Quantum energy inequalities in integrable models with several particle species and bound states

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough-Dodd, the Federbush, and the $O(n)$-nonlinear sigma models.


Introduction
It is well known that the energy operator in quantum field theory (QFT) is positive, while the energy density T 00 may be locally negative.However, for physically reasonable theories, bounds on this negativity are expected when local averages are taken: quantum energy inequalities (QEIs).They may, for example, take the form ϕ, dtg(t) 2 T 00 (t, x)ϕ ≥ −c g ϕ 2 , (1.1) where the constant c g does not depend on ϕ, and the inequality holds for a suitably large set of vectors ϕ.In Minkowski space, the bound is also uniform in x.Without these bounds, accumulation of negative energy might lead to violations of the second law of thermodynamics [For78].They also have significant importance in semiclassical gravity, where the expectation value of T µν appears on the righthand side of the Einstein equations.In this context, QEIs can yield constraints on exotic spacetime geometries and lead to generalized singularity theorems extended from classical results in general relativity; see [KS20, Sec.5] for a review.
QEIs have been established quite generically in linear QFTs, including QFTs on curved spacetimes; see [Few12] for a review.They are also known in 1+1-dimensional conformal QFTs [FH05].However, their status is less clear in self-interacting models, i.e., models with a nontrivial scattering matrix between particles.Some generic results, weaker than (1.1), can be obtained from operator product expansions [BF09].Concrete results in models with self-interaction are rare, though.
The situation is somewhat better in 1+1-dimensional integrable models.In these models, the scattering matrix is constrained to be factorizing but nonetheless allows for a large class of interactions; see, e.g., [KW78;ZZ79;AAR01].A QEI in this context was first established in the Ising model [BCF13].Also, a QEI at oneparticle level (i.e., where (1.1) holds for one-particle states ϕ) has been obtained more generally for models with one scalar particle type and no bound states [BC16].
The class of integrable models is much richer, though-they can also describe several particle species with a more complicated scattering matrix between them or particles with inner degrees of freedom; further, these particles may form bound states1 .This article aims to generalize the results of [BCF13; BC16] to these cases.
As an a priori problem, one may ask what form the energy density operator T 00 takes in these models, even at one-particle level.The classical Lagrangian is often used as heuristic guidance; however, if one takes an inverse scattering approach to integrable models, starting by prescribing the two-particle scattering function, then a classical Lagrangian may not even be available in all cases.Instead, we will restrict the possible form of the energy density starting from generic physical assumptions (such as the continuity equation, but initially disregarding QEIs); see Theorem 3.2 below.
We then ask whether QEIs can hold for these energy densities.There are two main results: First, we confine ourselves to a class of models with rapidityindependent scattering function, i.e., where the scattering matrix is independent of the particle momenta.In this setup, for a canonical choice of energy density, we establish a QEI in states of arbitrary particle number (Theorem 4.3).Second, for generic scattering functions, we give necessary and sufficient criteria for QEIs to hold at one-particle level (Theorem 5.1).Here it turns out that the existence of QEIs critically depends on the large-rapidity behaviour of the two-particle form factor F 2 of the energy density.
We apply our results to several concrete examples, namely, to the Bullough-Dodd model (Sec.7.1) which has bound states, to the Federbush model (Sec.7.2) as an interacting model with rapidity-independent scattering function, and to the O(n) nonlinear sigma model (Sec.7.3) which features several particle species.In particular, we investigate how QEIs further restrict the choice of the stress-energy tensor in these models, sometimes fixing it uniquely.
In short, the remainder of this article is organized as follows.We recall some background on integrable QFTs in Section 2 and discuss the possible form of the energy density in Section 3. Section 4 establishes a QEI in models with constant scattering function, and Section 5 for more generic scattering functions but only at one-particle level.For controlling the large-rapidity asymptotics of F 2 , critically important to our results in Section 5, we first explain the relation between the scattering function and the so-called "minimal solution" in Section 6, with technical details given in the appendix (which contains known facts as well as original results).This is then applied to examples in Section 7. Conclusion and outlook follow in Section 8.
This article is based on the PhD thesis of one of the authors [Man23a].

General notation
We will work on 1+1-dimensional Minkowski space M. The Minkowski metric g is conventionally chosen to be diag(+1, −1) and the Minkowski inner product will be denoted by p.x = g µν p µ x ν .A single parameter, called rapidity, conveniently parametrizes the mass shell on M. In this parameterization, the momentum at rapidity θ is given by p 0 (θ; m) := m ch θ and p 1 (θ; m) := m sh θ, where m > 0 denotes the mass.We will use θ, η, λ to denote real and respectively, the region S[0, π] will be of particular significance and is referred to as the physical strip.
In the following, let K be a finite-dimensional complex Hilbert space with inner product (•, •), linear in the second position.We denote its extension to K ⊗2 as (•, •) K ⊗2 and the induced norm as • K ⊗2 ; i.e., for v i , w i ∈ K, i = 1, 2 we have (v 1 ⊗ v 2 , w 1 ⊗ w 2 ) K ⊗2 = (v 1 , w 1 )(v 2 , w 2 ).For computations, it will be convenient to choose an orthonormal basis {e α }, α ∈ {1, . . ., dim K}.In this basis, we denote v ∈ K ⊗m and w ∈ B(K ⊗m , K ⊗n ) in vector and tensor notation by v α := (e α , v), w α β := (e α , we β ). (2.1) Operators on K or K ⊗2 will be denoted by uppercase Latin letters.This also applies to vectors in K ⊗2 , which are identified with operators on K as follows: For an antilinear involution J ∈ B(K) (to be fixed later), the map A → Â defined by ∀u, v ∈ K : (u, Âv) := (u ⊗ Jv, A) K ⊗2 (2.2) yields a vector space isomorphism between K ⊗2 and B(K).In particular, we consider the special element I ⊗2 ∈ K ⊗2 defined by I ⊗2 = ½ K .For an arbitrary orthonormal basis {e α } α of K, it is explicitly given by (2.3) Remark 2.1.I ⊗2 is invariant under the action of U ⊗2 for any U ∈ B(K) with U unitary or anti-unitary and [U, J] = 0.

