Showing papers in "Communications in Mathematical Physics in 1977"

Existence of solitary waves in higher dimensions

Abstract: The elliptic equation Δu=F(u) possesses non-trivial solutions inR n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.

Simple $C^*$-algebras generated by isometries

Abstract: We consider theC*-algebra\(\mathcal{O}_n \) generated byn≧2 isometriesS1,...,Sn on an infinite-dimensional Hilbert space, with the property thatS1S*1+...+SnS*n=1. It turns out that\(\mathcal{O}_n \) has the structure of a crossed product of a finite simpleC*-algebra ℱ by a single endomorphism scaling the trace of ℱ by 1/n. Thus,\(\mathcal{O}_n \) is a separableC*-algebra sharing many of the properties of a factor of typeIIIλ with λ=1/n. As a consequence we show that\(\mathcal{O}_n \) is simple and that its isomorphism type does not depend on the choice ofS1,...,Sn.

Zeta function regularization of path integrals in curved spacetime

Abstract: This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.

Dimensional Renormalization and the Action Principle

Abstract: Dimensional renormalization is defined in such a way that the renormalized action principle holds. It is shown that this leads to a minimal, additive renormalization. The derivation of Ward-Takahashi indentities and Callan-Symanzik equations from the action principle is exemplified.

All unitary ray representations of the conformal group SU(2,2) with positive energy

Abstract: We find all those unitary irreducible representations of the ∞-sheeted covering group\(\tilde G\) of the conformal group SU(2,2)/ℤ4 which have positive energyP0≧0. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j1,j2) of the Lorentz group SL(2ℂ). They all decompose into a finite number of unitary irreducible representations of the Poincare subgroup with dilations.

The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles

Abstract: For classicalN-particle systems with pair interactionN−1\(\mathop \Sigma \limits_{1 \leqq i \leqq j \leqq N} \) o(qi−qi) the Vlasov dynamics is shown to be thew*-limit asN→∞. Propagation of molecular chaos holds in this limit, and the fluctuations of intensive observables converge to a Gaussian stochastic process.

Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$

Abstract: Using dilation invariance and dilation analytic techniques, and with the help of a new virial theorem, we give a detailed description of the spectral properties of the operator (p2+m2)1/2−Ze2/r. In the process the norm of the operator ∣x∣α∣p∣−α is calculated explicitly inLp(® N ).

A sketch of Lie superalgebra theory

Abstract: This article deals with the structure and representations of Lie superalgebras (ℤ2-graded Lie algebras). The central result is a classification of simple Lie superalgebras over ℝ and ℂ.

Relative Entropy and the Wigner-Yanase-Dyson-Lieb Concavity in an Interpolation Theory

Abstract: We show that the Wigner-Yanase-Dyson-Lieb concavity is a general property of an interpolation theory which works between pairs of (hilbertian) seminorms. As an application, the theory extends the relevant work of Lieb and Araki to positive linear forms of arbitrary *-algebras. In this context a “relative entropy” is defined for every pair of positive linear forms of a *-algebra with identity. For this generalized relative entropy its joint convexity and its decreasing under identity-preserving completely positive maps is proved.

Instantons and algebraic geometry

Abstract: Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsatzeA l forl≧1.

The Hartree-Fock Theory for Coulomb Systems

Abstract: For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.

Hausdorff measure and the Navier-Stokes equations

Abstract: Solutions to the Navier-Stokes equations are continuous except for a closed set whose Hausdorff dimension does not exceed two.

The back reaction effect in particle creation in curved spacetime

Abstract: The problem of determining the changes in the gravitational field caused by particle creation is investigated in the context of the semiclassical approximation, where the gravitational field (i.e., spacetime geometry) is treated classically and an effective stress energy is assigned to the created particles which acts as a source of the gravitational field. An axiomatic approach is taken. We list five conditions which the renormalized stress-energy operatorTμv should satisfy in order to give a reasonable semiclassical theory. It is proven that these conditions uniquely determineTμv, i.e. there is at most one renormalized stress-energy operator which satisfies all the conditions. We investigate existence by examining an explicit “point-splitting” type prescription for renormalizingTμv. Modulo some standard assumptions which are made in defining the prescription forTμv, it is shown that this prescription satisfies at least four of the five axioms.

Construction of a selfadjoint, strictly positive transfer matrix for euclidean lattice gauge theories

Abstract: It is shown that physical positivity holds in Wilson's lattice gauge theories, i.e. transition probabilities between gauge invariant states are non-negative and the quantum mechanical Hamiltonian has real eigenvalues only.

Stationary non-equilibrium states of infinite harmonic systems

Abstract: We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not “stable” towards perturbations in the potential. We are particularly concerned here with states in which there is a non-vanishing steady heat flux passing through “every point” of the infinite system. Such “superheat-conducting” states are of course only possible in systems in which Fourier's law does not hold: the perfect harmonic crystal being an example of such a system. For a one dimensional system, we find such states (explicitely) as limits, whent→∞, of time evolved initial states μ i in which the “left” and “right” parts of the infinite crystal are in “equilibrium” at different temperatures, β − ≠β −1 , and the “middle” part is in an arbitrary state. We also investigate the limit of these stationary (t→∞) states as the coupling strength λ between the “system” and the “reservoirs” goes to zero. In this limit we obtain a product state, where the reservoirs are in equilibrium at temperatures β −1 and β −1 and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit.

