# Upper Bounds on the Degeneracy of the Ground State in Quantum Field Models

Authors: Asao Arai; Daiju Funakawa
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Axiomatic abstract formulations are presented to derive upper bounds on the degeneracy of the ground state in quantum field models including massless ones. In particular, given is a sufficient condition under which the degeneracy of the ground state of the perturbed Hamiltonian is less than or equal to the degeneracy of the ground state of the unperturbed one. Applications of the abstract theory to models in quantum field theory are outlined.


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## Body

## 1. Introduction

LetH be a complex Hilbert space with inner product ·,· (complex linear in the right variable) and norm ·. For a linear operator A on H, we denote its domain by D(A) and its spectrum by σ(A).LetH be a self-adjoint operator on H and bounded below. Then, by abstract use of word, we call the infimum of σ(H)(1)E0H≔inf⁡σH thelowest orminimal energy of H (this name originally comes from the context in quantum physics where H denotes the Hamiltonian of a quantum system). If E0(H) is an eigenvalue of H, then H is said tohave ground state and a nonzero vector in the eigenspace ker(H-E0(H)) is called aground state of H. In that case, the dimension(2)mH≔dim⁡ker⁡H-E0H of the eigenspace ker(H-E0(H)) is called thedegeneracy or themultiplicity of the ground state of H. If m(H)=1, then the ground state is said to beunique. If m(H)≥2, then the ground state is said to bedegenerate.As is well known, it has been an important issue to determine the degeneracy of the ground state of a given Hamiltonian in quantum physics. For Schrödinger type Hamiltonians in quantum mechanics with finite degrees of freedom andmassive Bose field Hamiltonians in quantum field theory, general theorems on the uniqueness of the ground state and upper bounds on the degeneracy of the ground state have been established (see, e.g., [1, §XIII.12] and references in Notes for §XIII.12). For fermion systems, Faris [2] discussed conditions which ensure the uniqueness of the ground state. Faris’s ideas and methods have been extended by Miyao [3, 4] to obtain general criteria on the uniqueness of the ground state in bosonic quantum field models as well as fermionic ones.As for models in which a massless quantum field appears, estimation of the degeneracy of the ground state is highly nontrivial, because, in that case, one has to treat an embedded eigenvalue problem so that the regular perturbation theory or the min-max principle cannot be used (for a review of this aspect, see, e.g., [5]). A first breakthrough result on this problem was given by Bach et al. [6]. They considered a model in nonrelativistic quantum electrodynamics and proved that, under suitable hypotheses, the degeneracy of the ground state of the total Hamiltonian of the model is less than or equal to the degeneracy of the ground state of the unperturbed Hamiltonian [6, Theorem I.1(f)]. The methods used in [6] to estimate the degeneracy of the ground state have been generalized by Hiroshima [7] to be applied to a class of quantum field models whose Hamiltonian is of the following form:(3)Hg=A⊗I+I⊗dΓS+gHintacting in the tensor product Hilbert space(4)F≔h⊗FbWof a complex Hilbert space h and the boson Fock space Fb(W) over a complex Hilbert space W (see (80) for the definition). Here A is a self-adjoint operator on h which is bounded below, I denotes identity, S is a nonnegative self-adjoint operator on W, dΓ(S) is the second quantization operator of S on Fb(W) (see, e.g., [8, p.302] and [9, §X.7]), g∈R is a coupling constant, and Hint is a symmetric operator on F which describes an interaction between a Bose field and a quantum system whose Hilbert space of state vectors is h. In [7], it is assumed that Hintis relatively bounded with respect to the unperturbed operatorH(0)≔A⊗I+I⊗dΓ(S). It is proved in [7] that, under a suitable condition, m(H(g))≤c(g)m(H(0)) with c(g)>0 being a constant depending on g and, in particular, m(H(g))≤m(H(0)) for all sufficiently small |g| in an abstract framework and in the case where W=⨁DL2(Rd), the D-direct sum of L2(Rd) (D,d∈N). Moreover, these results were applied to the generalized spin-boson model [10], the Pauli-Fierz model, and a model in relativistic quantum electrodynamics with cutoffs [7].One of the motivations of the present work comes from extending results in [7] to the case where Hint is not necessarily relatively bounded with respect to H(0). But we find that, before going on analyzing such models, it is better to construct an abstract theory on the degeneracy of ground state with the requirement that it formulates general aspectsindependent of concrete models. From this point of view, we construct in this paper such an abstract theory in axiomatic manners. Careful investigations and structural analyses on results on the existence and the degeneracy of ground state which have been established so far (e.g., [6, 7, 10–12]) make it possible. Applications of the abstract theory to concrete models will be discussed in a separate paper. We hope that the abstract theory given in the present paper not only clarifies general structures behind the theory on the degeneracy of ground state in [7] but also makes the range of applications wider, because the abstract results established in the present paper show what are general independently of models and what should be proved in each concrete model.The present paper is organized as follows. In Section2, we consider a bounded below and self-adjoint operator H0 and a symmetric operator H1 on a Hilbert space H. The sum H≔H0+H1 is supposed to be self-adjoint and bounded below. We formulate a sufficient condition which yields inequality m(H)≤cm(H0) with a constant c>0 being computed from the given data. In particular, an additional condition for c=1 is given so that m(H)≤m(H0). In Section 3, we derive an integral equation for any ground state of H in terms of a linear operator A on H which has some characteristic properties (Theorem 11). In Section 4 we state and prove the main theorem in the present paper (Theorem 12). In the last section, we give remarks for applications of the main theorem to concrete models in quantum field theory.


