50 research outputs found
Deformation Quantization of Nambu Mechanics
Phase Space is the framework best suited for quantizing superintegrable
systems--systems with more conserved quantities than degrees of freedom. In
this quantization method, the symmetry algebras of the hamiltonian invariants
are preserved most naturally, as illustrated on nonlinear -models,
specifically for Chiral Models and de Sitter -spheres. Classically, the
dynamics of superintegrable models such as these is automatically also
described by Nambu Brackets involving the extra symmetry invariants of them.
The phase-space quantization worked out then leads to the quantization of the
corresponding Nambu Brackets, validating Nambu's original proposal, despite
excessive fears of inconsistency which have arisen over the years. This is a
pedagogical talk based on hep-th/0205063 and hep-th/0212267, stressing points
of interpretation and care needed in appreciating the consistency of Quantum
Nambu Brackets in phase space. For a parallel discussion in Hilbert space, see
T Curtright's contribution in these Proceedings [hep-th 0303088].Comment: Invited talk by the first author at the Coral Gables Conference
(C02/12/11.2), Ft Lauderdale, Dec 2002. 14p, LateX2e, aipproc, amsfont
Umbral Vade Mecum
In recent years the umbral calculus has emerged from the shadows to provide
an elegant correspondence framework that automatically gives systematic
solutions of ubiquitous difference equations --- discretized versions of the
differential cornerstones appearing in most areas of physics and engineering
--- as maps of well-known continuous functions. This correspondence deftly
sidesteps the use of more traditional methods to solve these difference
equations. The umbral framework is discussed and illustrated here, with special
attention given to umbral counterparts of the Airy, Kummer, and Whittaker
equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de
Vries, and Toda systems.Comment: arXiv admin note: text overlap with arXiv:0710.230
Quantum Mechanics in Phase Space
Ever since Werner Heisenberg's 1927 paper on uncertainty, there has been
considerable hesitancy in simultaneously considering positions and momenta in
quantum contexts, since these are incompatible observables. But this persistent
discomfort with addressing positions and momenta jointly in the quantum world
is not really warranted, as was first fully appreciated by Hilbrand Groenewold
and Jos\'e Moyal in the 1940s. While the formalism for quantum mechanics in
phase space was wholly cast at that time, it was not completely understood nor
widely known --- much less generally accepted --- until the late 20th century.Comment: A brief history of deformation quantization, ca 1930-1960, with some
elementary illustrations of the theor