Negative energy densities in integrable quantum field theories at one-particle level

# Negative energy densities in integrable quantum field theories at one-particle level

Henning Bostelmann University of York, Department of Mathematics, York YO10 5DD, United Kingdom; e-mail: henning.bostelmann@york.ac.uk    Daniela Cadamuro University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, United Kingdom. Present address: Mathematisches Institut, Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany; e-mail: daniela.cadamuro@mathematik.uni-goettingen.de
2 March 2016
###### Abstract

We study the phenomenon of negative energy densities in quantum field theories with self-interaction. Specifically, we consider a class of integrable models (including the sinh-Gordon model) in which we investigate the expectation value of the energy density in one-particle states. In this situation, we classify the possible form of the stress-energy tensor from first principles. We show that one-particle states with negative energy density generically exist in non-free situations, and we establish lower bounds for the energy density (quantum energy inequalities). Demanding that these inequalities hold reduces the ambiguity in the stress-energy tensor, in some situations fixing it uniquely. Numerical results for the lowest spectral value of the energy density allow us to demonstrate how negative energy densities depend on the coupling constant and on other model parameters.

## 1 Introduction

The energy density is one of the fundamental observables in classical as well as quantum field theories. It has a special significance in field theories on curved backgrounds, since it enters Einstein’s field equation as a source term, and is therefore linked to the structure of space-time. But also on flat Minkowski space, as well as in low-dimensional conformal field theories, it plays an important role.

In the transition from classical to quantum field theories, some of the distinctive properties of the energy density are lost. In particular, the classical energy density is positive at every point, which in General Relativity implies certain stability results, such as the absence of wormholes [FR96]. This is however not the case in quantum field theory, not even on flat spacetime: While the global energy operator is still nonnegative, the energy density can have arbitrarily negative expectation values [EGJ65]. However, a remnant of positivity is still expected to hold. When one considers local averages of the energy density, for some fixed smooth real-valued function , then certain lower bounds – quantum energy inequalities (QEIs) – should be satisfied. In the simplest case, one finds for any given averaging function a constant such that

 ⟨φ,T00(g2)φ⟩≥−cg∥φ∥2 (1.1)

for all (suitably regular) vector states of the system; this is a so-called state-independent QEI. In general, (1.1) may need to be replaced with a somewhat weaker version (a state-dependent QEI) where the right-hand side can include a slight dependence on the total energy of the state .

This raises the question under which conditions the QEI (1.1) can be shown to hold rigorously. The inequality has in fact been established for various linear quantum fields, on flat as well as curved spacetime, and in conformal QFTs (e.g., [Fla97, PF98, Few00, FV02, FH05]; see [Few12] for a review). However, dropping the restriction to linear fields, that is, allowing for self-interacting quantum field theories, few results are available. This is not least due to the limited availability of rigorously constructed quantum field theoretical models; see however our recent proof of (1.1) in the massive Ising model [BCF13]. In a model-independent setting, one can establish state-dependent inequalities for certain “classically positive” expressions [BF09], based on a model-independent version of the operator product expansion [Bos05], but the relation of these expressions to the energy density remains unknown.

In the present paper, we will investigate the inequality (1.1) in a specific class of self-interacting models on 1+1 dimensional Minkowski space, so-called quantum integrable models, which have recently become amenable to a rigorous construction. Specifically, we consider integrable models with one species of massive scalar boson and without bound states.

An a priori question is what form the stress-energy tensor takes in these models. There is a straightforward answer in models derived from a classical Lagrangian, such as the sinh-Gordon model, where a candidate for the operator can be computed [FMS93, KM93, MS94]. However, we also consider theories where no associated Lagrangian is known; and more generally, we aim at an intrinsic characterization of the quantum theory without referring to a “quantization process”. In fact, there will usually be more than one local field that is compatible with generic requirements on , such as covariance, the continuity equation, and its relation to the global Hamiltonian.

Given , one can ask whether the QEI (1.1) holds for this stress-energy tensor, or rather for which choice of stress-energy tensor. In fact, QEIs may hold for some choices of but not for others, as the nonminimally coupled free field on Minkowski space shows [FO08]. Ideally, one would hope that requiring a QEI fixes the stress-energy tensor uniquely.

