Quantum inequalities in two dimensional Minkowski spacetime

Eanna E. FlanaganCornell University, Newman Laboratory, Ithaca, NY 14853-5001.

We generalize some results of Ford and Roman constrain-ing the possible behaviors of the renormalized expected stress-energy tensor of a free massless scalar eld in two dimensionalMinkowski spacetime. Ford and Roman showed that the en-ergy density measured by an inertial observer, when aver-aged with respect to the observers proper time by integratingagainst some weighting function, is bounded below by a nega-tive lower bound proportional to the reciprocal of the squareof the averaging timescale. However, the proof required aparticular choice for the weighting function. We extend theFord-Roman result in two ways: (i) We calculate the opti-mum (maximum possible) lower bound and characterize thestate which achieves this lower bound; the optimum lowerbound diers by a factor of three from the bound derived byFord and Roman for their choice of smearing function. (ii)We calculate the lower bound for arbitrary, smooth positiveweighting functions. We also derive similar lower bounds onthe spatial average of energy density at a xed moment oftime.

04.62.+v, 03.70.+k, 42.50.Dv

I. INTRODUCTION AND SUMMARY

In classical physics, the energy densities measured byall observers are non-negative, so that the matter stress-energy tensor Tab obeys Tabu

aub 0 for all timelikevectors ua. This \weak energy condition" strongly con-strains the behavior of solutions of Einsteins eld equa-tion: once gravitational collapse has reached a certaincritical stage, the formation of singularities becomes in-evitable [1]; traversable wormholes are forbidden [2]; andthe asymptotic gravitational mass of isolated objectsmust be positive [3].

However, as is well known, in quantum eld theorythe energy density measured by an observer at a point inspacetime can be unboundedly negative [4]. Examples ofsituations where observers measure negative energy den-sities include the Casimir eect [5] and squeezed statesof light [6], both of which have been probed experimen-tally. In addition, the theoretical prediction of black holeevaporation [7] depends in a crucial way on negative en-ergy densities. If nature were to place no restrictions onnegative energies, it might be possible to violate cosmiccensorship [8,9], or to produce traversable wormholes orclosed timelike curves [10]. As a consequence, in recentyears there has been considerable interest in constraintson negative energy density that follow from quantum eldtheory. For reviews of recent results and their ramica-tions see, e.g, Refs. [11{14].

In this paper we shall be concerned with so-called\quantum inequalities", which are constraints on themagnitude and duration of negative energy fluxes anddensities measured by inertial observers, rst introducedby Ford [15] and extensively explored by Ford and Ro-man [9,11,12,16,17].

A. Quantum Inequalities

Consider a free, massless quantum scalar eld intwo dimensional Minkowski spacetime. We consider thefollowing three dierent spacetime-averaged observables.Fix a smooth nonnegative function = () withZ 1

1()d = 1; (1.1)

which we will call the smearing function. Let T^ab be thestress tensor, and let (x; t) be coordinates such that themetric is ds2 = dt2 + dx2. Dene

ES [] =

Z 11

dx (x)Ttt(x; 0); (1.2)

ET [] =

Z 11

dt (t)Ttt(0; t); (1.3)

and

EF [] =

Z 11

dt (t)T xt(0; t): (1.4)

The quantity ES [] is the spatial average of the energydensity over the spacelike hypersurface t = 0, while ET []is the time average with respect to proper time of the en-ergy density measured by an inertial observer, and EF []is the time average with respect to proper time of the en-ergy flux measured by an inertial observer. Of these threeobservables, ES and ET are classically positive, while EFis classically positive when only the right moving sectorof the theory contains excitations.

In the quantum theory, let ES;min[] and ET;min[] de-note the minimum over all states of the expected valueof the observables ES[] and ET [] respectively. Simi-larly, let EF;min[] denote the minimum over all states inthe right moving sector of the expected value of EF [].Ford and Roman have previously derived lower boundson ET;min[] and EF;min[], for a particular choice ofthe smearing function . Specically, they showed that[11,17]

1

ET;min[0] 1

82(1.5)

and [15]

EF;min[0] 1

162; (1.6)

where

0(t)

1

t2 + 2: (1.7)

The main result of this paper is that

ET;min[] = ES;min[] = 2 EF;min[]

= 1

24

Z 11

dv0(v)2

(v); (1.8)

for arbitrary smearing functions (v). Equation (1.8)generalizes the Ford-Roman results and shows that thequalitative nature of those results does not depend ontheir specic choice of smearing function (which waschosen to facilitate the proofs of the inequalities), asone would expect. Equation (1.8) also gives the opti-mum, maximum possible lower bound on the averagedenergy density, in contrast to the lower bounds (1.5) and(1.6). For the particular choice (1.7) of smearing func-tion, Eq. (1.8) shows that the optimum lower boundsare a factor of three smaller in absolute value than thebounds (1.5) and (1.6).