## 2. Comparison Theorem on Degeneracy of Ground States

For nonnegative self-adjoint operatorsA0,A1,…,An on H (n=1,2,…), we write “A0⪯∑j=1n‍Aj” if ⋂j=1nD(Aj1/2)⊂D(A01/2) and (5)A01/2ψ,A01/2ψ≤∑j=1nAj1/2ψ,Aj1/2ψ,ψ∈⋂j=1nDAj1/2.LetH0 be a bounded below and self-adjoint operator on H and let H1 be a symmetric operator on H such that(6)H≔H0+H1 is self-adjoint and bounded below (D(H)≔D(H0)∩D(H1)). We assume the following.Hypothesis 1.
(i) The operatorsH0 and H have ground state with m(H0)<∞.
(ii) There exists a nonnegative self-adjoint operatorL on H such that (7)I⪯L+PH0, where, for a bounded below self-adjoint operator T on H, we denote by PT the orthogonal projection onto ker(T-E0(T)): PTH=ker(T-E0(T)).
A vectorψ∈H is said tooverlap with a subset D⊂H if there exists a vector η∈D such that ψ,η≠0.The following theorem, which is a comparison theorem onm(H) and m(H0), and Corollary 2 below are more abstract versions of Lemma 3.3 and Corollaries 3.4-3.5 in [7].Theorem 1.
Assume Hypothesis1 and suppose that there exists a constant δ with 0<δ<1 such that(8)supϕ∈DL1/2∩PHH,ϕ=1⁡L1/2ϕ2≤δ.Then, (i)
anyϕ∈PHH with ϕ=1 overlaps with PH0H;(ii)
let(9)mLH≔dim⁡DL1/2∩PHH;then(10)mLH≤11-δmH0;(iii)
ifPHH⊂D(L1/2) in addition, then(11)mH≤11-δmH0.Proof.
(i) Letϕ∈D(L1/2)∩PHH with ϕ=1. Then, by Hypothesis 1(ii), we have (12)1≤L1/2ϕ2+ϕ,PH0ϕ≤δ+ϕ,PH0ϕ. Hence(13)1≤1-δ-1ϕ,PH0ϕ.In particular, ϕ,PH0ϕ≠0. Therefore ϕ overlaps with PH0H.
(ii) We first show thatD(L1/2)∩PHH is finite-dimensional. So suppose that D(L1/2)∩PHH were infinite-dimensional. Then there is an orthonormal system {ϕn}n=1∞ with ϕn∈D(L1/2)∩PHH,n≥1. Taking ϕn as ϕ in (13) and summing over n, we obtain ∑n=1∞‍ϕn,PH0ϕn=∞. But the left hand side is less than or equal to Tr⁡PH0=m(H0)<∞, where Tr means trace. Hence one is led to a contradiction. Therefore D(L1/2)∩PHH is finite-dimensional; that is, mL(H)<∞. Let {ϕn}n=1mL(H) be an orthonormal basis of D(L1/2)∩PHH. Then, by (13), we have (14)mLH≤11-δ∑n=1mLHϕn,PH0ϕn≤11-δTr⁡PH0=11-δmH0. Thus (10) follows.
(iii) This follows from (10).Corollary 2.
Assume Hypothesis1 and suppose that there exists a constant δ such that(15)0<δ<1mH0+1and (8) holds. Then(16)mLH≤mH0.In particular, if PHH⊂D(L1/2) in addition, then(17)mH≤mH0.Proof.
Condition (15) implies that 0<δ<1. Hence the conclusion of Theorem 1 holds. Therefore (18)mLH≤11-δmH0<mH0+1, where, in the second inequality, we have used (15). Since mL(H) and m(H0) are natural numbers, (16) follows.Corollary 3 (uniqueness).
Assume Hypothesis1 with m(H0)=1 and suppose that the following (i) and (ii) hold: (i)
PHH⊂D(L1/2).(ii)
There exists a constantδ with 0<δ<1/2 such that (8) holds. Then m(H)=1.Proof.
In the present case, (15) holds with m(H0)=1. Hence m(H)≤1. Since m(H) is a natural number, it follows that m(H)=1.In applications to quantum field models, it may be convenient to considerH in the form (19)Hg≔H0+gH1 with a constant parameter g>0, as in the case of H(g) given by (3).Remark 4.
The conditiong>0 is taken so that Hg is bounded below in the case where H1 is not H0-bounded but bounded below. In the case where H1 is H0-bounded, g needs not to be positive, being allowed to be a negative number as well, and, by the Kato-Rellich theorem, Hg is self-adjoint and bounded below for all sufficiently small g (g∈R) [7].Corollary 5.
Assume Hypothesis1 with H=Hg, 0<g<a (a>0 is a constant), and suppose that there exists a continuous function δ≔δ(g) on 0,a such that δ(0)=0 and(20)supϕ∈DL1/2∩PHgH,ϕ=1⁡L1/2ϕ2≤δg,g∈0,a.Then there exists a constant g0∈(0,a) such that, for all g∈(0,g0), any ϕ∈PHgH with ϕ=1 overlaps with PH0H and(21)mLHg≤mH0.In particular, if PHgH⊂D(L1/2) for all g∈(0,g0) in addition, then(22)mHg≤mH0for all g∈(0,g0).Proof.
By the continuity ofδ(g) in g∈[0,a) with δ(0)=0, there exists a constant g0∈(0,a) such that, for all g∈[0,g0), (23)0≤δg<1mH0+1. Hence, for each g∈(0,g0), one can apply Theorem 1 and Corollary 2 to obtain the desired results.Remark 6.
Condition (20) with g=0 implies that D(L1/2)∩PH0H⊂kerL1/2=kerL. In some models in quantum field theory, L can be taken in such a way that this property is satisfied (see Section 5).


