Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology
In this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are ''associativity'' or ''factorization'' conditions on the operator product expansion (OPE) of the theory, and...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2009-09-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2009.090 |
| _version_ | 1819089096576860160 |
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| author | Stefan Hollands |
| author_facet | Stefan Hollands |
| author_sort | Stefan Hollands |
| collection | DOAJ |
| description | In this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are ''associativity'' or ''factorization'' conditions on the operator product expansion (OPE) of the theory, and are proposed to be the defining property of any quantum field theory. Our framework is presented in the Euclidean setting, and is applicable in principle to any quantum field theory, including non-conformal ones. In our framework, we obtain a characterization of perturbations of a given quantum field theory in terms of a certain cohomology ring of Hochschild-type. We illustrate our framework by the free field, but our constructions are general and apply also to interacting quantum field theories. For such theories, we propose a new scheme to construct the OPE which is based on the use of non-linear quantized field equations. |
| first_indexed | 2024-12-21T22:02:30Z |
| format | Article |
| id | doaj.art-6a6e659e6979412ab89cd9cd9b38b28f |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-21T22:02:30Z |
| publishDate | 2009-09-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-6a6e659e6979412ab89cd9cd9b38b28f2022-12-21T18:48:47ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592009-09-015090Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild CohomologyStefan HollandsIn this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are ''associativity'' or ''factorization'' conditions on the operator product expansion (OPE) of the theory, and are proposed to be the defining property of any quantum field theory. Our framework is presented in the Euclidean setting, and is applicable in principle to any quantum field theory, including non-conformal ones. In our framework, we obtain a characterization of perturbations of a given quantum field theory in terms of a certain cohomology ring of Hochschild-type. We illustrate our framework by the free field, but our constructions are general and apply also to interacting quantum field theories. For such theories, we propose a new scheme to construct the OPE which is based on the use of non-linear quantized field equations.http://dx.doi.org/10.3842/SIGMA.2009.090quantum field theoryoperator product expansionquantum algebraHochschild cohomology |
| spellingShingle | Stefan Hollands Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology Symmetry, Integrability and Geometry: Methods and Applications quantum field theory operator product expansion quantum algebra Hochschild cohomology |
| title | Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology |
| title_full | Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology |
| title_fullStr | Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology |
| title_full_unstemmed | Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology |
| title_short | Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology |
| title_sort | axiomatic quantum field theory in terms of operator product expansions general framework and perturbation theory via hochschild cohomology |
| topic | quantum field theory operator product expansion quantum algebra Hochschild cohomology |
| url | http://dx.doi.org/10.3842/SIGMA.2009.090 |
| work_keys_str_mv | AT stefanhollands axiomaticquantumfieldtheoryintermsofoperatorproductexpansionsgeneralframeworkandperturbationtheoryviahochschildcohomology |