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Particles and Quantum Symmetries

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

A system of interacting particles equipped with quantum symmetry is described in an abstract algebraic way. The concept of quantum commutativity is used for description of the algebra of quantum states of the system. The graded commutativity and particles in singular magnetic field is considered as an example. The generalized Pauli exclusion principle is also mentioned. 1.

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•Jan 1, 2016•4 citations
A system of interacting particles equipped with quantum symmetry is described in an abstract algebraic way. The concept of quantum commutativity is used for description of the algebra of quantum states of the system. The graded commutativity and particles in singular magnetic field is considered as an example. The generalized Pauli exclusion principle is also mentioned. 1.
The connection between the exclusion principle and the principle of indistinguishability of identical particles in quantum mechanics is discussed. It is shown with the aid of a density matrix defined for states with arbitrary permutation symmetry that the particles are indistinguishable only for one-dimensional representations of the permutation group (symmetrical and antisymmetrycal), and are distinguishable in all the states that are degenerate in the permutations. It is demonstrated by the same token that the exclusion principle follows from the principle of indistinguishability of identical particles and the Pauli theorem on the connection between spin and statistics.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
Since typically the physics of many body quantum systems is solely described and determined by pairwise interactions, the concept of reduced particle information is fundamentally relevant, particularly for indistinguishable particles. Determining the compatibility of r-particle reduced density operators (r-RDOs) with some N-particle state is known as the N-representability problem. The Pauli exclusion principle provides a set of necessary and sufficient conditions for the N-representability of fermionic 1-RDOs. However, pure N-representability requires a set of stricter conditions which are known as Generalised Pauli constraints (GPCs). In this thesis we investigate the influence and significance of the GPCs beyond the well-explored and established relevance of the Pauli exclusion principle. In particular, we analytically study the (quasi-)pinning effect which corresponds to a saturation of the GPCs and imposes structural simplifications on the N-fermion state. So far, nothing is known about its origin and mechanisms. We will close this gap by analytically examining the dependence of quasipinning on the particle number, the dimensionality and the spin polarisation in a representative physical model system. Our results suggest the notion of a microscopic Pauli pressure which is a consequence of the competition between energy minimisation and antisymmetry, and drives the system towards the boundary of the allowed region. Furthermore, the relevance of quasipinning beyond the wellknown Pauli exclusion principle is evaluted by the Q-measure. The outcomes provide first direct evidence for the need to incorporate GPCs into the physical description. To explore the role of the fermionic exchange symmetry beyond the Pauli exclusion principle from a more general perspective, we consider systems of hardcore bosons. Indeed, they exhibit a (lattice-based) exclusion principle, although their exchange symmetry is bosonic. Occupation numbers on oneparticle states other than the lattice basis can exceed one. We prove that the maximal occupation number is given by λmax = N/d(d - N + 1) where N, d are the numbers of particles and lattice sites. In addition, we derive necessary conditions for the N-representability of hardcore boson 1-RDOs and investigate the one-particle entropy boundaries. Finally, we present one-dimensional physical systems which macroscopically saturate λmax and exhibit (fractional) Bose-Einstein condensation.