In the present article, we will consider the above questions in integrable models, but with one important restriction: We will consider the energy density at one-particle level only. That is, we will ask whether the inequality (1.1) holds for all (sufficiently regular) one-particle states .

This case might seem uninteresting at first: One might argue that in integrable models, where the particle number is a conserved quantity, the effect of interaction between particles is absent at one-particle level. But this is not the case, as already the massive Ising model shows [BCF13]: in this model of scalar bosons, one-particle states with negative energy density exist, whereas in a model of free bosons, the energy density is positive at one-particle level. We will demonstrate in this paper that self-interaction in our class of models leads to negative energy density in one-particle states, and that this effect increases with the strength of the interaction.

Our approach is as follows. Having recalled the necessary details of the integrable models considered (Sec. 2), we ask what form the stress-energy tensor can take at one-particle level. This will lead to a full characterization of the integral operators involved, with the energy density fixed up to a certain polynomial expression (Sec. 3).

In Sec. 4, we will show that under very generic assumptions, states of negative energy density exist, and that for certain choices of the stress-energy tensor, the energy density becomes so negative that the QEI (1.1) cannot hold even in one-particle states. For other choices of the energy density operator, we demonstrate in Sec. 5 that the QEI does hold. In consequence (Sec. 6), we find in the massive Ising model that one-particle QEIs hold for exactly one choice of energy density, whereas in other models (including the sinh-Gordon model), the choice of energy density is at least very much restricted by a QEI.

All this is based on rigorous estimates for the expectation values of . However, the best possible constant in (1.1) – in other words, the lowest spectral value of restricted to one-particle matrix elements – can only be obtained by numerical approximation. We discuss the results of an approximation scheme in Sec. 7, thus demonstrating how the effect of negative energy density varies with the coupling constant and with the form of the scattering function. The program code used for this purpose is supplied with the article [Cod]. We end with a brief outlook in Sec. 8.

## 2 Integrable models

For our investigation, we will use a specific class of quantum field theoretical models on 1+1 dimensional spacetime, a simple case of so-called integrable models of quantum field theory. These models describe a single species of scalar massive Bosons with nontrivial scattering. The scattering matrix is factorizing: When two particles with rapidities and scatter, they exchange a phase factor ; and multi-particle scattering processes can be described by a sequence of two-particle scattering processes.

There are several approaches to constructing such integrable quantum field theories. Conventionally, one starts from a classical Lagrangian, derives the two-particle scattering function from there, and then constructs local operators (quantum fields) by their matrix elements in asymptotic scattering states; this is the form factor programme [Smi92, BFK06]. A more recent, alternative approach [SW00, Lec08] starts from the function as its input, then constructs quantum field localized in spacelike wedges (rather than at spacetime points), and uses these to abstractly obtain observables localized in bounded regions.

We will largely follow the second mentioned approach here; in particular, we set out from a function rather than from a classical Lagrangian. In all what follows, we will assume that a scattering function is given, which we take to be a meromorphic function on which fulfils the symmetry properties

 S(−ζ)=S(ζ)−1=S(ζ+iπ)=¯¯¯¯¯¯¯¯¯¯¯S(¯ζ). (2.1)

A range of examples for such functions can easily be given, in particular because the properties (2.1) are preserved under taking products of functions; see Table 1. This includes the sinh-Gordon model, depending on a coupling parameter , which is normally constructed from a Lagrangian [FMS93]; but for other examples (e.g., the generalized sinh-Gordon models mentioned in Table 1), no corresponding Lagrangian is known.

We will not enter details of the construction of the associated quantum field theory based on here, but will recall only the general concepts as far as relevant to the present analysis. The single particle space of the theory is given by , where the variable of the wave function is rapidity, linked to particle two-momentum by ; here is the particle mass. On , the usual representation of the Poincaré group acts. One then constructs an “-symmetric” Fock space over , on which “interacting” annihilation and creation operators and act; instead of the CCR, they fulfil the Zamolodchikov-Faddeev relations [Lec08], depending on . The quantum field theory is constructed on this Fock space. If is an operator of the theory (of a certain regularity class, including smeared Wightman fields), localized in a bounded spacetime region, then it can be written in a series expansion [BC13, BC15]

 A=∞∑m,n=0∫dθdηm!n!F[A]m+n(θ+i0,η+iπ−i0)z†(θ1)⋯z†(θm)z(η1)⋯z(ηn), (2.2)

where are meromorphic functions with certain analyticity, symmetry and growth properties (which we will recall where we need them). Examples of such local observables would include smeared versions of the energy density, supposing they fall into the regularity class mentioned.

