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The Weyl functional calculus Cressman, Ross Eric

Abstract

The Weyl functional calculus for a family of n self-adjoint operators acting on a Hilbert space provides a map from spaces of functions on R[sup n] into the set of bounded operators. The calculus is not multiplicative under point-wise multiplication of functions unless the self-adjoint operators commute. However, if the operators happen to generate a strongly continuous unitary representation of a Lie group, we can hope to define a "skew product" on the function spaces under which the calculus is multiplicative. In part I, we show that, for exponential groups, a natural skew product exists by using the exponential map to pull the convolution on the group back to the Lie algebra. Moreover, whenever a skew product is defined in part I, it depends only on the underlying Lie group and not on the particular representation. We then examine when the skew product of two functions is again in the original function space. For compact Lie groups, the theory becomes more complex. A skew product is constucted but by a rather artificial method. The explicit calculations for SU(2) demonstrates the difficulties. In part II, a unique skew product is developed for the position and momentum operators of one dimensional quantum mechanics. The dynamics of quantum mechanics on phase space can be formulated through this skew product whenever the underlying Hamiltonian corresponds to a tempered distribution on the plane. The resulting evolution operator on phase space is shown to be equivalent to the difference of two "singular" integral operators obtained from the usual configuration space formulation. The evolution and configuration operators are then bounded with appropriate domains for the same set of tempered distributions. The skew product on this set of distributions is used to define noncommutative Banach algebras and to determine the multipliers on these spaces. For real, compactly supported distributions, it is shown that the phase space formulation has a unique solution if and only if there is a unique solution on configuration space. On the other hand, we observe that the symmetries of the evolution operator seem to imply that the two formulations of quantum mechanics are not equivalent for all real tempered distributions.

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