## 3. An Abstract Integral Equation for Ground States

In applications of Theorem1, we need to prove (8). In concrete models in quantum field theory, this has been done by using operators which have some characteristic relations to L and H0. In this section we introduce an abstract version of such operators.LetA be a densely defined closed linear operator on H and(24)A0t≔eitH0Ae-itH0,At≔eitHA0-te-itH,t∈R. We assume the following:(A.1)
(i)
For someα∈(0,1], A is H0α-bounded; that is, D(H0α)⊂D(A), and there exist constants a,b≥0 such that(25)Aψ≤aH0αψ+bψ,ψ∈DH0α.(ii)
There exists a coreD of H0α such that(26)limt→∞⁡AeitH0ψ=0,ψ∈D.Remark 7.
Under condition (A.1)(i), the functional calculus of the self-adjoint operatorH0 gives that, for all ψ∈D(H0α) and t∈R, eitH0ψ∈D(H0α) and H0αeitH0ψ=eitH0H0αψ. Hence D(H0α)⊂D(A0(t)) for all t∈R. In particular, D(H0)⊂D(A0(t)) for all t∈R. Since D(H)⊂D(H0) and e-itHD(H)=D(H) for all t∈R, it follows that D(H)⊂D(A(t)) for all t∈R. In what follows, we use these facts without mentioning.Lemma 8.
Assume (A.1). Then, for allϕ∈PHH, limt→∞A(t)ϕ=0.Proof.
Sincee-itHϕ=e-itE0(H)ϕ, we have A(t)ϕ=AeitH0ϕ. By (A.1)(ii), for any ɛ>0, there exists a vector ψɛ∈D such that (27)ψɛ-ϕ<ɛ,H0αψɛ-H0αϕ<ɛ. Hence (28)AeitH0ϕ≤AeitH0ϕ-ψɛ+AeitH0ψɛ≤aH0αϕ-ψɛ+bϕ-ψɛ+AeitH0ψɛ<a+bɛ+AeitH0ψɛ. Taking the limit t→∞, we have (29)0≤limsupt→∞⁡AeitH0ϕ≤a+bɛ. Then the limit ɛ→0 yields that limt→∞AeitH0ϕ=0. Thus limt→∞A(t)ϕ=0.To state additional assumptions, we recall the concept of weak commutator [13]. Let T and S be densely defined linear operators on H and let E be a dense subspace in H such that E⊂D(T)∩D(S)∩D(T∗)∩D(S∗), where, for a densely defined linear operator A on H, A∗ denotes the adjoint of A. Then the pair (T,S) is said to have weak commutator on E if there exists a linear operator C on H such that E⊂D(C) and T∗ψ,Sϕ-S∗ψ,Tϕ=ψ,Cϕ, ψ,ϕ∈E. In this case, the operator C restricted to E is written as [T,S]wE so that (30)T∗ψ,Sϕ-S∗ψ,Tϕ=ψ,T,SwEϕ,ψ,ϕ∈E. We call [T,S]wE theweak commutator of (T,S) on E.We also need the following assumption:(A.2)
(i)
For eacht∈R and j=0,1, (Hj,A0(t)) has weak commutator [Hj,A0(t)]wD(H) on D(H). Moreover, for all ψ∈D(H), [H1,A0(t)]wD(H)ψ is strongly continuous in t.(ii)
For allψ∈D(H), A0(t)ψ is strongly differentiable in t and its strong derivative dA0(t)ψ/dt is of the form (31)dA0tψdt=iH0,A0twDHψ,t∈R.(iii)
There exist aσ-finite measure space (M,Σ,μ), a nonnegative Σ-measurable function u on M with 0<u(k)<∞ for μ-a.e. k∈M, and a linear operator S(k) on H (k∈M) with D(S(k))⊃D(H) such that the following (a)–(c) hold:
(a)
for allψ,χ∈D(H), ∫M‍|χ,S(k)ψ|dμ(k)<∞ and(32)χ,H1,A0-twDHψ=∫Meitukχ,Skψdμk;(b)
for allϕ∈PHH, ∫M‍S(k)ϕdμ(k)<∞ and ∫MS(k)ϕ/u(k)dμ(k)<∞;(c)
for allψ∈D and ϕ∈PHH, (33)∫0∞∫Mψ,eisH-E0H+ukSkϕdμkds<∞.Remark 9.
Condition (a) in (A.2)(iii) may be regarded as an abstract form of “sum rules” in quantum mechanics [13]. In fact, under stronger additional conditions, one can prove (32) (cf. [13]), where μ is determined by the spectral measure of H0.
Under condition (A.2),(H-E0(H)+u(k))-1 is bounded for μ-a.e. k∈M with (34)H-E0H+uk-1≤1uk,μ-a.e.k∈M. Hence, for all ϕ∈PHH, (35)∫MH-E0H+uk-1Skϕdμk≤∫MSkϕukdμk<∞.It is easy to see that, for allψ∈D(H), eis(H-E0(H))[H1,A0(-s)]wD(H)ψ is strongly continuous in s. Hence, for each t∈R, the strong Riemann integral ∫0t‍eis(H-E0(H))·[H1,A0(-s)]wD(H)ψds exists.Lemma 10.
Assume (A.1) and (A.2). Then, for allϕ∈PHH and t∈R,(36)Atϕ=Aϕ+i∫0teisH-E0HH1,A0-swDHϕds.Moreover, for all ψ∈D(H),(37)ψ,Atϕ=ψ,Aϕ+i∫0tds∫Mdμkψ,eisH-E0H+ukSkϕ.Proof.
