Precanonical quantization, quantum gravity and Clifford analysis

Igor Kanatchikov, National Center for Quantum Information in Gdansk, Poland

I will outline the precanonical approach to field quantization which leads to a generalization of quantum theoretic formalism were the space-time Clifford algebra replaces the complex numbers in quantum mechanics: both the wave functions and operators are Clifford-valued. The approach also leads to a Dirac-like generalization of the Schroedinger equation with the mass term replace by the De Donder-Weyl Hamiltonian operator. This formulation leads to an interesting class of problems which can be studied using the techniques of Clifford analysis. I will also discuss how the precanonical approach leads to a nonperturbative quantization of gravity where the quantum geometry is described by Clifford-valued transition amplitudes between different values of spin connection at different points. I show that the fundamental equation which describes the quantum geometry in this approach is a Clifford-algebraic matrix multi-variable generalization of the confluent hypergeometric equation. Thus the needs of quantum gravity call for a study of generalizations of hypergeometric functions within the Clifford analysis.

On the structure of Standard Model from the point of view of precanonical quantization

Igor Kanatchikov, National Center for Quantum Information in Gdansk, Poland

The approach of precanonical quantization of fields leads to a generalization of quantum theoretic formalism from mechanics to field theory. where the space-time Clifford algebra (which arises from quantization of differential forms) plays as important role as the complex numbers in quantum mechanics do. In this approach the space-time variables are treated on the equal footing and they play a role of the multidimensional analogue of the time parameter in quantum mechanics. I briefly discuss the relation between precanonical quantization and the standard QFT in the functional Schroedinger representation and few interesting consequences of the application of precanonical quantization in the context of perturbative QFT (an application to quantum gravity will be discussed in the talk at the general session). I also demonstrate that precanonical quantization of fermions naturally leads to the restriction of the number of fermionic fields which we can be incorporated into the (precanonically quantized) theory simultaneously in the way compatible with the standard functional Schroedinger representation. The number of fermions can not exceed the ability of the most general Clifford number on the space-time to embed the fermions in a linear way. This fact may have consequences for our understanding of the origin of symmetries of the Standard Model, as the latter should be related to the linear automorphisms of the space-time Clifford algebra and the dimensionality of the space-time, and also shed light on the possible nature of the generations of fundamental fermions.