Volume 39, Issue 3, March 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Quantization of diffeomorphisminvariant theories with fermions
View Description Hide DescriptionWe extend ideas developed for the loop representation of quantum gravity to diffeomorphisminvariant gauge theories coupled to fermions. Let be a principal bundle over space and let be a vector bundle associated to whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group each representation appearing together with its dual. We consider theories whose classical configuration space is where is the space of connections on and is the space of sections of regarded as a collection of Grassmannvalued fermionic fields. We construct the “quantum configuration space” as a completion of Using this, we construct a Hilbert space for the quantum theory on which all automorphisms of act as unitary operators, and determine an explicit “spin network basis” of the subspace consisting of gaugeinvariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on We also construct a Hilbert space of diffeomorphisminvariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.

Onset of superconductivity in decreasing fields for general domains
View Description Hide DescriptionGinzburg–Landau theory has provided an effective method for understanding the onset of superconductivity in the presence of an external magnetic field. In this paper we examine the instability of the normal state to superconductivity with decreasing magnetic field for a closed smooth cylindrical region of arbitrary crosssection subject to a vertical magnetic field. We examine the problem asymptotically in the boundary layer limit (i.e., when the Ginzburg–Landau parameter, is large). We demonstrate that instability first occurs in a region exponentially localized near the point of maximum curvature on the boundary. The transition occurs at a value of the magnetic field associated with the halfplane at leading order, with a small positive correction due to the curvature (which agrees with the transition problem for the disc), and a smaller correction due to the second derivative of the curvature at the maximum.

Hidden algebras of the (super) Calogero and Sutherland models
View Description Hide DescriptionWe propose to parametrize the configuration space of onedimensional quantum systems of identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the and Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed—for arbitrary values of the coupling constants—as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra or the Lie superalgebra for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by Turbiner, and implies that the Calogero and Jack–Sutherland polynomials, as well as their supersymmetric generalizations, are related to finitedimensional irreducible representations of the Lie algebra and the Lie superalgebra .

BRST symmetries for the tangent gauge group
View Description Hide DescriptionFor any principal bundle one can consider the subspace of the space of connections on its tangent bundle given by the tangent bundle of the space of connections on The tangent gauge group acts freely on . Appropriate BRST operators are introduced for quantum field theories that include as fields elements of , as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the socalled Yang–Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological theories is also included and the relevant Batalin–Vilkovisky operator is described.

Discrete time adiabatic theorems for quantum mechanical systems
View Description Hide DescriptionThe theory of adiabatic asymptotics is adapted to systems with discrete time evolution. The corresponding theorems about the approximation of physical time evolution by the adiabatic time evolution are shown to hold true in a discrete setting.

Microscopic interpretation of a two triaxial rotor model
View Description Hide DescriptionA description of the relative motion of neutrons and protons in strongly deformed eveneven nuclei is reduced to that of two coupled triaxial rotors, one representing the neutrons and the other the protons, by using generating function methods. The two rotor theory is interpreted as a generalization of the Elliott SU(3) model with a neutron–proton direct product substructure. The second order Casimir operator for this system is constructed and some of its eigenfunctions are found in explicit form in the Fock–Bargmann space representation. Analytical results for isovector transitions probabilities are obtained and the results compared with experimental data and pseudoSU(3) calculations. Graphs picturing the relative orientation of neutronproton rotors in are presented.

Generalized annihilation operator coherent states
View Description Hide DescriptionA set of generalized coherent states as eigenstates of the annihilation operator is proposed. These states are analytic functions of a complex variable and admit a resolution of identity with positive measure. Guided by the classical action angle transformation and the correspondence principle a formalism is developed for the construction of the annihilation operator for a given Hamiltonian.

Radon–Nikodym derivatives of quantum instruments
View Description Hide DescriptionA convenient representation for Radon–Nikodym derivatives of completely positive (c.p.) instruments on with respect to a scalar measure is suggested, similar to the Stinespring–Kraus representation for c.p. maps, but involving possibly nonclosable unbounded operators. The structure of covariant c.p. instruments is studied in detail. In particular, an exhaustive description is given to instruments covariant with respect to shifts or rotations, corresponding to “position” or “angle” measurements.

adic stochastic hidden variable model
View Description Hide DescriptionWe propose stochastic hidden variables model in which hidden variables have a adic probability distribution and at the same time conditional probabilistic distributions are ordinary probabilities defined on the basis of the Kolmogorov measuretheoretical axiomatics. A frequency definition of adic probability is quite similar to the ordinary frequency definition of probability. adic frequency probability is defined as the limit of relative frequencies but in the adic metric. We study a model with adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell’s inequality is violated.

