By Johan G. F. Belinfante

During this reprint version, the nature of the ebook, in particular its concentrate on classical illustration concept and its computational elements, has now not been replaced

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**Example text**

The correspondence principle was used to guess the quantum analogues of classical dynamics in the early days of quantum mechanics. Any such method of assigning quantum analogues to classical dynamical variables may be called a quantization procedure. Starting with the methods proposed in 1926-1927 by P. A. M. Dirac, J. von Neumann and H. Weyl, a large variety of quantization procedures have been studied over the years [210]. One of the most useful of these quantization methods, due to Dirac, makes use of Lie algebraic ideas.

To prove that the various classical matrix groups are Lie groups, we must show that they are analytic manifolds and that in each case the group operation is analytic. Let us consider first the general linear groups GL(n, R) and GL(n, C). To see that they are manifolds, we embed them in a Euclidean space and simultaneously erect coordinate systems by noting that the n2 entries of a matrix can be used as the coordinates of a point in a Euclidean space of dimension n2. For the group GL(n, C), we treat real and imaginary parts of the matrix entries as separate coordinates and so use a real Euclidean space of dimension 2n2 rather than a complex space of dimension n2.

The real affine group is therefore the only connected non-Abelian two-dimensional Lie group. Thus, the only possible connected two-dimensional Lie groups are the plane, the cylinder and the torus. The real Cartesian plane R2 admits two group structures making it a Lie group, one Abelian and one non-Abelian. The torus and the cylinder admit only an Abelian Lie group structure. For higher dimensions, finding all connected Lie groups becomes a much more complicated problem. A part of the solution, finding all the possible real Lie algebras of a given dimension, has been studied [174] , [175] , [176], [177].