In the construction of functions that fulfil these properties, an important ingredient is the so-called minimal solution of the model [KW78]. We consider it here with the following conventions.

###### Definition 2.1.

Given a scattering function , a minimal solution is a meromorphic function on which has the following properties.

1. has neither poles nor zeros in the strip , except for a first-order zero at in the case that ,

2. ;

3. ;

4. ;

5. There are constants such that if , .

Note that properties (a) and (e) automatically hold analogously for the strip by property (b). The first-order zero at (and analogously ) must necessarily occur in the case due to (c).

The properties (a)(e) actually fix uniquely if it exists, so that we can speak of the minimal solution. We prove this in our context; cf. [KW78, p. 459].

###### Lemma 2.2.

For given fulfilling (2.1), there exists at most one minimal solution .

###### Proof.

Given two minimal solutions , , define . By property (a), this function is analytic in a neighbourhood of the strip – the possible zeros of at and cancel – and by (b) and (c), we have

 G(ζ+2πi)=G(ζ)=G(−ζ). (2.3)

We can therefore find an entire function such that : We observe that is bijective from the region to the upper, respectively lower, half-plane, and use the properties (2.3) to accommodate the branch cuts of the inverse hyperbolic function. From property (d), this function fulfils the estimate

 log|P(z)|≤a′|Rearcoshz|+b′ (2.4)

with certain constants and for large . Since grows like for large , this means that is polynomially bounded at infinity, and hence a polynomial. But due to property (a), has no zeros, and is therefore constant. Now from (d), . Hence . ∎

Let us note a simple consequence: One checks that together with , also fulfils properties (a)(e). Thus the lemma yields . Together with (b), this shows that is symmetric and real-valued on the line . We will use this fact frequently in the following.

In this article, we will always assume that a minimal solution exists. In fact, for the examples we mentioned, they are listed in Table 1. For the sinh-Gordon and related models, this involves the integral expression

 JB(ζ+iπ):=8∫∞0dxxsinhxB4sinhx(2−B)4sinhx2sinh2xsin2xζ2πfor B∈(0,2)+iR (2.5)

which is known from [FMS93] (but note that we use a different normalization for ). Due to the Riemann-Lebesgue lemma, converges to a constant as , with fixed.

For the generalized sinh-Gordon and Ising models in Table 1, can essentially be obtained as a product of the minimal solutions of the corresponding sinh-Gordon or Ising factors, since properties (b)(e) in Def. 2.1 are again preserved under products. However, in order to satisfy property (a), any possible double zeros of the product function at need to be cancelled by dividing by appropriate powers of .

## 3 Energy density at one-particle level

The first question we want to consider is which form the energy density operator can take in our models. More specifically, we ask what the functions in the expansion (2.2) can be if

 A=Tαβ(f)=∫dtf(t)Tαβ(t,0) (3.1)

is a component of the stress-energy tensor smeared with a real-valued test function in time direction. The answer may appear obvious in models such as the free field or the sinh-Gordon model, where the energy density is linked to the classical Lagrangian and well studied. However, we aim at an intrinsic characterization of the energy density within the quantum theory, and therefore we are looking for the most general form of the stress-energy tensor compatible with generic assumptions on this operator, which will be detailed below.

As announced in the introduction, we will consider only one-particle states of the theory, and evaluate the stress-energy tensor only in these. More precisely, we will consider the stress-energy tensor only in matrix elements of the form

 ⟨φ,Tαβ(f)ψ⟩with φ,ψ∈D(R)⊂H1. (3.2)

(The restriction of the quadratic form to smooth functions of compact support, i.e., , is perhaps too cautious – the form can easily be extended to non-smooth and to sufficiently rapidly decaying wave functions, and we will in fact use piecewise continuous functions in the numeric evaluation in Sec. 7; but for the moment we restrict to for simplicity.)

