Massive supersymmetric quantum gauge theory

We continue the study of the supersymmetric vector multiplet in a purely quantum framework. We obtain some new results which make the connection with the standard literature. First we construct the one‐particle physical Hilbert space taking into account the (quantum) gauge structure of the model. Then we impose the condition of positivity for the scalar product only on the physical Hilbert space. Finally we obtain a full supersymmetric coupling which is gauge invariant in the supersymmetric sense in the first order of perturbation theory. By integrating out the Grassmann variables we get an interacting Lagrangian for a massive Yang‐Mills theory related to ordinary gauge theory; however the number of ghost fields is doubled so we do not obtain the same ghost couplings as in the standard model Lagrangian.     

A generalization of Fermat's principle for classical and quantum systems

The analogy between dynamics and optics had a great influence on the development of the foundations of classical and quantum mechanics. We take this analogy one step further and investigate the validity of Fermat's principle in many-dimensional spaces describing dynamical systems (i.e., the quantum Hilbert space and the classical phase and configuration space). We propose that if the notion of a metric distance is well defined in that space and the velocity of the representative point of the system is an invariant of motion, then a generalized version of Fermat's principle will hold. We substantiate this conjecture for time-independent quantum systems and for a classical system consisting of coupled harmonic oscillators. An exception to this principle is the configuration space of a charged particle in a constant magnetic field; in this case the principle is valid in a frame rotating by half the Larmor frequency, not the stationary lab frame.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Relativistic quantum mechanics of spin-zero and spin-half particles

We consider the quantum mechanics of directly interacting relativistic particles of spin-zero and spin-half. We introduce a scalar product in the vector space of physical states which is finite, positive definite and relativistically invariant and keeps orthogonal eigenstates of total four momentum belonging to different eigenvalues. This allows us to show that the vector space of physical states is, in fact, a Hilbert space. The case of two particles is explicitly considered and the Cauchy problem of physical wave function illustrated. The problem of a spin-1/2 particle interacting with a spin-zero particle is considered and a new equation is proposed for two spin-1/2 particles interacting via the most general form of interaction possible. The restrictions due to Hermiticity, space inversion and time reversal invariance are also considered.     

Stochastic Dynamics of Discrete Curves and Multi-Type Exclusion Processes

This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has in general a non Gibbs form. The corresponding steady-state regime is analyzed in detail, by using a tagged particle together with a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka–Volterra equations—which describe the fluid limits of stationary states—can be traced back directly at the discrete level to tagged particle cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained via an iterative scheme. The related Hamilton–Jacobi equation, which leads to the large deviation functional, is investigated and solved in the reversible case, just for the sake of checking.     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Freezing Transitions in Non-Fellerian Particle Systems

Non-Fellerian particle systems are characterized by nonlocal interactions, somewhat analogous to non-Gibbsian distributions. They exhibit new phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the trivial invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state.     

Noninvariance of space- and time-scale ranges under a Lorentz Transformation and the implications for the study of relativistic interactions

We present an analysis which shows that the ranges of space and time scales spanned by a system are not invariant under Lorentz transformation. This implies the existence of a frame of reference which minimizes an aggregate measure of the range of space and time scales. Such a frame is derived, for example, for the following cases: free electron laser, laser-plasma accelerator, and particle beams interacting with electron clouds. The implications for experimental, theoretical, and numerical studies are discussed. The most immediate relevance is the reduction by orders of magnitude in computer simulation run times for such systems.     

On the decomposition of wightman functionals in the euclidean framework

We consider Wightman functionals having the property that the Schwinger functions can be represented by a Euclidean invariant measure on the space of tempered distributions. The strong form of the Osterwalder-Schrader positivity condition is shown to imply that the measure is positive and some restrictions on the Schwinger functions are discussed which guarantee that this condition holds. The ergodic decomposition of the measure leads to a decomposition of the Wightman functional into such with cluster property. We discuss also the role of the positivity condition in connection with a general criterion for the existence of a decomposition.     

