Calculation of superpropagators in non-linear quantum field theories

A new method of constructing the superpropagators, i.e. the Fourier transforms of the expressions of the form is suggested. The method makes it possible to derive by use of the same technique explicit analytic expressions for the superpropagators for a wide class of field theories — from strictly local up to essentially non-local. The essence of the method is the construction of a differential equation for the superpropagator which in general is of an infinite order. By use of the boundary condition atp ²=0 we find the solution of this equation depending on one arbitrary real parameter. Simple examples are given to illustrate the method.     

The Chern classes of Sobolev connections

AssumeF is the curvature (field) of a connection (potential) onR ⁴ with finiteL ² norm . We show the chern number (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds forF satisfying the Yang-Mills equations. We actually prove the natural general result in all even dimensions larger than 2.     

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

Some Remarks on the Smoluchowski–Kramers Approximation

According to the Smoluchowski–Kramers approximation, solution q _t of the equation , where is the White noise, converges to the solution of equation as µ 0. Many asymptotic problems for the last equation were studied in recent years. We consider relations between asymptotics for the first order equation and the original second order equation. Homogenization, large deviations and stochastic resonance, approximation of Brownian motion W _t by a smooth stochastic process, stationary distributions are considered.     

On the time evolution of the states of infinitely extended particles systems

We consider an infinite classical system of interacting particles in , . We study the time evolution of a particular class of nonequilibrium states. More precisely, the states we consider are Gibbs with respect to a Hamiltonian which differs from the Hamiltonian governing the motion by an external field (possibly not localized), satisfying certain conditions. It is proved that the time-evolved states satisfy superstable estimate and are described by correlation functions obeying the BBGKY hierarchy in a weak form.     

Space-time structure near particles and its influence on particle behavior

An interrelation between the properties of the space-time structure near moving particles and their dynamics is discussed. It is suggested that the space-time metric near particles becomes a curved one depending on a random vectorb _E =(b ₄,b) with a distributionw(b _E ² /l ²); the averaged space-time metric over this distribution gives the general effect on particle behavior. As a result the particle motion in our scheme is described by a nonlinear equation. It turns out that the nonrelativistic limit of this equation gives a simple connection between the space-time structure at small distances and the dynamical behavior of particles. Different types of particle motion (nearly rectilinear, stochastic, and solitonlike) caused by some concrete forms of the averaged conformally flat space-time metric are considered.     

Analyticity properties and a convergent expansion for the inverse correlation length of the low-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(z) of the truncated spin-spin correlation function of theZ ^d Ising model with + or — boundary conditions admits the representationm(z) = –(4d–4)ln z(1–_d1) + r(z) for smallz=e ^–, i.e., large inverse temperatures is ad-dependent analytic function atz = 0, already known in closed form ford = 1 and 2; ford = 3 b_n can be computed explicitly from a finite number of the Z^d limits of z = 0 Taylor series coefficients of the finite lattice correlation function at a finite number of points ofZ ^d.     

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, with finite, in which spin flips (i.e., changes of S ^t _x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.     

Gauge-invariant on-shellZ 2 in QED,QCD and the effective field theory of a static quark

We calculate theon-shell fermion wave-function renormalization constantZ ₂ of a general gauge theory, to two loops, inD dimensions and in an arbitrary covariant gauge, and find it to be gauge-invariant. In QED this is consistent with the dimensionally regularized version of the Johnson-Zumino relation: d logZ ₂/da ₀=i(2)^–D e ₀ ² d ^D k/k ⁴=0. In QCD it is, we believe, a new result, strongly suggestive of the cancellation of the gauge-dependent parts of non-abelian UV and IR anomalous dimensions to all orders. At the two-loop level, we find that the anomalous dimension _F of the fermion field in minimally subtracted QCD, withN _L light-quark flavours, differs from the corresponding anomalous dimension of the effective field theory of a static quark by the gauge-invariant amount

A note on the boson-fermion correspondence and infinite dimensional groups

We show how to construct irreducible projective representations of the infinite dimensional Lie group Map (S ¹, ), by embedding it into the group of Bogoliubov automorphisms of the CAR. Using techniques of G. Segal for extending certain representations of Map (S ¹, SU(2)) we show that our representations extend to give representations of a certain infinite dimensional superalgebra. We relate our work to the well known boson-fermion correspondence which exists in 1+1 dimensions.     

The Heisenberg ferromagnet as a quantum field theory

We consider a lattice of spin 1/2 ions, described by the discrete form of the current commutation relationsJ _i J _(i) =1/2, [J _i ,J _i ]=i _ij J _i where =1, 2, 3 andi label the lattice sites. The algebra is realized as the Clifford algebra over a Hilbert space. The equations of motion are specified by a formal Hamiltonian of the Heisenberg form: , wheref _ij 0 and only a finite numberQ of ions are linked to any given lattice site. We prove that the Hamiltonian is non-negative in a representation of , and has a ground state exhibiting ferromagnetism. The time displacement group acts continuously on , inducing automorphisms. is asymptotically abelian with respect to the space translations of the lattice.The model is an example of an algebraic quantum field theory and possesses a broken symmetry, the rotation group 0(3). The consequent Goldstone theorem is proved, namely, there is no energy gap in the spectrum ofH.     

