Heisenberg-Weyl algebra revisited: Combinatorics of words and paths
P. Blasiak (1), A. Horzela (1), G. H. E. Duchamp (2), K. A. Penson, (3), A. I. Solomon (3), ((1) Polish Academy of Sciences, Krakow, Poland, (2), Institut Galilee, University of Paris, France, (3) LPTMC, University of Paris, VI, France)
TL;DR
This paper explores the Heisenberg-Weyl algebra through combinatorial models involving lattice paths and Ferrers boards, offering new perspectives and methods relevant to quantum theory representations.
Contribution
It introduces a combinatorial framework for the Heisenberg-Weyl algebra using paths and rook problems, providing novel insights and tools for quantum algebra analysis.
Findings
Model of the algebra via lattice paths
Connection between rook problem and normally ordered calculus
New combinatorial methods for quantum algebra
Abstract
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.
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Taxonomy
TopicsDNA and Biological Computing · Advanced Combinatorial Mathematics