthand symbol

H ¼ Rþ1�1 .

In their initial analysis, Ford and Roman used aLorentzian test function to yield to a lower bound whichdepend on modified Bessel functions [23]. However a moregeneral inequality was found by Flanagan in (1þ 1)-dimensional flat spacetime [26], where he used quantumfield theory for a massless scalar field minimally coupled togravity to find the optimum lower bound independent of thespecific choice of the test function,

Ic ;w � � 1

24�

I ðw0ðtÞÞ2wðtÞ dt: (3)

Fewster and Eveson used a distinct approach [27] to yieldinequalities in (1þ 1) dimensions,

Ic ;w � � 1

16�

I ðw0ðtÞÞ2wðtÞ dt; (4)

and in (3þ 1) dimensions,

Ic ;w � � 1

16�

Ið½w1=2�00ðtÞÞ2dt: (5)

It is obvious that the better bound in (1þ 1) dimensions isgiven by Flanagan’s result rather than the one by Fewsterand Eveson. However the most general and optimumbound in (3þ 1)-dimensional flat spacetime, for a mass-less scalar field minimally coupled to gravity, is shown inEq. (5).Combining the results of Flanagan, and Fewster and

Eveson, it is possible to rewrite the QIs in2m-dimensional spacetime in the more general form [29][writing wðtÞ ¼ fðtÞ2 to automatically enforce the positiv-ity constraint],I

hTren00 ðt; 0Þic ½fðtÞ�2dt � � 1

cm

I½DmfðtÞ�2dt; (6)

whereD is the derivative operator, and the various differentconstants are given by

cm ¼�6� m ¼ 1;m�m�1=222m�ðm� 1

2Þ m � 2;(7)

and where the weighting function is now normalized toRþ1�1 jfðxÞj2dx ¼ 1. The case m ¼ 1 is equivalent to

Eq. (3), while Eq. (5) is equivalent to m ¼ 2. After inte-grating Eq. (6) by parts, the QIs become a statementregarding the nonexistence of negative eigenvalues for aone-dimensional pseudo-Hamiltonian,

hfjHjfi � 0; (8)

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where

H ¼ ð�1ÞmD2m þ cmhTren00 ðt; 0Þic : (9)

The quantity cmhTren00 ðt; 0Þic can effectively be viewed as a

‘‘potential’’ V for a quantum mechanical system,

V � cmhTren00 ðt; 0Þic ; (10)

and the QIs become the statement

hfjfð�1ÞmD2m þ Vgjfi � 0: (11)

We shall now use the framework of quantum mechanics toanalyze the QIs and the QIC as an eigenvalue problem.

B. The QIC as an eigenvalue problem

As adopting this point of view allows us to use themathematical background of one-dimensional quantummechanics, we will change to the standard quantum me-chanical notation. For instance, the differential operator (9)becomes the quantum pseudo-Hamiltonian

H ¼ P2m þ V; (12)

where P and V are operators on the usual Hilbert space ofsquare-integrable functions. The eigenvalue problem forthis Hamiltonian, in coordinates, is the ordinary differen-tial equation (we have effectively set @=2m ! 1 to simplifythe algebra),

ð�1Þm d2m

dx2m’ðxÞ þ VðxÞ’ðxÞ ¼ E’ðxÞ; (13)

for the eigenfunctions’ðxÞwhich again belong to the usualHilbert space of square-integrable functions. (If desired, amore rigorous approach can be followed, as in [29].)Taking m ¼ 1 (i.e., (1þ 1) spacetime dimensions), werecover the one-dimensional time-independent SDE

� d2

dx2’ðxÞ þ VðxÞ’ðxÞ ¼ E’ðxÞ; (14)

while, for m ¼ 2 (i.e., (3þ 1) spacetime dimensions), weobtain the one-dimensional time-independent bSDE,

d4

dx4’ðxÞ þ VðxÞ’ðxÞ ¼ E’ðxÞ: (15)

For the case of the SDE, there is a theorem by Simon [31]which, by imposing a constraint on the potential, ensuresthe existence of a bound state for the Hamiltonian. In asimple form it reads as follows:

Simon’s Theorem: Let VðxÞ obeyR1�1ð1þ

x2ÞjVðxÞjdx <1, with VðxÞ not zero almost everywhere.Then H ¼ �d2=dx2 þ VðxÞ has a negative eigenvalue ifZ 1