Morse and Melnikov functions for NLS Pde's

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat linear operator , is developed in sufficient detail for later use in studies of perturbations of the NLS equation. Counting lemmas for the non-selfadjoint operator , are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discD in the complex eigenvalue plane. The radius of the discD is controlled by theH ¹ norm of the potential . For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds-whiskered tori for the NLS pde.The Floquet discriminant is used to introduce a natural sequence of NLS constants of motion, [ , where _c ^j denotes thej ^th critical point of the Floquet discriminant ()]. A Taylor series expansion of the constants , with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of , which themselves are expressed in terms of quadratic products of eigenfunctions of . The second variation permits identification, within the disc D, of important bifurcations in the spectral configurations of the operator . The constant , as the height of the Floquet discriminant over the critical point _c ^j , admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, , which is associated with critical points outside the discD. Within this disc, the interpretation is only valid locally, with the same obstruction to its global validity as to a global ordering of the spectrum. Nevertheless, this local Morse theory, together with the Backlund representations of the whiskered tori, produces extremely clear pictures of the stratification of NLS invariant sets near these whiskered tori-pictures which are useful in the study of perturbations of NLS. Finally, a natural connection is noted between the constants of the integrable theory and Melnikov functions for the theory of perturbations of the NLS equation. This connection generates a simple, but general, representations of the Melnikov functions.Funded in part by AFOSR-90-0161 and by NSF DMS 8922717 A01     

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Equilibrium states of gravitational systems

We formulate the equilibrium correlation functions for local observables of an assembly of non-relativistic, neutral gravitating fermions in the limit where the number of particles becomes infinite, and in a scaling where the region , to which they are confined, remains fixed. We show that these correlation functions correspond, in the limit concerned, to states on the discrete tensor product , where the are copies of the gauge invariantC*-algebra of the CAR overL ²(R ³). The equilibrium states themselves are then given by , where , is the Gibbs state on for an infinitely extended ideal Fermi gas at density , and where ₀ is the normalised density function that minimises the Thomas-Fermi functional, obtained in [2], governing the equilibrium thermodynamics of the system.     

Hearing the zero locus of a magnetic field

We investigate the ground state of a two-dimensional quantum particle in a magnetic field where the field vanishes nondegenerately along a closed curve. We show that the ground state concentrates on this curve ase/h tends to infinity, wheree is the charge, and that the ground state energy grows like (e/h)^2/3. These statements are true for any energy level, the level being fixed as the charge tends to infinity. If the magnitude of the gradient of the magnetic field is a constantb ₀ along its zero locus, then we get the precise asymptotics(e/h) ^2/3 (b ₀) ^2/3 E _* +O(1) for every energy level. The constantE _* .5698 is the infimum of the ground state energiesE() of the anharmonic oscillator family .     

Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

A locally finite, causal, and quantal substitute for a locally Minkowskian principal fiber bundle of modules of Cartan differential forms over a bounded region X of a curved C -smooth spacetime manifold M with structure group G that of orthochronous Lorentz transformations L ⁺ := SO(1,3), is presented. is usually regarded as the kinematical structure of classical Lorentzian gravity when the latter is viewed as a Yang-Mills type of gauge theory of a sl(2, {})-valued connection 1-form . The mathematical structure employed to model this replacement of is a principal finitary spacetime sheaf of quantum causal sets with structure group G _n, which is a finitary version of the continuous group G of local symmetries of General Relativity, and a finitary Lie algebra g _n-valued connection 1-form on it, which is a section of its subsheaf . is physically interpreted as the dynamical field of a locally finite quantum causality, whereas its associated curvature as some sort of finitary and causal Lorentzian quantum gravity.     

Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields

Rigorous upper bounds are derived for large-scale turbulent flame speeds in a prototypical model problem. This model problem consists of a reaction-diffusion equation with KPP chemistry with random advection consisting of a turbulent unidirectional shear flow. When this velocity field is fractal with a Hurst exponentH with 0<H<1, the almost sure upper bounds suggest that there is an accelerating large-scale turbulent flame front with the enhanced anomalous propagation lawy=C _H t ^1+H for large renormalized times. In contrast, a similar rigorous almost sure upper bound for velocity fields with finite energy yields the turbulent flame propagation law within logarithmic corrections. Furthermore, rigorous theorems are developed here which show that upper bounds for turbulent flame speeds with fractal velocity fields are not self-averaging, i.e., bounds for the ensemble-averaged turbulent flame speed can be extremely pessimistic and misleading when compared with the bounds for every realization.     

Exact time correlation function for a nonlinearly coupled vibrational system

The time correlation function (t)=Re<[c(t), c (0)]>, which is related to the dipole spectrum and is the main focus of quantum molecular time scale generalized Langevin equation theory, is calculated for the Hamiltonian system in which a single oscillator is coupled by a nonlinear Davydov term to a chain of oscillators comprising a phonon heat bath. An exact expression for (t) is obtained. At long times we find that the time correlation function decays as a small power law atT=0K, but switches to exponential decay at higher temperature. This is a new result and bears on the long-standing issue of the existence of long-time tails.     

On Modules over Matrix Quantum Pseudo-Differential Operators

We classify all the quasifinite highest-weight modules over the central extension of the Lie algebra of matrix quantum pseudo-differential operators, and obtain them in terms of representation theory of the Lie algebra (, R _m ) of infinite matrices with only finitely many nonzero diagonals over the algebra R _m = [t]/(t ^m+1). We also classify the unitary ones.     

Nonlinear superposition for Liouville's equation in three spatial dimensions

We derive in 3+0 dimensions exact solutions of Liouville's equation ²=exp , by applying the Bäcklund transformation technique in conjunction with the principle of nonlinear superposition. The procedure, which is later extended to 3+1 dimensions, yield, as a byproduct, particular solutions of ² and ² =exp (² +² ).