On Dirac Physical Measures for Transitive Flows

We discuss some examples of smooth transitive flows with physical measures supported at fixed points. We give some conditions under which stopping a flow at a point will create a Dirac physical measure at that indifferent fixed point. Using the Anosov-Katok method, we construct transitive flows on surfaces with the only ergodic invariant probabilities being Dirac measures at hyperbolic fixed points. When there is only one such point, the corresponding Dirac measure is necessarily the only physical measure with full basin of attraction. Using an example due to Hu and Young, we also construct a transitive flow on a three-dimensional compact manifold without boundary, with the only physical measure the average of two Dirac measures at two hyperbolic fixed points.     

Renormalisation group flows and conserved vector currents

Irreversibility of RG flows in two dimensions is shown using conserved vector currents. Out of a conserved vector current, a quantity decreasing along the RG flow is built up such that it is stationary at fixed points where it coincides with the constant coefficient of the two-current correlation function. For Wess-Zumino-Novikov-Witten models this constant coefficient is the level of their associated affine Lie algebra. Extensions to higher dimensions using the spectral decomposition of the two-current correlation function are studied.     

Free probability and quantum electrodynamics

The stochastic limit of a free particle coupled to the quantum electromagnetic field without dipole approximation leads to many new features such as: interacting Fock space, Hilbert module commutation relations, disappearance of the crossing diagrams, etc. In the present paper we begin to study how the situation is modified if a free particle is replaced by a particle in a potential which is the Fourier transform of a bounded measure.We prove that the stochastic limit procedure converges and that the overall picture is similar to the free case with the important difference that the structure of the limit Hilbert module is strongly dependent on the wave operator of the particle.     

Comment on “Entanglement of two interacting bosons in a two-dimensional isotropic harmonic trap” [Phys. Lett. A 373 (2009) 3833]

The correct form of the Schmidt decomposition of the stationary wave functions for a system of two interacting particles trapped in a two-dimensional harmonic potential is given.     

Superdiffusive Behaviour of a Passive Ornstein–Uhlenbeck Tracer in a Turbulent Shear Flow

In this paper, we are interested in the study of the diffusion of a passive particle with positive mass by a divergence free velocity field. We consider here the very simple turbulent shear flow case, in which we will prove the superdiffusive behaviour of the motion for large enough values of the energy spectrum of the velocity field. For small values, the proof of the diffusive behaviour of the model is also new, and it is shown that this diffusion is strictly greater than the one obtained with a non-massive particle. One interesting point to insist on is that we are able to obtain explicit hydrodynamic equations without even having the stationary measure of the studied processes     

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state

In this paper we consider a one-dimensional model of interacting particles in a bounded interval with (possibly not homogeneous) diffusive boundary conditions. We prove that, when the number of particlesN goes to infinity and the interaction is suitably rescaled (the Boltzmann-Grad limit), the one-particle distribution function of the unique invariant measure for the particle system, converges to the unique solution of the Boltzmann equation of the model, provided that the mean free path is sufficiently large.     

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.     

Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration.     

Electrodynamics in an LTB scenario

In this paper we investigate the Yokoyama gaugeon formalism for perturbative quantum gravity in a general curved spacetime. Within the gaugeon formalism, we extend the configuration space by introducing vector gaugeon fields describing a quantum gauge degree of freedom. Such an extended theory of perturbative gravity admits quantum gauge transformations leading to a natural shift in the gauge parameter. Further we impose the Gupta–Bleuler type subsidiary condition to remove the unphysical gaugeon modes. To replace the Gupta–Bleuler type condition by a more acceptable Kugo–Ojima type subsidiary condition we analyze the BRST symmetric gaugeon formalism. Further, the physical Hilbert space is constructed for the perturbative quantum gravity which remains invariant under both the BRST symmetry and the quantum gauge transformations.     

On conformal invariance in quantum field theory

We study the transformation law of interacting fields under the universal convering group of the conformal group. It is defined with respect to the representations of the discrete series. These representations are field representations in the sense that they act on finite component fields defined over Minkowski space. The conflict with Einstein causality is avoided as in the case of free fields with canonical dimension. Furthermore, we determine the conformal invariant two-point function of arbitrary spin. Our result coincides with that obtained by Rühl. In particular, we investigate the two-point function of symmetric and traceless tensor fields and give the explicit form of the trace terms.