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.     

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.     

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 (φ4)2 field theory as a classical Ising model

We approximate a (spatially cutoff) (⁴)₂ Euclidean field theory by an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions. This approximation is used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (⁴)₂ theory. Application of these results are presented.     

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.     

The continuous-spin Ising model,g0∶φ4∶d field theory,and the renormalization group

We have used the method of high-temperature series expansions to investigate the critical point properties of a continuous-spin Ising model and g₀⁴_d Euclidean field theory. We have computed through tenth order the hightemperature series expansions for the magnetization, susceptibility, second derivative of the susceptibility, and the second moment of the spin-spin correlation function on eight different lattices. Our analysis of these series is made using integral and Padé approximants. In three dimensions we find that hyperscaling fails for sufficiently Ising-like systems; the strong coupling limit of g₀⁴₃ depends on how the ultraviolet cutoff is removed. The level contours of the renormalized coupling constant for this model in theg ₀, correlation-length plane exhibit a saddle point. If the ultraviolet cutoff is removed beforeg ₀ , the usual field theory results and the renormalization-group fixed point with hyperscaling is obtained. If the order of these limits is reversed, the Ising model limit where hyperscaling fails and the field theory is trivial is obtained. In four dimensions, we find that hyperscaling fails completely; g₀⁴₄ is trivial for all g₀ when the ultraviolet cutoff is removed.Work supported in part by the U.S. Department of Energy.     

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.     

BRST analysis of physical states for 2D gravity coupled toc≦1 matter

We consider 2D gravity coupled toc1 conformal matter in the conformal gauge. The Liouville system is represented by a free scalar field, ^L , with background charge such that the BRST operator imposing reparametrization invariance is nilpotent. We compute the cohomology of this BRST charge on the product of the Fock space of ^L with those of the ghosts and one other free scalar field, ^M representing the matter system. From this calculation the physical states of the full theory are determined. For thec<1 case the further projection from the Fock space of ^M to the irreducible representation, using Felder's resolution, reproduces the results of Lian and Zuckerman.Supported by the NSF Grant # PHY-88-04561Supported in part by the Department of Energy Contract # DE-FG03-84ER-40168 and by the USC Faculty Research and Innovation Fund     

Critical acceleration of lattice gauge simulations

We present a stochastic cluster algorithm that drastically reduces critical slowing down forZ ₂ lattice gauge theory in three dimensions. The dynamical exponentz is reduced fromz>2 (standard Metropolis algorithm) tozO.73. The Monte Carlo pseudodynamics acts on the gauge-invariant flux tubes that are known to be the relevant large-scale low-energy excitations. A comparison of our results with known results for the 3D Ising model and ⁴ model supports the conjecture of universality classes for stochastic cluster algorithms.     

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.     

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.     

Remarks on the quantum field theory in lattice space. I

We calculate the Gelfand functionsE(f,g;a) for quantized field in lattice space,a being the lattice constant. In the limita 0 the functionals take on two different forms depending upon the potentialF[] of the lattice Hamiltonian (coupling between different lattice sites not included). IfF[] is of a short-range type (see text for definition) the limit functional is Gaussian. The corresponding representation of CCR is reducible and its realization apparently non-unique unlessF[] is quadratic. The most natural realization is to represent the field as a linear combination of Fock fields whose masses are given by the excitation energies of the lattice Hamiltonian. IfF[] is of a long-range type, the limit functional takes the more general form once studied byAraki.     

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.     

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.     

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t) ^d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeL ^d of the system. This lack of self-averaging is tested for both the Ising model and the ⁴ model on the square lattice. Both models exhibit the same lawl(t)=(Rt) ^x withx=1/2, although the ⁴ model has soft walls. However, spurious results withx1/2 are obtained if bad pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 10³). We also predict a critical singularity of the rateR(1–T/T _c) ^v(z–1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureT _c itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.     

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.     

Analytic structure of gauge fields

The analytic structure of gauge fields in the presence of fermions is studied in arbitrary symmetry. A Hamiltonian formalism is developed which relates Cauchy-Riemann equations to the symmetry. The formalism is applied to three problems in (2+1)-dimensional Euclidean space: (1) a free fermion, (2) a fermion interacting with a massless scalar field, and (3) a fermion interacting with a vector field. We find that the Hamiltonian for the free fermion is analytic and single-valued in a finite region of momentum space. With the addition of an auxiliary field, the Hamiltonian can be analytic in the entire momentum space. The scalar field then acquires spin-dependent coordinates by interaction with the fermion; the interactions break the Abelian symmetry of so that ₁ 1/(x₁ –-im ₁ ^–1 (x₁ –-im ₁ ^–1 ), wherem ₁ are spin-dependent and multivalued. There are four solutions for each chirality eigenvalue of the fermion. For spinless fermions gives the Jackiw-Nohl-Rebbi solution and is separable into Coulomb-like 1/x analytic functions on the first and fourth quadrants. For a vector field the results are similar except that the coordinates are not spindependent or multivalued; interactions break the initial symmetry andA (x )A ¹ (x ) and theA ¹ have a non-Abelian algebra. Thel indices represent directions fixed by spin matrices in a spin-dependent color space.     

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.     

Phase transformations in quaternionic quantum field theory

Phase transformations of a scalar quaternionic quantum field are examined as unitarily implemented symmetries. Under very general quantization conditions it is shown, in both global and local cases, that the only sensible phase invariance that has been suggested is pp ^–1, wherep is a quaternion and a quaternionic scalar field.     

Renormalization Group study of a critical lattice model

The Renormalization Group is used to study the correlation functions of a nonlocal hierarchical model mimicking the ()⁴ model, dipole gas and the like. It is shown that the infrared behaviour of the correlations is that of the massless gaussian 1/2c()()².