#### 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.

#### 1. Introduction

Let be a complex Hilbert space with inner product (complex linear in the right variable) and norm . For a linear operator on , we denote its domain by and its spectrum by .

Let be a self-adjoint operator on and bounded below. Then, by abstract use of word, we call the infimum of the* lowest* or* minimal energy* of (this name originally comes from the context in quantum physics where denotes the Hamiltonian of a quantum system). If is an eigenvalue of , then is said to* have ground state* and a nonzero vector in the eigenspace is called a* ground state* of . In that case, the dimension of the eigenspace is called the* degeneracy* or the* multiplicity of the ground state* of . If , then the ground state is said to be* unique*. If , then the ground state is said to be* degenerate*.

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 and* massive* 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 (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:acting in the tensor product Hilbert spaceof a complex Hilbert space and the boson Fock space over a complex Hilbert space (see (80) for the definition). Here is a self-adjoint operator on which is bounded below, denotes identity, is a nonnegative self-adjoint operator on , is the second quantization operator of on (see, e.g., [8, p.302] and [9, X.7]), is a coupling constant, and is a symmetric operator on which describes an interaction between a Bose field and a quantum system whose Hilbert space of state vectors is . In [7], it is assumed that * is relatively bounded with respect to the unperturbed operator *. It is proved in [7] that, under a suitable condition, with being a constant depending on and, in particular, for all sufficiently small in an abstract framework and in the case where , the -direct sum of (). 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 is not necessarily relatively bounded with respect to . 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 aspects* independent 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 Section 2, we consider a bounded below and self-adjoint operator and a symmetric operator on a Hilbert space . The sum is supposed to be self-adjoint and bounded below. We formulate a sufficient condition which yields inequality with a constant being computed from the given data. In particular, an additional condition for is given so that . In Section 3, we derive an integral equation for any ground state of in terms of a linear operator on 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 operators on (), we write “” if and

Let be a bounded below and self-adjoint operator on and let be a symmetric operator on such that is self-adjoint and bounded below (). We assume the following.

*Hypothesis 1. *(i) The operators and have ground state with .

(ii) There exists a nonnegative self-adjoint operator on such that where, for a bounded below self-adjoint operator on , we denote by the orthogonal projection onto : .

A vector is said to* overlap* with a subset if there exists a vector such that .

The following theorem, which is a comparison theorem on and , and Corollary 2 below are more abstract versions of Lemma and Corollaries in [7].

Theorem 1. *Assume Hypothesis 1 and suppose that there exists a constant with such thatThen, *(i)*any with overlaps with ;*(ii)*let* *then*(iii)*if in addition, then*

*Proof. *(i) Let with . Then, by Hypothesis 1(ii), we have HenceIn particular, . Therefore overlaps with .

(ii) We first show that is finite-dimensional. So suppose that were infinite-dimensional. Then there is an orthonormal system with . Taking as in (13) and summing over , we obtain . But the left hand side is less than or equal to , where means trace. Hence one is led to a contradiction. Therefore is finite-dimensional; that is, . Let be an orthonormal basis of . Then, by (13), we have Thus (10) follows.

(iii) This follows from (10).

Corollary 2. *Assume Hypothesis 1 and suppose that there exists a constant such thatand (8) holds. ThenIn particular, if in addition, then*

*Proof. *Condition (15) implies that . Hence the conclusion of Theorem 1 holds. Therefore where, in the second inequality, we have used (15). Since and are natural numbers, (16) follows.

Corollary 3 (uniqueness). *Assume Hypothesis 1 with and suppose that the following (i) and (ii) hold: *(i)*.*(ii)*There exists a constant with such that (8) holds.** Then .*

* Proof. *In the present case, (15) holds with . Hence . Since is a natural number, it follows that .

In applications to quantum field models, it may be convenient to consider in the form with a constant parameter , as in the case of given by (3).

*Remark 4. *The condition is taken so that is bounded below in the case where is not -bounded but bounded below. In the case where is -bounded, needs not to be positive, being allowed to be a negative number as well, and, by the Kato-Rellich theorem, is self-adjoint and bounded below for all sufficiently small () [7].

