QuantumMechanics as Approximation of Statistical Mechanics of Classical Fields
Andrei Yu. Khrennikov
In Section 3.1, we showed that QM can be represented as a classical field model in which physical quantities are given by quadratic functionals of fields. In this chapter we shall show that in complete accordance with Einstein’s views QM is an approximation of CSM of fields, i.e., classical statistical mechanics with the infinite- dimensional phase space H = L2(R3;R) × L2(R3;R), where L2(R3;R) is the space of real-valued square-integrable functions. (In the previous considerations the symbol L2(R3) was used to denote the space of complex-valued square-integrable functions.) QM can be represented as the quadratic Taylor approximation of CSM of fields. By expanding classical quantities, functionals of the prequantum field, φ → f (φ), into the Taylor series up to the second term, we obtain quantum quantities. By averaging these expansions we establish a coupling between classical average (so to say, prequantum) and quantum average. The latter appears as the main term in the expansion of the classical phase space average with respect to a small parameter α, the dispersion of the prequantum fluctuations, α = σ 2(μ). Here μ is the probability distribution of a prequantum random field. In the limit α → 0 the classical prequantum and quantum averages coincide. Thus QM can be interpreted as the limit (as α → 0) of the theory of classical random fields, cf. Section 3.11 for an experimentwhichmay show a deviation from the prediction of QM, namely, violation of Born’s rule.
•Mar 19, 2014•0 citations