Writing in expanded form as in (2.2), we see that only the coefficients and contribute to this one-particle matrix element. Since the coefficient equals the vacuum expectation value of the operator, we can assume without loss that ; in fact, this is necessary for the energy density, since it would otherwise not integrate to the Hamiltonian . This leaves us with

 ⟨φ,Tαβ(f)ψ⟩=∫dθdη¯¯¯¯¯¯¯¯¯¯φ(θ)F2[Tαβ(f)](θ+i0,η+iπ−i0)ψ(η). (3.3)

Since we expect to be a translation-covariant operator-valued distribution in , we will assume that

 F2[Tαβ(f)](θ,η+iπ)=Fαβ(θ,η)~f(μcoshθ−μcoshη) (3.4)

with a function independent of , where we take the Fourier transform with the convention .

The task is therefore to determine the possible form of , starting from physical properties. We will list these assumptions one by one and explain their motivation, but we skip details of how they are derived; that is, we will take these assumptions as axioms in our context.

The first set of conditions follows from the general properties of the expansion coefficients of a local operator in integrable models, as derived in [BC15].

1. are meromorphic functions on , analytic in a neighbourhood of the region .

This is due to the general analyticity properties of the coefficients [BC15, property (FD1)], together with the absence of “kinematic poles” in the specific case of the coefficient (from property (FD4) there).

2. They have the symmetry properties

 Fαβ(θ,η)=S(θ−η)Fαβ(η+iπ,θ−iπ)=Fαβ(η+iπ,θ+iπ). (3.5)

This is a rewritten form of the properties of “-symmetry” and “-periodicity” – properties (FD2) and (FD3) in [BC15].

3. Hermiticity of the observable is expressed as

 Fαβ(θ,η)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Fαβ(η,θ)%forallθ,η∈R. (3.6)
4. We demand that there exist constants such that

 |Fαβ(θ,η)|≤ℓ(coshReθ)k(coshReη)k whenever −π

This is motivated by the bounds for discussed in [BC15]; the condition guarantees that the smeared version for every Schwartz function will fulfil, at one-particle-level, polynomial high-energy bounds in the form of [BC15, property (FD6)].

Further conditions are derived from properties that one expects specifically of the stress-energy tensor .

1. The stress-energy tensor is a symmetric tensor, which rewrites in our context as

 Fαβ(θ,η)=Fβα(θ,η). (3.8)
2. It is covariant under Lorentz transformations, which means in our terms, cf. [BC13, Prop. 3.9],

 Fαβ(θ−λ,η−λ)=Λαα′Λββ′Fα′β′(θ,η), (3.9)

where is the boost matrix with rapidity parameter .

3. It is invariant under spacetime reflections, which by [BC15, Thm. 5.4] translates to

 Fαβ(θ+iπ,η+iπ)=Fαβ(θ,η). (3.10)
4. It fulfils the continuity equation (), which reads in our terms,

 (pα(θ)−pα(η))Fαβ(θ,η)=0, (3.11)

where .

5. The (0,0)-component of the tensor integrates to the Hamiltonian, , which translates to

 F00(θ,θ)=μ22πcosh2θ. (3.12)

Taking these properties as our starting point, we ask what functions are compatible with them. The answer is given in the following proposition.

###### Proposition 3.1.

Functions fulfil the properties (T1)(T9) if, and only if, there exists a real polynomial with such that

 Fαβ(θ,η)=Fαβfree(θ,η)P(cosh(θ−η))Fmin(θ−η+iπ), (3.13)

where

 (3.14)

Note that is the well-known one-particle expression of the “canonical” stress-energy tensor of the free Bose field.

###### Proof.

It is straightforward to check that as given in (3.13) fulfils all conditions (T1)(T9), knowing that fulfils them in the case .