Corollary 5. *Assume Hypothesis 1 with , ( is a constant), and suppose that there exists a continuous function on such that andThen there exists a constant such that, for all , any with overlaps with andIn particular, if for all in addition, thenfor all .*

* Proof. *By the continuity of in with , there exists a constant such that, for all , Hence, for each , one can apply Theorem 1 and Corollary 2 to obtain the desired results.

*Remark 6. *Condition (20) with implies that . In some models in quantum field theory, 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 Theorem 1, 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 and . In this section we introduce an abstract version of such operators.

Let be a densely defined closed linear operator on and We assume the following:(A.1) (i)For some , is -bounded; that is, , and there exist constants such that(ii)There exists a core of such that

*Remark 7. *Under condition (A.1)(i), the functional calculus of the self-adjoint operator gives that, for all and , and . Hence for all . In particular, for all . Since and for all , it follows that for all . In what follows, we use these facts without mentioning.

Lemma 8. *Assume (A.1). Then, for all , .*

* Proof. *Since , we have . By (A.1)(ii), for any , there exists a vector such that Hence Taking the limit , we have Then the limit yields that . Thus .

To state additional assumptions, we recall the concept of weak commutator [13]. Let and be densely defined linear operators on and let be a dense subspace in such that , where, for a densely defined linear operator on , denotes the adjoint of . Then the pair is said to have weak commutator on if there exists a linear operator on such that and , . In this case, the operator restricted to is written as so that We call the* weak commutator* of on .

We also need the following assumption:(A.2) (i)For each and , has weak commutator on . Moreover, for all , is strongly continuous in .(ii)For all , is strongly differentiable in and its strong derivative is of the form (iii)There exist a -finite measure space , a nonnegative -measurable function on with for -a.e. , and a linear operator on () with such that the following (a)–(c) hold:(a)for all , and(b)for all , and ;(c)for all and ,

*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 is determined by the spectral measure of .

Under condition (A.2), is bounded for -a.e. with Hence, for all ,

It is easy to see that, for all , is strongly continuous in . Hence, for each , the strong Riemann integral exists.

Lemma 10. *Assume (A.1) and (A.2). Then, for all and ,Moreover, for all ,*

*Proof. *Let and Then it is easy to see that is differentiable in with Using the identity , we obtain Hence By a property of strong Riemann integral, we have Thus (36) follows. Formula (37) follows from (36) and (A.2)(iii).

For all , Hence the Bochner integral exists. The main theorem in this section is as follows.

Theorem 11. *Assume (A.1) and (A.2). Then, for all ,*

*Proof. *Let . Then, taking the limit in (37) and using Lemma 8, we obtain For all , we have By (c) in (A.2)(iii), the right hand side is integrable in . Hence, by the Lebesgue dominated convergence theorem, we have Note that Hence, by Fubini’s theorem, we have Therefore It is easy to see that, for -a.e., Hence we can apply the Lebesgue dominated convergence theorem to obtain 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 Hypothesis 1. Let be a -finite measure space and let be a nonnegative -measurable function on such that , -a.e. . We set and, for each , introduce a subspace

Suppose that there exists a family of densely defined closed linear operators on such that, for all , and, for all ,where is the complex conjugate of . We introduce an operator:For the operator , we assume conditions similar to (A.1) and (A.2) for in the preceding section.

*Hypothesis 2. *(i) For some and , and for each , there exist constants such that(ii) There exists a core of such that(iii) There exists a dense subspace such that, for all and , has weak commutator on and, for all , is strongly continuous in .

(iv) For all and , is strongly differentiable in and

(v) For all , , has weak commutator on and, for all , is strongly continuous in . Moreover, for -a.e. , there exists a densely defined linear operator on with such that the following hold:(a)for all and , and for all ,(b)for all and , (c)there exists a dense subspace such that, for all , , and ,

We need an assumption on a relation between and .

*Hypothesis 3. *There exist a complete orthonormal system (CONS) of with , and a constant such that

The main theorem in the present paper is as follows.

