On quantization of field theories in polymomentum variables

Abstract

Abstract. Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex generalization of quantum mechanics to field theory in which the algebra of complex numbers and a time parameter are replaced by the spacetime Clifford algebra and space-time variables treated in a manifestly covariant fashion. The corresponding covariant generalization of the Schrödinger equation is shown to be consistent with several aspects of the correspondence principle such as a relation to the De Donder-Weyl Hamilton-Jacobi theory in the classical limit and the Ehrenfest theorem. A relation of the corresponding wave function (over a finite dimensional configuration space of field and space-time variables) with the Schrödinger wave functional in quantum field theory is examined in the ultra-local approximation.

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“…An identification of κ with the ultra-violet cutoff or a fundamental length scale quantity was discussed in [47,49]. The realization (2.5) is essentially inspired by the relation between the Clifford algebra and the endomorphisms of the exterior algebra [60].…”

“…This philosophy leads to the following (covariant, "multi-temporal", hypercomplex) generalization of the Schrödinger equation to the precanonical framework [47][48][49] …”

“…The main argument in favour of a generalized Schrödinger equation (2.6) is that it satisfies at least two important aspects of the correspondence principle [48,49]:…”

“…The elements of field quantization based on the aforementioned PoissonGerstenhaber brackets on differential forms have been discussed in [26,[47][48][49] and will be briefly summarized in Section 2.2. Unfortunately, many fundamental aspects of the corresponding precanonical approach to field quantization, as we suggest to call it, so far remain poorly understood (see [47,49] and Section 4 for a discussion) and require a further analysis.…”

“…Unfortunately, many fundamental aspects of the corresponding precanonical approach to field quantization, as we suggest to call it, so far remain poorly understood (see [47,49] and Section 4 for a discussion) and require a further analysis. This particularly concerns an interplay with the standard formalism and notions of quantum field theory (see [50] for a recent progress).…”