Equation (1.8) also shows that the lower bounds onthe temporal averages and spatial averages of energy areidentical, which is not surprising in a two dimensionaltheory.

II. DERIVATION OF THE QUANTUMINEQUALITY

We start by showing that the minimum values of thethree observables that we have dened are not indepen-dent of each other, cf. the rst part of Eq. (1.8) above.To see this, introduce null coordinates u = t+x, v = tx,so that the eld operator can be decomposed as

^(x; t) = ^R(v) + ^L(u): (2.1)

Here ^R(v) acts on the right-moving sector and ^L(u)on the left-moving sector of the theory. The non-zerocomponents of the stress tensor in the (u; v) coordinates

are T^uu(u) =: (@u^L)2 : and

T^vv(v) =: (@v^R)2 :; (2.2)

where the colons denote normal ordering. Dene theright-moving and left-moving energy flux observables

E(R)[]

Zdv (v) T^vv(v) (2.3)

and

E(L)[]

Zdu (u) T^uu(u): (2.4)

Then we have ES [] = ET [] = E(R)[] + E(L)[], whileEF [] = E(R)[] E(L)[], from which the rst part ofEq. (1.8) follows.

Thus, it is sucient to consider the right-moving sectorof the theory and to calculate

E(R)min[] minstates hE(R)[]i; (2.5)

from which we can obtain ET;min[] = ES;min[] =

E(R)min[]=2.

A. Bogilubov transformation

In this subsection we derive our main result, which isthat

E(R)min[] =

1

48

Z 11

dv0(v)2

(v); (2.6)

for any smooth smearing function (v). The key idea inour proof is to make a Bogilubov transformation whichtransforms the quadratic form (2.3) into a simple form.In general spacetimes such a Bogilubov transformation isdicult to obtain, but in flat, two dimensional spacetimesit can be obtained very simply by using a coordinatetransformation, as we now explain.

We can write the mode expansion of the right-movingeld operator as

^R(v) =1p

2

Z 10

d!1p

2!

ei!va^! + h:c:

; (2.7)

where h:c: means Hermitian conjugate. The Hamiltonianof the right-moving sector is

H^R =

Z 10

d! !a^y!a^!: (2.8)

Consider now a new coordinate V which is a mono-tonic increasing function of v, V = f(v) say, so thatv = f1(V ). We dene a mode expansion with respectto the V coordinate:

^R(v) = ^R[f1(V )]

=1p

2

Z 10

d!1p

2!

hei!V b^! + h:c:

i: (2.9)

The operators b^! can be expressed as linear combinationsof the a^!s and a^

y!s. Note that we are using a notation

where the argument of ^R is always the v-coordinate ofthe spacetime point, never the V coordinate; in otherwords we shall not use the notation ^R(V ) to denote the

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eld operator at the spacetime point with V -coordinatevalue V and v coordinate value f1(V ).

We dene the unitary operator S^ by

S^ a^!S^y = b^!; (2.10)

from which it follows that

S^y ^R(v) S^ = ^R[f(v)]: (2.11)

Consider now the transform S^yT^vv(v)S^ of the operator

T^vv(v). Using Eq. (2.2) this can be written as

S^yT^vv(v)S^ = limv0!v

S^y@v0@v

h^R(v

0)^R(v)H(v v0)iS^;

(2.12)

where

H(v) = 1

4[ln jvj+ i(v)] (2.13)

is the distribution that the normal ordering procedureeectively subtracts o. Here is the step function.Equations (2.2), (2.11) and (2.12) now yield

S^yT^vv(v)S^ = limv0!v

@v0@v

h^R[f(v

0)]^R[f(v)]H(v v0)i

= limv0!v

V 0(v)2@V 0^R(V0)@V ^R(V )

@v0@vH(v v0)

= V 0(v)2 : [@V ^R(V )]2 : (v);

= V 0(v)2T^vv(V ) (v); (2.14)

where primes denote derivatives with respect to v and

(v) = limv0!v

@v@v0

H(v v0)H[f(v) f(v0)]

:

(2.15)

Using Eq. (2.13) we nd

(v) =1

4

V 000(v)

6V 0(v)V 00(v)2

4V 0(v)2

= 1

12

pV 0(v)

1pV 0(v)

!00: (2.16)

Equation (2.14) is the key result that we shall use.Note that taking the expected value of Eq. (2.14) in thevacuum state yields

h j T^vv(v) j i = (v); (2.17)

where j i = S^ j0i is the natural vacuum state associatedwith the V coordinate, which satises b^! j i = 0. Thisreproduces the standard formula for the expected stresstensor in the vacuum state associated with a given nullcoordinate, see, e.g., Ref. [18].