One-particle space and scattering function
Definition 2.2.A one-particle little space (with a global symmetry) (K, V, J, M ) is given by a finite-dimensional Hilbert space K, a unitary representation V of a compact Lie group G on K, an antiunitary involution J on K, and a linear operator M on K with strictly positive spectrum.We further assume that M , V (g) and J commute with each other.
Given such a little space (K, V, J, M ), we define the one-particle space (2.6) This defines a unitary strongly continuous representation of the proper Poincare group P + and of G, where the antiunitary U 1 (j) is the PCT operator, representing spacetime reflection.
We will denote the spectrum of the mass operator M as M ⊂ (0, ∞) and its spectral projections as E m , m ∈ M.Moreover, introduce the total energy-momentum operator P µ on H ⊗2 1 by Definition 2.3.Let (K, V, J, M ) be a one-particle little space.A meromorphic function S : C → B(K ⊗2 ) with no poles on the real line is called S-function iff for all ζ, ζ ′ ∈ C the following holds: (S5) Crossing symmetry: An S-function is called regular iff (S8) Regularity: ∃κ > 0 : S ↾ S(−κ,κ) is analytic and bounded.In this case, κ(S) denotes the supremum of such κ's.
Remark 2.4.The S-function is the central object to define the interaction of the model.It is also referred to as auxiliary scattering function [Bab+99, Eq. (2.7)] and closely related to the two-by-two-particle scattering matrix of the model, differing from it only by a "statistics factor", namely −1 on a product state of two fermions and +1 on fermion-boson-or boson-boson-vectors.The full scattering matrix is given as a product of two-by-two-particle scattering matrices of all participating combinations of one-particle states; see, e.g., [Bab+99, Sec.2] and [BC21, Secs.5-6].

