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Classical mechanics on Grassmannian and disc. (English) Zbl 1062.37051

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 181-207 (2001).
Summary: We discuss from a purely geometric point of view classical mechanics on certain type of Grassmannians and discs. We briefly discuss a superversion which in some sense combines these two models, and corresponds to the large-\(N_c\) limit of \(\text{SU} (N_c)\) gauge theory with fermionic and bosonic matter fields, both in the fundamental representation, in \(1+1\) dimensions [see the first and third author, J. Math. Phys. 43, 2988–3010 (2002; Zbl 1059.81124)].
This result is a natural extension of ideas due to the second author [Int. J. Mod. Phys. A 9, 5583–5624 (1994; Zbl 0985.81721)]. There it has been shown that the large-\(N_c\) phase space of \(1+1\) dimensional QCD is given by an infinite-dimensional Grassmannian. The complex scalar field version of this theory is worked out in [the second and the third author, Commun. Math. Phys. 192, 493–517 (1998; Zbl 0937.37039)] and it is shown that the phase space is an infinite-dimensional disc.
For the entire collection see [Zbl 0957.00038].

MSC:

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
70H05 Hamilton’s equations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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