Scalar Product for a Version of Minisuperspace Model with the Grassmann Variables

Sergey L. Cherkas; Vladimir L. Kalashnikov

doi:10.20944/preprints202304.0508.v2

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preprints.org > physical sciences > astronomy and astrophysics > doi: 10.20944/preprints202304.0508.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Scalar Product for a Version of Minisuperspace Model with the Grassmann Variables

Sergey L. Cherkas

^* and

Vladimir L. Kalashnikov

Version 1 : Received: 17 April 2023 / Approved: 18 April 2023 / Online: 18 April 2023 (08:49:14 CEST)
Version 2 : Received: 2 June 2023 / Approved: 9 June 2023 / Online: 9 June 2023 (05:45:13 CEST)

A peer-reviewed article of this Preprint also exists.

Cherkas, S.L.; Kalashnikov, V.L. Scalar Product for a Version of Minisuperspace Model with Grassmann Variables. Universe 2023, 9, 508. Cherkas, S.L.; Kalashnikov, V.L. Scalar Product for a Version of Minisuperspace Model with Grassmann Variables. Universe 2023, 9, 508.

Abstract

he Grassmann variables are used to formally transform a system with constraints into an unconstraint system. As a result, the Schr\"{o}dinger equation arises instead of the Wheeler-DeWitt one. The Schrodinger equation describes a system's evolution, but a definition of the scalar product is needed to calculate the mean values of the operators. We suggest an explicit formula for the scalar product related to the Klein-Gordon scalar product. The calculation of the mean values is compared with an etalon method, in which a redundant degree of freedom is excluded. Nevertheless, we could note that a complete correspondence with the etalon picture is not found. Apparently, the picture with the Grassmann variables requires a further understanding of the underlying Hilbert space.

Keywords

minisuperspace model; quantum evolution; ghost variables; operator mean values

Subject

Physical Sciences, Astronomy and Astrophysics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 9 June 2023
Commenter: Sergey Cherkas
Commenter's Conflict of Interests: Author

Comment: References and Appendix about "extended Zitterbewegung'' have been added.

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