Integrable models, form factors, and the stress-energy tensor
From the preceding data-one-particle little space (K, V, J, M ) and S-function S-it is well-known how to construct an integrable model of quantum field theory (inverse scattering approach).This can be done at the level of n-point functions of local fields [Smi92; BFK08] or more rigorously in an operator algebraic setting, at least provided that S is regular, analytic in the physical strip, and satisfies an intertwining property [AL17].We give a brief overview of the construction here, focussing only on aspects that will be relevant in the following.A detailed account can be found in [Man23a].
The interacting state space H, on which our local operators will act, is an Ssymmetrized Fock space generated by S-twisted creators z † and annihilators z known as ZF operators [LM95; LS14].They are defined as operator-valued distributions (2.9) Here Ψ n is the n-particle component of Ψ, and Symm S denotes S-symmetrization: For n = 2 (other cases will not be needed here) and a K ⊗2 -valued function f in two arguments, it can be defined as (2.10) Products of z † and z can be linearly extended to arguments in tensor powers of (2.13) To define locality in our setup, it is helpful to introduce two auxillary fields, for arbitrary f ∈ S(M, K) and where f ± (θ) := f (±p(θ; M )) and U (j) implements the CPT transformation on all of H. Φ(f ) and Φ ′ (f ) may be understood as being localized in a left, resp., right wedge containing the support of f .An operator A is then referred to as localized in some bounded spacetime region O ⊂ M, given as the intersection of a left and a right wedge, if it is relatively local to Φ and Φ ′ ; for more details on this, we refer to [BC15, Sec.2.4], [Lec15] and references therein.Now, any such local operator A can be expanded into a series of the form (2.15) (see [BC15] for the case dim K = 1).Here the n are meromorphic functions of n variables depending linearly on A which are known as the form factors of A; they satisfy a number of well-known properties, the form factor equations [BFK08].In line with the literature, we will call F n the n-particle form factor, though note that expectation values in n-particle states generically have contributions from all zeroto 2n-particle form factors.The symbols O n are given by (2.17) (2.18) and analogously for higher n, but only n ≤ 2 will be needed in the following.Conversely, given F n that fulfil the form factor equations and suitable regularity conditions, (2.15) defines a local operator A. The series (2.15) is to be read in the sense of quadratic forms on D × D with a dense domain D ⊂ H, which we can take to consist of elements Ψ = (Ψ n ) ∈ H, where each Ψ n is smooth and compactly supported and Ψ n = 0 for large enough n.With suitably chosen F n , we can also regard each O n [F n ] as an operator on D, for example for n = 1 if F 1 and F 1 (•+ iπ) are square-integrable.
In the following, we are interested in a specific local operator, the stress-energy tensor, i.e., we study averaged in time with a nonnegative test function g 2 , g ∈ S R (R), and at x 1 = 0 without loss of generality; the integral is to be read weakly on D × D. Also, we will focus on its two-particle coefficient 2 ; this is because: (a) In some models, the energy density has only these coefficients, i.e., F [A] n = 0 for n = 2 (see Sec. 4).
(b) One-particle expectation values, which will partly be our focus, are determined solely by the coefficients n for n ≤ 2.
The above analysis applies to T µν under quite generic assumptions on the highenergy behaviour; for a detailed derivation we refer to [Man23a, Sec.5.2].For our present purposes, details of the technical setup are not needed; in fact, we will proceed in the opposite way: We will select a suitable form factor F µν 2 (ζ; x) for the stress-energy tensor, then use Eq.(2.21) to define T µν (g 2 ) as a quadratic form at one-particle level, i.e., on H 1 ∩ D, or more generally, the expansion (2.15) to define it for arbitrary particle numbers, as a quadratic form on D.
3 The stress-energy tensor at one-particle level This section analyses what form the stress-energy tensor T µν and, in particular, the energy density T 00 can take in our setup.Since our models do not necessarily arise from a classical Lagrangian, we study the stress-energy tensor using a "bootstrap" approach: We require a list of physically motivated properties for T µν and study which freedom of choice remains.
Here we restrict our attention to the one-particle level, where the stress-energy tensor is determined by its form factor F µν 2 , as explained in Section 2.3.We will impose physically motivated axioms directly for the function F µν 2 ; see properties (T1)-(T12) in Definition 3.1 below.Without making claims on the existence of a full stress-energy tensor T µν , we motivate these axioms by the expected features of T µν as follows: First, T µν (x) should be a local field, i.e., commute with itself at spacelike separation.This property is well-studied in the form factor programme to integrable systems and is expected to be equivalent to the form factor equations [Smi92, Sec.2].The same relations can be justified rigorously in an operator algebraic approach, at least for a single scalar field (dim K = 1) without bound states [BC15], with techniques that should apply as well for more general K [Man23a] and in the presence of bound states2 .At one-particle level, where the form factor equations simplify, this yields properties (T1)-(T4) below, with hermiticity of T µν (x) implying (T5); confer also [Man23a, Thm.3.2.1,Prop.5.2.2].The pole set P appearing below is directly connected to the bound state poles of the S-function[KW78; Bab+99; BK02] and will be specified in the examples (Sec.7).
For the following definition (and later on) we use the notation Definition 3.1.Given a little space (K, V, J, M ), an S-function S, and a subset P ⊂ S(0, π), a stress-energy tensor at one-particle level (with poles P) is formed by functions , where the poles within S(0, π) are all first-order and P denotes the set of poles in that region.
Theorem 3.2.Given a little space (K, V, J, M ), an S-function S, and a subset P ⊂ S(0, π), then F 2 is a stress-energy tensor at one-particle level (with poles P) iff it is of the form where F : C → K ⊗2 is a meromorphic function which satisfies for all ζ ∈ C that (a) F ↾ S[0, π] has exactly the poles P; It is parity covariant iff, in addition, Remark 3.3.As can be seen from the proof, it is sufficient to require (T10) for µ = 0; the case µ = 1 is automatic.
Equivalently, X commutes with M .On such X, all of M 1 , M 2 and (M ⊗ M ) 1/2 act the same and in a slight abuse of notation we will use M to denote any of these.If F has this property, i.e., F (ζ) has it for all ζ ∈ C, then the above result simplifies: The energy density, in particular, becomes Proof.On X ∈ K ⊗2 which is diagonal in mass we can simplify and yields the proposed form of F 2 .
Remark 3.5.In some models, the one-particle form factor of the stress-energy tensor, F 1 , is nonzero; in particular in models with bound states, where (ζ) can be determined analogous to Theorem 3.2.In this case the continuity equation, where The analogues of the other conditions in Theorem 3.2 are automatically satisfied.
It is instructive to specialize the above discussion to free models: For a single free particle species of mass m, either a spinless boson (S = 1) or a Majorana fermion (S = −1), we have The canonical expressions for the stress-energy tensor at one-particle level are for the bosonic and the fermionic case, respectively; these conform to Definition 3.1, including parity covariance.Theorem 3.2 applies with 4 A state-independent QEI for constant scattering functions In this section, we treat scattering functions S which are constant, i.e., independent of rapidity.In this case, (S1) and (S2) imply that S ∈ B(K ⊗2 ) is unitary and selfadjoint, hence has the form S = P + − P − in terms of its eigenprojectors P ± for eigenvalues ±1.Further, we require that S has a parity-invariant diagonal, which is to be understood as [S, F]I ⊗2 = 0. (4.1) This setup yields two important simplifications.First, for constant S with parity-invariant diagonal, one easily shows that satisfies the conditions (a) to (h) from Theorem 3.2 with respect to S. Thus, F µν 2 as given in Eq. (3.3) is a parity-covariant stress-energy tensor at one-particle level.
Second, for constant S, the form factor equations for F n , n > 2 simplify significantly; the residue formula connecting F n with F n−2 , see Item c in [BFK08, Sec.3], becomes trivial for even n.As a consequence, the expression reducing the usually infinite expansion (2.15) to a single term, is a local operator after time-averaging.In fact, locality may be checked by direct computation from (T1)-(T4).Moreover, properties (T5)-(T12) mean that T µν is hermitian, is a symmetric covariant two-tensor-valued field with respect to U 1 (x, λ) (properly extended from Eq. (2.4) to the full state space), integrates to the total energy-momentum operator P µ = ds T µ0 (t, s), and is conserved, ∂ µ T µν = 0. Hence, T µν is a valid candidate for the stress-energy tensor of the interacting model.While our axioms certainly do not fix T µν uniquely, T µν as given in (4.3) constitutes a minimal choice and agrees with the canonical one in models such as the free massive scalar and Majorana field as well as the Ising model [FE98;Daw06;BCF13].For this T µν , we aim to establish a QEI result.Our main technique is an estimate for two-particle form factors of a specific factorizing form, which can be stated as follows.
Lemma 4.1.Let h : S(0, π) → K be analytic with L 2 boundary values at R and R + iπ.For we have in the sense of quadratic forms on D × D, Proof.From the ZF algebra relations in (2.11)-(2.13),one verifies that The left-hand side is positive as a quadratic form, implying the result.
Our approach is to decompose F 00 2 into sums and integrals over terms of the factorizing type (4.4) with positive coefficients, then applying the estimate (4.5) to each of them.
To that end, we will call a vector This is equivalent to X being a sum of mutually orthogonal vectors of the form e ⊗ Je with positive coefficients. 3We also recall the notion of a vector diagonal in mass, Eq. (3.13).Now we establish our master estimate as follows: Lemma 4.2.Fix n ∈ {0, 1}.Suppose that X ∈ K ⊗2 is positive, diagonal in mass, and that SX = (−1) n X.Let h : S(0, π) → C be analytic with continuous boundary values at R and Then, in the sense of quadratic forms on D × D, it holds that where the integral is convergent and where Here, each E ⊗2 m X is positive, diagonal in mass and, by (S6), satisfies As a consequence, we may assume without loss of generality that X = E ⊗2 m X.Moreover, by positivity of X, we may decompose X = r α=1 c α e α ⊗ Je α with r ∈ N, c α > 0 and orthonormal vectors e α ∈ K, α = 1, . . ., r.Let and let f ± ν,α relate to h ± ν,α as in Eq. (4.4).Further define Now, in (4.8) use the convolution formula (n ∈ {0, 1}, p 1 , p 2 ∈ C), then split the integration region into the positive and negative halflines, and obtain (4.13) Noting that the h ± ν,α are square-integrable at the boundary of S[0, π], we can now apply Lemma 4.1 to each f ± ν,α ; then, rescaling ν → mν in the integral (4.13) yields the estimate (4.9).Note here that the integration in ν can be exchanged with taking the expectation value Ψ, O 2 [•]Ψ , since the integration regions in ζ are compact for Ψ ∈ D, and the series in (2.15) is actually a finite sum.
Lastly, we show that the r.h.s of Eq. (4.9) is finite.By the Cauchy-Schwarz inequality, the integrand is bounded by a constant times ν n times By assumption, |h(θ)| ≤ a(ch θ) b for some a, b > 0, and the resulting integrand , and using the rapid decay in s by the corresponding property of g.In conclusion, the θ-and ν-integrals converge by Fubini-Tonelli's theorem.