�1VðxÞdx � 0: (16)

h

With the help of this theorem, it is possible to generalizethe QIC to a wider set of energy pulses, and not only berestricted to �-function pulses as in [28].In order to better understand the QIC in this framework,

first we have to point out that a potential which fulfills theQIs must violate condition (16). Now let us rewrite thepotential as

VðxÞ ¼ VðxÞþ � VðxÞ�: (17)

This splits the potential into its positive part minus itsnegative contributions, with VðxÞ� � 0. Then, if the po-tential violates condition (16), from (17) we haveI

VðxÞþdx >I

VðxÞ�dx: (18)

This means, if we go back to the GR terminology, thatthe QIs imply that the expectation value of the renormal-ized stress-energy tensor must satisfyI

hTren00 ðt; 0Þic dt > 0; (19)

which is slightly stronger than the averaged weak energycondition (AWEC). Splitting the energy density into itspositive and negative parts

hTren00 ðt; 0Þi ¼ hTren

00 ðt; 0Þiþ � hTren00 ðt; 0Þi�; (20)

this impliesIhTren

00 ðt; 0Þiþdx >IhTren

00 ðt; 0Þi�dx: (21)

That is, the positive part of a stress tensor that satisfies theQIs must always overcompensate its negative part; this isone version of QIC. (Of course, there are significant ele-ments missing from this form of the conjecture, as com-pared with the original form and Pretorius’ scalingargument [32]. Specifically there is no immediate way toextract a bound for the time separation between the energypulses and the quantum interest as a function of thisseparation. However, a benefit of the current discussion isthat we are not limited to �-function pulses of compactsupport.)In the next section we will prove Simon’s theorem for

the SDE by expanding an appropriate class of test func-tions in a power series; this strategy will be used again forthe bSDE in Sec. IV, and in the Appendix a similar resultwill be obtained in arbitrary even-dimensional Minkowskispacetime.

III. 1þ 1-DIMENSIONAL MINKOWSKI SPACE

In order to prove Simon’s theorem, avoiding most of thetechnical issues arising in the formal proof [31], and keep-ing in mind the idea of generalizing the theorem to thebSDE, and even higher-derivative ‘‘Hamiltonians,’’ weshall use a power series expansion of the test function inthe SDE to find the conditions for the potential V to bind.

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A. Gaussian wave function

Let us start, for simplicity and clarity, with a Gaussiantest function,

’test ¼�expf�ðx��Þ2=ð2�2Þgffiffiffiffiffiffiffi

2�p

�

�1=2

; (22)

which automatically enforces the unit normalizationH j’ðxÞj2dx ¼ 1. Therefore, a variational argument im-

plies that for the lowest eigenvalue of the SDE one has

E �I½’0ðxÞ2 þ VðxÞ’ðxÞ2�dx: (23)

Then the kinetic term isH’0ðxÞ2 ¼ 1=�2, whereas [assum-

ing all the moments jH x2nVðxÞdxj<1 so that the expan-

sion makes sense]

IVðxÞ’ðxÞ2dx ¼ 1ffiffiffiffiffiffiffi

2�p

�

X1n¼0

ð�1Þnn!ð2�2Þn

�I

VðxÞðx��Þ2ndx: (24)

Interchanging the integral and the summation is justifiedsince the Gaussian function ’ðxÞ2 is an analytic function ofxwhose Taylor series has an infinite radius of convergence.

For a fixed � we now see that

E�2 � 1þ �ffiffiffiffiffiffiffi2�

pI

VðxÞdxþOð1=�Þ: (25)

From this last expression, we can see that ifIVðxÞdx < 0; (26)

then for � sufficiently large we can guarantee E�2 < 0(hence E< 0 and the potential will bind, and so violate theQIs). This is close to Simon’s condition. (We currentlyhave a <, rather than Simon’s � . The only ‘‘difficult’’thing about Simon’s theorem is proving the existence of abound state in the marginal ¼ case. Note that the weakerresult we have here is already enough, in GR language, toassure that the stress tensor satisfies the AWEC.)

