Schrödinger wave 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 κ introduced by precanonical quantization is replaced by γ 0 δ inv (0), where γ 0 is the time-like tangent space Dirac matrix and δ inv (0) is an invariant spatial (n − 1)-dimensional Dirac's delta function whose regularized value at 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.

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

“…An important aspect of realization of the potential of precanonical quantization is understanding of how it could be related to more familiar and already working concepts of standard QFT. In this paper, we extend our previous results on the relationship between precanonical quantization and the functional Schrödinger picture in QFT [27][28][29][30][31][32] to scalar field theory on arbitrary curved space-times.…”

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

“…The study of this conditions shows [41] that it can be fulfilled if the third term in (10) vanishes and the limiting condition (13) is satisfied. Using the fact that √ g = √ g 00 h, where h := | det(g ij )|, and √ g 00 γ 0 = γ 0 , the condition (13) takes the form…”

“…More specifically, we would like to generalize the relation found in flat space-time in the case of quantum scalar field theory [34,37,38] and quantum YM theory [29,31] to curved space-time. Our presentation here follows [41][42][43].…”

“…However, the terms I + III + IV in (12) still have played no role. One can argue based on the specific form of Φ(x) found in (18) and the covariant Stokes theorem that I + III have vanishing contribution to the time evolution of Ψ (see [41] for details). Then the effective equation for ∂ t Ψ Σ in (8) reads…”

“…On the quantum level, the connection between precanonical quantization and the functional Schrödinger representation of QFT [36] was found for scalar fields [34,37,38] and YM fields [29][30][31] in flat space-time. As a step towards understanding the connections between precanonical quantization of gravity [20][21][22][23][24][26][27][28] and the existing approaches based on canonical quantization [39,40] we have also explored a possible extension of those results to curved space-time [41][42][43]. The present discussion is a concise presentation of those papers.…”