Now integrate Eq. (2.14) against the smearing function(v). From Eq. (2.3) this yields

S^yE^(R)[]S^ =

Zdv(v)V 0(v)2T^vv[V (v)]

Zdv(v)(v): (2.18)

If we now choose the coordinate V to be such that(v)V 0(v) = 1, then the rst term on the right hand

side of Eq. (2.18) becomesRdV T^vv(V ), which is just the

Hamiltonian H^R, c.f. Eq. (2.8) above. Inserting the re-lation V 0(v) = 1=(v) into Eqs. (2.16) and (2.18) gives

S^yE^(R)[]S^ = H^R ; (2.19)

where

= 1

12

Zdvp(v)

p(v)

00=

1

48

Zdv

0(v)2

(v): (2.20)

On the second line we have integrated by parts, and as-sumed that 0(v)! 0 as v ! 1.

It is clear from Eq. (2.19) that E(R)min[] = , since

H^R is a positive operator with minimum eigenvalue zero.Equation (2.6) then follows from Eq. (2.20). Also, thestate which achieves the minimum value of E(R)[] isjust the vacuum state j i = S^ j0i associated with the Vcoordinate; this is a generalized (multi-mode) squeezedstate. The V coordinate is given in terms of (v) by

V (v) =

Zdv

(v): (2.21)

B. Algebraic reformulation

The derivation just described suers from the technicaldrawback that in certain cases the \scattering matrix" S^will fail to exist. This operator S^ will exist when [19]Z 1

0

d!

Z 10

d!0 j!!0 j2 1=2 (2.24)

everywhere. Therefore, for smearing functions whichsatisfy the normalization condition (1.1), the Bogilubov

3

transformation to the mode basis associated with the newcoordinate (2.21) does not yield a well dened scattering

operator S^. Thus, strictly speaking, the proof outlined inSec. II A above is not valid except for non-normalizablesmearing functions satisfying (2.24).

However, it is straightforward to generalize the proofto arbitrary smearing functions using the algebraic for-mulation of quantum eld theory [19], as we now outline.For any algebraic state on Minkowski spacetime, let

Fg;(v) = hTvv(v)i (2.25)

denote the expected value of the vv component of thestress tensor in the state . Here g = gab denotes the flatMinkowski metric

gabdxadxb = dt2 + dx2 = dudv; (2.26)

where xa = (x; t). Now, for any coordinate V = V (v),consider the conformally related metric gab given by

gabdxadxb = dudV = V 0(v)dudv: (2.27)

We can naturally associate with the state onMinkowski spacetime (M; gab) a state on the space-time (M; gab) which has the same n point distributions

h^R(v1) : : : ^R(vn)i. It can be checked that the resultingalgebraic state obeys the Hadamard and positivity con-ditions on the spacetime (M; gab) and so is a well denedstate. If we dene

Fg;(v) = hTvv(v)i ; (2.28)

then a straightforward point-splitting computation ex-actly analogous to that outlined in Sec. II A above yields

Fg;(v) = V0(v)2Fg;[V (v)](v); (2.29)

where (v) is the quantity dened by Eq. (2.16) above.Now choosing V 0(v) = 1=(v) yields, in an obvious nota-tion,

hE(R)[]i = hH^Ri : (2.30)

Finally we use the fact that the quadratic form H^R ispositive indenite for all algebraic states (not just forstates in the folium of the vacuum state). The remainderof the proof now follows just as before.

III. CONCLUSION

We have derived a very general constraint on the be-havior of renormalized expected stress tensors in free eldtheory in two dimensions, generalizing earlier results ofFord and Roman. Our result conrms the generality ofthe Ford-Roman time-energy uncertainty-principle-typerelation [11]: that the amount E of energy measuredover a time t is constrained by

E > h

t: (3.1)

Our result also shows that the amount of negative en-ergy that can be contained in a nite region of space intwo dimensions is innite, by taking the limit where thesmearing function approaches a step function. However,this innity is merely an ultraviolet edge-eect, in thesense that states which have large total negative energiesinside the nite region will have most of the energy den-sity concentrated near the edges, and furthermore willhave compensating large positive energy densities justoutside the nite region.

ACKNOWLEDGMENTS

The author thanks Robert Wald, Tom Roman andLarry Ford for helpful discussions. This research wassupported in part by NSF grants PHY 95{14726 andPHY{9408378, and by an Enrico Fermi fellowship.

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