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Journal of Mathematical Physics

Quantization of spinor fields. IV. Joint Bose–Fermi spectral problems

Piotr Garbaczewski

Published Apr 1, 1984• 10 citations

Abstract

In this continued study of the connection between classical c-number spinor models and their quantized Fermi partners, we elaborate further necessary consequences of the bosonization. The genuine (c-number) path integral representation of tr exp(−iHt) is derived for the Fermi oscillator and simple lattice Fermi models. We find that the underlying Hamiltonian of the Fermi system HF can be equivalently written as PHBP, where HB is the related Bose Hamiltonian, P is an appropriate projection in the state space of the Bose system, and [P, HB]−=0, HF=PHBP. Grassmann algebras are not used. We prove further that both for the massive Thirring model (MT) and the chiral invariant Gross–Neveu model (CN), the Bethe ansatz eigenstates for the Fermi Hamiltonians are exact eigenstates of the Bose MT and CGN Hamiltonians, so introduced that HF=PHBP, [P, HB]−=0. As a consequence, through studying the c-number path integral representation for tr exp(−iHFt), we establish a class of classical (c-number) spinor solutions of the underlying field equations, which at the same time stationary both the c-number Bose and c-number Fermi actions.

DOI: 10.1063/1.526237

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In this continued study of the connection between classical c-number spinor models and their quantized Fermi partners, we elaborate further necessary consequences of the bosonization. The genuine (c-number) path integral representation of tr exp(−iHt) is derived for the Fermi oscillator and simple lattice Fermi models. We find that the underlying Hamiltonian of the Fermi system HF can be equivalently written as PHBP, where HB is the related Bose Hamiltonian, P is an appropriate projection in the state space of the Bose system, and [P, HB]−=0, HF=PHBP. Grassmann algebras are not used. We prove further that both for the massive Thirring model (MT) and the chiral invariant Gross–Neveu model (CN), the Bethe ansatz eigenstates for the Fermi Hamiltonians are exact eigenstates of the Bose MT and CGN Hamiltonians, so introduced that HF=PHBP, [P, HB]−=0. As a consequence, through studying the c-number path integral representation for tr exp(−iHFt), we establish a class of classical (c-number) spinor solutions of the underlying field equations, which at the same time stationary both the c-number Bose and c-number Fermi actions.

An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c*iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.

Influenced by Klauder’s investigations on the same subject, we study the question of correspondence principle for Dirac fields, looking for its formulation without use of Grassman algebras. We prove that with each Fermi operator (the series with respect to asymptotic free fields): Ω (ψ,ψ̄): one can associate the functional ΩC(ψC, ψ̄C) with respect to classical spinor fields. Here the projector 1F and the Hilbert (Fock) space ℱF=1FℱB are given such that the identity 1F: ΩC(ψB, ψ̄B): 1FℱF = :Ω (ψ, ψ̄):ℱF defines the mediating boson level, where coherent state expectation values of operator expressions are in order: 〈:ΩC(ψB, ψ̄B):〉=ΩC(ψC, ψ̄C). For proofs we employ functional differentiation (resp. integration) methods, especially in connection with the use of functional representations of the CCR and CAR algebras.

We demonstrate that Bose systems can be used to simulate properties of their Fermi partners in thermal bath, provided the spectral property HB =PHBP +(1−P)HB(1−P), HF =PHBP can be proved in the state space of the Bose system. The approximation accuracy can be made arbitrarily good by varying the (free) coupling parameter λ∈(0,∞), and permits studying of fermionic partition or correlation functions in a finite volume by means of the standard techniques (Bose:path integral, Monte Carlo, etc).

In previous papers a Bose representation of fermions and parafermions was constructed in such a way that (1) the only fundamental quantum fields were Bose fields and (2) no property of the Fermi or para-Fermi systems (as e.g., the Pauli principle for fermions) be lost. The use of that construction and also the proofs of the theorems were made difficult by the complexity of many of the mathematical expressions we used previously. In the present paper we simplify much the above-mentioned construction. Moreover no results (concerning the Bose representation) proved in the previous papers are lost in the present reformulation, but all results are more general: In the previous papers we made the construction of the Bose representation of fermions for the case of pure Fock representations of Quantum Field Theory. The present work is consistent with Fock as well as with non-Fock representations of the Fermi anticommutation relations.

Abstract We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi–Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing first on the one-dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU ( 2 ) chiral Gross–Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross–Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.