Now we can formulate:
Theorem 4.3 (QEI for constant S-functions).Consider a constant S-function S ∈ B(K ⊗2 ) with a parity-invariant diagonal, i.e., [S, F]I ⊗2 = 0 and denote its eigenprojectors with respect to the eigenvalues ±1 by P ± .Suppose that P ± I ⊗2 are both positive.Then for the energy density T 00 (x) in Eq. (4.3) and any g ∈ D R (R), one has in the sense of quadratic forms on D × D: where ) In the scalar case, dim K = 1, this bound agrees with the previously known bounds for the free massive scalar and Majorana field as well as the Ising model; see Remark 4.4 below.
Proof.We use Lemma 4.2 five times: with 2 (these with n = 1 and X = P − I ⊗2 ); note that P ± I ⊗2 are positive by assumption and diagonal in mass by (S6).Summation of Eq. (4.8) for all these five terms and multiplication with 1 4π M ⊗2 yields the expression dt g 2 (t)F 00 2 (•; (t, 0)) for the energy density in Eq. (4.3).From Lemma 4.2 we obtain Here s i := (−1) n i .Now we compute Thus, the QEI applies to all such models.This does not only include the known QEI results for the free Bose field [FE98], the free Fermi field [Daw06], the Ising model [BCF13], and combinations of those, but also the symplectic model, a fermionic variant of the Ising model (see, e.g., [Las94] or [BC21]).
It also applies to the Federbush model (and generalizations of it as in [Tan14]): Although the Federbush model's S-function is not parity invariant, it has a parity invariant diagonal and Eq.(4.2) yields a valid (parity covariant) candidate for the stress-energy tensor, i.e., it satisfies all the properties (a) to (h).The candidate is in agreement with [CF01, Sec.4.2.3].For further details on the Federbush model, see Section 7.2.
Remark 4.5.The QEI result is independent of the statistics of the particles; it depends only on the mass spectrum and the S-function.The aspect of particle statistics comes into play when computing the scattering function from the S-function (see Remark 2.4); it also enters the form factor equations for local operators (see, e.g., [BC21, Sec.6]).However, in the equations for F 2 relevant for our analysis, the "statistics factors" occur only in even powers, so that our assumptions on the stressenergy tensor-specifically, properties (T3) and (T4) in Def.3.1-are appropriate in both bosonic and fermionic cases.
Remark 4.6.In the short-distance scaling limit, corresponding to m → 0 with M = m½ at fixed g [BLM11], the QEI bound simplifies and becomes proportional to the number of degrees of freedom in the model, r.h.s. of (4.15) Comparing with optimal bounds in the known cases, the free massless scalar and Majorana field, this bound is not optimal: It is larger by a constant factor 3 2 (scalar), resp., 3 (Majorana), as has been noted before in [BCF13].

QEI at one-particle level for general integrable models
This section aims to give necessary and sufficient conditions for QEIs at one-particle level in general integrable models, including models with several particle species and bound states.The conditions are expressed in Theorem 5.1 in cases (a) and (b), respectively.
Given a stress-energy tensor F µν 2 at one-particle level, including diagonality in mass, the expectation values of the averaged energy density are, combining Eq. (2.21) with Corollary 3.4, given by ϕ, T 00 (g 2 )ϕ = dθ dη ch 2 θ + η 2 ϕ(θ), M 2 2π g 2 (p 0 (θ; M )−p 0 (η; M )) F (η−θ+iπ)ϕ(η) (5.1) for ϕ ∈ D ∩ H 1 .We ask whether this quadratic form is bounded below.In fact, this can be characterized in terms of the asymptotic behaviour of F : Theorem 5.1.Let F µν 2 be a parity-covariant stress-energy tensor at one-particle level which is diagonal in mass and F be given according to Corollary 3.4.Then: (a) Suppose there exists u ∈ K with u K = 1, and c > 1 4 such that ∃r > 0 ∀|θ| ≥ r : (5.2) Then for all g ∈ S R (R), g = 0 there exists a sequence (ϕ j ) j in D(R, K), Then for all g ∈ S R (R) there exists c g > 0 such that for all ϕ ∈ D(R, K), ϕ, T 00 (g 2 )ϕ ≥ −c g ϕ 2 2 . (5.5) The two cases are mutually exclusive.While case (b) establishes a QEI at oneparticle level (5.5), case (a) implies that no such QEI can hold.Before we proceed to the proof, let us comment on the scope of the theorem.Remark 5.2.We require parity-covariance of F µν 2 .In absence of this property, at least the parity-covariant part F µν 2,P of F µν 2 , which is given by replacing F with F P := 1 2 (1 + F)F , has all features of a parity-covariant stress-energy tensor at one-particle level except possibly for S-symmetry (T3), which requires the extra assumption [S, F]F = 0.In any case, since S-symmetry will not be used in the proof, Theorem 5.1 still applies to F µν 2,P .Now, if (5.2) holds for F with u satisfying Ju = ηu with η ∈ C and |η| = 1, it holds for F P due to (u, F (θ)u) = (Ju, F (θ)Ju) = (u, FF (θ)u).As a consequence, case (a) applies and no QEI can hold for F µν 2 .On the other hand, if (5.4) is fulfilled for F (hence for F P ), then a one-particle QEI for F µν 2 of the form (5.5) holds at least in parity-invariant one-particle states.Remark 5.3.While Theorem 5.1(b) establishes a QEI only at one-particle level, the result usually extends to expectation values in vectors where is the one-particle form factor of the energy density.This F 1 may be nonzero.However, due to Remark 3.5, it is of the form F 1 (ζ; 0) = F 1 (0) sh 2 ζ; thus, the rapid decay of g 2 and the Cauchy-Schwarz inequality imply that the additional summand is bounded in Ψ 1 2 , hence in Ψ 2 .The rest of this section is devoted to the proof of Theorem 5.1, which we develop separately for the two parts (a) and (b).We first note that from Theorem 3.2, the operators F (ζ) fulfil (5.8) (5.9) In more detail, these equations are implied by S-periodicity and parity-invariance for (5.7), by S-periodicity and CPT-invariance for (5.8), and by normalization for (5.9).Now the strategy for part (a) closely follows [BC16, Proposition 4.2], but with appropriate generalizations for matrix-valued rather than complex-valued F .
Proof of Theorem 5.1(b).For fixed ϕ ∈ D(R, K) and g ∈ S R (R), we introduce X ϕ := ϕ, T 00 (g 2 )ϕ .Our aim is to decompose . Since [M, F (ζ)] = 0 from diagonality in mass, we have X ϕ = m∈M X Emϕ and can treat each E m ϕ, m ∈ M, separately.Therefore in the following, we assume M = m½ K without loss of generality.