Letχ∈D(H) and (38)ft≔χ,Atϕ=e-itH-E0Hχ,A0-tϕ. Then it is easy to see that f(t) is differentiable in t with (39)dftdt=iH-E0He-itHχ,A0-te-itHϕ+e-itHχ,-iH0,A0-twDHe-itHϕ. Using the identity A0(-t)∗e-itHχ,(H-E0(H))e-itHϕ=0, we obtain (40)dftdt=ie-itHχ,H1,A0-twDHe-itHϕ=iχ,eitH-E0HH1,A0-twDHϕ. Hence (41)ft=χ,Aϕ+∫0tiχ,eisH-E0HH1,A0-swDHϕds. By a property of strong Riemann integral, we have (42)∫0tiχ,eisH-E0HH1,A0-swDHϕds=χ,i∫0teisH-E0HH1,A0-swDHϕds. Thus (36) follows. Formula (37) follows from (36) and (A.2)(iii).For allϕ∈PHH, (43)∫MH-E0H+uk-1Skϕdμk≤∫MSkϕukdμk<∞. Hence the Bochner integral ∫M‍(H-E0(H)+u(k))-1S(k)ϕdμ(k) exists. The main theorem in this section is as follows.Theorem 11.
Assume (A.1) and (A.2). Then, for allϕ∈PHH,(44)Aϕ=∫MH-E0H+uk-1Skϕdμk.Proof.
Letψ∈D. Then, taking the limit t→∞ in (37) and using Lemma 8, we obtain (45)ψ,Aϕ=-i∫0∞ds∫Mdμkψ,eisH-E0H+ukSkϕ. For all ɛ>0, we have (46)e-ɛs∫Mdμkψ,eisH-E0H+ukSkϕ≤∫Mdμkψ,eisH-E0H+ukSkϕ. By (c) in (A.2)(iii), the right hand side is integrable in s∈[0,∞). Hence, by the Lebesgue dominated convergence theorem, we have (47)ψ,Aϕ=-ilimɛ→0⁡∫0∞dse-ɛs∫Mdμkψ,eisH-E0H+ukSkϕ. Note that (48)∫0∞dse-ɛs∫Mψ,eisH-E0H+ukSkϕ≤∫0∞dse-ɛs∫MψSkϕdμk<∞. Hence, by Fubini’s theorem, we have (49)∫0∞dse-ɛs∫Mdμkψ,eisH-E0H+ukSkϕ=∫Mdμkψ,iH-E0H+uk+iɛ-1Skϕ. Therefore (50)ψ,Aϕ=limɛ→0⁡∫Mdμkψ,H-E0H+uk+iɛ-1Skϕ. It is easy to see that, for μ-a.e.k, (51)limɛ→0⁡ψ,H-E0H+uk+iɛ-1Skϕ=ψ,H-E0H+uk-1Skϕ,ψ,H-E0H+uk+iɛ-1Skϕ≤ψSkϕuk. Hence we can apply the Lebesgue dominated convergence theorem to obtain (52)ψ,Aϕ=ψ,∫MH-E0H+uk-1Skϕdμk. Thus (44) holds.


## 4. Main Theorem

We now state and prove the main theorem in the present paper. For this purpose, we first rewrite the theory in the preceding section in a form suitable for applications to models in quantum field theory.We assume Hypothesis1. Let (K,Σ,μ) be a σ-finite measure space and let ω be a nonnegative Σ-measurable function on K such that 0<ω(k)<∞, μ-a.e. k∈K. We set (53)X≔L2K,dμ and, for each s∈R, introduce a subspace (54)Xs≔f∈X∣∫Kωksfk2dμk<∞.Suppose that there exists a familyAf∣f∈X of densely defined closed linear operators on H such that, for all f∈X, D(L1/2)⊂D(A(f)) and, for all ψ∈D(L1/2),(55)Azf+wgψ=z∗Afψ+w∗Agψ,f,g∈X,z,w∈C,where z∗ is the complex conjugate of z∈C. We introduce an operator:(56)A0f,t≔e-itH0AfeitH0,t∈R,f∈X.For the operator A(f), we assume conditions similar to (A.1) and (A.2) for A in the preceding section.Hypothesis 2.
(i) For someα∈(0,1] and β≥0, D(H0α)⊂D(A(f)),f∈X-β and for each f∈X-β, there exist constants af,bf≥0 such that(57)Afψ≤afH0αψ+bfψ,ψ∈DH0α.
(ii) There exists a coreD of H0α such that(58)limt→∞⁡AfeitH0ψ=0,ψ∈D,f∈X-β.
(iii) There exists a dense subspaceY⊂X-β such that, for all t∈R and f∈Y, (H0,A0(f,t)) has weak commutator on D(H) and, for all ψ∈D(H), [H0,A0(f,t)]wD(H)ψ is strongly continuous in t.
(iv) For allψ∈D(H) and f∈Y, A0(f,t)ψ is strongly differentiable in t and (59)dA0f,tψdt=iH0,A0f,twDHψ,t∈R.(v) For allt∈R, f∈Y, (H1,A0(f,t)) has weak commutator on D(H) and, for all ψ∈D(H), [H1,A0(f,t)]wD(H)ψ is strongly continuous in t. Moreover, for μ-a.e. k∈K, there exists a densely defined linear operator T(k) on H with D(T(k)∗)∩D(T(k))⊃D(H) such that the following hold:(a)
for allψ,χ∈D(H) and f∈Y, ∫K‍|f(k)χ,T(k)ψ|dμ(k)<∞ and for all t∈R,(60)χ,H1,A0f,twDHψ=∫Keitωkfk∗χ,Tkψdμk;(b)
for allϕ∈PHH and f∈Y, (61)∫KfkTkϕdμk<∞,∫KfkTkϕωkdμk<∞;(c)
there exists a dense subspaceY0⊂Y such that, for all ψ∈D, f∈Y0, and ϕ∈PHH, (62)∫0∞∫Kfk∗ψ,eisH-E0H+ωkTkϕdμkds<∞.We need an assumption on a relation betweenA(f) and L.Hypothesis 3.
There exist a complete orthonormal system (CONS){en}n=1∞ of X with en∈Y,n∈N, and a constant ɛ0∈[0,1) such that (63)supϕ∈DL1/2∩PHH,ϕ=1⁡L1/2ϕ2-∑n=1∞Aenϕ2≤ɛ0.The main theorem in the present paper is as follows.Theorem 12.
Assume Hypotheses1–3. Suppose that there exist nonnegative functions f1 and f2 on (K,Σ) such that f1+ω-1f2∈X with(64)γ≔f1+ω-1f22<1-ɛ0,(65)Tk∗ψ≤f1kH-E0H+ωkψ+f2kψ,ψ∈DH,μ-a.e.k∈K.Then, (i)
anyϕ∈PHH with ϕ=1 overlaps with PH0H;(ii)
(66)mLH≤11-γ-ɛ0mH0,wheremL(H) is defned by (9); moreover, if(67)γ+ɛ0<1mH0+1in addition, then(68)mLH≤mH0;(iii)
suppose thatPHH⊂D(L1/2) in addition; then(69)mH≤11-γ-ɛ0mH0;in particular, if (67) holds, then(70)mH≤mH0;(iv)
ifPHH⊂D(L1/2), m(H0)=1, and γ+ɛ0<1/2, then m(H)=1.To prove Theorem12, we need some lemmas.Lemma 13.
Assume Hypothesis2. Then, for all ϕ∈PHH and f∈Y,(71)Afϕ=∫Kfk∗H-E0H+ωk-1Tkϕdμk.Proof.
One can apply Theorem11 with A=A(f). Then (71) follows.Lemma 14.
Assume Hypothesis2. Let ϕ∈PHH and (72)∫KH-E0H+ωk-1Tkϕ2dμk<∞. Then, for any CONS {en}n=1∞ of X with en∈Y (n∈N),(73)∑n=1∞Aenϕ2=∫KH-E0H+ωk-1Tkϕ2dμk.Proof.
Define a mappingT:X→H by (74)Tf≔∫KfkH-E0H+ωk-1Tkϕdμk,f∈X. Then, in quite the same way as in the proof of [12, Lemma 2.6], one can show that T is Hilbert-Schmidt and (75)T22=∫KH-E0H+ωk-1Tkϕ2dμk, where T2 is the Hilbert-Schmidt norm of T. In general, for any CONS {fn}n=1∞ of X, T22=∑n=1∞‍Tfn2. The set {en∗}n=1∞ also is a CONS of X. By (71), we have Ten∗=A(en)ϕ. Thus (73) holds.We are now ready to prove Theorem12.Proof of Theorem12.
Letk∈K be such that 0<ω(k)<∞. Then (65) implies that T(k)∗(H-E0(H)+ω(k))-1 is bounded with operator norm (76)Tk∗H-E0H+ωk-1≤f1k+f2kωk. Hence (H-E0(H)+ω(k))-1T(k) is bounded with domain D(T(k)) and (77)H-E0H+ωk-1Tk≤f1k+f2kωk. Hence, for all ψ∈D(H), (78)∫KH-E0H+ωk-1Tkψ2dμk≤γψ2. Therefore, by Lemma 14 and Hypothesis 3, we have (79)L1/2ϕ2≤γ+ɛ0,ϕ∈DL1/2∩PHH,ϕ=1. Thus, by Theorem 1, we obtain (66). The other parts of Theorem 12 easily follow from Theorem 1 and Corollaries 2 and 3.From the purely operator theoretical point of view, Theorem12 can be regarded as a comparison theorem on the degeneracy of ground states in the framework given by the quintuple (H,H0,H1,L,{A(f)∣f∈X}).


## 5. Remarks for Applications

As for applications of Theorem12 to quantum field models, we describe only basic aspects, because full descriptions of applications need many pages and it may be suitable to present them in a separate paper.The Hilbert spaceH in the abstract theory may have different concrete realizations depending on quantum field models. Here we present a unified treatment of various models in quantum field theory, taking as H a general Hilbert space. This will make less work in applications.LetWb and Wf be complex Hilbert spaces. Then the boson Fock space Fb(Wb) over Wb is defined by(80)FbWb≔⨁n=0∞⁡⨂sn⁡Wb=Ψ=Ψnn=0∞∣Ψn∈⨂sn⁡Wb,n≥0,Ψ2≔∑n=0∞Ψn2<∞,where ⨂snWb denotes the n-fold symmetric tensor product Hilbert space of Wb with convention ⨂s0Wb≔C. On the other hand, the fermion Fock space Ff(Wf) over Wf is defined by (81)FfWf≔⨁n=0∞⁡⨂asn⁡Wf, where ⨂asnWf is the n-fold antisymmetric tensor product Hilbert space of Wf with convention ⨂as0Wf≔C. Theboson-fermion Fock spaceFBF(Wb,Wf) over (Wb,Wf) is defined by (82)FBFWb,Wf≔FbWb⊗FfWf. This Hilbert space is a Hilbert space for a quantum system in which a Bose field interacts with a Fermi field.A general Hilbert space unifying Hilbert spaces for various quantum field models is given by(83)H=h⊗FBFWb,Wf,where h is a complex Hilbert space. Indeed, H includes, as special cases, three types of Hilbert spaces which appear typically in quantum field theory: (i)
In the case whereWf={0}, then Ff(Wf)=C. Hence, in this case, H is identified with h⊗Fb(Wb), which is a Hilbert space for a general quantum system interacting with a Bose field. In particular, if h=C, then H is identified with Fb(Wb).(ii)
In the case whereWb={0}, H is identified with h⊗Ff(Wf), which is a Hilbert space for a general quantum system interacting with a Fermi field. In particular, if h=C, then H is identified with Ff(Wf).(iii)
In the case whereh=C, H is identified with FBF(Wb,Wf). In this sense, at least for applications to quantum field theory, the above choice of the Hilbert space H is general enough.LetA be a nonnegative self-adjoint operator on h having ground state with E0(A)=0 and m(A)<∞. In what follows, # denotes either b or f. Let S# be a nonnegative injective self-adjoint operator on W# and denote by dΓ#(S#) the second quantization of S# on F#(W#). Let H1 be a symmetric operator on H. Then the following operator H serves as unification of Hamiltonians of various quantum field models: (84)H=H0+H1, where (85)H0≔A⊗I⊗I+I⊗dΓbSb⊗I+I⊗I⊗dΓfSf, and H1 is not necessarily H0-bounded.SinceS# is nonnegative and injective, it follows that dΓ#(S#) has a unique ground state with zero ground state energy and (86)ker⁡dΓbSb=Fb0≔cΩb∣c∈C,ker⁡dΓfSf=Ff0≔cΩf∣c∈C, where Ω#≔{1,0,0,…} is the Fock vacuum in F#(W#). We denote by P# the orthogonal projection onto F#(0) from F#(W#).The operatorH0 has ground state with E0(H0)=0 and (87)PH0H=ker⁡A⊗Fb0⊗Ff0. We have (88)PH0=PA⊗Pb⊗Pf. Hence (89)mH0=mA.We denote byN# the number operator on F#(W#): N#≔dΓ#(I). The operator (90)QA≔I-PA is the orthogonal projection onto (ker⁡A)⊥ (the orthogonal complement of ker⁡A). For each ɛ>0, we define an operator Lɛ by(91)Lɛ≔1+ɛI⊗Nb⊗I+1+14ɛI⊗I⊗Nf+QA⊗Pb⊗Pf.Lemma 15.
