Perturbation theory of Wightman functions

A perturbative expansion of the Wightman functions, and more generally of vacuum expectation values of products of time-ordered and anti-time-ordered products, is derived for ₄ ⁴ field theory. The result is expressed as a sum over generalized Feynman graphs. The derivation is based exclusively on the equation of motion and the Wightman axioms. Neither canonical commutation relations nor asymptotic conditions are needed at any point. In the zero-mass case the individual graphs are infrared divergent, but the sum over all graphs of a given order is convergent.     

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities ² and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions E. The functions E are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. ³ These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.     

Local existence of the Borel transform in Euclidean φ 4 4

We bound rigorously the large order behaviour of ₄ ⁴ euclidean perturbative quantum field theory, as the simplest example of renormalizable, but non-super-renormalizable theory. The needed methods are developed to take into account the structure of renormalization, which plays a crucial role in the estimates. As a main thorem, it is shown that the Schwinger functions at ordern are bounded byK ⁿ n!, which implies a finite radius of convergence for the Borel transform of the perturbation series.     

Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.     

A first-order equation for spin in a manifestly relativistically covariant quantum theory

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time . The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector n; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of p. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to n (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over n required by unitarity results in timelike motion (as in the expectation value of Dirac's ). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance.     

Divergence of perturbation theory for bosons

Perturbation theory is studied in two dimensional space-time. There all non-derivative boson self-interactions are renormalizable and in each order of perturbation theory there are no divergences, that is all renormalizations are finite in perturbation theory. Thus the unrenormalized perturbation series may be studied and it is shown that any interaction of the general form leads to Green's functions which are not analytic in at =0. This result holds in momentum space at a large set of points, enough to show that the Green's functions are not distributions in the momenta which are analytic in at =0. Furthermore the proper self energy and the two-particle scattering amplitude are shown not to be analytic in at =0 for certain momenta on or below the bare mass shell. In the course of this analysis we use the integral representations for Feynman graphs to derive a minorization of the form |I)p ₁,...,p _e )|>A B ⁿ for the contribution from alln ^th order connected graphs in a theory with an interaction of the form . Then the constantsA andB depend only on the momentap _i , and not on the structure of a particular graph.     

Consequences of the complex character of the internal symmetry in supersymmetric theories

The consequences of the invariance of the superpotential under the complexificationG ^c of the internal symmetry group on the determination of the possible patterns of symmetry and supersymmetry breaking are established in a globally supersymmetric theory. In particular, in the case of global internal symmetry we show that a vacuum associaated to a pointz, whereG _z ^c G _z ^c is always degenerate with a vacuum associated to a pointz, whereG _z ^c =G _z ^c ; all the other degeneracies of the minimum of the potential on an orbit ofG ^c are also determined and shown to be completely removed when the internal symmetry is gauged. The zeroes of theD-term of a supersymmetric gauge theory are characterized as the points of the closed orbits ofG ^c which are at minimum distance from the origin; at these pointsG _z ^c =G _z ^c . It is rigorously proved that the minimum of the potential is zero if the gradient of the superpotential vanishes somewhere. It is also shown that theD-term necessarily vanishes at the minimum of the potential if the direction of spontaneous supersymmetry breaking is invariant byG.Partially supported by the Swiss National Science Foundation and INFN, Sezione di PadovaOn leave of absence from the Department of Physics of the University of Padova, Italy     

Theory of ultrasonic attenuation in metallic systems containing Rare Earth impurities

A theory of ultrasonic attenuation in metallic systems containing crystalline-field split Rare Earth ions is presented. It is shown that two types of absorption mechanisms can occur. The first one is of the well known Pippard type and describes the energy dissipation of the ultrasonic waves directly into the conduction electron system. However in addition to that mechanism there is another one which is of importance and which describes the energy dissipation into the Rare-Earth ion system.In order to distinguish between both mechanisms the change of the attenuation in an applied magnetic field is investigated. For the purpose of demonstration the theory is applied to a ₇– ₈ crystal field level scheme as it applies for Ce³⁺ or Sm³⁺ in cubic symmetry. An estimation of the relative strength of both absorption mechanisms is given.     

jj-Type of coupling in the scheme to construct the energy levels of a shallow acceptor in semiconductors

There is examined the classification of shallow acceptor energy levels by jj-type coupling, whose wave functions are converted by sets of irreducible representations by one of the subgroupsD _4h ^/ ,D _3d ^/ ,D _2h ^/ of the group o _h ^/ . The representations ₅ ⁺ of the groupD _4h ^/ are obtained in the one-function approximation by using a variational method, and systems of eight radical second-order differential equations are solved for two functions ₅ ⁺ in the case of a shallow acceptor in germanium by the method of orthogonal differential factorization. A comparison is made of the two low levels found and their radial functions for each of the functions ₅ ⁺ with the computed levels and the functions of LS-type coupling.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 103–107, September, 1981.The authors are grateful to N. P. Konyukhova for great assistance in the numerical solution of the system of radial equations.     

One-, Two-, and Three-Dimensional Ising Model in the Static Fluctuation Approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator _f ^z withquadratic fluctuations is replaced with its mean value [( _f ^z )² ( _f ^z )²]. This means that the true quantum mechanical spectrum of the operator _f ^z is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.     

