Ehrenfest Theorem in Precanonical Quantization

Abstract

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 precanonical wave functions.

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

“…Instead, precanonical quantization of a scalar field φ(x) on a curved space-time background (cf. [10,11]) given by the metric tensor g µν (x) leads to the description in terms of a Clifford-algebra-valued wave function Ψ(φ, x µ ) which satisfies the partial derivative precanonical Schrödinger equation on the finite-dimensional bundle with the coordinates (φ, x µ ):…”

“…The operator H on the right-hand side of (2) is called the De Donder-Weyl (DW) Hamiltonian operator and it is constructed according to the procedure of precanonical quantization [10,11]. It contains an ultraviolet parameter κ of the dimension of the inverse spatial volume which typically appears in the representations of precanonical quantum operators already in flat space-time [10,[14][15][16]. It is interesting to note that the DW Hamiltonian operator H coincides with the expression in flat space-time (cf.…”

“…It is interesting to note that the DW Hamiltonian operator H coincides with the expression in flat space-time (cf. [10,[14][15][16]): the metric tensor components appear in the classical expression of the DW Hamiltonian function but they disappear in the quantum operator corresponding to it. The only manifestation of curved spacetime is through the curved space-time Dirac matrices (3) and the spin-connection on the left-hand side of Eq.…”

Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

“…The Poisson bracket operation defined by the weight +1 density valued x-dependent polysymplectic structure Ω = dp [1,9,10]…”

“…Precanonical quantization [1][2][3][4][5] is the approach to field quantization based on the De Donder-Weyl (DW) generalization of Hamiltonian formalism to field theory [6] which does not require the space+time decomposition and treats all space-time variables on equal footing. Despite the DW theory has been known since the 1930-es and it was considered as a possible basis of field quantization by Hermann Weyl himself [7], its various mathematical structures have been studied starting from the late 1960-es (with the relevant notion of the multisymplectic structure coined in Poland [8]), it is the structure of the Poisson-Gerstenhaber algebra of Poisson brackets defined on differential forms found within the DW Hamiltonian formulation in [5,9,10] which has proven to be suitable for a new approach to field quantization.…”