The construction of the field theory on the noncommutative spaces and its application
Project/Area Number  09640331 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
素粒子・核・宇宙線

Research Institution  TOHOKU UNIVERSITY 
Principal Investigator 
WATAMURA Satoshi Tohoku University, Graduate School of Science, associate professor, 大学院・理学研究科, 助教授 (00201252)

CoInvestigator(Kenkyūbuntansha) 
ISHIKAWA Hiroshi Tohoku University, Graduate School of Science, assistant, 大学院・理学研究科, 助手 (20291247)

Project Fiscal Year 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  Noncommutative geometry / Quantum space / Field theory / Matrix theory / Noncommutative sphere / Dbrane / 非可換幾何学 / 量子空間 / 場の理論 / マトリクス理論 / 非可換球面 / 反ドジッター時空 
Research Abstract 
The aim of this research is to construct the field theory over the noncommutative space and to analyze its physical properties. In the last 4 years research supported by the present GrantinAid we have been especially investigating the field theory on the noncommutative sphere as a concrete example of a noncommutative curved space. The noncommutative sphere is defined by quantizing the algebra over the 2sphere. As a first step in this direction we have constructed the noncommutative differential algebra according to the approach given by Connes (paper 1), and as its application we formulated the scalar field on the noncommutative sphere (paper 2). Our main interest is to define the gauge theory on the noncommutative sphere. However, for this end we had to analyze further detailed structure of the differential calculus. In paper 3 we succeeded to formulated the U (1) gauge theory, and we have analyzed its commutative limit. Especially in gauge theory we obtained a new term and the structure of this new term, which does not have a correspondence in the commutative case, gives us a tool to classify the gauge theory on the noncommutative sphere. Recently, it is found that the Dbrane which wraps around the torus possesses a noncommutative structure in the background of an antisymmetric tensor field. It is an interesting problem to investigate the ralation between string theory and noncommutative geometry. From this point of view we investigated the Dbrane in the group manifold and constructed the boundary state in the SU (2) manifold (paper 5). The effective theory of this Dbrane in SU (2) is given by the gauge thoery on the noncommutative sphere described above and we are presently researching those relations.

Report
(6results)
Research Output
(24results)