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...

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Дата:2013
Автор: Oeckl, R.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2013
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/149232
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1492322019-02-20T01:23:46Z Free Fermi and Bose Fields in TQFT and GBF Oeckl, R. 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. 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149232 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Oeckl, R.
spellingShingle Oeckl, R.
Free Fermi and Bose Fields in TQFT and GBF
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Oeckl, R.
author_sort Oeckl, R.
title Free Fermi and Bose Fields in TQFT and GBF
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
publisher Інститут математики НАН України
publishDate 2013
url http://dspace.nbuv.gov.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.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT oecklr freefermiandbosefieldsintqftandgbf
first_indexed 2023-05-20T17:32:17Z
last_indexed 2023-05-20T17:32:17Z
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