On the precanonical structure of the Schrödinger wave functional

Abstract

We show that the Schrödinger wave functional may be obtained as the product integral of precanonical wave functions on the space of field and space-time variables. The functional derivative Schrödinger equation underlying the canonical field quantization is derived from the partial derivative covariant analogue of the Schrödinger equation, which appears in the precanonical field quantization based on the De Donder-Weyl generalization of the Hamiltonian formalism for field theory. The representations of precanonical quantum operators typically contain an ultraviolet parameter $\varkappa$ of the dimension of the inverse spatial volume. The transition from the precanonical description of quantum fields in terms of Clifford-valued wave functions and partial derivative operators to the standard functional Schrödinger representation obtained from canonical quantization is accomplished if $\frac{1}{\varkappa }\to 0$ and $\frac{1}{\varkappa }{\gamma }_0$ is mapped to the infinitesimal spatial volume element $\mathrm{d}\mathbf{x}$. Thus the standard QFT obtained via canonical quantization corresponds to the quantum theory of fields derived via precanonical quantization in the limiting case of an infinitesimal value of the parameter $\frac{1}{\varkappa }$.

19 pages. v2: a misprint in Arxiv abstract is corrected. v3: slightly improved linguistically, accepted by ATMP

References in corpus (1)

• A Lie Infinity Algebra of Hamiltonian Forms in n-plectic Geometry

Cited by in corpus (5)

• On precanonical quantization of gravity
• Ehrenfest Theorem in Precanonical Quantization
• A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity
• 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