1994 Fiscal Year Final Research Report Summary
Gauge Field Theory on Quantum Groups
| Project/Area Number |
04804014
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
核・宇宙線・素粒子
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| Research Institution | Toyama University |
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Principal Investigator |
HIRAYAMA Minoru Toyama University, Department of Physics, Professor, 理学部, 教授 (80018986)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAKOSHI Hitoshi Toyama National College of Technology, Lecturer, 講師 (70249770)
HOSONO Shinobu Toyama University, Department of Mathematics, Associate Professor, 理学部, 助教授 (60212198)
HAMAMOTO Shinji Toyama University, Department of Physics, Associate Professor, 理学部, 助教授 (80018994)
MATUMOTO Keniti Toyama University, Department of Physics, Professor, 理学部, 教授 (90019456)
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| Project Period (FY) |
1992 – 1994
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| Keywords | Quantum group / Gauge field |
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| Research Abstract |
Although we have tried to formulate the quantum field theory which possesses a quantum-group gauge symmetry, we have not succeeded in it until now.
On the other hand, since the discovery of the geometric phase by Berry, it has been clarified that some quantum mechanical and quantum field theoretic models possess gauge-theoretic structures. Berry found it in the case of quantum mechanics with adiabatically changing environment. The similar structure exists, however, in more general systems where a fast mode of motion and a slow one coexist. In the case of Berry's phase factor, the gauge potential A is given by a parameter-dependent state vector and its time-derivative. The field strength F constructed from A can also be described as the imaginary part of a gauge-invariant complex quantity T.The real part G of T has not been investigated so intensively. It can be interpreted, however, as the metric of the space of parameter-dependent state vectors (projective Hilbert space). By making use of this fact, Anandan and Aharonov succeeded in deriving a new type of uncertainly relation.
By the research in this year, we generalized the above-mentioned relation. The Grassmann manifold is a natural generalization of the projective Hilbert space and can be regarded as the space of sets of some orthonormal state vectors. Obtaining the distance formula for the Grassmann manifold, we succeeded in deriving the time-energy uncertainty relation satisfied by a set of orthonormal vectors.
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