Thermal Lie Groups, Classical Mechanics, and Thermofield Dynamics

Abstract

The concept of thermoalgebra, a kind of representation for the Lie-symmetries developed in connection with thermal quantum field theory, is extended to study unitary representations of the Galilei group for thermal classical systems. One of the representations results in the first-quantized Schönberg formalism for the classical statistical mechanics. Furthermore, the close analogy between thermal classical mechanics and thermal quantum field theory is analysed, and such an analogy is almost exact for harmonic oscillator systems. The other unitary representation studied results in a field-operator version of the Schönberg approach. As a consequence, in this case the counterpart of the thermofield dynamics (TFD) in classical theory is identified as both the first and second-quantized form of the Liouville equation. Non-unitary representations are also studied, being, in this case, the Lie product of the thermoalgebra identified as the Poisson brackets. A representation of the thermalSU(1, 1) is analysed, such that the tilde variables (introduced in TFD) are functions in a double phase space. As a result the equations of motion for dissipative classical oscillators are derived.