By Victor Guillemin
Multiplicity diagrams may be seen as schemes for describing the phenomenon of "symmetry breaking" in quantum physics. the topic of this publication is the multiplicity diagrams linked to the classical teams U(n), O(n), and so on. It offers such issues as asymptotic distributions of multiplicities, hierarchical styles in multiplicity diagrams, lacunae, and the multiplicity diagrams of the rank 2 and rank three teams. The authors take a singular method, utilizing the recommendations of symplectic geometry. The e-book develops intimately a few topics which have been touched on within the hugely winning Symplectic innovations in Physics by means of V. Guillemin and S. Sternberg (CUP, 1984) , together with the geometry of the instant map, the Duistermaat-Heckman theorem, the interaction among coadjoint orbits and illustration idea, and quantization. scholars and researchers in geometry and mathematical physics will locate this booklet interesting.
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Additional resources for Symplectic Fibrations and Multiplicity Diagrams
Example text
Although we know from the inverse function theorem that the map S is one-to-one on some neighborhood of the zero section of the bundle G X G, n°, we don't know whether this neighborhood can be chosen to be G-invariant. On the other hand there exists a G-invariant neighborhood of the zero section consisting of the points at which the Jacobian of E is bijective. Indeed, since the map E is G-equivariant and since it is an embedding on each fiber, a point [g, v] is a regular point of the map if and only if the orbit through the image S([g, v]) is transverse to the image of the fiber g (A + n°).
4 (base change) Let,-r: X -+ B be a symplectic fibration, let B1 be a manifold, and let f: BI B be a smooth map. 10) B If r is a connection on X we can pull it back to get a connection r, on X, which will be symplectic if r is. Furthermore the forms pull back consistently to give wr, = g*wr. 4 let us take B1 = T*B. Let wB be the standard symplectic form on T* B. 9) to obtain the form wr, + rri WB. 12) This is clearly closed and r l -compatible. 12) is always symplectic. 12) is used in physics to adjoin "internal variables" to a classical dynamical system.
In other words, in the weak coupling limit there is a unique symplectic structure on M compatible with the symplectic structure on fiber and base. 2 Examples of Symplectic Fibrations: The Coadjoint Orbit Hierarchy As we pointed out in the introduction the purpose of this monograph is to explore connections between two subjects which seem to have, on the face of it, little to do with each other: symplectic fibrations and the multiplicity diagrams associated with representations of Lie groups. What will supply the bridge between these two topics is the coadjoint orbit hierarchy.