Thus, let fulfil conditions (T1)(T9). We first use (T8) with and with , along with (T5), to obtain

 F11(θ,η)=tanh2θ+η2F00(θ,η). (3.15)

Now we consider the functions , which are meromorphic by (T1). Note that (T8) with implies

 (p0(ζ)−p0(−ζ))=0G00(2ζ)+(p1(ζ)−p1(−ζ))=−2μsinhζG10(2ζ)=0 (3.16)

and hence . Then by (T5). Also, (3.15) leads to , and the only nonzero component of is . From (T2) and (T7) one concludes

 G00(−ζ)=G00(ζ),G00(ζ−iπ)=S(ζ)G00(ζ+iπ). (3.17)

Now set

 Q(ζ):=2πμ2G00(ζ)/Fmin(ζ+iπ). (3.18)

This is analytic in a neighbourhood of the strip . (Note that in the case , the zeros of the denominator at are cancelled by corresponding zeros of the numerator which exist due to (3.17).) The symmetry relations (3.17) and Def. 2.1(b),(c) imply that

 Q(−ζ)=Q(ζ),Q(ζ+iπ)=Q(ζ−iπ). (3.19)

Arguing as in the proof of Lemma 2.2, we can therefore find an entire function such that . From property (T4) and Def. 2.1(e), this fulfils the estimates

 |P(z)|≤2πμ2∣∣G00(arcoshz)∣∣|Fmin(arcosh(z)+iπ)|≤ℓ′(cosh12Rearcoshz)2k(coshRearcoshz)k′≤ℓ′(2|z|+1)k−k′ (3.20)

with some constants . That is, is a polynomially bounded entire function, and hence a polynomial. From (T3), we can conclude that for all , thus the coefficients of are real. Also, (T9) implies and hence . Thus has the properties claimed in the proposition. Combining our results, we have shown that

 Fαβ(ζ2,−ζ2)=μ22π(1000)P(coshζ)Fmin(ζ+iπ). (3.21)

This means that (3.13) holds in the case . But then it holds for any value of , since both sides of (3.13) fulfil the covariance condition (T6). ∎

In the following, when we speak of the stress-energy tensor of a model, we will always refer to one of the form (3.4), with as in Proposition 3.1. We will often abbreviate

 FP(ζ):=P(coshζ)Fmin(ζ+iπ), (3.22)

noting that enjoys most of the defining properties of (namely, Def. 2.1(b)(e) with shifted argument, but not (a)). Also, is symmetric and real-valued on the real line. The expectation value of the energy density now becomes

 ⟨φ,T00(f)φ⟩=μ22π∫dθdη¯¯¯¯¯¯¯¯¯¯φ(θ)φ(η)cosh2θ+η2FP(θ−η)~f(μcoshθ−μcoshη). (3.23)

Proposition 3.1 shows that on the one-particle level, we recover the well-known “canonical form” of the energy density of the free field and of the sinh-Gordon model [FMS93], and the considered for the massive Ising model in [BCF13], up to a possible polynomial factor in . We emphasize that, while one might expect , all our assumptions so far are perfectly compatible with a more general polynomial . However, we will see later (in Sec. 6) that the choice of is restricted, in some cases uniquely to , if we demand that quantum energy inequalities hold.

## 4 States with negative energy density

As the next question about properties of the energy density, we will ask whether single-particle states with negative energy density exist at all; more specifically, whether can be negative if and is a real-valued Schwartz function, i.e., . The example of the free field with canonical energy density (, ) shows that this is not guaranteed: In this specific case, is known to be positive between one-particle states. However, as we shall see in a moment, the introduction of interaction quite generically leads to negative energy densities.

We will exhibit these negative energy densities by explicitly constructing corresponding states . In preparation, we fix a nonnegative, smooth, even function with support in . For , , we set , so that has support in and is normalized with respect to the norm .

###### Proposition 4.1.

Suppose that there is such that . Then there exist and such that .

###### Proof.

We will show that, with suitable choice of , one has

 0>∫dθdη¯¯¯¯¯¯¯¯¯¯φ(θ)φ(η)cosh2θ+η2FP(θ−η)=:X. (4.1)

( is the expectation value of up to a factor.) Rewriting (3.23) as

 ⟨φ,T00(g2)φ⟩=μ22π∫dtg2(t)∫dθdη¯¯¯¯¯¯¯¯¯¯φ(θ)φ(η)cosh2θ+η2FP(θ−η)eitμ(coshθ−coshη), (4.2)

and noticing that the inner integral expression is real, continuous in , and gives at , it is then clear that we can choose so that (4.2) becomes negative.

