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Deformation of Symmetries |
It turns out that, at least in the case of non-dynamical systems (such as algebraic hyper-surfaces and statistical Hamiltonians): 1) The deformed algebra always (for any value of deformational parameter) has an equal number of generators as does initial undeformed algebra. 2) In the case of an algebraic hyper-surface, if the topology of the underlying object (that hyper-surface) is not changed, then the deformed algebra (symmetry algebra of the deformed object) is isomorphic to the algebra of the initial, undeformed object. 3) Explicit expressions for deformed generators exist. 4) Part of the initial symmetry that is considered broken in the standard terminology can be represented in a nonlinear fashion. A paper on these results is in preparation.
In the application of these results to statistical Hamiltonians, some interesting points are worth mentioning: If the initial Hamiltonian $H_0$ has a certain continuous symmetry (with the algebra $g_0$), then on adding an additional term $t H_1$ the symmetry algebra of new Hamiltonian is s deformation $g_t$ of $g_0$; but, as in case of coupled spin chains, the majority (all that are not linear symmetries of total Hamiltonian) of deformed generators become nonlinear and (in the continuous limit) nonlocal.
My goal is to understand equivalent phenomena arising in case of dynamical systems (ODEs, PDEs, classical field theories), and their change upon quantization. Results that I have obtained for such systems are still not transparent, and are not in the form suitable for publication.
Updated on June 24th 1997
http://www.het.brown.edu/people/stojkov/research/def/def.index.html