For anyɛ>0, (92)I⪯Lɛ+PH0.Proof.
It is easy to see that(93)I⪯Nb1/2+Pb,I⪯Nf1/2+Pf. Hence (94)I⪯I⊗Nb1/2+Pb⊗Nf1/2+Pf. By this inequality and the fact that P#≤I and QA+PA=I, we obtain (95)I⪯I⊗Nb1/2⊗Nf1/2+I⊗Nb1/2⊗I+I⊗I⊗Nf1/2+QA⊗Pb⊗Pf+PH0. By the functional calculus, (96)I⊗Nb1/2⊗Nf1/2⪯ɛI⊗Nb⊗I+14ɛI⊗I⊗Nf. Since σ(N#)={0}∪N, it follows that N#1/2⪯N#. Hence (92) is obtained.In the present framework, the operatorLɛ will be the operator L in Hypothesis 1.Remark 16.
The parameterɛ>0 in Lɛ is introduced to maintain the best possibility of (92) when the Hilbert space H is reduced to h⊗Fb(Wb) (the case where Wf={0} so that Nf=0 and Pf=I) or Ff(Wf) (the case where Wb={0} so that Nb=0 and Pb=I).We next describe a candidate for the operatorA(f) in Section 4. For this purpose, we use an isomorphism between a separable Hilbert space and an L2 space. Hence we assume that W# is separable. Then, by the multiplication operator form of the spectral theorem on a self-adjoint operator [8, Theorem VIII.4], there exist a finite measure space (K#,Σ#,μ#), a unitary operator U#:W#→L2(K#,dμ#), and a nonnegative function ω# on K# satisfying 0<ω#(k)<∞, μ#-a.e. k∈K# so that (97)US#U-1=ω#, where the right hand side denotes the multiplication operator by the function ω# on L2(K#,dμ#). The isomorphism U# induces the isomorphism (98)ΓU#:F#W#⟶F#L2K#,dμ# defined by (99)ΓU#≔⨁n=0∞⁡⨂n⁡U# with ⨂0U#≔1.In what follows we freely use the identification ofW# and F#(W#) with L2(K#,dμ#) and F#(L2(K#,dμ#)), respectively.Let(100)K≔Kb∪Kf with Kb∩Kf≔∅ and let μ be the measure on K such that the restriction of μ to K# is equal to μ#. For each element (f1,f2)∈L2(Kb,dμb)⊕L2(Kf,dμf), we define a function f on K by (101)fk≔f1k,ifk∈Kb,f2k,ifk∈Kf. Then it is easy to see that the correspondence (f1,f2)↦f gives an isomorphism between L2(Kb,dμb)⊕L2(Kf,dμf) and L2(K,dμ). In this sense, we write f=(f1,f2). Below we see that, in the present case, the Hilbert space X in Section 4 is given by (102)X≔L2K,dμ. We freely use the identification of L2(Kb,dμb)⊕L2(Kf,dμf) with X.In the present case, we takeω in Section 4 as follows: (103)ωk≔ωbk,ifk∈Kb,ωfk,ifk∈Kf.We are now ready to describe a candidate forA(f) in Section 4. Let u∈Wb and let a(u) be the boson annihilation operator on Fb(Wb), which is the densely defined closed linear operator on Fb(Wb) such that its adjoint a(u)∗ is of the form (104)au∗Ψ0=0,au∗Ψn=nSnu⊗Ψn-1,Ψ∈Dau∗, where Sn is the symmetrization operator on the n-fold tensor product ⨂nWb of Wb. The following facts are well known: (a.1)
Canonical commutation relations: for allu,u′∈Wb, (105)au,au′∗=u,u′,au,au′=0on the subspace(106)Fb,0Wb≔Ψ∈FbWb∣∃n0∈N,∀n≥n0,Ψn=0,where[A,B]≔AB-BA.(a.2)
For allu∈Wb, D(Nb1/2)⊂D(a(u)) and auΨ≤uNb1/2Ψ,Ψ∈D(Nb1/2).(a.3)
For all CONS{un}n=1∞ of Wb, (107)Nb1/2Ψ2=∑n=1∞aunΨ2,Ψ∈DNb1/2.(a.4)
For allu∈D(Sb-1/2), D(dΓb(Sb)1/2)⊂D(a(u)) and (108)auΨ≤Sb-1/2udΓbSb1/2Ψ,Ψ∈DdΓbSb1/2.(a.5)
