Precanonical quantization and the Schrödinger wave functional revisited

Abstract

We address the issue of the relation between the canonical functional Schrödinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schrödinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schrödinger equation under the formal limiting transition ${\gamma }^0\varkappa \to \delta (0)$, where the constant $\varkappa$ naturally appears within the precanonical quantization as the inverse of a small "elementary volume" of space. We obtain a formal explicit expression of the Schrödinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.

13 pages. v2: few linguistic improvements, ref. 15 updated. v3: the published version with few tiny corrections

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• On a generalization of the Dirac bracket in the De Donder-Weyl Hamiltonian formalism
• On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras
• Multisymplectic formalism and the covariant phase
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Cited by in corpus (7)

• On precanonical quantization of gravity
• Ehrenfest Theorem in Precanonical Quantization
• On the precanonical structure of the Schrödinger wave functional
• 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
• $n$-plectic Maxwell Theory