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arXiv:0801.1571

Classical and quantum constraints in spin physics

X. Artru

Published Jan 10, 2008• 1 citation
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Abstract

Constraints on spin observables coming from discrete symmetries such as P, C, T and identical particles may be divided in two types: 1) classical ones, which insure the invariance of the cross sections under the symmetry operation; 2) non-classical ones, which can only be obtained at the level of amplitudes. Similarly, positivity constraints can be divided into classical and non-classical constraints. The former insure the positivity of the cross section for arbitrary individual polarisations of the external particles, the latter extend this requirement to the case of entangled external spins. The domain of classical positivity is shown to be dual to the domain of separability

DOI
10.48550/arxiv.0801.1571

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Constraints on spin observables coming from discrete symmetries such as P, C, T and identical particles may be divided in two types: 1) classical ones, which insure the invariance of the cross sections under the symmetry operation; 2) non-classical ones, which can only be obtained at the level of amplitudes. Similarly, positivity constraints can be divided into classical and non-classical constraints. The former insure the positivity of the cross section for arbitrary individual polarisations of the external particles, the latter extend this requirement to the case of entangled external spins. The domain of classical positivity is shown to be dual to the domain of separability
The spin correlations in a reaction involving initial and final polarized particles are encoded in a cross section matrix which is taken as an example of multipartite density matrix. Quantum positivity stipulates that the cross section is positive for any spin states - separable or entangled - of the particles. A weaker condition, classical positivity only requires that the cross section is positive for separable spin state. Classical and Quantum positivity are represented by diagrams. The domains of classical positivity and of separability are dual under a reciprocal polar transformation. This property can be usefull in quantum information theory.
Elementary particles appear in different ways.Particles of the same kind (similar particles) share all state-independent properties like mass, charge, and spin quantum number; there are many similar particles, but they are indistinguishable with respect to a certain class of properties.However, the case might even be stronger: similar particles apparently share all state-dependent properties, like spatial location and spin projection.This threatens the validity of Leibniz's famous Principle of the Identity of Indiscernibles (PII; see Rodriguez-Pereyra (2014) for a historical reconstruction).As it is usually understood, quantum mechanics requires for similar particles to be in permutation invariant states.Take the following state, for example:
In these continuation papers (VI and VII) we are interested in approach the problem of spin from a classical point of view. In this first paper we will show that the spin is neither basically relativistic nor quantum but reflects just a simmetry property related to the Lie algebra to which it is associated. The classical approach will be paraleled with the usual quantum one to stress their formal similarities and epistemological differences. The important problem of Einstein-Bose condensation for fermions will also be addressed.
We extend the concept of classicality in quantum optics to spin states. We call a state ``classical'' if its density matrix can be decomposed as a weighted sum of angular momentum coherent states with positive weights. Classical spin states form a convex set C\mathcal{C}C, which we fully characterize for a spin 1∕21∕21∕2 and a spin 1. For arbitrary spin, we provide ``nonclassicality witnesses.'' For bipartite systems, C\mathcal{C}C forms a subset of all separable states. A state of two spins 1∕21∕21∕2 belongs to C\mathcal{C}C if and only if it is separable, whereas for a spin 1∕21∕21∕2 coupled to a spin 1, there are separable states which do not belong to C\mathcal{C}C. We show that in general the question whether a state is in C\mathcal{C}C can be answered by a linear programming algorithm.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the kkk component of position can represent segments of the kkk axis with length of the order of 1m\frac{1}{m}m1​; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin \textonehalf{}. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin \textonehalf{}. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin \textonehalf{}. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.