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Journal of the Physical Society of Japan

Covariant Exchange Algebras and Quantum Groups. I

Masao Nomura

Published May 15, 1992• 4 citations
Publisher

Abstract

Co-variant quantum group algebras, exploited as q-tensor formalism in previous papers, are rewritten in viewpoint of exchange algebras which obey co- and contravariant transformations. The formalism emphasizes unified treatment of transformation matrices, constituting quantum matrix groups, and basis functions.
DOI
10.1143/jpsj.61.1485

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Quantum matrix algebras s u q (2) are realized in terms of single-state boson operators and their conjugates, ( a + , a ), ( b + , b ), etc., specified as quantum analogs of symplectons. The boson operators obey the linear transformation of S U q (2) such as a ′ + = a + x + a v and a ′ = a + u + a y . Here, x , u , v , y are non-commutative elements which commute with a + and a . Commutation relations for creation and annihilation operators are determined so as to be invariant under the linear transformation. Exchange algebras with the universal R j ′ j ′′ matrix are realized in terms of P ( j ′ , m ′ ; b ), P ( j ′′ , m ′′ ; a ), etc., where P ( j ′ , m ′ ; b ) is the j z ′ = m ′ component of rank- j ′ tensor written as an order-2 j ′ polynomial in b + and b .
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world.
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3, 2)O(3,\,2)O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein's Lorentz-covariant world.
Abstract Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.