To achieve (4.1), we will choose the wave function as

 φ(θ)=2∑j=1βjχ1,ρ(θ−γj), (4.3)

where , and will be specified later; the quantity then depends on these parameters. To show that for some , it suffices to show that converges to a negative limit as . Noting that in this limit, one obtains from (4.1) that

 limρ↘0X=β∗Mβ,%whereMjk=cosh2γj+γk2FP(γj−γk). (4.4)

This expression is negative for suitable if the determinant of the matrix is negative. Setting , , with still to be chosen, one computes

 detM=cosh2(γ+θP/2)cosh2(γ−θP/2)−cosh4γF2P(θP). (4.5)

Since as , and since by assumption, does indeed become negative for sufficiently large , which concludes the proof. ∎

In particular, the conditions of Proposition 4.1 are met in the sinh-Gordon and the Ising models for any choice of , as well as in the free model if . Thus single-particle states with negative energy density exist in generic situations.

Under stricter assumptions on the function , we can in fact show a significantly stronger result: If grows stronger then a certain rate, then the negative expectation values of become so large that quantum energy inequalities cannot hold.

###### Proposition 4.2.

Suppose there exist and such that

 ∀θ≥θ0:FP(θ)≥ccoshθ. (4.6)

Let , . Then, there exists a sequence in , , such that

 ⟨φj,T00(g2)φj⟩→−∞as j→∞. (4.7)
###### Proof.

We set

 φj(θ):=βj,1χ2,ρj(θ−j)+βj,2χ2,ρj(θ+j), (4.8)

where fulfil (but are otherwise arbitrary), and where is a null sequence to be specified later. With this choice, we have , and one computes from (3.23) that

 ⟨φj,T00(g2)φj⟩=μ22πβ∗jMjβj, (4.9)

where is the matrix

 Mj,mn=∫dθdηhj,mn(θ,η)˜g2(μkj(θ,η))χ2,ρj(θ)χ2,ρj(η) (4.10)

with the functions

 hj,11(θ,η)=hj,22(θ,η) =cosh2(j+θ+η2)FP(θ−η), (4.11) hj,12(θ,η)=hj,21(θ,η) =cosh2(θ−η2)FP(2j+θ+η), (4.12) kj(θ,η) =2sinh(j+θ+η2)sinh(θ−η2). (4.13)

It enters here that is even. One of the eigenvalues of is , and we will show that , proving that (4.7) holds for a suitable choice of .

To that end, we establish estimates on , and for , that is, in the region where the integrand of (4.10) is nonvanishing. First, continuity of and imply that if is small; note that enters here. Estimating (), we then have in the relevant range for ,

 (4.14)

Further, the growth condition (4.6) implies for ,

 hj,12(θ,η)≥ccosh(2j+θ+η)≥c2e2je−2ρj. (4.15)

Now (4.14) and (4.15) combine to give

 hj,11(θ,η)−hj,12(θ,η)≤c+12+14e2j(e2ρj(c+12)−2ce−2ρj)→12−c<0≤2c−c′e2j<0 (4.16)

with some constant and for large . Finally, since in the integrand,

 |kj(θ,η)|≤2(eje(θ+η)/2)(2|θ−η|2)≤12ejρj. (4.17)

Now setting specifically , with still to be specified, we have independent of . Noting that , we can achieve with a suitable choice of that

 ˜g2(μkj(θ,η))≥12˜g2(0)>0. (4.18)

Using (4.16) and (4.18) in the integrand of (4.10), we then obtain

 λj=Mj,11−Mj,12≤(2c−c′e2j)12˜g2(0)(∥χ2,ρj∥1)2=a2˜g2(0)(2ce−j−c′e+j)(ρ−1/2j∥χ2,ρj∥1)2. (4.19)

Here is actually independent of . Hence as , which concludes the proof. ∎

## 5 Quantum energy inequalities

We now turn to the existence of quantum energy inequalities, i.e., we want to show that the operator is bounded below at one-particle level. As we have seen in Proposition 4.2, this can be true only if the function does not grow too fast. The main goal of the section is the following theorem, which establishes a QEI under certain bounds on .

