Showing papers in "Communications in Mathematical Physics in 1982"

Orbital stability of standing waves for some nonlinear Schrödinger equations

Abstract: We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrodinger equations. For example, we treat the cases of nonlinear Schrodinger equations arising in laser beams, of time-dependent Hartree equations ....

Calculation of norms of Bethe wave functions

Abstract: A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schrodinger equation and the HeisenbergXXZ model.

Connections with l**p bounds on curvature

Abstract: We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR n when the integralL n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL p integral norms bounded,p>n/2.

The unpredictability of quantum gravity

Abstract: Quantum gravity seems to introduce a new level of unpredictability into physics over and above that normally associated with the uncertainty principle. This is because the metric of spacetime can fluctuate from being globally hyperbolic. In other words, the evolution is not completely determined by Cauchy data at past or future infinity. I present a number of axioms that the asymptotic Green functions should obey in any reasonable theory of quantum gravity. These axioms are the same as for ordinary quantum field theory in flat spacetime, except that one axiom, that of asymptotic completeness, is omitted. This allows pure quantum states to decay into mixed states. Calculations with simple models of topologically non-trivial spacetime indicate that such loss of quantum coherence will occur but that the effect will be very small except for fundamental scalar particles, if any such exist.

The rotation number for almost periodic potentials

Abstract: We define and analyze the rotation number for the almost periodic Schrodinger operatorL= −d 2/dx 2+q(x). We use the rotation number to discuss (i) the spectrum ofL; (ii) its relation to the Korteweg-de Vries equation.

Geometric analysis of φ4 fields and Ising models. Parts I and II

Abstract: We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.

Locality and the structure of particle states

Abstract: Starting from the principle of locality of observables we derive localization properties of massive particle states which hold in all models of relativistic quantum theory, including gauge theories. It turns out that particles may always be regarded as well localized distributions of matter, although their mathematical description might require the introduction of non-local (unobservable) fields, which are assigned to infinite string-like regions. In spite of the non-locality of these fields one can show that such particles obey Bose- or Fermi (para) statistics, that to each particle there exists an antiparticle and that collision states of particles exist. A selfcontained exposition of the underlying physical ideas is given in the Introduction, and some perspectives for the structure of field-theoretic models arising from our analysis are discussed in the Conclusions.

Monopoles and geodesics

Abstract: Using the holomorphic geometry of the space of straight lines in Euclidean 3-space, it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the Atiyah-Ward Ansatz\(A_k \).

Removable singularities in Yang-Mills fields

Abstract: We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR 4 with bounded functional (L 2 norm) may be obtained from a field onS 4=R 4∪{∞}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.

Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study

Abstract: We consider a two-parameter family of maps of the plane to itself. Each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point. For certain parameter values there is a Hopf bifurcation to an invariant circle, which is smooth for parameter values in a neighborhood of the bifurcation point. However, computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point. This paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type.

On Witten's proof of the positive energy theorem

Abstract: This paper gives a mathematically rigorous proof of the positive energy theorem using spinors. This completes and simplifies the original argument presented by Edward Witten. We clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant Dirac operator.

The random-walk representation of classical spin systems and correlation inequalities

Abstract: Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ4)2 quantum field model. We also establish new upper bounds on critical temperatures.

Modular structure of the local algebras associated with the free massless scalar field theory

Abstract: The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII1 factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.

Topology of lattice gauge fields

Abstract: Non-Abelian gauge fields on a four-dimensional hypercubic lattice with small action density [Tr{U( $$\dot p$$ )} for SU(2) gauge fields] are shown to carry an integer topological chargeQ, which is invariant under continuous deformations of the field. A concrete expression forQ is given and it is verified thatQ reduces to the familiar Chern number in the classical continuum limit.

Spacelike hypersurfaces with prescribed boundary values and mean curvature

Abstract: We consider the boundary-value problem for the mean curvature operator in Minkowski space, and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions. Our main results hold for non-constant mean curvature, and make no assumptions about the smoothness of the boundary or boundary data.

The phase transition in the one-dimensional Ising Model with 1/ r 2 interaction energy

Abstract: We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r2 interaction energy.

Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators

Abstract: For a large class ofN-body potentialsV we prove that if ϕ is an eigenfunction of −δ+V with eigenvalueE then sup{α2+E:α≧0, exp(α|x|)ϕ∈L2} is either a threshold or +∞. Consequences of this result are the absence of positive eigenvalues and “optimal”L2-exponential lower bounds.

On the positivity of the effective action in a theory of random surfaces

Abstract: It is shown that the functional $$S[\eta ] = \frac{1}{{24\pi }}\int {\left( {\frac{1}{2}\left| { abla \eta } \right|^2 + 2\eta } \right)d\mu _0 }$$ , defined onC ∞ functions on the two-dimensional sphere, satisfies the inequalityS[η]≧0 if η is subject to the constraint $$\int {(e^\eta - 1)d\mu _0 = 0}$$ . The minimumS[η]=0 is attained at the solutions of the Euler-Lagrange equations. The proof is based on a sharper version of Moser-Trudinger's inequality (due to Aubin) which holds under the additional constraint $$\int {e^\eta xd\mu _0 = 0}$$ ; this condition can always be satisfied by exploiting the invariance ofS[η] under the conformal transformations ofS 2. The result is relevant for a recently proposed formulation of a theory of random surfaces.