For allt∈R and u∈Wb, eitdΓb(Sb)a(u)e-itdΓb(Sb)=a(eitSbu).On the other hand, the fermion annihilation operatorb(v) (v∈Wf) on Ff(Wf) is the everywhere defined bounded linear operator on Ff(Wf) such that (109)bv∗Ψ0=0,bv∗Ψn=nAnv⊗Ψn-1,Ψ∈FfWf, where An is the antisymmetrization operator on ⨂nWf. The following facts are well known: (b.1)
Canonical anticommutation relations: for allv,v′∈Wf, (110)bv,bv′∗=v,v′,bv,bv′=0,where{A,B}≔AB+BA.(b.2)
b(v)=b(v)∗=v, v∈Wf.(b.3)
For all CONS{vn}n=1∞ of Wf, (111)Nf1/2Ψ2=∑n=1∞bvnΨ2,Ψ∈DNf1/2.(b.4)
For allt∈R and v∈Wf, eitdΓf(Sf)b(v)e-itdΓf(Sf)=b(eitSfv).In the present case, we see that a candidate forA(f) in Section 4 may be the closure of the following operator: (112)Aɛf≔1+ɛI⊗af1⊗I+1+14ɛI⊗I⊗bf2,f=f1,f2∈X, where ɛ>0 is arbitrary. We denote the closure of Aɛ(f) by the same symbol. The parameter ɛ in Aɛ is introduced in correspondence to Lɛ defined by (91). It is easy to see that (113)DLɛ1/2=DI⊗Nb1/2⊗I∩DI⊗I⊗Nf1/2. Hence D(Lɛ1/2)⊂D(Aɛ(f)) for all f∈X and (55) holds.It follows from (a.4) that, for allf=(f1,f2)∈D(Sb-1/2)⊕Wf, D(H01/2)⊂D(Aɛ(f)) and (114)AɛfΨ≤1+ɛI⊗af1⊗IΨ+1+14ɛf2Ψ≤1+ɛSb-1/2f1I⊗dΓbSb1/2⊗IΨ+1+14ɛf2Ψ≤1+ɛSb-1/2f1H01/2Ψ+1+14ɛf2Ψ. Hence Hypothesis 2(i) holds with β=1 and α=1/2.In the present case, we have(115)A0f,t=Aɛe-itωbf1,e-itωff2,f=f1,f2∈X,t∈R,where, for notational simplicity, the dependence of A0(f,t) on ɛ is not explicitly written.To ensure that Hypothesis2(ii) holds, we consider an additional condition:(S)
The self-adjoint operatorsSb and Sf are purely absolutely continuous.For a subspaceD# of W#, we define F#,fin(D)⊂F#(W#) by (116)F#,finD#≔span⁡Ω#,c#g1∗⋯c#gn∗Ω#∣n∈N,gj∈D#,j=1,…,n, where, for a subset V of a vector space, spanV denotes the subspace algebraically spanned by all the vectors in V and cb(·)=a(·), cf(·)=b(·). For a subspace G# of F#(W#), we denote by h⊗^Gb⊗^Gf the algebraic tensor product of h, Gb, and Gf.Lemma 17.
Assume (S). Then, for allΨ∈h⊗^Fb,fin(Wb)⊗^Ff,fin(Wf),(117)limt→±∞⁡AɛfeitH0Ψ=0.Proof.
It is sufficient to prove the assertion for vectorsΨ of the form (118)Ψ=ψ⊗au1∗⋯aun∗Ωb⊗bv1∗⋯bvm∗Ωf,ψ∈h,uj∈Wb,vl∈Wf,j=1,…,n,l=1,…,m,n,m∈N. Let f=(f1,f2)∈X. Then, using (115), (a.1), and (b.1), we have (119)e-itH0AɛfeitH0Ψ=∑k=1ne-itSbf1,ukΨb,k+∑l=1me-itSff2,vlΨf,l, where (120)Ψb,k≔ψ⊗au1∗⋯auk-1∗auk+1∗⋯aun∗Ωb⊗bv1∗⋯bvm∗Ωf,Ψf,l≔-1l-1ψ⊗au1∗⋯aun∗Ωb⊗bv1∗⋯bvl-1∗bvl+1∗⋯bvm∗Ωf. By assumption (S), limt→±∞⁡e-itSbf1,uk=0 and limt→±∞⁡e-itSff2,vl=0. Hence limt→±∞⁡e-itH0Aɛ(f)eitH0Ψ=0. Since e-itH0 is unitary, (117) follows.SinceD(A)⊗^Fb,fin(D(Sb))⊗^Ff,fin(D(Sf)) is a core of H01/2, Lemma 17 implies that Hypothesis 2(ii) holds in the present case.It is well known that, for allu∈D(Sb)∩D(Sb-1/2), (dΓb(Sb),a(u)) has weak commutator on D(dΓb(Sb)) and (121)dΓbSb,auwDdΓbSb=-aSbuonDdΓbSb. Also, for all v∈D(Sf), (dΓf(Sf),b(v)) has weak commutator on D(dΓf(Sf)) and (122)dΓfSf,bvwDdΓfSf=-bSfv. Hence, by (115), (H0,A0(f,t)) has weak commutator on D(H0) and(123)H0,A0f,twDH0=-Aɛe-itSbSbf1,e-itSfSff2onDH0.Lemma 18.
Letf=(f1,f2)∈X with f1∈D(Sb)∩D(Sb-1/2) and f2∈D(Sf). Then, for all Ψ∈D(H0), [H0,A0(f,t)]wD(H0)Ψ is strongly continuous in t∈R.Proof.
