Free Fermi and Bose Fields in TQFT and GBF

Free Fermi and Bose Fields in TQFT and GBF

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterizatio...

Full description

Saved in:
Bibliographic Details
Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2013
Main Author: Oeckl, R.
Format: Article
Language:English
Published: Інститут математики НАН України 2013
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/149232
Tags: Add Tag
No Tags, Be the first to tag this record!
Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Free Fermi and Bose Fields in TQFT and GBF / R. Oeckl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 19 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-149232
record_format dspace
spelling Oeckl, R.
2019-02-19T19:03:50Z
2019-02-19T19:03:50Z
2013
Free Fermi and Bose Fields in TQFT and GBF / R. Oeckl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 19 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 57R56; 81T70; 81P16; 81T20
DOI: http://dx.doi.org/10.3842/SIGMA.2013.028
https://nasplib.isofts.kiev.ua/handle/123456789/149232
We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.
This work was supported in part by UNAM–DGAPA–PAPIIT through project grant IN100212.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Free Fermi and Bose Fields in TQFT and GBF
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Free Fermi and Bose Fields in TQFT and GBF
spellingShingle Free Fermi and Bose Fields in TQFT and GBF
Oeckl, R.
title_short Free Fermi and Bose Fields in TQFT and GBF
title_full Free Fermi and Bose Fields in TQFT and GBF
title_fullStr Free Fermi and Bose Fields in TQFT and GBF
title_full_unstemmed Free Fermi and Bose Fields in TQFT and GBF
title_sort free fermi and bose fields in tqft and gbf
author Oeckl, R.
author_facet Oeckl, R.
publishDate 2013
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/149232
citation_txt Free Fermi and Bose Fields in TQFT and GBF / R. Oeckl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 19 назв. — англ.
work_keys_str_mv AT oecklr freefermiandbosefieldsintqftandgbf
first_indexed 2025-12-01T15:27:30Z
last_indexed 2025-12-01T15:27:30Z
_version_ 1850860552526495744

Similar Items