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Ehrenfest theorem in precanonical quantization. (English) Zbl 1316.81085

Summary: 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.

MSC:

81T70 Quantization in field theory; cohomological methods
81T20 Quantum field theory on curved space or space-time backgrounds
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
15A66 Clifford algebras, spinors
81T13 Yang-Mills and other gauge theories in quantum field theory
81S10 Geometry and quantization, symplectic methods
53D42 Symplectic field theory; contact homology
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