B. Generic class of test functions

To deal with the marginal case, we proceed by using thefact that we are free to choose the test function in a more-or-less arbitrary manner, and so to use this freedom obtaintighter constraints. Let us start by picking some functiongðxÞ that is analytic on the entire real line (infinite radius ofconvergence) and then constructing the piecewise analyticfunction

hðxÞ ¼ gðjxjÞ: (27)

Note that hðxÞ need not, and typically will not, be analyticat zero. We then enforce the normalization

IhðxÞdx¼ 1;

IxhðxÞdx¼ 0;

Ix2hðxÞdx¼ 1:

(28)

Suitable examples of such functions are

hðxÞ ¼ expð� ffiffiffi2

p jxjÞffiffiffi2

p ; (29)

and

hðxÞ ¼ XNi¼1

ci expð�jxj=diÞ; (30)

subject to

2XNi¼1

cidi ¼ 1; 4XNi¼1

cid3i ¼ 1: (31)

We now choose

’test ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihð½x���=�Þ

�

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðjx��j=�Þ

�

s; (32)

whenceIj’ðxÞj2dx ¼ 1;

Ixj’ðxÞj2dx ¼ �; (33)

and Iðx��Þ2j’ðxÞj2dx ¼ �2: (34)

Moreover,H’0ðxÞ2dx ¼ �=�2, with

� � 1

4

I h0ðxÞ2hðxÞ dx: (35)

The numeric value of � depends on the choice of the testfunction and further optimizations may be useful. For now,let us focus on the Taylor expansion of gðxÞ:

gðxÞ ¼ X1n¼0

anxn: (36)

This is assumed to exist (by analyticity), and to convergeon the entire real line. We now deduce the existence of asimilar power series expansion for hðxÞ:

hðxÞ ¼ gðjxjÞ ¼ X1n¼0

anjxjn: (37)

This is now an expansion in the variable jxj. Since itdepends on jxj, this is not a Taylor series for hðxÞ, but byconstruction it is convergent over the entire real line. Thenfor the shifted and rescaled test function we have

j’testj2 ¼ 1

�

X1n¼0

an

�jx��j�

�n: (38)

Assuming all the moments jH jxjnVðxÞdxj<1, so that the

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expansion makes sense, we furthermore have

E � �

�2þ 1

�

X1n¼0

an�n

IVðxÞjx��jndx: (39)

In interchanging the summation and the limit we have firstsplit

H � Rþ1�1 ¼ R�

�1 þRþ1� , and then appealed to the

analyticity of gðxÞ and gð�xÞ on the appropriate ranges,finally recombining the two sets of integrals to run over theentire real line.

We now have the potential for additional relevant termsto appear in the series expansion for the variational boundon the lowest eigenvalue E, and it is convenient to placesome restrictions on the coefficients an appearing in theseries expansion.

First, if a0 ¼ 0, then

hðxÞ � a1jxj þ ; (40)

so that

½h0ðxÞ�2hðxÞ � a1

jxj þ : (41)

Thus in order for the integral defining � [Eq. (35)] toconverge at x ¼ 0 it is necessary to chose a1 ¼ 0. So infact

hðxÞ � a2x2 þ ; (42)

and

½h0ðxÞ�2hðxÞ � 4a2 þ : (43)

In this situation we then cannot extract much information,since the SDE now merely gives

E�2 � �þOð��1Þ: (44)

Hence, to extract useful information, we need a0 � 0, andso (since probability densities are always positive), a0 > 0.

In contrast, for a0 > 0, the integral defining � [Eq. (35)]always converges at x ¼ 0, since

hðxÞ � a0 þ a1jxj þ ; (45)

and so

½h0ðxÞ�2hðxÞ � a21

a0þ : (46)

Thus we have

E�2 � �þ �a0I

VðxÞdxþ a1I

jx��jVðxÞdxþOð1=�Þ: (47)

Therefore, with � sufficiently large the only importantcontributions come from the first two terms (n ¼ 0, andn ¼ 1) of the power series expansion.

