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Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization. (English) Zbl 1407.81121

Summary: 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 \(\kappa\) 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.

MSC:

81T20 Quantum field theory on curved space or space-time backgrounds
81T70 Quantization in field theory; cohomological methods
81T99 Quantum field theory; related classical field theories
83C47 Methods of quantum field theory in general relativity and gravitational theory
15A66 Clifford algebras, spinors
11E88 Quadratic spaces; Clifford algebras
35R10 Partial functional-differential equations
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