Negative energy densities in integrable quantum field theories at oneparticle level
Abstract
We study the phenomenon of negative energy densities in quantum field theories with selfinteraction. Specifically, we consider a class of integrable models (including the sinhGordon model) in which we investigate the expectation value of the energy density in oneparticle states. In this situation, we classify the possible form of the stressenergy tensor from first principles. We show that oneparticle states with negative energy density generically exist in nonfree situations, and we establish lower bounds for the energy density (quantum energy inequalities). Demanding that these inequalities hold reduces the ambiguity in the stressenergy 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 spacetime. But also on flat Minkowski space, as well as in lowdimensional 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 realvalued 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
(1.1) 
for all (suitably regular) vector states of the system; this is a socalled stateindependent QEI. In general, (1.1) may need to be replaced with a somewhat weaker version (a statedependent QEI) where the righthand 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 selfinteracting 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 modelindependent setting, one can establish statedependent inequalities for certain “classically positive” expressions [BF09], based on a modelindependent 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 selfinteracting models on 1+1 dimensional Minkowski space, socalled 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 stressenergy tensor takes in these models. There is a straightforward answer in models derived from a classical Lagrangian, such as the sinhGordon 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 stressenergy tensor, or rather for which choice of stressenergy 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 stressenergy 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 oneparticle level only. That is, we will ask whether the inequality (1.1) holds for all (sufficiently regular) oneparticle 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 oneparticle level. But this is not the case, as already the massive Ising model shows [BCF13]: in this model of scalar bosons, oneparticle states with negative energy density exist, whereas in a model of free bosons, the energy density is positive at oneparticle level. We will demonstrate in this paper that selfinteraction in our class of models leads to negative energy density in oneparticle 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 stressenergy tensor can take at oneparticle 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 stressenergy tensor, the energy density becomes so negative that the QEI (1.1) cannot hold even in oneparticle 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 oneparticle QEIs hold for exactly one choice of energy density, whereas in other models (including the sinhGordon 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 oneparticle 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
model  parameters  
free field  
Ising  
sinhGordon  
(generalized sinhGordon) 
, either real or in complexconjugate pairs  
(generalized Ising) 
For our investigation, we will use a specific class of quantum field theoretical models on 1+1 dimensional spacetime, a simple case of socalled 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 multiparticle scattering processes can be described by a sequence of twoparticle scattering processes.
There are several approaches to constructing such integrable quantum field theories. Conventionally, one starts from a classical Lagrangian, derives the twoparticle 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
(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 sinhGordon model, depending on a coupling parameter , which is normally constructed from a Lagrangian [FMS93]; but for other examples (e.g., the generalized sinhGordon 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 twomomentum 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 ZamolodchikovFaddeev 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]
(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 socalled 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.

has neither poles nor zeros in the strip , except for a firstorder zero at in the case that ,

;

;

;

There are constants such that if , .
Note that properties (a) and (e) automatically hold analogously for the strip by property (b). The firstorder 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
(2.3) 
We can therefore find an entire function such that : We observe that is bijective from the region to the upper, respectively lower, halfplane, and use the properties (2.3) to accommodate the branch cuts of the inverse hyperbolic function. From property (d), this function fulfils the estimate
(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 realvalued 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 sinhGordon and related models, this involves the integral expression
(2.5) 
which is known from [FMS93] (but note that we use a different normalization for ). Due to the RiemannLebesgue lemma, converges to a constant as , with fixed.
For the generalized sinhGordon and Ising models in Table 1, can essentially be obtained as a product of the minimal solutions of the corresponding sinhGordon 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 oneparticle 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
(3.1) 
is a component of the stressenergy tensor smeared with a realvalued test function in time direction. The answer may appear obvious in models such as the free field or the sinhGordon 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 stressenergy tensor compatible with generic assumptions on this operator, which will be detailed below.
As announced in the introduction, we will consider only oneparticle states of the theory, and evaluate the stressenergy tensor only in these. More precisely, we will consider the stressenergy tensor only in matrix elements of the form
(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 nonsmooth 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 oneparticle 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
(3.3) 
Since we expect to be a translationcovariant operatorvalued distribution in , we will assume that
(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].

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).

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

Hermiticity of the observable is expressed as
(3.6) 
We demand that there exist constants such that
(3.7)
Further conditions are derived from properties that one expects specifically of the stressenergy tensor .

The stressenergy tensor is a symmetric tensor, which rewrites in our context as
(3.8) 
It is covariant under Lorentz transformations, which means in our terms, cf. [BC13, Prop. 3.9],
(3.9) where is the boost matrix with rapidity parameter .

It is invariant under spacetime reflections, which by [BC15, Thm. 5.4] translates to
(3.10) 
It fulfils the continuity equation (), which reads in our terms,
(3.11) where .

The (0,0)component of the tensor integrates to the Hamiltonian, , which translates to
(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.
Note that is the wellknown oneparticle expression of the “canonical” stressenergy 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
(3.15) 
Now we consider the functions , which are meromorphic by (T1). Note that (T8) with implies
(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
(3.17) 
Now set
(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
(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
(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
(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 stressenergy 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
(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 realvalued on the real line. The expectation value of the energy density now becomes
(3.23) 
Proposition 3.1 shows that on the oneparticle level, we recover the wellknown “canonical form” of the energy density of the free field and of the sinhGordon 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 singleparticle states with negative energy density exist at all; more specifically, whether can be negative if and is a realvalued 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 oneparticle 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
(4.1) 
( is the expectation value of up to a factor.) Rewriting (3.23) as
(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
(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
(4.4) 
This expression is negative for suitable if the determinant of the matrix is negative. Setting , , with still to be chosen, one computes
(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 sinhGordon and the Ising models for any choice of , as well as in the free model if . Thus singleparticle 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
(4.6) 
Let , . Then, there exists a sequence in , , such that
(4.7) 
Proof.
We set
(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
(4.9) 
where is the matrix
(4.10) 
with the functions
(4.11)  
(4.12)  
(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 ,
(4.15) 
Now (4.14) and (4.15) combine to give
(4.16) 
with some constant and for large . Finally, since in the integrand,
(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
(4.18) 
Using (4.16) and (4.18) in the integrand of (4.10), we then obtain
(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 oneparticle 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
(5.1) 
Further, let . Then, there exists such that
(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 halfaxis. Setting for , we can rewrite our expectation value as
(5.3) 
where the matrix is given by
(5.4) 
The eigenvectors of are and , independent of , and the corresponding eigenvalues are
(5.5) 
Denoting by the components of in the direction of , we thus have
(5.6) 
We will compare to the following related integral expression:
(5.7) 
where
(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
(5.9) 
we can rewrite the integral as
(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
(5.11) 
Then, there exists such that for all and ,
(5.12) 
Proof.
One notes that and that is symmetric in . A Taylor expansion of in around then yields that
(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
(5.15)  
In the region , , we therefore have due to (5.1), (5.14),
(5.16) 
with some . For the derivative of the second term in , we obtain
(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