Here, as for the Gaussian test functions, the conditionHVðxÞdx < 0 implies binding, which means that the ex-

pectation value of the renormalized stress-energy tensormust fulfill the AWEC. However, if we consider the bor-derline of AWEC violation, i.e.,

HVðxÞdx ¼ 0, then the

next term in the expansion is now no longer neglectable:

E�2 � �þ a1I

jx��jVðxÞdxþOð1=�Þ: (48)

But recall that a1 is completely arbitrary, both in sign andin magnitude. That is, if for any value of �

Ijx��jVðxÞdx � 0; (49)

then this implies the existence of a bound state for the SDE.This is a very strong constraint on the potential and itquickly yields a proof of Simon’s theorem. Taking theconverse of the above, if

HVðxÞdx ¼ 0 then to prevent

the occurrence of a bound state we must have

8�:I

jx��jVðxÞdx ¼ 0: (50)

Differentiating twice with respect to � we get

8�: 2I

�ðx��ÞVðxÞdx ¼ 0 ) Vð�Þ ¼ 0: (51)

Thus any nonzero potential binds if its integral is null.Combining this with the fact that

HVðxÞdx < 0 also im-

plies binding, now provides a simple variational proof ofSimon’s theorem, Eq. (16), though under the muchstronger technical conditions that all the moments existjH jxjnVðxÞdxj<1. It is these stronger technical condi-

tions—which are still physically quite reasonable and arecertainly satisfied by isolated pulses of stress energy—thatwill make it easy for us to extend Simon’s theorem to thebiharmonic Schrodinger equation, and so lead to a (3þ 1)-dimensional version of the QIC.

IV. 3þ 1-DIMENSIONAL MINKOWSKI SPACE

To generalize the condition given by Simon’s theorem tothe bSDE, we proceed as before, expanding appropriatetest functions in power series. Applying a variational argu-ment to the bSDE we find that the lowest eigenvaluesatisfies

E �I½’00ðxÞ2 þ VðxÞ’ðxÞ2�dx; (52)

assuming that the test functions are normalized. We nowuse this relation to probe for the existence of a bound statefor the pseudo-Hamiltonian for the bSDE, which is ulti-mately related to the 4-dimensional flat spacetime QIs andthe QIC.

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A. Gaussian wave function

We start with the normalized Gaussian test functionpreviously used for the SDE, Eq. (22). The kinematicterm yields

H’00ðxÞ2dx ¼ �2=�4, with � a numeric con-

stant. Assuming all appropriate moments exist, we againexpand the test function in a power series, whence

IVðxÞ’2ðxÞdx ¼ 1ffiffiffiffiffiffiffi

2�p

�

X1n¼0

ð�1Þnn!ð2�2Þn

�I

VðxÞðx��Þ2ndx: (53)

Then, from (52)

E�4 � �2 þ �3ffiffiffiffiffiffiffi2�

pI

VðxÞdx� �

2ffiffiffiffiffiffiffi2�

pI

VðxÞðx��Þ2dx

þOð1=�Þ: (54)

As for the SDE, in this case alsoHVðxÞdx < 0 implies that

the potential binds for a sufficiently large �. This gives usmost of Simon’s theorem, apart from the extremal casewhere the integral vanishes.

However, even with the Gaussian test function, in thisbSDE case we recover more information than for the SDE.Specifically if

HVðxÞdx ¼ 0, then the next term of the

expansion is important, and to guarantee the absence of abound state we must also enforce:

8�:Iðx��Þ2VðxÞdx � 0: (55)

This constraint, while certainly significant, is not quitestrong enough to imply Simon’s theorem.

B. Generic class of test functions

Finally, for the bSDE, we consider the same normaliza-tion, (28), and the same choice for the generic test function,as we used for the SDE, (32). NoteI

’00ðxÞ2dx ¼ �

�4; (56)

with

� �If½ ffiffiffi

hp �00ðxÞg2dx (57)

¼ 1

16

I ½2hðxÞh00ðxÞ � h0ðxÞ2�2hðxÞ3 dx: (58)

In view of the fact that we have chosen hðxÞ ¼ gðjxjÞ thisbecomes (note the �-function arising from twice differ-entiating the absolute value)

� � 1

16

I f2gðjxjÞ½g00ðjxjÞ þ g0ðjxjÞ�ðxÞ� � g0ðjxjÞ2g2gðjxjÞ3 dx:

(59)

In order to derive a useful bound wewill want � to be finite,

which means that we want the coefficient of the � functionto be zero, that is a1 ¼ 0. To make the integral converge atzero we also want a0 � 0, and in fact must have a0 > 0. Incontrast, a2 and a3 are unconstrained as to sign and mag-nitude and we have

E�4 � �þ a0�3I

VðxÞdxþ a2�I

VðxÞjx��j2dx

þ a3I

VðxÞjx��j3dxþOð1=�Þ: (60)

As before, by suitably choosing a sufficiently large �, wesee that

HVðxÞdx < 0 implies binding.