The connection between the S-function and the minimal solution
For the purpose of analysing particular examples, it is helpful to introduce the minimal solution of a model, a well-known concept in the form factor programme [KW78] which plays an essential role in the description and classification of the observables of the model.We will here give a brief summary of necessary facts for the examples in Section 7 and a recipe for obtaining QEIs for other models.For technical details and full proofs, we refer to Appendix A. Given an S-function, in generic cases including diagonal models and all our examples we can perform an eigenvalue decomposition into meromorphic complex-valued functions S i and meromorphic projection-valued functions P i such that (see Proposition A.1).For each eigenfunction S ≡ S i (omitting the index i for the moment), the minimal solution is a meromorphic function F min : C → C which is the most regular solution of the form factor equations at one-particle level (or Watson's equations), subject to the normalization condition F min (iπ) = 1 (see Appendix A.2).A general solution to (6.2) is then of the form where q is a rational function which is fixed by the pole-and zero-structure of F q , and q(−1) = 1 if F q (iπ) = 1 (Lemma A.5). Uniqueness of F min follows under mild growth conditions (Lemma A.2). Existence can be proved for a large class of (eigenvalues of) S-functions by employing a well-known integral representation.For this class, the function is well-defined and referred to as the characteristic function of S. In the case S(0) = 1, the minimal solution is then obtained from f = f [S] as the meromorphic continuation of For S(0) = −1, an additional factor needs to be included (see Theorem A.6).
For our analysis of QEIs, it will be crucial to control the large-rapidity behaviour of F min using properties of the characteristic function f [S].This is in fact possible as follows (Proposition A.11): For a continuous function f : [0, ∞) → R, which is exponentially decaying at large arguments and second-order differentiable on some interval [0, δ], δ > 0, and where With this said, we have a recipe for a large class of models to determine whether a one-particle QEI in the sense of Theorem 5.1 holds, or no such QEI can hold: According to Theorem 3.2 and Corollary 3.4, we know that Then F can be decomposed into the eigenbasis with respect to S, namely Let us restrict to parity-invariant F and constant eigenprojectors P i , i.e., having F = FF and P i = const.Then (in some orthonormal basis) the components of each F i will satisfy Watson's equations and take the form as in Eq. (6.3).Therefore, each F i will be of the form , where Q i is a rational function that takes values in K ⊗2 and F i,min is the minimal solution with respect to S i .In case of symmetries, the choice of Q i is further restricted by G-invariance.The asymptotic growth of the F i will be bounded by the growth of the Q i and the bound (6.6) for the F i,min .In summary, depending on the growth of the Q i and the F i,min , we can determine the asymptotic growth of F and thus decide whether a one-particle QEI holds or not.

QEIs in examples
We now discuss some examples of integrable models which illustrate essential features of the abstract results developed in Sections 4 and 5.These include a model with bound states (Bullough-Dodd model, Sec.7.1), an interacting model with a constant scattering function (Federbush model, Sec.7.2), and a model with several particle species (O(n)-nonlinear sigma model, Sec.7.3).
As a first step, we review in our context the known results for models of one scalar particle type and without bound states [BC16].That is, we consider K = C, J the complex conjugation, M = {m} for the one-particle space, and P = ∅ for the stress-energy tensor, with a scattering function of the form  This can be derived in the following way: Since g(t) := (t sh t) −1 (1 + f (t; b)) is exponentially decaying and regular (in particular at t = 0), it is integrable and In the limit |ℜζ| → ∞ the parts which are non-constant with respect to ζ vanish due to the Riemann-Lebesgue lemma for |ℑζ| < π; uniformity follows from g(t) exp(± tℑζ π ) being uniformly L 1bounded in |ℑζ| ≤ δ (see, e.g., proof of Thm.IX.7 in [RS75]).
Next, according to Corollary A.4, the minimal solution with respect to S is given by For the stress-energy tensor at oneparticle level, we obtain (using Corollary 3.4, Lemma A.5, and Corollary A.3) that ) with q a polynomial having real-valued coefficients and q(−1) = 1. Let , where c q is the leading coefficient of q.By the preceding remarks we find that for some c ′ , c ′′ with 0 < c ′ < c < c ′′ and δ, r > 0: where c ′ and c ′′ can be chosen arbitrarily close to c for large enough r.We can therefore conclude by Theorem 5.1 that a QEI of the form (5.5) holds if deg q < 1 2 s(ǫ, n) + 1 and cannot hold if deg q > 1 2 s(ǫ, n) + 1.In case that deg q = 1 2 s(ǫ, n) + 1, details of q become relevant.This can only occur if s(ǫ, n) is even, i.e., ǫ = +1.If here c is less (greater) than 1 4 , then a QEI holds (cannot hold).

