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.     

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.     

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     

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.     

Anisotropic fiber with cylindrical polar axes

The newly proposed anisotropic fiber structures with cylindrical polar principal axes appear to be an interesting novel class of special lightguides. In this paper, some interesting results relating to such fibers are derived which, to the knowledge of this author, have not yet been reported in the literature. It is found that, ifn _c0 ( )^1/2>n _zc0( _r )^1/2, TE₀₁ will be the fundament mode with a range of single-mode operation given by 2.61n _zc0 (2 _r )^1/2 /a<2.61n _c0(2 )^1/2. On the other hand, ifn _zc0 ( _r )^1/2>n _c0( )^1/2, then TM₀₁ becomes the fundamental mode whose single-mode operation range is 2.61n _c0 (2 )^1/2 /a<2.61n _zc0(2 _r )^1/2.     

A discrete spectrum of solutions of the wave equation with strong cubic nonlinearity

The nonlinear wave equation, _tt –+³=0, has many solutions that are periodic in time and localized in space, all with infinte energies. The search for spherically symmetric solutions that are well represented by the simple approximation, (r, t)A(r) sin t, leads to a discrete spectrum of solutions{ _N (r, t; )}. The solutions are nonlinear wavepackets, and they can be regarded as particles. The asymptotic theory () of the motion of the guiding center of theNth wavepacket, in the presence of a specified potential, is characterized by an infinite mechanical mass and an infinte interaction mass, and they are compatible. The rest mass in the classical relativistic mechanics of guiding centers ism ₀ c ²= _N ; i.e. the spectrum { _N } determines a spectrum of Planck's constants.On leave (1972–73) Université de Paris VI, Département de Mécanique, 75 Paris 5^e, France.     

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.     

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.     

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

A subalgebraM ₀ of a von Neumann algebraM is called weakly sufficient with respect to a pair (,) of states if the relative entropy of and coincides with the relative entropy of their restrictions toM ₀. The main result says thatM ₀ is weakly sufficient for (,) if and only ifM ₀ contains the Radon-Nikodym cocycle [D,D] _t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.     

Theory of fiber bundle with local metric of internal space and gravitation with torsion

In this paper we propose a new theory of a fiber bundle provided with a local metric of internal space. The fibers differ from usual fibers, having an enlarged factor. The enlargement may be procured by a differential mapping(x) from structure groupG to the fiberF _x atx M, and(x)R. The torsion presented stems from the local metric of internal space and the local metric stems from a induced mapping _*(x) of(x). From the theory we can get the Brans-Dicke theory with torsion. If we assume the spin density of the gauge field determines the enlarged factor of the fiberF _x, our theory is an extended Cartan theory.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

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.     

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.     

Cosmological Dynamics of the Tachyon with an Inverse Power-Law Potential

We investigate tachyon dynamics with an inverse power-law potential V () ^–. We find global attractors of the dynamics leading to a dust behavior for > 2 and to an accellerating universe for 0 < 2. We study linear cosmological perturbations and we show that metric fluctuations are constant on large scales in both cases. In the presence of an additional perfect fluid, the tachyon with this potential behaves as dust or dark energy.     

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.     

The Volume Element of Space-Time and Scale Invariance

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S= L ₁ d ⁴ x+ L ₂ d ⁴ x where the volume element d ⁴ x is independent of the metric. For global scale invariance, a dilaton has to be introduced, with non-trivial potentials V()=f ₁ e in L ₁ and U()=f ₂ e ² in L ₂ . This leads to non-trivial mass generation and a potential for which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for Quintessential models, which are scale invariant but formulated with the use of volume element d ⁴ x alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants.     

Radiative kinks and fermionic zero modes in a (λφ6)1+1 model

We consider bidimensional scalar models including kink solutions _k (x). Using the hidden supersymmetric properties of the Dirac equation, we describe a general method to find normalizable fermionic zero modes. In particular, we apply the technique to a ( ⁶)₁₊₁ model. Going to the one-loop order of the effective potential, the emergence of a radiative kink provides an interesting scalar background in order to discuss the Dirac equation.     

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     

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.