Schrödinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization

Abstract

The functional Schrödinger representation of a scalar field on an $n$-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder-Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is replaced by $\underline{\gamma }{}_0{\delta }^{\mathrm{inv}}(0)$, where $\underline{\gamma }{}_0$ is the time-like tangent space Dirac matrix and ${\delta }^{\mathrm{inv}}(0)$ is an invariant spatial $(n-1)$-dimensional Dirac's delta function whose regularized value at $\mathbf{x}=0$ is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

15 pages. arXiv admin note: text overlap with arXiv:1805.05279

References in corpus (4)

• On precanonical quantization of gravity
• Covariant Hamiltonian formalism for the calculus of variations with several variables
• Ehrenfest Theorem in Precanonical Quantization
• On the Polysymplectic Integrator for the Short Pulse Equation

Cited by in corpus (1)

• Precanonical structure of the Schrödinger wave functional in curved space-time