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Application of second quantization methods to the classical statistical mechanics

par Schonberg, Mario
Référence Il Nuovo cimento, 9, 12, page (1139-1182)
Publication Publié, 1952-12

Article révisé par les pairs

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Résumé :

Methods similar to those of second quantization are applied to the Liouville equation of the classical statistical mechanics. An equation similar to the Schrödinger equation of a quantized field is given. It is shown that the interpretation rules of the quantal type of the field formalism lead to the rules of the classical statistical mechanics, the particles being treated as indistinguishable. The application of the Fock treatment of second quantization leads to the introduction of wave functions in phase space, the probability density of the classical statistical mechanics being the square of the absolute value of the wave function in phase space. The choice of the sign in the commutation rules of the field operators leads to symmetrical or anti-symmetrical wave functions in phase space. To each kind of symmetry corresponds a different statistics. Bose or Fermi, as in quantum theory. The Boltzmann statistics corresponds to phase space wave functions without symmetry conditions. The wave functionals for the states with zero particles of the fields in phase space are analogous to those of the quantized fields of quantum theory. © 1952 Società Italiana di Fisica.