Physical vertex operators in the Ramond sector
View Description Hide DescriptionIntroducing physical string coordinate’s modes as well as physical string coordino’s modes, we derive the generating functional of physical vertex operators (GFPVO) of fermionic particles (in superstring), which is just equal to the one already proposed by using superconformal mapping in the Ramond sector. Inversely solving the superconformal mapping problem, we can derive general formulas, which give physical vertex operators of various fermionic particles up to arbitrarily excited mass levels. As an example, we explicitly derive some (Gliozzi–Scherk–Oliveallowed) lowlying physical vertex operators.

On topology and quantum mechanics
View Description Hide DescriptionThe relationship between topology and quantum mechanics is considered in two different ways. We suggest regarding quantum mechanics as a firstorder approximation to topologytheory on a fixed set. In this spirit, the approach of quantizing the lattice of topologies on a set [C. J. Isham, “Quantum topology and quantization on the lattice of topologies,” Class. Quantum. Grav. 6, 1509–1534 (1989)] is shown to be easily applicable to a quantum mechanical lattice. In the main part of the paper we deal with the question of the quantization of the whole category of topological spaces and continuous injective maps. Ideas of the modern categorical approach to quantization, as known from topological quantum field theory, are invoked in this part. The final paragraphs are devoted to the problem of extending quantization, even to the level of the underlying set theory.

Similarity and intertwining of Dirac operators
View Description Hide DescriptionSimilarity and symmetry of quantum mechanical operators are the basic theoretical means for systematizing quantum dynamics. The classical counterparts of these concepts are canonical transformations and conservation laws. For operators, a concept called intertwining encompasses both similarity and symmetry. In this paper, we construct pairs of Dirac operators with noncentral potentials that are intertwined by multiplication operators.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Radiation principle for inhomogeneous media
View Description Hide DescriptionThe classical Sommerfeld radiation principle, well known for stationary oscillations in homogeneous media, is extended to a more general case where a smooth density depends both on and on angular coordinates of the point We assume that the density behaves like a power at infinity, and, moreover, that there exists a oneparameter family of closed asymptotic wave fronts. A Sommerfeld–Rellichtype radiation condition is obtained and a corresponding theorem of unique solvability of the stationary oscillation problem is proved under some conditions imposed on the asymptotic phase. Some examples are given which include the case of elliptic wave fronts.

On the electrodynamic group: The relativistic radiation reaction force
View Description Hide DescriptionThe relativistic force exerted by a classical electromagnetic field on an electron is examined from the point of view that associates the differential geometry of the space–time trajectory with a continuous group of transformations. The group relevant to this particular case is the group generated by the electromagnetic field tensor (electrodynamic group). This group is studied in some detail by treating separately the two parts of the Liénard–Wiechert expression for the tensor. The intrinsic differential equations of the space–time trajectories associated with the electrodynamic group are obtained by the Cartan method of the moving frame. These equations show that the curvature of the trajectory is associated only with the generalized Coulomb field and the hypertorsion with the radiation field. According to these equations, the relativistic force may have two components along orthogonal spacelike directions: The first, an apparent force, reduces to the Coulomb force in the nonrelativistic limit. The other is orthogonal to the fourmomentum and to the apparent force; this is the radiation reaction force. In the nonrelativistic limit it manifests itself as a frictional force reducing the momentum. The approach also offers a point of view of the conventional treatment of radiation reaction in which the nonlocality in time emerges from the geometry of the problem. The implications pertaining to other areas of physics of the causal connection found between the hypertorsion and the radiation field are briefly considered.