(Generalized) Bullough-Dodd model
We now consider a class of integrable models which treat a single neutral scalar particle that is its own bound state.The presence of the bound state requires the S-function to have a specific "bound state pole" in the physical strip with imaginary positive residue and to satisfy a bootstrap equation for the self-fusion process.In Lagrangian QFT, from a one-component field ϕ and a Lagrangian one obtains as Sfunction S BD (•; b).For more general elements of the described class, no Lagrangian is known [CT15].
In our context, we will consider the generalized variant of the model, but for simplicity restrict to finitely many CDD factors and do not include the exotic factor: Definition 7.1.The generalized Bullough-Dodd model is specified by the mass parameter m > 0 and a finite sequence (b k ) k∈{1,...,n} ⊂ (0, 1) + iR, n ∈ N, which has an odd number of real elements and where the non-real b k appear in complex conjugate pairs.The one-particle little space is given by K = C, G = {e}, V = 1 C , and M = m1 C .J corresponds to complex conjugation.The S-function S gBD is of the form Clearly, S BD is obtained from S gBD for n = 1 and b 1 = b.Since S gBD is defined as a product of a finite number of factors of the form S(•; b), its minimal solutions exists and is given by, see Corollary A.4, It enters here that S gBD (0) = −1.
The presence of bound states in the model implies the presence of poles in the form factors of local operators [BFK08], in particular also for F µν 2 .For F µν 1 = 0 we expect a single first-order pole of Lemma 7.2 (Stress tensor in the generalized BD model).A tensor-valued function F µν 2 : C 2 × M → K ⊗2 is a stress-energy tensor at one-particle level with respect to S gBD and P ⊂ {i 2π 3 } iff it is of the form where F gBD,min is the unique minimal solution with respect to S gBD and where q is a polynomial with real coefficients and q(−1) = 1.
Proof.By Theorem 3.2 and Corollary 3.4, F µν 2 is given by (7.10), where F : C → C satisfies properties (a)-(g) of Theorem 3.2 with respect to S gBD .According to Lemma A.5, F is of the form F q (7.11); the factor (−2 ch ζ − 1) −1 takes the one possible first-order pole within S[0, π], namely at i 2π 3 , into account.That q has real coefficients is a consequence of property (e) and Corollary A.3.
Conversely, it is clear that F µν 2 , respectively F = F q , as given above has the properties (a)-(g).
Theorem 7.3 (QEI for the generalized BD model).Let the stress-energy tensor at one-particle level be given by F µν 2 as in Eq. (7.10).Then a QEI of the form (7.12) where c q denotes the leading coefficient of q.If here c is less (greater) than 1 4 then a QEI holds (cannot hold).
Proof.As the minimal solution F gBD,min is given as a finite product of factors ζ → (−i sh ζ 2 ) and F b,min , the asymptotic growth can be estimated analogously to the procedure in the introduction of Section 7. Similar to the estimate (7.6), one obtains for some c ′ and c ′′ with 0 < c ′ < c < c ′′ and some ǫ, r > 0: where c ′ and c ′′ can be chosen arbitrarily close to c for large enough r.
Noting that parity covariance is trivial for K = C and applying Theorem 5.1 yields the conclusions from above depending on deg q and c.

Federbush model
The Federbush model is a well-studied integrable QFT model with a constant, but non-trivial, scattering function; see [Fed61; STW76; Rui81; Rui82; CF01] and references therein.In Lagrangian QFT, the traditional Federbush model is described in terms of two Dirac fields Ψ 1 , Ψ 2 by a Lagrangian density 4 The Federbush model obeys a global U (1) ⊕2 symmetry since L Fb is invariant under The fields Ψj take values in C 2 .ǫµν denotes the antisymmetric tensor with ǫ01 = −ǫ10 = 1.Other standard notations are ψj := ψ † j γ0 and / ∂ = γ µ ∂µ with anticommuting matrices γ 0 , γ The stress-energy tensor of the model has been computed before [SH78] and its trace (Eq.(44) in the reference) is given by m j : Ψj Ψ j : (7.17) which agrees with the (trace of the) stress-energy tensor of two free Dirac fermions.Note in particular that it is parity-invariant.
In our framework, the model can be described in the following way: Definition 7.4.The Federbush model is specified by three parameters, the particle masses m 1 , m 2 ∈ (0, ∞) and the coupling parameter λ ∈ (0, ∞).The symmetry group is G = U (1) ⊕2 .The one-particle little space is given by L = (K, V, J, M ) with L = L 1 ⊕ L 2 and where for j = 1, 2 we define K j = C 2 and as operators on K j where J j is antilinear and for the choice of basis {e ≡ (0, 1) t }.The S-function is denoted by S Fb ∈ B(K ⊗2 ).Its only nonvanishing components, enumerated as α, β = 1+, 1−, 2+, 2− corresponding to e 1/2 , are given by S αβ := (S Fb ) βα αβ with Note that S Fb is a constant diagonal S-function; e.g., S αβ = S * βα = S −1 βα imply that S Fb is self-adjoint and unitary.Note also that, S αβ = S ᾱ β = S βα , where ᾱ corresponds to α ∈ {1+, 1−, 2+, 2−} by flipping plus and minus.These relations correspond to the fact that S Fb is C-, PT-and CPT-but not P-or T-symmetric.However, S Fb has a P-invariant diagonal (in the sense of Eq. (4.1)) due to S α ᾱ = S ᾱα (or Remark 4.4).

Lemma 7.5 (Stress tensor for the Federbush model). A tensor-valued function
is a stress-energy-tensor at one-particle level with respect to S Fb , is diagonal in mass (Eq.(3.13)), and has no poles, P = ∅, iff it is of the form and where each q s/as j is a polynomial with real coefficients and q s j (−1) = 1.The stress-energy tensor at one-particle level is parity-covariant iff q as 1 = q as 2 ≡ 0.
Proof.By Theorem 3.2 and Corollary 3.4, we have that Eq. (7.20) holds with F satisfying properties (a)-(g).U (1) ⊕2 -invariance, property (f), is equivalent to As a consequence, ( k , F (ζ)) = 0 unless j = k and r = −s.On the remaining components, S acts like −F; thus, which implies ⊗ as e (−) j (7.23) for some functions f s/as j , where we have factored out the necessary zeroes due to the relations (7.22).Then from the properties of F we find f s/as j : C → C to be analytic and to satisfy and f as j (iπ) unconstrained.Moreover, f s/as j are regular in the sense of Eq. (A.4) of Lemma A.5; the lemma implies that f s/as j j , and by the antilinearity of J, we find that q s/as j (ζ + iπ) = q s/as j ( ζ + iπ) such that q s/as j have real coefficients.Parity invariance of F , i.e., FF = F , is equivalent to q as j = −q as j ; thus, q as j = 0, because of (½ ∓ F) e We see that the stress-energy tensor does not need to be parity-covariant.Concerning QEIs we state: Theorem 7.6 (QEI for the Federbush model).The parity-covariant part of the stress-energy tensor at one-particle level, given by F 2 in Eq. (7.20) with q as 1 = q as 2 ≡ 0, satisfies a one-particle-QEI of the form (7.25) iff q s 1 = q s 2 ≡ 1.The candidate stress-energy tensor given by Eq. (4.3) (i.e. for q s 1 = q s 2 = 1, q as 1 = q as 2 = 0) satisfies a QEI of the form and in the sense of a quadratic form on D × D.
We see that for the Federbush model, requiring a one-particle QEI fixes a unique (parity-covariant part of the) stress-energy tensor at one-particle level that extendssince S Fb is constant-to a dense domain of the full interacting state space.The parity-covariant part is in agreement with preceding results for the stress-energy tensor at one-particle level [CF01,Sec. 4.2.3].This indicates that the parity-violating part of our expression is indeed not relevant for applications in physics.Our candidate for the full stress-energy tensor has the same trace as in [SH78].That the respective energy density satisfies a generic QEI is no surprise after all, as the QEI results are solely characterized in terms of the trace of the stress-energy tensor which here agrees with that of two free Dirac fermions (as was indicated also by Eq. (7.17)).