To extract further information from (60) it is necessaryto check what happens if we set

HVðxÞdx ¼ 0. Thereafter

the next two terms in the expansion become relevant,however both the sign and magnitude of a2 and a3 arefreely specifiable, hence (for any �) eitherI

jx��j2VðxÞdx � 0; (61)

or Ijx��j3VðxÞdx � 0; (62)

is a sufficient condition for the Hamiltonian to have abound state.Conversely, if

HVðxÞdx ¼ 0, a necessary condition for

the absence of a bound state is that

8�:I

jx��j2VðxÞdx ¼ 0; (63)

and

8�:I

jx��j3VðxÞdx ¼ 0: (64)

Now if we differentiate the last expression with respect to� then

8�:I

jx��j2signðx��ÞVðxÞdx ¼ 0; (65)

whence, combining with Eq. (63), we have

8�:Z 1

�jx��j2VðxÞdx ¼ 0: (66)

If we repeatedly differentiate the last expression withrespect to � then

8�:Z 1

�jx��jVðxÞdx ¼ 0; (67)

8�:Z 1

�VðxÞdx ¼ 0; (68)

and finally VðxÞ ¼ 0. That is, ifHVðxÞdx ¼ 0, a necessary

condition for the absence of a bound state is that VðxÞ ¼ 0.Conversely if

HVðxÞdx ¼ 0 and VðxÞ 6�0, then this is a

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sufficient condition for the presence of a bound state. Thisproves the equivalent of Simon’s theorem for the 4th-orderbSDE and, furthermore proves the QIC in (3þ 1)-dimensional Minkowski space.

Note that the version of the QIC that we have proved isthis: The QI’s imply that either hTren

00 ðt; 0Þi � 0 everywherealong the world line, orI

hTren00 ðt; 0Þic dt > 0; (69)

which we emphasize is slightly stronger than the AWEC.Splitting the energy density into its positive and negativeparts, this implies that as long as hTren

00 ðt; 0Þi is not identi-cally zero along the world line, thenI

hTren00 ðt; 0Þiþdx >

IhTren

00 ðt; 0Þi�dx: (70)

That is, any negative energy ‘‘loan’’ is overcompensatedelsewhere along the world line.

V. DISCUSSION

With the variant of Simon’s theorem that we have nowproved for the bSDE, it is possible to reformulate the QICfor a more general range of energy pulses in (3þ 1)-dimensional flat spacetime, as has been already done forthe (1þ 1)-dimensional case. In flat spacetime, an energypulse which satisfies the QIs (and so the version of the QICdiscussed above), must also fulfill an AWEC-like inequal-ity. That is, the expectation value of the renormailzedstress-energy tensor shall violate the condition providedby Eq. (16), and so satisfy (69).

Moreover, using the argument derived from Eq. (20), theQIC can be extended to (3þ 1)-dimensional Minkowskispace. Furthermore, from (21) and (70), we know that thepositive contributions of the expectation value of the re-normalized stress-energy tensor must overcompesate itsnegative parts. The key trade-off in the current argumentis that while we have been able to deduce a general resultfor arbitrary even-dimensional Minkowski space, andwhile we are not limited to �-function pulses or pulses ofcompact support, we have on the other hand lost some ofthe precision information that can be deduced whenstronger assumptions are made regarding the temporaldistribution of the stress energy.

Furthermore, this proof of the QIC that we have given inflat four-dimensional spacetime gives only a partial pictureof the nature of negative energies and the constraints thatcan be placed on them, as it does not (yet) include theeffects of curved spacetime. Indeed, for many technicalreasons it would be preferable to work with the null energycondition, rather than the weak energy condition [18]. Thatis, a truly complete formulation of the QIs and QIC shouldreally include curved spacetimes, at the very least (3þ 1)-dimensional curved spacetimes, and work with some ver-sion of the null energy condition—this is the key arena

wherein the possibility of exotic phenomena such as tra-versable wormholes, warp drives, and time machines arerelated to the existence of negative energies and the con-straints thereupon.