Deriving the Hamilton equations of motion for a nonconservative system using a variational principle
View Description Hide DescriptionThe classical derivation of the canonical transformation theory [H. Goldstein, Classical Mechanics, 2nd ed. (Addison–Wesley, Reading, 1981)] is based on Hamilton’s principle which is only valid for conservative systems. This paper avoids this principle by using an approach that is basically reversed compared to the classical derivation. The Lagrange equations of motion are formulated in the undefined and general variable set and the general Hamilton equations of motion are derived from the Lagrange equations by using a variational principle. The undefined general variables are defined through a transformation to a special (defined) variable set The transformation equations connecting the two sets are derived by using the invariants property of the value of the Lagrangian. This approach results in a more general interpretation of the generator function.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


A classical large hierarchical vector model in three dimensions: A nonzero fixed point and canonical decay of correlation functions
View Description Hide DescriptionWe consider a hierarchical component classical vector model on a threedimensional lattice for large The model differs from the usual one in that the kernel of the inverse Laplace operator is nontranslational invariant but has matrix elements which are positive and exhibit the same falloff as the inverse Laplacian in We introduce a renormalization group transformation and for corresponding to the leading order of the expansion, we construct explicitly a nonzero fixed point for this transformation and also obtain some correlation functions. The twopoint function has canonical decay. For we obtain the fixed point and the twopoint function in the first approximation. Canonical decay is still verified, in contrast to what is reported for the full model.

Theory of nonequilibrium firstorder phase transitions for stochastic dynamics
View Description Hide DescriptionA dynamic definition of a firstorder phase transition is given. It is based on a master equation description of the time evolution of a system. When the operator generating that time evolution has an isolated near degeneracy there is a firstorder phase transition. Conversely, when phenomena describable as firstorder phase transitions occur in a system, the corresponding operator has near degeneracy. Estimates relating degree of degeneracy and degree of phase separation are given. This approach harks back to early ideas on phase transitions and degeneracy, but now enjoys greater generality because it involves an operator present in a wide variety of systems. Our definition is applicable to what have intuitively been considered phase transitions in nonequilibrium systems and to problematic near equilibrium cases, such as metastability.

A stochastic method for Brownianlike optical transport calculations in anisotropic biosuspensions and blood
View Description Hide DescriptionA generic stochastic method is presented that rapidly evaluates numerical bulk flux solutions to the onedimensional integrodifferential radiative transport equation, for coherent irradiance of optically anisotropicsuspensions of nonspheroidal bioparticles, such as blood. As Fermat rays or geodesics enter the suspension, they evolve into a bundle of random paths or trajectories due to scattering by the suspended bioparticles. Overall, this can be interpreted as a bundle of Markov trajectories traced out by a “gas” of Brownianlike point photons being scattered and absorbed by the homogeneous distribution of uncorrelated cells in suspension. By considering the cumulative vectorial intersections of a statistical bundle of random trajectories through sets of interior data planes in the space containing the medium, the effective equivalent information content and behavior of the (generally unknown) analytical flux solutions of the radiative transfer equation rapidly emerges. The fluxes match the analytical diffuse flux solutions in the diffusion limit, which verifies the accuracy of the algorithm. The method is not constrained by the diffusion limit and gives correct solutions for conditions where diffuse solutions are not viable. Unlike conventional Monte Carlo and numerical techniques adapted from neutron transport or nuclear reactor problems that compute scalar quantities, this vectorial technique is fast, easily implemented, adaptable, and viable for a wide class of biophotonic scenarios. By comparison, other analytical or numerical techniques generally become unwieldy, lack viability, or are more difficult to utilize and adapt. Illustrative calculations are presented for blood medias at monochromatic wavelengths in the visible spectrum.
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 RELATIVITY AND GRAVITATION


A note on Tauber’s expanding universe in conformally flat coordinates
View Description Hide DescriptionThe solutions given by Tauber, based on conformally flat space–time, of Einstein’s field equations for the Friedmann–Robertson–Walker, closed topology, dust model are corrected. The corrected solutions are shown to reduce to the standard textbook solutions obtained directly in Friedmann–Robertson–Walker coordinates, with the parameter identified as a component of the transformation to conformally flat space–time coordinates. The general significance of Tauber’s work and its relationship to more recent papers by Endean on this subject are discussed.

Fast and slow solutions in General Relativity: The initialization procedure
View Description Hide DescriptionWe apply recent results in the theory of PDE, specifically in problems with two different time scales, to Einstein’s equations near their Newtonian limit. The results imply a justification to postNewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned, the first postNewtonian equations and their solutions in terms of Green’s functions are presented in order to indicate how to proceed in calculations with this approach.
