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linear Representations of a Group

Fl. Stancu

Published Nov 21, 1996• 0 citations
Publisher

Abstract

Abstract In quantum mechanics we are interested in the properties of eigenstates under various transformations (Chapter 1 ). Group theory offers a systematic way of finding these properties from the symmetries of the Hamiltonian. The eigenstates form linear vector spaces which in terms of group theory provide matrix representations of the group of transformationsG under consideration. Let us denote byD(g) such a representation where g is an element ofG.

DOI
10.1093/oso/9780198517429.003.0003

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It is instructive to note that the principal advantage of linear operators is that their action on any vector is defined purely by their action on a set of basis vectors. Let us consider the linear operator F ̂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqni2_1.jpg"/> transforming the vector a via its components a k : 2.1 F ̂ a i = ∑ k F i k a k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn2_1.jpg"/>
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The current form of quantum mechanics is very successful and is almost certainly correct. It is remarkable, however, that the entire structure-from the mass, spin and charge labels on particlelike states to antisymmetry to broken internal symmetries to gauge transformations to the equations of motion-is built upon concepts from group representation theory. That is, the theory is constructed exactly as if it were a representational form of an underlying theory. Our proposed form for the underlying theory is that it is based on a linear equation, OF(V)=0. F is a function of some set of independent, currently unknown variables V, with O being a linear, partial differential operator in those variables. The operator is assumed to be invariant under a group of transformations of the Vs, homomorphic to the direct product of the inhomogeneous Lorentz group and the internal symmetry group. In such a theory, a state vector, denoted by a ket with group-theoretic labels, would represent a function of the independent variables. In addition to explaining the group representational structure of quantum mechanics, an underlying theory offers insight into gauge theory.