Abstract
In this Letter we propose two path integral approaches to describe the classical mechanics of spinning particles. We show how these formulations can be derived from the associated quantum ones via a sort of geometrical dequantization procedure proposed in a previous paper.
Highlights
Feynman’s path integral is one of the most fruitful methods to study quantum mechanics
Feynman himself said that “path integrals suffer most grievously from a serious defect. They do not permit a discussion of spin operators”. The reason for this difficulty is that the path integral formulation needs as an ingredient the Lagrangian of the system, which is a classical concept, and nothing like that existed for the spin in the Forties and the Fifties
The first one goes as follows: since spinning particles are described by Pauli matrices, which are anticommuting operators, the underlying classical mechanics must be formulated via anticommuting or Grassmann numbers
Summary
Feynman’s path integral is one of the most fruitful methods to study quantum mechanics. Casalbuoni and independently Berezin and Marinov went into this direction in [3]-[4] and their path integral for spinning particles involves a functional integration over Grassmann variables Another quantum path integral formulation for particles with spin, described in Refs. By minimizing the action associated to these Lagrangians one gets two “classical” descriptions of the spin This may sound quite strange because most people think that spin is an intrinsically quantum concept. [10] a dequantization procedure to pass from the QPI to the CPI has been put forward This dequantization procedure will be our way of getting a classical description of spin starting from the quantum one.
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