For notational simplicity, we omit identityI in tensor products of operators (e.g., I⊗a(u)⊗I (u∈Wb) is simply written as a(u)). For all t,t′∈R and Ψ∈D(H01/2), we have by (a.4) (124)ae-it′SbSbf1Ψ-ae-itSbSbf1Ψ≤e-it′SbSb1/2f1-e-itSbSb1/2f1H01/2Ψ⟶0t′⟶t. Similarly one can show that limt′→t⁡be-it′SfSff2Ψ=b(e-itSfSff2)Ψ. Hence Aɛ(e-itSbSbf1,e-itSfSff2)Ψ is strongly continuous in t. By this fact and (123), we obtain the desired result.Lemma 19.
For allΨ∈D(H0) and f∈X-1∩X2, A0(f,t)Ψ is strongly differentiable in t and (125)dA0f,tΨdt=iH0,A0f,twDH0Ψ,t∈R.Proof.
This follows from (115) and (123).As for the weak commutator of(H1,A0(f,t)), the following form gives unification of some models: (126)H1,A0f,twDHΨ=∫Keitωkfk∗TkΨdμk,Ψ∈DH,f∈Dωb∩Dωb-1/2⊕Dωf, where T(k) is a densely defined linear operator on H defined for μ-a.e. k∈K, satisfying D(T(k))∩D(T(k)∗)⊃D(H) and (127)∫KfkTkΨdμk<∞,∫KfkωkTkΨdμk<∞. Note that one can write (128)H1,A0f,twDHΨ=∫Kbeitωbkf1k∗T1kΨdμbk+∫Kfeitωfkf2k∗T2kΨdμfk, where T1(k)≔T(k),k∈Kb and T2(k)≔T(k),k∈Kf.To give an example ofH1 which has the abovementioned properties and is not relatively bounded with respect to H0, we recall a basic object in Fb(Wb)(129)ϕu≔12au+au∗,u∈Wb, which is called theSegal field operator with test vector u [9, §X.7].Example 20.
Consider the case whereWb=L2(Rd) (d∈N) and Wf={0} so that H=h⊗Fb(L2(Rd)). In this case, H0 becomes (130)H0=A⊗I+I⊗dΓbωb and Aɛ(f) takes the form (131)Af≔I⊗af,f∈L2Rd, where we set ɛ=0. Let F be a nonnegative bounded continuous function on R and λ∈L2(Rd)∖{0} such that λ/ωb∈L2(Rd). Let n≥1 and (132)H1≔FA⊗ϕλ2n. Then H1 is a symmetric operator. It is proved that H is self-adjoint (this is nontrivial and will be discussed elsewhere) and bounded below (this is trivial). If n≥2, then H1 is not H0-bounded.
One can show that(H1,A0(f,t)) has weak commutator on D(H) with (133)H1,A0f,twDH=-2ne-itωbf,λFA⊗ϕλ2n-1 on D(H). Hence, in the present example, we have (134)Tk=-2nλkFA⊗ϕλ2n-1. Under additional conditions, one can show that Hypothesis 2(v) and (65) hold (these facts also will be discussed elsewhere).Example 21.
Consider the case whereh=C and Wb=Wf=L2(Rd)(d∈N) so that H=Fb(L2(Rd))⊗Ff(L2(Rd)). We set Kb=Rd, Kf=Rd, and K=Kb∪Kf. We denote by μ the measure on K such that the restriction of μ to K# is the d-dimensional Lebesgue measure. In this case, H0 becomes (135)H0=dΓbωb⊗I+I⊗dΓfωf,Aɛf=1+ɛaf1⊗I+1+14ɛI⊗bf2,f=f1,f2∈L2Rd⊗L2Rd. Let λ∈L2(Rd) be as in Example 20 and let v,w∈L2(Rd) be fixed and (136)B≔bv∗+bw. Then, for n≥2, we define (137)H1≔ϕλ⊗B∗B+ϕλ2n⊗I. The operator H1 is not H0-bounded. It is easy to see that H is bounded below (the self-adjointness of H will be shown elsewhere). One can show that (H1,A0(f,t)) has weak commutator on D(H) and (138)H1,A0f,twDH=-1+ɛ2e-itωbf1,λI⊗B∗B-21+ɛne-itωbf1,λϕλ2n-1⊗I+1+14ɛe-itωff2,vϕλ⊗B∗-e-itωff2,wϕλ⊗B. Hence, in the present example, we have (139)
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.Let{un}n=1∞ (resp., {vn}n=1∞) be a CONS of Wb (resp., Wf) and define en∈Wb⊕Wf as follows: (140)e2n-1≔un,0,e2n≔0,vn,n∈N. Then {en}n=1∞ is a CONS of Wb⊕Wf. By (a.3) and (b.3), we have (141)∑n=1∞AɛenΨ2=1+ɛI⊗Nb1/2⊗IΨ2+1+14ɛI⊗I⊗Nf1/2Ψ2 for all Ψ∈D(I⊗Nb1/2⊗I)∩D(I⊗I⊗Nf1/2). Hence, by (91), we obtain (142)supϕ∈DLɛ1/2∩PHH,ϕ=1⁡Lɛ1/2ϕ2-∑n=1∞Aɛenϕ2=supϕ∈DLɛ1/2∩PHH,ϕ=1⁡QA⊗Pb⊗Pfϕ2. Therefore we need only to show that there exists a constant ɛ0∈[0,1) such that(143)supϕ∈DLɛ1/2∩PHH,ϕ=1⁡QA⊗Pb⊗Pfϕ2≤ɛ0.Then Hypothesis 3 is satisfied. Estimate (143) can be obtained by extending the methods in [6, 7, 10] to the present case.In this way, for quantum field models within the class under consideration, one can obtain results (i)–(iii) in Theorem12 (under additional conditions). The details will be given in a separate paper.



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