ACKNOWLEDGMENTS

This research was supported by the Marsden Fund ad-ministered by the Royal Society of New Zealand. G.A.was additionally supported by the Victoria University ofWellington.

APPENDIX: HIGHER DIMENSIONS

Having now seen the argument in (1þ 1) and (3þ 1)dimensions it is clear how to generalize to any even num-ber of dimensions. Consider a 2m-dimensional [that is,(½2m� 1� þ 1) dimensional] spacetime. Then the QIs areequivalent to the statement:

69 bound state: H ¼ ð�1Þm d2m

dx2mþ VðxÞ: (A1)

Picking the generic class of test functions discussed pre-viously, and applying a variational argument, the groundstate energy is bounded by

E �If½’ðmÞðxÞ�2 þ VðxÞ½’ðxÞ�2gdx: (A2)

The kinetic term is

I½’ðmÞðxÞ�2dx ¼ �2

�2m(A3)

(where the precise numerical value of � is not important aslong as it is finite), whereas (assuming all appropriatemoments exist)

IV½’2�dx ¼ 1

�

X1n¼0

an�n

IVðxÞjx��jndx: (A4)

The interchange of the summation and integration is againjustified by the assumed piecewise analytic nature of thetest function. Therefore

E�2m � �2 þ �2m�1X2m�1

n¼0

an�n

IVðxÞjx��jndx

þOð1=�Þ; (A5)

where there will be some set of constraints on the coef-ficients an to keep � finite.Now since ’ðxÞ is a test function we are always free to

choose a0 � 0, and since probability densities are alwayspositive, this forces a0 > 0. We are also free to (temporar-ily) choose all the odd a2nþ1 ¼ 0, which is a convenience(since it implies the absence of squared � functions comingfrom differentiations of the absolute value function) toguarantee � finite. Then

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E�2m � �2 þ �2m�1Xm�1

n¼0

a2n�2n

IVðxÞjx��j2ndx

þOð1=�Þ; (A6)

But then ifHVðxÞdx < 0, it follows that for � suffi-

ciently large we can guarantee E�2m < 0, whence E< 0,and the potential will bind (thus violating the QIs). This isthe easy bit.

In particular, 69 bound state ) HVðxÞdx � 0. Now con-

sider the borderline caseHVðxÞdx ¼ 0. We are again free

to choose a0 � 0 (and so a0 > 0), and are also free tochoose both the sign and magnitude of all the a2n. But thisimplies the potential will bind unless

8�:I

VðxÞjx��j2ndx ¼ 0; 8 n 2 ð0; m� 1Þ:(A7)

In particular,

8�:I

VðxÞjx��j2m�2dx ¼ 0: (A8)

Now ‘‘turn on’’ one of the odd a2nþ1. Specifically,consider a2m�1. Since for m � 2 we have

dm

dxmðjxj2m�1Þ ¼ 0; (A9)

we can switch on this a2m�1 coefficient without risk ofdeveloping a squared � function in the evaluation of �, and

so keep � finite. But then since a2m�1 is arbitrary as to signand magnitude, to prevent binding we must have

8�:I

VðxÞjx��j2m�1dx ¼ 0: (A10)

Differentiating with respect to �

8�:I

VðxÞjx��j2m�2signðx��Þdx ¼ 0; (A11)

whence, combining the two preceding equations, we see

8�:Z 1

�VðxÞjx��j2m�2dx ¼ 0: (A12)

Repeated differentiations with respect to � will now even-tually yield VðxÞ ¼ 0.That is, if

HVðxÞdx ¼ 0 then to prevent a bound state we

must have VðxÞ ¼ 0. Conversely, ifHVðxÞdx ¼ 0 and

VðxÞ 6�0, then there will be a bound state.Consequently, 69 bound state ) either VðxÞ � 0 orHVðxÞdx > 0. This now is a version of Simon’s theorem

appropriate to the multiharmonic SDE, leading to a generaleven-dimensional version of the QIC: Either hTren

00 ðt; 0Þi �0 everywhere along the world line, or

IhTren

00 ðt; 0Þic dt > 0; (A13)

this last inequality being slightly stronger than the AWEC.

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