O(n)-nonlinear sigma model
The O(n)-nonlinear sigma model is a well-studied integrable QFT model of n scalar fields φ j , j = 1, . . ., n, that obey an O(n)-symmetry.For a review see [AAR01, Secs.6-7] and references therein.In Lagrangian QFT, it can be described by a combination of a free Lagrangian and a constraint where g ∈ (0, ∞) is a dimensionless coupling constant.Clearly, L NLS is invariant for Φ transforming under the vector representation of O(n), i.e., Note that the model-other than one might expect naively from L NLS -describes massive particles.This is known as dynamical mass transmutation; the resulting mass of the O(n)-multiplet can take arbitrary positive values depending on a choice of a mass scale and corresponding renormalized coupling constant; see, e.g., [AAR01, Sec.7.2.1] and [JN88].
In our framework, the model can be described in the following way: Definition 7.7.The O(n)-nonlinear sigma model is specified by two parameters, the particle number n ∈ N, n ≥ 3, and the mass m > 0. The one-particle little space (K, V, J, M ) is given by K = C n with the defining/vector representation V of G = O(n), M = m½ C n , and where J is complex conjugation in the canonical basis of C n .The S-function is given by where in the canonical basis of C n S NLS is the unique maximally analytic element of the class of O(n)-invariant Sfunctions [ZZ78].Maximal analyticity means here that in the physical strip S(0, π), the S-function has no poles and the minimal amount of zeroes which are compatible with the axioms for an S-function, i.e., (S1)-(S7).Its eigenvalue decomposition is given by As a first step, we establish existence of the minimal solution with respect to S 0 and an estimate of its asymptotic growth: Lemma 7.8.The minimal solution with respect to S 0 exists and is given by Lemma 7.9 (Stress tensor in NLS model).A tensor-valued function F µν 2 : C 2 ×M → K ⊗2 is a parity covariant stress-energy tensor at one-particle level with respect to S NLS with no poles, P = ∅, iff it is of the form where F 0,min is the unique minimal solution with respect to the S-matrix eigenvalue S 0 and q is a polynomial with real coefficients with q(−1) = 1.In summary, Lemma A.5 can be applied to λ, so that λ(ζ) = q(ch(ζ))F 0,min (ζ), and thus, F has the form (7.39).That q has real coefficients is a consequence of (e) and Corollary A.3.
Conversely, it is clear that F µν 2 as in Eq. (7.38), respectively F , has the properties (a)-(g).
Proof.Given F 2 as in Lemma 7.9 and using I ⊗2 = ½ K , we have F We have established QEIs in a larger class of 1+1d integrable models than previously known in the literature.In particular, QEIs for generic states hold in a wide class of models with constant scattering functions, including not only the Ising model, as known earlier, but also the Federbush model.Moreover, the class includes combinations and bosonic or fermionic variants of these models.In all of these situations, the form factor F 2 of the energy density determines the entire operator.Furthermore, we have established necessary and sufficient conditions for QEIs to hold at one-particle level in generic models, which may include bound states or several particle species.Also in this case, only F 2 contributes to expectation values of the energy density, and the conditions for QEIs are based on the large-rapidity behaviour of F 2 .At the foundation of both results was a characterization by first principles of the form of the energy density.However, we found that those principles do not constrain polynomial prefactors (in ch ζ) added to a viable candidate for the energy density (at one-particle level).As seen in the case of the Bullough-Dodd, the Federbush, and the O(n)-nonlinear sigma model, one-particle QEIs can then fix the energy density at one-particle level partially or entirely, in analogy to [BC16].
Our results suggest a number of directions for further investigation, of which we discuss the most relevant ones: What is the nature of the freedom in the form of the stress-energy tensor?
The factor q(ch ζ) in the energy density was partially left unfixed by our analysis.At least in the scalar case (K = C), it can be understood as a polynomial in the differential operator = g µν ∂ µ ∂ ν acting on T µν : Given a stress-energy tensor T µν , define T µν := q(−1 − 2 M 2 )T µν for some polynomial q.Then at one-particle level and, provided that q(−1) = 1, F [ T µν (x)] 2 defines another valid candidate for the stressenergy tensor at one-particle level.However, for generic models, q may depend on the particle types and cannot be understood in terms of derivatives only.
In the physics literature, given a concrete model, a few standard methods exist to check the validity of a specific choice of q: In case the model admits a Lagrangian, perturbation theory checks are used, e.g., [BK02;BFK10;BFK13].In case the model can be understood as a perturbation of a conformal field theory model, a scaling degree for the large-rapidity behaviour (conformal dimension) of the stress-energy tensor can be extracted, which fixes the large-rapidity behaviour of F 2 , e.g., [Zam86; DSC96; CF01].The large-rapidity scaling degree is also related to momentum-space clustering properties, which were studied for some integrable models, e.g., [Smi92; KM93; BFK21].But in the general case, none of these methods may be available, and other constraints-perhaps from QEIs in states of higher particle numbermight need to take their place.
Which other models can be treated with these methods?We performed our analysis of one-particle QEIs in a very generic setting; there are nevertheless some limitations.For one, we employed the extra assumption of parity covariance of the stress-energy tensor.While parity invariance of the scattering function (and therefore covariance of the stress-energy-tensor) is satisfied in many models, it is not fully generic.Nevertheless, a non-parity-covariant stress-energy tensor is still subject to constraints by our results; in particular, the necessary condition we gave for a oneparticle QEI to hold remains unmodified (see Remark 5.2).We expect a sufficient condition for a one-particle QEI, similar to the one presented in Theorem 5.1(b), to apply also in a parity-breaking situation.Some numerical tests indicate this; however, an analytic proof remained elusive to us.
Another point is the decomposition of the two-particle form factor of the (trace of the) stress-energy tensor F into polynomials and factors which are fixed by the model (including the minimal solutions and pole factors).For generic models, multiple polynomial prefactors can appear (at least one for each eigenvalue of the S-function).In typical models, these are few to begin with, and symmetries exclude many of those prefactors (as was presented for the Federbush or the O(n)-nonlinear sigma model).In other situations, however, there might be too many unfixed factors for the QEI to meaningfully constrain them.
Lastly, we should remark that also in the presence of higher-order poles in the scattering function, the poles in the form factors are expected to be of first-order [BK02; BFK06] so that such models should be tractable with our methods.This includes for instance the Z(n)-Ising, sine-Gordon, or Gross-Neveu model.Also generic Toda field theories don't seem to pose additional problems.Do QEIs hold in states with higher particle numbers?Apart from the case of constant S-functions, we treated only one-particle expectation values of the energy density in this paper.At n-particle level, generically the form factors F 1 , ..., F 2n all enter the expectation values; these are more challenging to handle since the number of rapidity arguments increases and since additional ("kinematic") poles arise at the boundary of the analyticity region that were absent in the case n = 1.Therefore, treating higher particle numbers requires new methods; a result in either direction, validity or non-validity of QEIs, is by no means straightforward and we leave this analysis to future work.While we conducted a few promising numerical tests for the sinh-Gordon model at two-particle level [Man23b], these can only serve as an indication.We do not expect to obtain numerical results at much higher particle numbers due to computational complexity scaling exponentially with n.

