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On the probabilistic treatment of fields

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Abstract

Some basic problems of the probabilistic treatment of fields are considered, proceeding from the fundamentals of the complete probability theory. Two essentially equivalent definitions of random fields related to continuous objects are suggested. It is explained why the conventional classical probabilistic treatment generally is inapplicable to fields in principle. Two types of finite-dimensional random variables created by random fields are compared. Some general regularities related to Lagrangian and Hamiltonian partial equations, obtainable proceeding from the corresponding sets of ordinary differential equations, are revealed by using the functional derivative defined anew. It is shown that Hamiltonian random fields give rise to two types of Hamiltonian random variables, variables of the second type being those considered in the author's previous paper and immediately suited to the quantum approach. The results obtained are illustrated by some general examples. Critical remarks concerning second quantization are made, demonstrating the artificiality of this method. It is emphasized that the given probabilistic consideration of fields cannot be directly applied to, for instance, the electromagnetic field, which needs a special treatment.

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Mayants, L.S. On the probabilistic treatment of fields. Found Phys 11, 57–76 (1981). https://doi.org/10.1007/BF00715196

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Keywords

  • Differential Equation
  • Ordinary Differential Equation
  • Electromagnetic Field
  • Probability Theory
  • Special Treatment