Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 6

On the precanonical structure of the Schrödinger wave functional

Pages: 1377 – 1396

DOI: http://dx.doi.org/10.4310/ATMP.2016.v20.n6.a3

Author

Igor V. Kanatchikov (National Quantum Information Centre in Gdańsk (KCIK), Sopot, Poland; and Research Center of Einstein Physics, Free University of Berlin, Germany)

Abstract

We show that the Schrödinger wave functional may be obtained as the product integral of precanonical wave functions on the space of field and space-time variables. The functional derivative Schrödinger equation underlying the canonical field quantization is derived from the partial derivative covariant analogue of the Schrödinger equation, which appears in the precanonical field quantization based on the De Donder–Weyl generalization of the Hamiltonian formalism for field theory. The representation of precanonical quantum operators typically contains an ultraviolet parameter $\varkappa$ of the dimension of the inverse spatial volume. The transition from the precanonical description of quantum fields in terms of Clifford-valued wave functions and partial derivative operators to the standard functional Schrödinger representation obtained from canonical quantization is accomplished if $\frac{1}{\varkappa} \to 0$ and $\frac{1}{\varkappa} \gamma_0$ is mapped to the infinitesimal spatial volume element $\mathrm{dx}$. Thus the standard QFT obtained via canonical quantization corresponds to the quantum theory of fields derived from the precanonical quantization in the limiting case of an infinitesimal value of the parameter $\frac{1}{\varkappa}$.

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