Quantum field theory and the coloring problem of graphs

The ^k theory is compared with the multilinear theory of scalar fields ₁, ₂, ..., _k having the same mass as that of . In particular, it is shown that Feynman integrals encountered in the ³ theory are not necessarily present also in the _{1 2 3} theory, but they are if they correspond to planar Feynman graphs having no tadpole part. Furthermore, a necessary and sufficient condition for the presence of a ³ Feynman integral in the _{1 2} ² theory is found. Those considerations are applications of graph theory, especially of the coloring problem of graphs, to Feynman graphs.     

Boson fields under a general class of cut-off interactions

We consider a boson field (x) under an interaction of the form V( _k (x))dx, where _k (x) is the momentum cut-off field, andV() is a continuous bounded function. Under a weak regularity condition onV(), we prove that the total energy operator is self adjoint, that the asymptotic fields exist and that the scattering operator exists.     

Strings and the Gauge Theory of Spacetime Defects

We present a new topological invariant todescribe space-time defects which is closely related tothe torsion tensor in a Riemann–Cartan manifold.By virtue of the topological current theory and-mapping method, we show that there must existmultistring objects generated from the zero points ofthe -mapping. These strings are topologicallyquantized. The topological quantum numbers are thewinding numbers described by the Hopf indices and the Brouwerdegrees of the -mapping.     

The physical properties of linear and action-angle coordinates in classical and quantum mechanics

The quantum harmonic oscillator is described in terms of two basic sets of coordinates: linear coordinates x, p_x and angular coordinates eⁱ, P (action-angle variables). The angular coordinate eⁱ is assumed unitary, the conjugate momentum p is assumed Hermitian, and eⁱ and p are assumed to be a canonical pair. Two transformations are defined connecting the angular coordinates to the linear coordinates. It is found that x, p_x can be physical, i.e., Hermitian and canonical, only under constraints on the p eigenvalue spectrum. The conclusion is that eⁱ can be a unitary operator. A parallel analysis of the classical harmonic oscillator is done with equivalent results.     

A rigorous block spin approach to massless lattice theories

The renormalization group technique is used to study rigorously the ()⁴ perturbation of the massless lattice field in dimensionsd2. Asymptoticity of the perturbation expansion in powers of is established for the free energy density. This is achieved by using Kadanoff's block spin transformation successively to integrate out high momentum degrees of freedom and by applying ideas previously used by Gallavotti and Balaban in the context of the ultraviolet problems. The method works for arbitrary semibounded polynomials in and .Supported in part by the National Science Foundation under Grant No. PHY 79-16812On leave from Department of Mathematical Methods of Physics, University, PL-00-682 Warsaw, Poland     

Maximal violation of Bell's inequalities is generic in quantum field theory

Under weak technical assumptions on a net of local von Neumann algebras {A(O)} in a Hilbert space , which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector state in there exist observables localized in complementary wedge-shaped regions in Minkowski space-time that maximally violate Bell's inequalities in the state . If, in addition, the algebras corresponding to wedge-shaped regions are injective (which is known to be true in many examples), then the maximal violation occurs in any state on () given by a density matrix.     

Adjoint solutions of the Brans-Dicke equations

It is shown that if the Brans-Dicke equations have the solution,g _ij generated by the trace free sourceT _n (T-O) then there exists an adjoint solution ^–1, ²g_ij of these equations generated by the source ^-2 T _u. An example is considered.     

Some topological configurations in gauge theories

A Higgs vacuum field is characterized by the set of conditionsD =0, which lead to a generalized Meissner effect and partially determine the vector potentialA in terms of . Applying this method to the Weinberg-Salam theory, we assert that there exist stringlike configurations in which a pair of magnetic poles are bound by a flux string of theZ ⁰ field, with an energy scale in the TeV range. We also point out that pure gauges in non-Abelian gauge theories are not well-defined due to topological singularities. In order to be meaningful, they must be enlarged to a class of almost pure gauges which include the various known topological configurations.Work supported in part by the U.S. Department of Energy, San Francisco Office, and by the Sherman Fairchild Distinguished Scholars Program.On leave from the University of Chicago.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

Traces,Dispersions of States and Hidden Variables

No Heading The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let be a state on a C^*-algebra A with the projection structure P(A). The dispersion () is defined as () = sup{(p) – (p)² | p P(A)}. It is proved that () 2/9 whenever is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.     

Further extension of the Gauss-Hertz principle

The continuum form of the Gauss-Hertz principle is extended to include the time domain as well as space. The Schrödinger equation and general relativity are derived by this method. The equivalence of the principle is shown to that of the Hamiltonian method where the energy is the expression –[ ² +A·² A], with being the difference between the acceleration potential and potential energy density, andA being the difference between the vector potentials of the acceleration field and the force field. The goal of Hertz to demonstrate a third arrangement of the principles of mechanics...which starts with... time, space and mass has apparently been achieved for relativity and for quantum mechanics, in addition to those classical equations previously found.     

Bounds on the coupling constant in two dimensional boson models

We consider two dimensional boson field theories with an interaction potentialV (). We show how to define a cut-off, renormalised Hamiltonian for a certain class of non-polynomialV (), which are defined via an integral transform. We formulate precisely a variational argument devised by Coleman, obtaining a constraint on the coupling constant of the theory with generalV (), and illustrate the argument with several examples.     