The Newtonian limit in a family of metric affine theories of gravitation

A brief review of a first order theory with a quadratic LagrangianL=R+₀R² is presented. It is shown that a test particle follows a geodesic of the metric connection. The theory behaves in the Newtonian limit as the Newtonian theory with a correction which is proportional to the matter density at the field point. This behavior can be produced by a Yukawa potential with an atomic scale characteristic range and a coupling constant proportional to 1/ ². This type of potential is not excluded by the present experimental data.     

Hydrodynamic correlation functions of hard-sphere fluids at short times

The short-time behavior of the coherent intermediate scattering function for a fluid of hard-sphere particles is calculated exactly through ordert ⁴, and the other hydrodynamic correlation functions are calculated exactly through ordert ². It is shown that for all of the correlation functions considered the Enskog theory gives a fair approximation. Also, the initial time behavior of various Green-Kubo integrands is studied. For the shear-viscosity integrand it is found that at densityn³=0.837 the prediction of the Enskog theory is 32% too low. The initial value of the bulk viscosity integrand is nonzero, in contrast to the Enskog result. The initial value of the thermal conductivity integrand at high densities is predicted well by Enskog theory.     

The cluster expansion for potentials with exponential fall-off

Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH ₀+V need be stable in the extended sense thatH ₀+4V+BN0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther ^–1 potential replaced bye ^–ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.This work was supported in part by NSF Grant MPS 75-10751Michigan Junior Fellow     

Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG _(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T _C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk ₂>0. In this equation,k ₁ andk ₂ are functions of the control parameters, temperatureT and pressureP, andk ₁ is allowed to have a positive or negative values as (T, P). Thenk ₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk ₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk ₁ are related to a mutual inverse function. From the above relation, the rate of change ink ₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk ₁, [f(k ₁ ^± )] andf(k ₁ ^± ) is described approximately by ±₁ k ₁ ^± in the transient process thatk ₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok ₁ ^± and that the system is regulated by competition of the forces. an approximate function fork ₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992     

Kinetics of the Late Stage of Disintegration of Supersaturated Solid Solutions with Pronounced Crystalline Structure

The kinetics of disintegration of a supersaturated solid solution is studied in the context of the Lifshits–Slezov–Wagner (LSW) theory for a quasi-coherent interface between the new phase precipitates and the matrix. It is shown that the particle size distribution lies within a very narrow range of relative particle size (r _m/r _c = 6/5) and that the critical or average particle size varies with time as t^1/6 (r _c t ^1/6).     

Theu=0 structure theorem

The previous theorem of the author on the analytic structure of the bubble diagram functions that occur in unitary equations (and are kernels of products of connected scattering operatorsS _m,n ^c or (S ^–1) _m,n ^c , and related quantities), is extended to a class of situations, called here in generalu=0 points, that were not covered by this earlier result.This new theorem, which is proved on the basis of a refined macrocausality condition, resolves one of the remaining crucial problems in the derivation of discontinuity formulae and related results inS-matrix theory: all points are in factu=0 points for some of the bubble diagram functions, such as ((S ^–1) _3,3 ^c S _3,3 ^c ), that are encountered even in the simplest cases. In all previous approaches, ad hoc technical assumptions with no a priori physical basis were required for these terms.The origin of theu=0 problem is the absence of information, in general, on a product of distributions that are boundary values of analytic functions from opposite directions, and more generally on the essential support, or singular spectrum, of a product of distributions whose essential supports contain opposite directions. On the other hand, the recent results obtained by Kashiwara-Kawai-Stapp in the framework of hyperfunction theory apply mainly to phase-space factors, whose bubbles are constants times conservation -functions rather than actual scattering operators. The present work has basically required the development of new physical and mathematical ideas and methods. In particular, a new general result on the essential support of a product of bounded operators is presented inu=0 situations, under a general regularity property on individual terms. The latter follows in the application from the refined macrocausality condition, in the same time as information on the essential support ofS-matrix kernels.     

Post-Newtonian estimation in relativistic optics

A post-Newtonian analysis of the theory of gravity based on the metricg _ij(x,y)= _ij(x)+/c ²(1–1n ²)y _iy_j with the index of refractionn(x, y) is given. A generalized Lagrange space endowed with this metric is used for the study of gravitational phenomena. The index of refractionn(x, y) is expanded in integer powers of the gravitational potentialU=GM/rc ² andv ²/c ². It is shown that solar system tests impose a constraint on a combination of the constant, the post-Newtonian parameters defining the index of refractionn(x, y), and the post-Newtonian parameter associated to the Riemannian metric _ij(x).     

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

An alternative model for the integral Quantum-Hall-Effect

A new model is proposed to explain the integral Quantum-Hall-Effect without invoking electron localisation. It is shown that this quantum effect is present in all two-dimensional systems, but can only be observed if the contributions to the Hall-voltage and total current arising from a possible internal electric potential are supressed. It is argued that this situation is indeed encountered in the devices used in the experiments. The time dependence of current and voltage are calculated and it is found that within linear response theory the Hall-resistanceR _H =U _y /I _x is quantised up to finite size corrections (l ²/L _y ² ) only for filling factors 0<v2, wherel ²=/eB andL _y is the width of the system. For larger filling factors additional small corrections are found. Experiments to test the competing theories are suggested.