Ehrenfest Theorem in Precanonical Quantization

TL;DR

This paper extends the Ehrenfest theorem to precanonical quantization of fields, showing how classical field equations emerge from quantum operators in a framework treating space and time equally.

Contribution

It introduces a consistent precanonical quantization scheme for fields based on the De Donder--Weyl formulation, unifying operator representation, Schrödinger-like equation, and expectation values.

Findings

Classical field equations derived as expectation values of quantum operators.

Precanonical quantization maintains consistency with classical dynamics.

Framework treats space and time variables on equal footing.

Abstract

We discuss the precanonical quantization of fields which is based on the De Donder--Weyl (DW) Hamiltonian formulation and treats the space and time variables on an equal footing. Classical field equations in DW Hamiltonian form are derived as the equations for the expectation values of precanonical quantum operators. This field-theoretic generalization of the Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: (i) the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, (ii) the Dirac-like precanonical generalization of the Schr\"odinger equation without the distinguished time dimension, and (iii) the definition of the scalar product for calculation of expectation values of operators using the Clifford-valued precanonical wave functions.