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 combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications.