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.
Highlights
Let H be a complex Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖
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
We hope that the abstract theory given in the present paper clarifies general structures behind the theory on the degeneracy of ground state in [7] and 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
Summary
Let H be a complex Hilbert space with inner product ⟨⋅, ⋅⟩ (complex linear in the right variable) and norm ‖ ⋅ ‖. In [7], it is assumed that Hint is relatively bounded with respect to the unperturbed operator H(0) fl 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). 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]. We give remarks for applications of the main theorem to concrete models in quantum field theory
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