Large volume limit of the distribution of characteristic exponents in turbulence

Abstract: For spatially extended conservative or dissipative physical systems, it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity. In the case of a turbulent viscous fluid, however, this simple idea is complicated by the phenomenon of intermittency. In the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation. These bounds have a reasonable large volume behavior. For two-dimensional fluids a particularly striking result is obtained: the total information creation is bounded above by a fixed multiple of the total energy dissipation (at fixed viscosity). The distribution of characteristic exponents is estimated in an intermittent model of turbulence (see [7]), and it is found that a change of behavior occurs at the valueD=2.6 of the self-similarity dimension.

The evolution of a turbulent vortex

Abstract: We examine numerically the evolution of a perturbed vortex in a periodic box. The fluid is inviscid. We find that the vorticity blows up. The support of theL2 norm of the vorticity converges to a set of Hausdorff dimension ∼2.5. The distribution of the vorticity seems to converge to a lognormal distribution. We do not observe a convergence of the higher statistics towards universal statistics, but do observe a strong temporal intermittency.

Proof of confinement of static quarks in 3-dimensionalU(1) lattice gauge theory for all values of the coupling constant

Abstract: We study the 3-dimensional pureU(1) lattice gauge theory with Villain action which is related to the 3-dimensional ℤ-ferromagnet by an exact duality transformation (and also to a Coulomb system). We show that its string tension α is nonzero for all values of the coupling constantg2, and obeys a bound α≧const·mDβ−1 for smallag2, with β=4π2/g2 and\(m_D^2 = (2{\beta \mathord{\left/ {\vphantom {\beta a}} \right. \kern- ulldelimiterspace} a}^3 )e^{ - \beta \upsilon _{Cb} {{(0)} \mathord{\left/ {\vphantom {{(0)} 2}} \right. \kern- ulldelimiterspace} 2}} (a = lattice spacing)\). A continuum limita→0,mD fixed, exists and represents a scalar free field theory of massmD. The string tension αmD−2 in physical units tends to ∞ in this limit. Characteristic differences in the behaviour of the model for large and small coupling constantag2 are found. Renormalization group aspects are discussed.

The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. II. Completion of proof

Abstract: In this paper we complete the proof of global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space by showing that an appropriate norm of the solutions cannot blow up in a finite time. A key step in the proof is the demonstration that theL∞ norm of the curvature is boundeda priori. Our results apply to any compact guage group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.

Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

Abstract: This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.

The physical state space of quantum electrodynamics

Abstract: Starting from the fact that electrically charged particles are massive, we derive a criterion which characterizes the state space of quantum electrodynamics This criterion clarifies the special role of the electric charge amongst the uncountably many superselection rules in quantum electrodynamics and provides a basis for a general analysis of the infrared problem Within this framework we establish the existence of asymptotic electromagnetic fields in all charge-sectors, find a general characterization of infra-particles and introduce a notion of asymptotic completeness

The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on ℝ 3 . Part I

Abstract: This paper (Part I) and the sequel (Part II) prove the existence of a smooth, non-trivial, finite action solution to the SU (2) Yang-Mills-Higgs equations on ℝ3 in the Bogomol'nyi-Prasad-Sommerfield limit. The proof uses a simple form of Morse theory known as Ljusternik-Snirelman theory. Part I establishes that a form of Lusternik-Snirelman theory is applicable to the SU (2) Yang-Mills-Higgs equations. Here, a sufficient condition for the existence of the aforementioned solution is derived. Part II contains the completed existence proof. There it is demonstrated that the sufficient condition of Part I is satisfied by the SU (2) Yang-Mills-Higgs equations.

Massless phases and symmetry restoration in abelian gauge theories and spin systems

Abstract: We give a new, elementary proof for the existence of a deconfining transition to a massless (QED) phase in the four-dimensionalU(1) lattice gauge theory and of an intermediate QED phase, accompanied by dynamical restoration of localU(1) invariance, in the four dimensional ℤ N models, withN large. Our methods can also be used to prove the existence of a phase transition in theXY model in three or more dimensions, in three- and four-dimensional abelian Higgs models, and in more general models admitting some local, abelian gauge invariance.

First-order phase transitions in large entropy lattice models

Abstract: We show the existence of a first-order phase transition in thev-dimensional Potts model forv≧2, when the number of states of a single spin is big enough. Low-temperature pure phases are proved to survive up to the critical temperature. Also the existence of a first-order transition in thev-dimensional Potts gauge model,v≧3, is obtained if the underlying gauge group is finite but large.

Some results for SU(N) gauge-fields on the Hypertorus

Abstract: We show how to prove and to understand the formula for the “Pontryagin” indexP for SU(N) gauge fields on the HypertorusT4, seen as a four-dimensional euclidean box with twisted boundary conditions. These twists are defined as gauge invariant integers moduloN and labelled byNμv (=−Nμv). In terms of these we can write (ν∈#x2124;) $$P = \frac{1}{{16\pi ^2 }}\int {Tr(G_{\mu v} \tilde G_{\mu v} )d_4 x = v + \left( {\frac{{N - 1}}{N}} \right) \cdot \frac{{n_{\mu v} \tilde n_{\mu v} }}{4}} $$ . Furthermore we settle the last link in the proof of the existence of zero action solutions with all possible twists satisfying\(\frac{{n_{\mu v} \tilde n_{\mu v} }}{4} = \kappa (n) = 0(\bmod N)\) for arbitraryN.

Einstein's equations near spatial infinity

Abstract: A new class of space-times is introduced which, in a neighbourhood of spatial infinity, allows an expansion in negative powers of a radial coordinate Einstein's vacuum equations give rise to a hierarchy of linear equations for the coefficients in this expansion It is demonstrated that this hierarchy can be completely solved provided the initial data satisfy certain constraints