Theorem 12. *Assume Hypotheses 1–3. Suppose that there exist nonnegative functions and on such that withThen, *(i)*any with overlaps with ;*(ii)* * *where is defned by (9); moreover, if* *in addition, then*(iii)*suppose that in addition; then* *in particular, if (67) holds, then*(iv)*if , , and , then .*

To prove Theorem 12, we need some lemmas.

Lemma 13. *Assume Hypothesis 2. Then, for all and ,*

*Proof. *One can apply Theorem 11 with . Then (71) follows.

Lemma 14. *Assume Hypothesis 2. Let and Then, for any CONS of with (),*

*Proof. *Define a mapping by Then, in quite the same way as in the proof of [12, Lemma ], one can show that is Hilbert-Schmidt and where is the Hilbert-Schmidt norm of . In general, for any CONS of , . The set also is a CONS of . By (71), we have . Thus (73) holds.

We are now ready to prove Theorem 12.

*Proof of Theorem 12. *Let be such that . Then (65) implies that is bounded with operator norm Hence is bounded with domain and Hence, for all , Therefore, by Lemma 14 and Hypothesis 3, we have 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, Theorem 12 can be regarded as a comparison theorem on the degeneracy of ground states in the framework given by the quintuple .

#### 5. Remarks for Applications

As for applications of Theorem 12 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 space 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 a general Hilbert space. This will make less work in applications.

Let and be complex Hilbert spaces. Then the boson Fock space over is defined bywhere denotes the -fold symmetric tensor product Hilbert space of with convention . On the other hand, the fermion Fock space over is defined by where is the -fold antisymmetric tensor product Hilbert space of with convention . The* boson-fermion Fock space * over is defined by 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 bywhere is a complex Hilbert space. Indeed, includes, as special cases, three types of Hilbert spaces which appear typically in quantum field theory: (i)In the case where , then . Hence, in this case, is identified with , which is a Hilbert space for a general quantum system interacting with a Bose field. In particular, if , then is identified with .(ii)In the case where , is identified with , which is a Hilbert space for a general quantum system interacting with a Fermi field. In particular, if , then is identified with .(iii)In the case where , is identified with . In this sense, at least for applications to quantum field theory, the above choice of the Hilbert space is general enough.

Let be a nonnegative self-adjoint operator on having ground state with and . In what follows, denotes either or . Let be a nonnegative injective self-adjoint operator on and denote by the second quantization of on . Let be a symmetric operator on . Then the following operator serves as unification of Hamiltonians of various quantum field models: where and is not necessarily -bounded.

Since is nonnegative and injective, it follows that has a unique ground state with zero ground state energy and where is the Fock vacuum in . We denote by the orthogonal projection onto from .

The operator has ground state with and We have Hence

We denote by the number operator on : . The operator is the orthogonal projection onto (the orthogonal complement of ). For each , we define an operator by

Lemma 15. *For any , *

*Proof. *It is easy to see that Hence By this inequality and the fact that and , we obtain By the functional calculus, Since , it follows that . Hence (92) is obtained.

In the present framework, the operator will be the operator in Hypothesis 1.

*Remark 16. *The parameter in is introduced to maintain the best possibility of (92) when the Hilbert space is reduced to (the case where so that and ) or (the case where so that and ).

We next describe a candidate for the operator in Section 4. For this purpose, we use an isomorphism between a separable Hilbert space and an space. Hence we assume that 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 , a unitary operator , and a nonnegative function on satisfying , -a.e. so that where the right hand side denotes the multiplication operator by the function on . The isomorphism induces the isomorphism defined by with .

In what follows we freely use the identification of and with and , respectively.

Let with and let be the measure on such that the restriction of to is equal to . For each element , we define a function on by Then it is easy to see that the correspondence gives an isomorphism between and . In this sense, we write . Below we see that, in the present case, the Hilbert space in Section 4 is given by We freely use the identification of with .

In the present case, we take in Section 4 as follows:

We are now ready to describe a candidate for in Section 4. Let and let be the boson annihilation operator on , which is the densely defined closed linear operator on such that its adjoint is of the form where is the symmetrization operator on the -fold tensor product of . The following facts are well known: (a.1)Canonical commutation relations: for all , on the subspace where .(a.2)For all