66 research outputs found
More on Rotations as Spin Matrix Polynomials
Any nonsingular function of spin j matrices always reduces to a matrix
polynomial of order 2j. The challenge is to find a convenient form for the
coefficients of the matrix polynomial. The theory of biorthogonal systems is a
useful framework to meet this challenge. Central factorial numbers play a key
role in the theoretical development. Explicit polynomial coefficients for
rotations expressed either as exponentials or as rational Cayley transforms are
considered here. Structural features of the results are discussed and compared,
and large j limits of the coefficients are examined.Comment: Additional references, simplified derivation of Cayley transform
polynomial coefficients, resolvent and exponential related by Laplace
transform. Other minor changes to conform to published version to appear in J
Math Phy
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
Euler Incognito
The nonlinear flow equations discussed recently by Bender and Feinberg are
all reduced to the well-known Euler equation after change of variables.Comment: 2 page
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
Branched Hamiltonians and Supersymmetry
Some examples of branched Hamiltonians are explored both classically and in
the context of quantum mechanics, as recently advocated by Shapere and Wilczek.
These are in fact cases of switchback potentials, albeit in momentum space, as
previously analyzed for quasi-Hamiltonian chaotic dynamical systems in a
classical setting, and as encountered in analogous renormalization group flows
for quantum theories which exhibit RG cycles. A basic two-worlds model, with a
pair of Hamiltonian branches related by supersymmetry, is considered in detail.Comment: Minor changes to conform to published version. PACS: 03.65.Ca,
03.65.Ta, 45.20.J
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
Elementary results for the fundamental representation of SU(3)
A general group element for the fundamental representation of SU(3) is
expressed as a second order polynomial in the hermitian generating matrix H,
with coefficients consisting of elementary trigonometric functions dependent on
the sole invariant det(H), in addition to the group parameter.Comment: In memoriam Yoichiro Nambu (1921-2015
Galileons and Naked Singularities
A simple trace-coupled Galileon model is shown to admit spherically symmetric
static solutions with naked spacetime curvature singularities.Comment: References and acknowledgements added, and corrections made to Figure
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