Invariant theory

Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebie-te, second edition, Springer-Verlag, New York Heidelberg Berlin 1982. 

First we need some generalities on the relationship between the geometric invariant theory and the moment map. General reference of this section is [60, Chapter 8]. (See also [71] for reference to the geometric invariant theory.)... 

This function plays a fundamental role in the relationship between the moment map and the geometric invariant theory as seen in the following proposition. 

D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Third Enlarged Edition ,... 

Methods for treating relativistic effects in molecular quantum mechanics have always seemed to me, if I may say so without appearing too impertinent to those who work in the field, a complete dog s breakfast. The difficulty is to know to what question they are supposed to be the answer, in the circumstances in which we find ourselves. We do not know what a relativistically invariant theory applicable to molecular behaviour might look like. As was pointed out to us at the last meeting, the Dirac equation certainly will not do to describe interacting electrons and even at the single particle level, where it seems to work, there is an inconsistency in interpreting its solutions in terms... 

When X = C2, X can be identified with the set of GLJl(C)-orbits of (Hi, B2, i) where Bi, B2 are commuting n x n-matrices and i is a cyclic vector (Theorem 1.14). Many properties of (C2) are derived from this description. In Chapter 3, we shall regard the description as a geometric invariant theory quotient and a hyper-Kahler quotient. This description is very similar to the definition of quiver varieties which were studied in [62]. 

Technical Appendix A Criticisms of the U(l) Invariant Theory of the Aharonov-Bohm Effect and Advantages of an 0(3) Invariant Theory... 

There has been an unusual amount of debate concerning the development of 0(3) electrodynamics, over a period of 7 years. When the 2 (3) field was first proposed [48], it was not realized that it was part of an 0(3) electrodynamics homomorphic with Barrett s SU(2) invariant electrodynamics [50] and therefore had a solid basis in gauge theory. The first debate published [70,79] was between Barron and Evans. The former proposed that B,3> violates C and CPT symmetry. This incorrect assertion was adequately answered by Evans at the time, but it is now clear that if B<3) violated C and CPT, so would classical gauge theory, a reduction to absurdity. For example, Barrett s SU(2) invariant theory [50] would violate C... 

Therefore, it has been shown convincingly that electrodynamics is an 0(3) invariant theory, and so the 0(3) gauge invariance must also be found in experiments with matter waves, such as matter waves from electrons, in which there is no electromagnetic potential. One such experiment is the Sagnac effect with electrons, which was reviewed in Ref. 44, and another is Young interferometry with electron waves. For both experiments, Eq. (584) becomes... 

In a U(l) invariant theory, the pure gauge vacuum is defined by a scalar internal gauge space in which there exist the independent complex scalar fields ... 

TECHNICAL APPENDIX A CRITICISMS OF THE U(l) INVARIANT THEORY OF THE AHARONOV-BOHM EFFECT AND ADVANTAGES OF AN 0(3)... 

In this appendix, the U(l) invariant theory of the Aharonov-Bohm effect [46] is shown to be self-inconsistent. The theory is usually described in terms of a holonomy consisting of parallel transport around a closed loop assuming values in the Abelian Lie group U(l) [50] conventionally ascribed to electromagnetism. In this appendix, the U(l) invariant theory of the Aharonov-Bohm effect is... 

It is well known that the change in phase difference of two electron beams in the Aharonov-Bohm effect is described in the conventional U(l) invariant theory by... 

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