MG15 - Talk detail

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We discuss precanonical quantization based on the De Donder-Weyl generalization of the Hamiltonian formalism, its generalization to curved space-time and the connection with the functional Schroedinger representation in the limiting case when the ultraviolet parameter $\varkapp$ introduced by precanonical quantization goes to infinity. We extend to curved space-time our earlier analysis of the relation between the precanonical quantization of scalar and Yang-Mills fields with their description in the functional Schroedinger representation. References: 1. B. Hatfield (1992) Quantum Field Theory of Point Particles and Strings. 2. I.V. Kanatchikov, Ehrenfest theorem in quantization, J. Geom. Symmetry Phys. 37 (2015) 43, arXiv:1501.00480. 3. I.V. Kanatchikov, On the precanonical structure of the Schr\"o\-dinger wave functional, Adv. Theor. Math. Phys. 20 (2016) 1377, arXiv:1312.4518 [hep-th]. 4. I. V. Kanatchikov, On the spectrum of DW Hamiltonian of quantum SU(2) gauge field, Int. J. Geom. Meth. Mod. Phys. 14 (2017) 1750123, arXiv:1706.01766 [hep-th]. 5. I. V. Kanatchikov, Schroedinger wave functional in quantum Yang-Mills theory from precanonical quantization, arXiv:1805. to appear.

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We discuss a possible relation between the constructions of canonical quantum gravity in various formulations and the recently proposed precanonical quantization approach to quantum gravity which avoids the procedure of space-time decomposition. We present our results on the relation between precanonical quantization and the functional Schroedinger representation in the case of scalar fields, Yang-Mills fields, and scalar fields in curved space-time in order to argue that the theories resulting from the canonical quantization in the functional Schroedinger representation can be viewed as a singular limiting case corresponding to the infinite value of the ultraviolet parameter $varkappa$ which is introduced in precanonical quantization when a differential form corresponding to an infinitesimal spatial volume is represented by an element of Clifford algebra. References: 1. I.V. Kanatchikov, On the precanonical structure of the Schr\"o\-dinger wave functional, Adv. Theor. Math. Phys. 20 (2016) 1377, arXiv:1312.4518 [hep-th], 2. I. V. Kanatchikov, Schroedinger wave functional in quantum Yang-Mills theory from precanonical quantization, arXiv:1805. to appear, 3. I.V. Kanatchikov, On the ``spin-connection foam" picture of quantum gravity from precanonical quantization, MG14 Proceedings, arXiv:1512.09137 [gr-qc], 4. I.V. Kanatchikov, De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity, arXiv:1302.2610 [gr-qc].

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