Discourses in Mathematics and Its Applications, No. 2 1994

Wigner Distribution Functions

F. J. Narcowich and S. A. Fulling

Contents

PART ONE (by F. J. Narcowich) 1

I. Motivation and background 1 Classical statistical mechanics . . . . . . . . . . . . . . . . . . . 1 Quantum statistical mechanics . . . . . . . . . . . . . . . . . . . 3

II. The phase-space formulation of quantum mechanics 4 The Wigner distribution function . . . . . . . . . . . . . . . . . . 4 The Weyl transform . . . . . . . . . . . . . . . . . . . . . . . 6 Twisted products . . . . . . . . . . . . . . . . . . . . . . . . . 9 An asymptotic expansion for the twisted product . . . . . . . . . . . 12 The canonical commutation relations . . . . . . . . . . . . . . . . 13 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . 13 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

III. Dynamics 17 Time evolution of distribution functions . . . . . . . . . . . . . . . 17 Expansion in powers of h̄ . . . . . . . . . . . . . . . . . . . . . 17 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

IV. Closing remarks 19 Characterization of states in the phase-space formulation . . . . . . . 19 Problems involving constraints . . . . . . . . . . . . . . . . . . . 20 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Miscellaneous problems . . . . . . . . . . . . . . . . . . . . . . 20 Acknowledgments and further references . . . . . . . . . . . . . . 21

PART TWO (by S. A. Fulling) 22

V. Boundaries and cutoffs 22 VI. Gauge fields and parallel transport 29

REFERENCES 39 Supplementary references . . . . . . . . . . . . . . . . . . . . . 41

INDEX 42