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International Conference on Mathematical Methods in Physics (ICMP)

Bosonization within Thermofield Dynamics Approach

R. L. P. G. Amaral

Published Apr 11, 2007• 0 citations
Publisher

Abstract

We present an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T .This is achieved in the framework of the real time formalism of Thermofield Dynamics.The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature.In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermioncurrent two-point function in the imaginary-time formalism and real time formalisms.

DOI
10.22323/1.031.0025

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We present an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T .This is achieved in the framework of the real time formalism of Thermofield Dynamics.The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature.In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermioncurrent two-point function in the imaginary-time formalism and real time formalisms.
The main objective of this paper is to obtain an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T. This is achieved in the framework of the real time formalism of Thermofield Dynamics. Formally the results parallel those of the T = 0 case. The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature. In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermion-current two-point function in the imaginary-time formalism and real time formalisms. The calculations also illustrate the very different ways in which the transmutation from Fermi-Dirac to Bose-Einstein statistics is realized.
It is shown that exact solvability of the finite temperature massless Thirring model, as well as of its zero temperature case, in canonical quantization scheme originates from the intrinsic hidden exact linearizability of Heisenberg equations by means of dynamical mapping onto the Schrödinger physical fields. The normal forms of different one- and two- parametric (thermo) field's solutions are obtained. They are connected with each other by making use of generalized conformal shift transformations. The sequential use of bosonic canonical transformations provides a correct renormalization, anticommutation and symmetry properties of these solutions. The dynamical role of inequivalent representations of 1+1-D free massless Dirac fields, that are induced by inequivalent representations of 1+1-D free massless (pseudo) scalar field, and the appearance of Schwinger terms are elucidated. The inequivalent vacuum is established as coherent state for SU(1,1) group. A new alternative sources of superselection rules are shown. A generalization of Ojima tilde conjugation rules is suggested, which reveals the properties of coherent state for SU(2) group for the fermionic thermal vacuum state and is useful for the thermofield bosonization. The notions of "hot" and "cold" thermofields are introduced to distinguish different thermofield representations giving the correct normal form of thermofield solution. The weak sense of definition of zero and finite temperature operator bosonization rules in the framework of thermofield dynamics is demonstrated.
Finite-temperature properties of a non-Fermi-liauid system is one of the most challenging problems in the current understanding of strongly correlated electron systems. The paradigmatic arena for studying non-Fermi liauids is in one dimension, where the concept of a Luttinger liauid has arisen. The existence of a critical point at zero temperature in one-dimensional systems, and the fact that experiments are all undertaken at finite temperatures, implies a need for these one-dimensional systems to be examined at finite temperatures. Accordingly, we extended the well-known bosonization method of one-dimensional electron systems to finite temperatures. We have used this new bosonization method to calculate finite-temperature asymptotic correlation functions for linear fermions, the Tomonaga-Luttinger model, and the Hubbard model.
The finite-temperature real-time field theories for spin (1/2 are derived with use of a functional-integral approach, and their application to problems of equilibrium and nonequilibrium statistical mechanics discussed. These formalisms have a standard Wick's theorem and linked-graph theorem, leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the calculation of equilibrium quantities requires the consideration of the imaginary-time branch, leading to a three-component formalism, instead of the usual two-component formalism, typical of real-time field theories.
The Gaussian wavefunctional approach is developed in thermofield dynamics. We construct the thermal vacuum wavefunctional, its creation as well as annihilation operators, and accordingly the thermo-particle excited states. For a ( D + 1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, we calculate the finite-temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one for some concrete models as calculated by the imaginary-time Green function.