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arXiv:hep-th/0610312 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Two-dimensional order and disorder thermofields

L. V. Belvedere

Published Nov 16, 2006• 2 citations
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Abstract

The main objective of this paper was to obtain the two-dimensional 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.

DOI
10.1103/physrevd.74.107701

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The main objective of this paper was to obtain the two-dimensional 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.
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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.
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators and are associated, in the paper we showed that this approach can also be formulated for systems of deformed bosons, associated with a pair of deformed operators and . In this context we built sets of temperature dependent coherent states, both of the Barut-Girardello and of the Klauder-Perelomov type, these two types being dual to each other. In calculations involving ladder operators, both independent and temperature-dependent, we used the Diagonal Operator Ordering Technique (DOOT), which is a generalization of the Integration Within an Ordered Product technique (IWOP) applicable to canonical operators associated with the one-dimensional harmonic oscillator. We also refer to the connection between TFD and thermodynamic quantities.
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators are associated, in the paper we showed that this approach can also be formulated for systems of deformed bosons, associated with a pair of deformed operators. In this context we built sets of temperature dependent coherent states, both of the Barut-Girardello and of the Klauder-Perelomov type, these two types being dual to each other. In calculations involving ladder operators, both independent and temperature-dependent, we used the Diagonal Operator Ordering Technique (DOOT), which is a generalization of the Integration Within an Ordered Product technique (IWOP) applicable to canonical operators associated with the one-dimensional harmonic oscillator. We also refer to the connection between TFD and thermodynamic quantities.
In this paper we review the critical Ising model by using the properties of conformal invariance. We use the known mapping of the Ising model to a theory of Majorana fermions and recognize that the self-duality property appears as a double degeneracy of the periodic ground state. The Ising model is doubled obtaining an Ashkin-Teller model (two Ising models coupled with a four-spin coupling) at the decoupling point. Using the bosonization technique this model is mapped onto the Gaussian model at a particular value of the temperature (K=1/\ensuremath{\pi}). This allows us to give the expressions for products of order, disorder, and energy operators of both Ising models in terms of operators in the Gaussian model. We compute several correlation functions of order, disorder, energy and spinor operators and show that they reproduce the operator product expansions predicted by conformal invariance. We explicitly discuss the physics by which the correlation functions obtained from the Gaussian model (a theory with conformal anomaly c=1) reproduce those of the Ising model (c=1/2). This provides a proof of Kadanoff and Brown's equivalences and conjectures. We provide a simple prescription to compute all n-point correlation functions at criticality, including mixed correlations of order, disorder, energy, and spinor operators. A bosonized form of the continuum limit of the transfer matrix for the critical Ising model is constructed. It is nonlocal and contains both spin-wave and vortex operators of the Gaussian model.
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.