Representation of permutation operators in quantum mechanics

Jonathan D. H. Smith

Richard Healey

Vladimir V. Kornyak

Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.

Huangjun Zhu

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

John C. Inkson

Ryan Springall, Manolo C. Per, Ian K. Snook

Proposed is an alternative method for permutational sampling in quantum gases using the path integral formulation of statistical mechanics. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.