Ehrenfest Theorem in Precanonical Quantization

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 Schrödinger 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.

24 pages. v2: adapted to the Publisher style + intro rewritten + few changes in the text & typos corrected + new refs. added. v3: few minor typos corrected

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• Quantization of Polysymplectic Manifolds
• Polysymplectic formulation for topologically massive Yang-Mills field theory
• Schrödinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization
• Precanonical structure of the Schrödinger wave functional in curved space-time
• Cosmology in the De Donder-Weyl Formulation of Einstein-Cartan Gravity