Operator Formalism of Quantum Field Theory and Operator Algebra
Project/Area Number  14540112 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
ABE Mitsuo Kyoto Univ., RIMS, Assistant Professor, 数理解析研究所, 助手 (80221729)

Project Period (FY) 
2002 – 2004

Project Status 
Completed(Fiscal Year 2004)

Budget Amount *help 
¥4,000,000 (Direct Cost : ¥4,000,000)
Fiscal Year 2004 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 2003 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 2002 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Quantum Field Theory / Operator Formalism / Exact Solution / Anomaly / Operator Algebra / Cuntz Algebra / CAR Algebra / Embedding / 量子重力 / 重力アノマリー / T^*積 / 場の方程式 / 無限次元対称性 / Noetherの定理 / フェルミオン / 自己準同型 / 自己同型 / KMS状態 
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 ArakiWoods 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 indefinitemetric fermion system in gauge theories, the pseudo Cuntz algebra plays the similar role as above. Therefore, the problem
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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 AlvarezGaume and Witten. Its drawback is the confusion of the T^*product quantities with the Tproduct ones. In contrast with the Tproduct, 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 AlvarezGaume 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 conformalgauge twodimensional quantum gravity, constructions of the infinitedimensional symmetry corresponding to the infinitenumber of conserved quantities are studied. Owing to the speciality of the field equation as the conformalgauge 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

Report
(4results)
Research Products
(6results)