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Progress of Theoretical Physics

Two Types of Schrodinger Time Evolution in the Formalism of Thermo Field Dynamics and Their Equivalence to the Liouville-von Neumann Equation

João da Providência, M. Yamamura, A. Kuriyama

Published May 1, 1991• 2 citations
Publisher

Abstract

It is proved that two types of the time-dependent Schrödinger equation in the formalation of thermo field dynamics, which may be regarded as equivalent to the Liouville-von Neumann equation, are connected with each other under a certain unitary transformation in the tilde-space.
DOI
10.1143/ptp/85.5.939

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We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant ααα with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t)Ψ(x,t)Ψ(x,t) can be expanded in a complete set of functions Φn(x)Φ_n(x)Φn​(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t)W(x,p,t)W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant ααα appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ\hbarℏ.
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