Two-Dimensional Order and Disorder Thermofields

The main objective of this paper was to obtain the twodimensional order and disorder thermal operators using the Thermofield Bosonization formalism. We show that the general property of the two-dimensional world according with the bosonized Fermi field at zero temperature can be constructed as a product of an order and a disorder variables which satisfy a dual field algebra holds at finite temperature. The general correlation functions of the order and disorder thermofields are obtained. The operator realization for bosonization of fermions in 1+1 dimensions at zero temperature (T = 0) corresponds to a mapping of a Fermi field algebra into a Bose field algebra. This algebraic isomorphism defines a one-to-one mapping of the corresponding Hilbert space of states. At T 6= 0, it is not evident that the operator bosonization survives at non zero temperatures. By one side, in the fermionic theory, a unitary operator depending on the Fermi-Dirac statistical weigh implements the transformation that promotes the Fermi field algebra into a Fermi thermofield algebra. On the other hand, in the bosonized version of the theory, another unitary operator depending on the Bose-Einstein statistical weight implements the transformation that promotes the Bose field algebra into a Bose thermofield algebra. In this way, the surviving of the operator bosonization at T 6= 0 dependes on the “transmutation” from Fermi-Dirac to BoseEinstein statistics. Indeed, in an amazing way this statistical transmutation occurs, as shown in Ref. [1]. The operator formulation for bosonization of massless fermions in 1 + 1 dimensions, at finite, non-zero temperature T is presented in Ref. [1]. The thermofield bosonization has been achieved in the framework of the real time formalism of Thermofield Dynamics. The well known Fermion-Boson correspondences in 1 + 1 dimensions at zero temperature are shown to hold also at finite temperature [1]. In Ref. [2] the two-dimensional Fermi field operator with generalized statistics at T = 0 is considered as a product of order and disorder variables. The main purpose of the present paper is to fill a gap in the literature, by providing the generalization for finite temperature of the twodimensional order and disorder operators within the Thermofield dynamics approach [3, 4, 5, 6, 7, 8]. We use the Thermofield Bosonization formalism, introduced in Ref. [1], to construct Fermi thermofields out of order and disorder thermal operators, which satisfy an algebra analogous to the dual algebra of order and disorder variables in statistical mechanics, as it was first discussed by Kadanoff and Ceva [9]. This streamlines the presentation of the thermofield bosonization discussed in Ref. [1]. Within the Thermofield Dynamics approach [3, 4, 5, 6, 7, 8] a Quantum Field Theory at finite temperature is constructed by doubling the numbers of degrees of freedom. This is performed by introducing the “tilde” operators corresponding to each of the operators describing the system considered. This fictitious system is an identical copy of the original system under consideration, which entails a doubling of the Hilbert space of states. To begin with, let us consider the Fermi field doublet ( ψ(x), ψ̃(x) ) of the twodimensional massless Thirring model at T = 0 [1, 10] which is defined by the Lagrangian density L = i ψ̄ γ ∂μ ψ + g 2 2 ( ψ̄γ μ ψ ) ( ψ̄γμψ ) − ( − i ̃̄ ψ γ ∂μ ψ̃ + g 2 2 ( ̃̄ ψγψ̃ ) ( ̃̄ ψγμψ̃ )) . (0.1) The bosonized Fermi field doublet ( ψ(x), ψ̃(x) ) at zero temperature, which provides the operator solution of the quantum equations of motion, is constructed as a product of an order and a disorder operators , ψ(x) = f(ε)σ(x)μ(x) , (0.2) ψ̃(x) = f(ε) σ̃(x) μ̃(x) , (0.3) where f(ε) is an appropriate normalization factor and the order and disorder operators at zero temperature (T = 0) [2] are given in terms of Wick-ordered exponentials σ(x) = : e i a γ 5 φ(x) : , (0.4) μ(x) = : e i b ∫∞ x1 d z ∂0φ(x , z) : , (0.5) 1We have suppressed constant multiplicative factors and Klein factors that are present in the bosonized form of ψ [1].