A The minimal solution
This appendix collects rigorous results on existence and uniqueness of minimal solutions for integrable models, as well as estimates for their asymptotic growth, which are central to the question whether a QEI holds in the model (see Sec. 5).
Some of these results are also contained in [BC16], whereas a less rigorous but informative treatment can be found in [KW78].Our existence result is based on an integral representation of the minimal solution which is well-known in principle and has been employed before in many concrete models, e.g., sinh-Gordon [FMS93], SU (N )-Gross-Neveu [BFK10], and O(N )-nonlinear-σ [BFK13].Existence of the integral representations was argued in [KW78], but without giving explicit assumptions.General results on the asymptotic growth of the minimal solution, based on this integral representation, are new to the best of the authors' knowledge.

A.1 Eigenvalue decomposition of the S-function
To begin with, we establish the eigenvalue decomposition of an S-function.Since S(θ) ∈ B(K ⊗2 ) is unitary for real arguments, it is diagonalizable; this extends to complex arguments by analyticity: In particular, P The decomposition is unique up to relabeling.
Proof.For the eigenvalue decomposition of a matrix-valued analytic function, see e.Note that the S i (within any domain D from above) satisfy all the properties of a scalar S-function except for crossing symmetry.Specifically, these are the properties (S1) and (S2), since (S3), (S4), (S6), and (S7) are trivially satisfied in the scalar setting.
In typical examples, the decomposition in Eq. (A.1) can be extended to all of C if one allows for meromorphic S i and P i .This applies particularly to models with constant eigenprojectors (e.g., all models with constant or diagonal S-functions) but also the other examples treated in Section 7.

A.2 Uniqueness of the minimal solution and decomposition of oneparticle solutions
Throughout the remainder of Appendix A, we intend to analyse eigenvalues S of some matrix-valued S-function; thus, S will denote a C-valued (not matrix-valued) function from now on.The content of the present section is taken from [BC16] with slight generalizations.Central to the section is: Thus q is polynomially bounded at infinity and therefore a polynomial.However, since q does not have zeroes, it must be a constant with q ≡ q(−1) = 1 due to G(iπ) = 1.Hence F A = F B .

B Computing a characteristic function
In this appendix, we present a method to explicitly compute characteristic functions (as defined in Appendix A.3) for a certain class of S. The method is known but only briefly described in [Kar+77].We illustrate it here using the eigenvalues of the S-function of the O(n)-nonlinear sigma model, i.e., S i for i = ±, 0 (see Definition 7.7 and below).First, we present the general method; second, we check the examples f [S ± ] against the literature; lastly, we compute f [S 0 ].The method applies to S-function eigenvalues which are given as a product of Gamma functions; see [Bab+99, Appendix C] for some typical examples.While this product can be infinite in general, we restrict here to finite products, which suffice for our purposes.Specifically, let S be of the form where we used z = Γ(z + 1)/Γ(z) in order to represent h ± in terms of Γ.As a result, .
Such S-functions are classified in [CT15, Appendix A].The Bullough-Dodd model itself (see [AFZ79; FMS93] and references therein) corresponds to the maximally analytic element of this class which is given by ζ → S BD (ζ; b) = S(ζ; − 2 3 )S(ζ; b 3 )S(ζ; 2−b 3 ) where b ∈ (0, 1) is a parameter of the model.The full class allows for so-called CDD factors [CDD56] and an exotic factor of the form ζ → e ia sh ζ , a > 0.
.33) with S ± = b ± c and S 0 = b + c + nd.The S-function is P-, C-, and T-symmetric and satisfies S NLS (0) = −F.
g. [Par78, Theorem 4.8] or [Kat95, Chapter 2].Restricting S to its domain of analyticity D(S) we can apply the theorem from the first-named reference: For some k ∈ N and a discrete set ∆(S) ⊂ D(S) and any simply connected domain D ⊂ D(S) \ ∆(S) we obtain pairwise distinct analytic functions S i : D → C and P i , D i : D → B(K ⊗2 ) for i = 1, . . ., k such that for each ζ ∈ D S(ζ) = k i=1 S i (ζ)P i (ζ) + D i (ζ) is the unique Jordan decomposition of S(ζ) with eigenvalues S i (ζ), eigenprojectors P i (ζ) and eigennilpotents D i (ζ), i = 1, . . ., k.Let us enlarge D to D within D(S) \∆(S) such that D ∩ R ⊂ R is open and non-empty and such that D is still simply connected; this is always possible since C\D(S) and ∆(S) are discrete, i.e., countable and without finite accumulation points.Since S(θ) for θ ∈ R is unitary and therefore semisimple we find that D i ↾ D ∩ R = 0. Since D i is analytic, this implies D i = 0. From the properties of the Jordan decomposition we further infer that P i (ζ)P j (ζ) = δ ij P i (ζ), i, j = 1, . . ., k.The properties (b)-(d) are implied by the corresponding properties of S.