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Sobre correspondências de símbolos para sistemas de spin e de quark

Alcântara, Pedro Antonio Soares de

Published Feb 18, 2022• 0 citations
OpenAlex

Abstract

In this work, we investigate the correspondence principle applied to mechanical systems symmetric under a compact Lie group by means of symbol correspondences. In the first part, we consider systems symmetric under SU(2), the so-called spin systems: we verify conditions under which symbol correspondences present localization of pure states at the semiclassical limit, which provides a new criterion for the emergence of classical dynamics; then, we introduce the notion of sequential quantization from which we get a new characterization of asymptotic localization. In the second part, we begin the study of symbol correspondences for systems symmetric under SU(3), the quark systems: we characterize the symbol correspondences for such systems and obtain some properties and formulas for twisted products induced on the spaces of symbols.

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In this work, we investigate the correspondence principle applied to mechanical systems symmetric under a compact Lie group by means of symbol correspondences. In the first part, we consider systems symmetric under SU(2), the so-called spin systems: we verify conditions under which symbol correspondences present localization of pure states at the semiclassical limit, which provides a new criterion for the emergence of classical dynamics; then, we introduce the notion of sequential quantization from which we get a new characterization of asymptotic localization. In the second part, we begin the study of symbol correspondences for systems symmetric under SU(3), the quark systems: we characterize the symbol correspondences for such systems and obtain some properties and formulas for twisted products induced on the spaces of symbols.
Quantum or classical mechanical systems symmetric under SU(2)SU(2)SU(2) are called spin systems. A SU(2)SU(2)SU(2)-equivariant map from (n+1)(n+1)(n+1)-square matrices to functions on the 222-sphere, satisfying some basic properties, is called a spin-jjj symbol correspondence (n=2j∈Nn=2j\in\mathbb Nn=2j∈N). Given a spin-jjj symbol correspondence, the matrix algebra induces a twisted jjj-algebra of symbols. In this paper, we establish a new, more intuitive criterion for when the Poisson algebra of smooth functions on the 222-sphere emerges asymptotically (n→∞n\to\inftyn→∞) from the sequence of twisted jjj-algebras of symbols. This new, more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [Rios&Straume], is now given in terms of a classical (asymptotic) localization of the symbols of projectors (quantum pure states). For some important kinds of symbol correspondence sequences, classical localization of all projector-symbols is equivalent to asymptotic emergence of the Poisson algebra. But in general, such a classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. Finally, we obtain some relations between asymptotic localization of a symbol correspondence sequence and its quantizations of the classical spin system.
If the state space of a quantum system is finite dimensional it may be considered as the carrier space of an irreducible unitary representation of some compact Lie group G. In this case the set of operators may be put in one-to-one correspondence with a class of functions defined on the group parameters (Q representation). For these functions a binary relation is derived that corresponds to the product of two operators. The general formalism is applied to G=SU(2) (spin coherent states) and this result is used to derive a product formula for the non-compact Heisenberg-Weyl group (coherent oscillator states.).
A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.
Neste terceiro artigo da série, o autor discute a transição da mecânica clássica para a velha teoria quântica de sistemas periódicos introduzindo as regras de quantização e o princípio de correspondência de Bohr.Palavras-chave: velha teoria quântica, regras de quantização, princípio da correspondência.In the third paper of the series, the author discusses the transition from classical mechanics to the old quantum theory of periodic systems by introducing the quantization rules and the Bohr's correspondence principle.
We use the mathematical structure of group algebras and H+H^{+}H+-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2){\rm SU}(2)SU(2) and its quotient SO(3,R){\rm SO}(3,\mathbb{R})SO(3,R). The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of "large" quantum numbers, i.e., "quickly oscillating" functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2){\rm SU}(2)SU(2).
This work studies the semiclassical behaviour of quantum models according to their dynamics. On the first part, we analyse quantum models with well defined classical limit and define a corresponding classical dynamics. We are intend to investigate here how those systems behave at this classical limit and how the quantum dynamics can be approximated. For that we make use of a semiclassical approximation as a systematic way of finding quantum corrections to the classical mean values of observables. We also make use of exact results for testing the semiclassical approximation for calculating typical quantum effects and the fidelity of the approximate state with the exact one. In the second part we deal with the classical limit of quantum mechanics through intercation with environment. The evolution of quantum and classical probability distributions are compared using the Wigner representation. We find then conditions under which the coupling of the system with the environment preserves two initially identical particles as identical and the we analyze the process of decoherence in that system.