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Journal of Physics A Mathematical and Theoretical

Heisenberg–Weyl algebra revisited: combinatorics of words and paths

Paweł Błasiak, Gérard Duchamp, A. Horzela, K. A. Penson, A. I. Solomon

Published Sep 18, 2008• 19 citations
Publisher

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

DOI
10.1088/1751-8113/41/41/415204

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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.
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.
This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables. The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word . Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).
Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.
We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 Wess–Zumino–Witten (WZW) model. The Hamiltonian of the WZW model is the invariant XXX quantum spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through a conjectured energy-preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated ( m , n ) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the thermodynamic Bethe ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.
The well-known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg–Weyl group. In particular, it is shown that the Poisson Summation Formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalized coherent state systems of the Heisenberg–Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on the properties of von Neumann and finer lattices based on standard and generalized coherent state systems.