2004 Fiscal Year Final Research Report Summary
Operator Formalism of Quantum Field Theory and Operator Algebra
| Project/Area Number |
14540112
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
ABE Mitsuo Kyoto Univ., RIMS, Assistant Professor, 数理解析研究所, 助手 (80221729)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Quantum Field Theory / Operator Formalism / Exact Solution / Anomaly / Operator Algebra / Cuntz Algebra / CAR Algebra / Embedding |
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| Research Abstract |
The structure of the CAR algebra describing a fermion system is clarified through a new approach using restrictions of the Cuntz algebra. First, embeddings, automorphisms, endomorphisms, and various representations of Cuntz algebras are explicitly constructed. Second, by restricting them to the embedded CAR algebra in the Cuntz algebra, the corresponding structures of the CAR algebra are obtained. Using a certain class of endomorphorphisms, it is shown that the KMS states as the mixed states of a fermion system are obtained from the Fock representation as the pure state. This result corresponds to the theory of Araki-Woods factors in the classification of von Neumann algebras. By restricting automorphisms, various nontrivial time evolutions of a fermion system are explicitly constructed, which give new constructions of solvable fermion systems. As for the indefinite-metric fermion system in gauge theories, the pseudo Cuntz algebra plays the similar role as above. Therefore, the problem
… More
of representation of the FP ghosts in string theory is clarified likewise.
It is shown that the existence of the gravitational anomaly in two dimensions cannot be established by the reasoning of Alvarez-Gaume and Witten. Its drawback is the confusion of the T^*-product quantities with the T-product ones. In contrast with the T-product, the T^*-product commutes with the time differentiation, however some field equations are apparently broken. This problem has nothing to do with anomaly. The result obtaied by Alvarez-Gaume and Witten is shown to be quantitatively the same as the apparent breakdown of the field equation in the T^*-product.
As concerns the solvability of the conformal-gauge two-dimensional quantum gravity, constructions of the infinite-dimensional symmetry corresponding to the infinite-number of conserved quantities are studied. Owing to the speciality of the field equation as the conformal-gauge condition, it is difficult to define the symmetry transformation of fields from a generic conserved quantitity as a symmetry generator. This procedure corresponds to the inverse problem of the Noether theorem. Less
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Research Products
(6 results)
