Abstract
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.
Highlights
Quantum field theory in curved space-time [1,2,3,4,5] is often considered as an opportunity to study an interplay between gravitation, space-time and quantum theory in order to gain insights and intuitions into the quantum geometry of space-time and gravitation
To proceed with the derivation of the Schödinger equation for the wave functional from the restriction of the precanonical Schrödinger equation to a field configuration, in Section 3.3, we evaluate the functional derivatives of the functional composed from precanonical wave function with respect to the field configurations φ(x)
A more general case of non-static space-times with non-vanishing zero-th component of the spin connection is considered in Section 5 where we show that the extraneous term in (39), which contains the commutator of the zero-th component of spin connection with precanonical wave function, disappears if the wave functional is expressed in terms of transformed precanonical wave functions with the transformation given by the time-ordered exponential of the zero-th component of spin connection
Summary
Quantum field theory in curved space-time [1,2,3,4,5] is often considered as an opportunity to study an interplay between gravitation, space-time and quantum theory in order to gain insights and intuitions into the quantum geometry of space-time and gravitation. This is the bracket operation on differential forms, which is derived from the polysymplectic structure and leads to the Poisson–Gerstenhaber algebra [9,34,35] as a generalization of Poisson algebra, which has led us to a generalization of the canonical quantization [6,7,8] and geometrical quantization [9,45] in the context of the DW Hamiltonian formulation of fields Further discussion of this bracket or its different treatments and generalizations can be found in [37,46,47,48,49]. We proceed as follows: in Section 2, we first recall the results of canonical quantization in the functional Schrödinger representation and the precanonical quantization of scalar field theory in curved space-time, and we discuss drastic differences between them.
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