Dimensionally renormalized Green's functions for theories with massless particles. II

Abstract: In the framework of dimensional renormalization the existence of Green's functions to all orders of perturbation theory is proved for theories with massless particles, provided all terms in the interaction Lagrangian have infrared degree Ω≧4. If the vanishing of masses is enforced by some symmetry and this symmetry is respected by dimensional regularization, Schwinger's action principle holds for these Green's functions as in the massive case.

Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory

Abstract: In a conformal invariant quantum field theory (in 4 space time dimensions) Wilson operator product expansions converge on the vacuum, because they are closely related to conformal partial wave expansions.

A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line

Abstract: Two theorems are proved—the first and the more important of them due to Sarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.

Spectral and scattering theory of Schrödinger operators related to the Stark effect

Abstract: We analyze the spectral properties and discuss the scattering theory of operators of the formH=H 0+V,H 0=−Δ+E·x. Among our results are the existence of wave operators, Ω±(H, H 0), the nonexistence of bound states, and a (speculative) description of resonances for certain classes of potentials.

Quantum Detailed Balance and KMS Condition

Abstract: A definition of detailed balance for quantum dynamical semigroups is given, and its close connection with the KMS condition is investigated.

The deterministic version of the Glimm scheme

Abstract: The Glimm scheme for solving hyperbolic conservation laws has a stochastic feature; it depends on a random sequence. The purpose of this paper is to show that the scheme converges for any equidistributed sequence. Thus the scheme becomes deterministic.

Irreducible quantum dynamical semigroups

Abstract: We study some irreducible and ergodic properties of quantum dynamical semigroups, and apply our methods to semigroups of Lindblad type.

On the decay of correlations in ${\rm SO}(n)$-symmetric ferromagnets

Abstract: We prove that for low temperaturesT the spin-spin correlation function of the two-dimensional classicalSO(n)-symmetric Ising ferromagnet decays faster than |x|−constT providedn≧2. We also discuss a nearest neighbor continuous spin model, with spins restricted to a finite interval, where we show that the spin-spin correlation function decays exponentially in any number of dimensions.

Phase space cell expansion and borel summability for the Euclidean φ{ 3 / 4 } theory

Abstract: The stability of the free energy is proved for complex values of the coupling constant by the way of a convergent expansion. As a consequence, one obtains the Borel summability of the perturbation series.

On the convergence of exponential operators---the Zassenhaus formula, BCH formula and systematic approximants

Abstract: The convergence of the Zassenhaus formula is proven under an appropriate condition as well as for other exponential operators such as the Baker-Campbell-Hausdorff formula.

Exact two-particle S-matrix of quantum sine-Gordon solitons

Abstract: The exact and explicit formulas for the quantumS-matrix elements of the soliton-antisoliton scattering which satisfy unitarity and crossing conditions and have correct analytical properties are constructed. ThisS-matrix is in agreement with the massive Thirring model perturbation theory and with the semiclassical sine-Gordon results.

Geometric methods in multiparticle quantum systems

Abstract: Technically simple proofs are given of the HVZ theorem on the bottom of the essential spectrum of multiparticle systems and of Combes' result on completeness below the lowest three body threshold. The first proof is a variant of a proof of Enss and a decendent of Zhislin's original proof. Finally, we apply our methods to the bound state spectrum.

Extension of KMS states and chemical potential

Abstract: We present an algebraic description of the concept of chemical potential for a general compact gauge groupG, as a first step in the classification of thermodynamical equilibrium states of a given temperature. Adopting first the usual setting of a field algebra ℱ containing the observable algebra\(\mathfrak{A}\) as its gauge invariant part, we establish the following results (i) the existence and uniqueness, up to gauge, of τ-weakly clustering states ф of ℱ extending a given such state ω of\(\mathfrak{A}\) (τ an asymptotically abelian automorphism group of ℱ commuting withG) (ii) in the case of an ω faithful and β-KMS for a time evolution commuting with τ, and of a time-invariant ф, the fact that ф is β-KMS for a one-parameter group of time and gauge whose gauge part lies in the center of the stabilizerGφ of ф. (iii) a description of the general case where ф is neither time invariant nor faithful: ф is then in general vacuum-like in directions of the gauge space governed by an “asymmetry subgroup”. We further analyze the representations and von Neumann algebras determined by ω, ф and the gauge average\(\bar \omega \) of ф. The covariant representation generated by\(\bar \omega \) is shown to be obtained by inducing up fromGφ toG the representation generated by ф. Finally we present, for the case whereG is ann-dimensional torus, an intrinsic description of the chemical potential in terms of cocycle Radon-Nicodym derivatives of the state ω w.r.t. its (quasi equivalent) transforms by localized automorphisms of\(\mathfrak{A}\). Our main result (ii) is established using two independant techniques, the first making systematic use of clustering properties, the second relying on the analysis of representations. Both proofs are basically concerned with Tannaka duality—the second with a version thereof formulated in Robert's theory of Hilbert spaces in a von Neumann algebra.

Collision theory for massless bosons

Abstract: We present a complete solution of the collision problem for massless Bosons in four space-time dimensions.