Characters and contact transformations

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Abstract
If V is a symplectic space, each affine contact transformation is shown to define an automorphism of a certain algebra of Schwartz functions on V. This automorphism turns out to be a generalized inner automorphism and allows one to associate to the contact transformation a tempered distribution which can be found explicitly. It is shown by means of various examples that these distributions contain much information about the corresponding quantized system. For example the energy levels and stationary states of the harmonic oscillator can be deduced from the general formulae. Generalizations to other locally compact abelian groups V are described. Some connections with the theory of Fourier integral operators and with the characters of certain projective representations are also outlined.

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