Group theory and solutions of classical field theories with polynomial nonlinearities

In this paper we investigate a number of analytical solutions to the polynomial class of nonlinear Klein-Gordon equations in multidimensional spacetime. This is done in the context of classical ⁴ and ⁶ field theory, the former with and without the inclusion of an external force field conjugate to . Both massive (m0) and massless (m=0) cases are considered, as well as tachyonic solutions allowed (v>c). We first present a complete set of translationally invariant solutions for the ⁴ model and demonstrate the role of external force fields in altering the form of these solutions. Next, spherically symmetric solutions are discussed in both ⁴ and ⁶ cases since they provide the most realistic models of elementary particles.     

Instantaneous Fluorescence Quantum Yield—A New Quantity with Characteristic “Fingerprints” for Excited-State Processes

The instantaneous fluorescence quantum yield _ins—a new quantity for fluorescence studies defined as the ratio of the fluorescence intensity to the optical density, both measured at the moment of the maximum of the exciting pulse—proves to be a very sensitive function for excited-state processes. Dependent on the excitation intensity _ins exhibits characteristic features (maxima/minima) indicating, for example, excited-state absorptions and annihilation processes. _ins is therefore more informative as the intensity dependence of the usually utilized fluorescence yield, the information content of which is restricted because this function is hardly structured. In the paper the influences of specific molecular parameters (excited-state absorption cross section, annihilation constant) on _ins are given, problems of the experimental accessibility of _ins are discussed, and an experimental setup for determination of this new quantity is presented. The application of the method is demonstrated for identification of excited-state absorptions of organic molecules in solution.     

Branched polymers in a wedge geometry in three dimensions

We investigate the statistical and dimensional properties of uniform star polymers attached by the branching vertex of degreef in a wedge geometry in three dimensions and described by the wedge angles and. We show that the growth constant is equal to ^f , where is the self-avoiding walk limit. Thef and (, ) dependences of the corresponding critical exponent _f (, ) are studied using Monte Carlo techniques. In the casef=1, our results are compared with existing predictions obtained from series expansion and renormalization group methods. We have also estimated the amplitudes for the mean square radius of gyration and the mean square end-to-end branch length. Our results for the ratio of the mean square radius of gyration of anf-star to that of a linear polymer of the same degree of polymerization attached in a similar wedge, and the analogous ratio for the mean square end-to-end branch length, are consistent with these ratios being lattice-independent quantities.     

The ɛ-expansion for the Hierarchical Model

In this paper, the Hierarchical Model is studied near a non-trivial fixed point of its renormalization group. Our analysis is an extension of work of Bleher and Sinai. We prove the validity of the -expansion for . We then show that the renormalization transformations around have an unstable manifold which is completely characterized by the tangent map and can be brought to normal form. We then establish relations between this result and the critical behaviour of the model in the thermodynamic limit.     

A remark on a paper of J. F. Aarnes

LetA be aC*-algebra on the separable Hilbert space , and let be the von Neumann generated byA. LetG be a topological group anda(a) a representation ofG into the group of *-automorphisms ofA. Suppose that each (a) extends to a *-automorphism of , and suppose thata(a)(T)x, y is continuous for eachT inA andx, y and . Then, for a large class of groupsG, one has automatically thata(a)(T)x,y is continuous for allT in andx, y in .Supported in part by NSF Grant GP-9141.     

Correlation functionals of infinite volume quantum spin systems

The existence and analyticity of the correlation functionals of a quantum lattice in the infinite volume limit is proved. The result is valid at sufficiently high temperatures and for a large class of interactions. Our method estimates the kernelK for a set of Kirkwood-Salzburg equations. While a naive estimate would indicate that K =, we take into account cancellations between different contributions toK in order to show that for sufficiently high temperatures K <1, and this estimate is independent of the volume of the system.Supported in part by the Air Force Office of Scientific Research.     

Über die an „Schwachen“ Grenzflächen Entstehenden Kopfwellen

Zusammenfassung In der vorliegenden Arbeit werden die Kopfwellen untersucht, die beim Auffallen einer harmonischen Schall-Kugelwelle auf die Ebene Grenzfläche zwischen zwei flüssigen Medien entstehen. Vorausgesetzt wird, daß die Grenzfläche schwach ist, d. h. daß die Geschwindigkeit der Schallwellen im Medium mit der Erregungsquelle nur um weniges kleiner ist als die Geschwindigkeit der Schallwellen im Medium ohne die Quelle.

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Topological Quantization of Magnetic Monopoles and Their Bifurcation Theory

Using SU(2) gauge field theory and the-mapping method, we quantize the magnetic monopolesat the topological level and determine their quantumnumbers by the Hopf indices and Brouwer degrees of the -mapping. Then, based on the implicitfunction theorem and Taylor expansion, we study theorigin and bifurcation theories of magnetic monopoles atthe limit points and bifurcation points (includingfirst-order and second-order degenerate points),respectively. We point out that a magnetic monopole cansplit into at most four particles at one time.