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Heisenberg–Weyl algebra revisited: Combinatorics of words and paths
The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications.
Urn models & operator ordering procedures
Ordering of operators is purely combinatorial task involving a number of commutators shuffling components of operator expression to desired form. Here we show how it can be illustrated by simple urn models in which normal ordering procedure is equivalent to enumeration of urn histories.