###### Theorem 5.1.

Suppose that there exist constants , , and such that

 |FP(ζ)|≤ccoshReζwhenever|Reζ|≥θ0,|Imζ|<λ0. (5.1)

Further, let . Then, there exists such that

 ∀φ∈D(R):⟨φ,T00(g2)φ⟩≥−cg∥φ∥2. (5.2)

The constant depends on (and on , hence on and ) but not on .

The idea of the proof is as follows. In (3.23), we split the integration region in both and into the positive and negative half-axis. Setting for , we can rewrite our expectation value as

 Xφ:=⟨φ,T00(g2)φ⟩=μ22π∫∞0dθ∫∞0dη˜g2(μcoshθ−μcoshη)φ(θ)∗M(θ,η)φ(η), (5.3)

where the matrix is given by

 M(θ,η)=⎛⎜⎝cosh2θ+η2FP(θ−η)cosh2θ−η2FP(θ+η)cosh2θ−η2FP(θ+η)cosh2θ+η2FP(θ−η)⎞⎟⎠. (5.4)

The eigenvectors of are and , independent of , and the corresponding eigenvalues are

 h±(θ,η)=cosh2θ+η2FP(θ−η)±cosh2θ−η2FP(θ+η). (5.5)

Denoting by the components of in the direction of , we thus have

 Xφ=μ22π∫∞0dθ∫∞0dη˜g2(μcoshθ−μcoshη)∑±¯¯¯¯¯¯¯¯¯¯¯¯¯¯φ±(θ)h±(θ,η)φ±(η). (5.6)

We will compare to the following related integral expression:

 Yφ:=μ22π∫∞0dθ∫∞0dη˜g2(μcoshθ−μcoshη)∑±¯¯¯¯¯¯¯¯¯¯¯¯¯¯φ±(θ)k±(θ)k±(η)φ±(η), (5.7)

where

 k±(θ):=√|h±(θ,θ)|=√|cosh2θ±FP(2θ)|. (5.8)

Specifically, we will show that and that is bounded in . We do this in several steps; the hypothesis of the theorem is always assumed.

###### Lemma 5.2.

For any , we have .

###### Proof.

Using the identity

 ˜g2(p−p′)=∫dq2π~g(q+p)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~g(q+p′), (5.9)

we can rewrite the integral as

 Yφ=μ24π2∑±∫dq∣∣∫∞0dηa±(η,q)∣∣2witha±(η,q):=k±(η)φ±(η)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~g(q+μcoshη). (5.10)

But this is clearly nonnegative. ∎

For estimating , we first need an estimate for the relevant integral kernels, into which the growth bound (5.1) for will crucially enter.

###### Lemma 5.3.

Set

 ℓ±(ρ,τ):=h±(ρ+τ2,ρ−τ2)−k±(ρ+τ2)k±(ρ−τ2). (5.11)

Then, there exists such that for all and ,

 |ℓ±(ρ,τ)|≤aτ2cosh2ρ. (5.12)
###### Proof.

One notes that and that is symmetric in . A Taylor expansion of in around then yields that

 |ℓ±(ρ,τ)|≤τ22sup|ξ|≤1∣∣∂2ℓ±∂τ2(ρ,ξ)∣∣. (5.13)

Thus our task is to estimate the derivative. As a first step, we remark that Cauchy’s formula allows us to deduce estimates for the derivatives of from (5.1): One finds constants such that

 (5.14)

Now we explicitly compute

 ∂2∂τ2h±(ρ+τ2,ρ−τ2) =∂2∂τ2(cosh2ρFP(τ)±cosh2τ2FP(2ρ)) (5.15) =cosh2ρd2FPdθ2(τ)±12coshτFP(2ρ).

In the region , , we therefore have due to (5.1), (5.14),

 ∣∣∂2∂τ2h±(ρ+τ2,ρ−τ2)∣∣≤a1cosh2ρ (5.16)

with some . For the derivative of the second term in , we obtain

 ∂2∂τ2k±(ρ+τ2)k±(ρ−τ2)=14d2k±dθ(ρ+τ2)k±(ρ−τ2)+14k±(ρ+τ2)d2k±dθ(ρ−τ2). (5.17)

Note here that the radicand in – cf. (5.8) – is actually positive for due to (5.1), thus the function is differentiable. ( enters here.) For