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

The functional Schrödinger representation of a nonlinear scalar quantum field theory in curved space-time is shown to emerge as a singular limit from the formulation based on precanonical quantization. The previously established relationship between the functional Schrödinger representation and precanonical quantization is extended to arbitrary curved space-times. In the limiting case when the inverse of the ultraviolet parameter ϰ introduced by precanonical quantization is mapped to the infinitesimal invariant spatial volume element, the canonical functional derivative Schrödinger equation is derived from the manifestly covariant partial derivative precanonical Schrödinger equation. The Schrödinger wave functional is expressed as the trace of the multidimensional spatial product integral of Clifford-algebra-valued precanonical wave function or the product integral of a scalar function obtained from the precanonical wave function by a sequence of transformations. In non-static space-times, the transformations include a nonlocal transformation given by the time-ordered exponential of the zero-th component of spin-connection.

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

The functional Schrödinger representation of a scalar field on an [Formula: see text]-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 [Formula: see text] introduced by precanonical quantization is replaced by [Formula: see text], where [Formula: see text] is the time-like tangent space Dirac matrix and [Formula: see text] is an invariant spatial [Formula: see text]-dimensional Dirac’s delta function whose regularized value at [Formula: see text] 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.

Zitterbewegung near a Schwarzschild-type black hole

In this paper, we present an approximate analytic solution to the Dirac equation in the Schwarzschild geometry to investigate the Zitterbewegung (ZB) effect. The analytical expression for the current density, which is induced by the motion of a wave packet of electrons, is obtained. In addition, the intensity of dipole radiation near the black hole is calculated. The proposed approach is based on the Schrödinger representation, thereby allowing to consider the ZB effect in the case of a curved space. The considered example demonstrates a possibility for applying this approach to astrophysical applications, in particular to problems of the electron radiation in the vicinity of real black holes.