Spin Multiplicities
The number of times spin s appears in the Kronecker product of n spin j representations is computed, and the large n asymptotic behavior of the result is obtained. Applications are briefly sketched.
hep-th↗
arXiv subjects
Cosmas Zachos
Publications and source records attributed to Cosmas Zachos.
Approximate Solutions of Functional Equations
Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by construction of approximate continuous functional iterates for x/(1-x), sin x, and λx(1-x). Simple functional conjugation by these functions, and their inverses, substantially improves the numerical accuracy of formal series approximations for their continuous iterates.
math-ph↗
Evolution profiles and functional equations
Time evolution is formulated and discussed in the framework of Schroeder's functional equation. The proposed method yields smooth, continuous dynamics without the prior need for local propagation equations.
math-ph↗
Classical and Quantal Ternary Algebras
We consider several ternary algebras relevant to physics. We compare and contrast the quantal versions of the algebras, as realized through associative products of operators, with their classical counterparts, as realized through classical Nambu brackets. In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal algebra, and then explicitly realized through classical brackets. We illustrate this classical-contraction method by the Virasoro-Witt example.
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Vertex Ring-Indexed Lie Algebras
Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers.
hep-th↗
Bounding Gauged Skyrmion Masses
Normally, standard (ungauged) skyrmion masses are proportional to the coupling of the Skyrme term needed for stability, and so can grow to infinite magnitude with increasing coupling. In striking contrast, when skyrmions are gauged, their masses are bounded above for any Skyrme coupling, and, instead, are of the order of monopole masses, O(v/g), so that the coupling of the Skyrme term is not very important. This boundedness phenomenon and its implications are investigated.
hep-th↗
Branes, Quantum Nambu Brackets, and the Hydrogen Atom
The Nambu Bracket quantization of the Hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu Brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the Quantum Nambu Brackets comprise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the Hydrogen atom, the results coincide with those furnished by Hamiltonian quantization--but the method is not limited to Hamiltonian systems.
math-ph↗
Semileptonic B Decays at BABAR
We present results on semileptonic B decays obtained with the BABAR detector. The large data set accumulated at the PEP-II asymmetric-energy B-Factory allows a new measurement technique, where the hadronic decay of one B meson is fully reconstructed and the semileptonic decay of the recoiling B meson is studied. Traditional analysis techniques of inclusive and exclusive B decays complement this approach with very high statistics data samples. These measurements play an important role in the tests of the description of CP violation in the Standard Model: The determinations of the Cabibbo-Kobayashi-Maskawa matrix elements |Vcb| and |Vub| provide constraints on the unitarity of the CKM triangle. Furthermore, the experimental measurement of parameters of Heavy Quark Effective Theory test the consistency of the theoretical description of semileptonic B decays.
hep-ex↗
Membranes and Consistent Quantization of Nambu Dynamics
The dynamics of even topological open membranes relies on Nambu Brackets. Consequently, such 2p-branes can be quantized through the consistent quantization of the underlying Nambu dynamical structures. This is a summary construction relying on the methods detailed in refs hep-th/0205063 and hep-th/0212267.
hep-th↗
Classical and Quantum Nambu Mechanics
The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtained by allowing for different time scales on different invariant sectors of a theory, the resulting non-Abelian approach to quantum Nambu mechanics is shown to be fully consistent.
hep-th↗
Quantizing Dirac and Nambu Brackets
We relate classical and quantum Dirac and Nambu brackets. At the classical level, we use the relations between the two brackets to gain some insight into the Jacobi identity for Dirac brackets, among other things. At the quantum level, we suggest that the Nambu bracket is the preferred method for introducing constraints, although at the expense of some unorthodox behavior, which we describe in detail.
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Negative Probability and Uncertainty Relations
A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p.
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Generating All Wigner Functions
In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
hep-th↗
A Survey of Star Product Geometry
A brief pedagogical survey of the star product is provided, through Groenewold's original construction based on the Weyl correspondence. It is then illustrated how simple Landau orbits in a constant magnetic field, through their Dirac Brackets, define a noncommutative structure since these brackets exponentiate to a star product---a circumstance rarely operative for generic Dirac Brackets. The geometric picture of the star product based on its Fourier representation kernel is utilized in the evaluation of chains of star products. The intuitive appreciation of their associativity and symmetries is thereby enhanced. This construction is compared and contrasted with the remarkable phase-space polygon construction of Almeida.
hep-th↗
Geometrical Evaluation of Star Products
The geometric picture of the star-product based on its Fourier representation kernel is utilized in the evaluation of chains of star-products and the intuitive appreciation of their associativity and symmetries. Such constructions appear even simpler for a variant asymmetric product, and carry through for the standard star-product supersymmetrization.
hep-th↗
Phase-space Quantization of Field Theory
In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple - indeed, classical - for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published in J Phys A32 (1999) 771 and Phys Rev D58 (1998) 025002, reported at the Yukawa Institute Workshop "Gauge Theory and Integrable Models", 26-29 January, 1999.
hep-th↗
Wigner Trajectory Characteristics in Phase Space and Field Theory
Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.
hep-th↗
Features of Time-independent Wigner Functions
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional ("star") eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ("supersymmetric") isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Poeschl-Teller potential, and the Liouville potential.
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