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Physical Review

Use of Field Theory Techniques in Quantum Statistical Mechanics

D. J. Thouless

Published Aug 15, 1957• 54 citations
Publisher

Abstract

It is shown that the normal product in quantum statistical mechanics, as defined by Matsubara, has zero expectation value. Another method for the treatment of the Bose particles is given.Received 1 March 1957DOI:https://doi.org/10.1103/PhysRev.107.1162©1957 American Physical Society

DOI
10.1103/physrev.107.1162

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It is shown that the normal product in quantum statistical mechanics, as defined by Matsubara, has zero expectation value. Another method for the treatment of the Bose particles is given.Received 1 March 1957DOI:https://doi.org/10.1103/PhysRev.107.1162©1957 American Physical Society
The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows. In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.