The probability operator in quantum theory and quantization of a system of free fields

Abstract

A theoretical framework for quantization, defined by the normalized positive-definite probability operator establishing dynamical correspondence between classical and quantum Poisson brackets, is presented. The resulting quantum theory, unlike the conventional one, admits consistent probabilistic interpretation. It is shown that, in the nonrelativistic case, quantization based on the probability operator leads to the theory known as ‘‘quantum mechanics with a non-negative quantum distribution function.’’ A generalization of the proposed framework to the case of the relativistic theory of fields is attempted. Four auxiliary problems of constructing probability operators of one-dimensional field oscillators in Bose and Fermi algebras are formulated and solved. On the basis of these solutions it is concluded that spinor fields are not quantizable in the Bose algebra with the help of the probability operator (the analog of Pauli’s theorem). An equation for the probability operator of a system of free fields is derived from the principles of dynamical correspondence and translational invariance. The physical meaning of the operators corresponding to classical field amplitudes, such as annihilation and creation operators of field quanta with definite energy-momentum, is shown to emerge as a consequence of this equation. It is shown that the quantization of a system consisting only of tensor fields or only of spinor fields in the formalism of the probability operator leads to difficulties. It is shown further that these difficulties can be removed by considering quantization of a system containing both tensor and spinor fields. As an illustration, quantization of a system consisting of a massive vector field and a massive spinor field is considered and it is found that a noncontradictory quantization requires the mass of the vector particle to be less than that of the spinor particle. The probability operator thus acts as a mechanism of selection of the fields to be quantized already at the level of free fields.