The Wigner Kernel of a Particle obtained from the Wigner Kernel of a Spin by Group Theoreti.pdf

a r X i v : q u a n t - p h / 0 0 1 1 0 2 9 v 1 8 N o v 2 0 0 0 The Wigner Kernel of a Particle obtained from the Wigner Ker- nel of a Spin by Group Theoretical Contraction? J.-P. Amiet and St. Weigert Institut de Physique, Universite? de Neucha?tel Rue A.-L. Breguet 1, CH-2000 Neucha?tel, Switzerland Outline The Moyal formalism for a particle can be derived from the Moyal formalism for a spin. This is done by contracting the group of rotations to the oscillator group. A new derivation is given for the contraction of the spin Wigner-kernel to the Wigner kernel of a particle. Introduction A symbolic calculus is a one-to-one correspondence between (self-adjoint) operators A? acting on a Hilbert space H of a quantum system, and (real) functions WA defined on the phase-space Γ of the corresponding classical system (see [1] for a summary). Represen- tating quantum mechanics in terms of c-number valued functions has various appealing properties since it allows one to situate the quantum mechanical description of a system in a familiar frame. The visualisation of quantum states and operators in classical phase space helps to develop an intuitive understanding of quantum features. Furthermore, it is interesting from a structural point of view: to calculate expectation values of operators by means of ‘quasi-probabilities’ in phase space is strongly analogous to the determination of mean values in classical statistical mechanics [2]. The quantum mechanics of spin and particle systems can be represented faithfully in terms of functions defined on the surface of a sphere with radius s, and on a plane, respectively. Intuitively, one expects these phase space-formulations to approach each other for increasing values of the spin quantum number since the surface of a sphere is then approximated by a plane with increasing accuracy. Therefore, appropriate Wigner functions of a spin, say, should go over smoothly into particle Wigner-functions in the limit of large s. Two differen