Discrete Mathematics & Theoretical Computer Science, Vol 12, No 2 (2010)

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Ladder Operators and Endomorphisms in Combinatorial Physics

Gerard H. E. Duchamp, Laurent Poinsot, Allan I. Solomon, Karol A Penson, Pawel Blasiak, Andrzej Horzela


Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we first show how the ordering of the non-commuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but row-finite, matrices, which may also be considered as endomorphisms of ℂ⟦x⟧. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladder-operators familiar in physics.

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