Field theory where all Γskjk/i invariants vanish

We investigate whether a nontrivial field theory exists for which all scalars formed from _jk ⁱ (whereg _ij is used to raise and lower indices) can be zero for all points of space and time. We find some examples for which the invariants are all zero for all points at which the field is finite. We also comment upon the problem of boundary conditions, in general.     

The Branch Processes of Chern-Simons (CS) <Emphasis Type="Italic">p</Emphasis>-Branes

In this paper, by making use of Duan’s topological current theory, the branch process of Chern-Simons (CS) p-branes is discussed in detail. Chern-Simons (CS) p-branes are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points and higher degenerated points systematically of the vector order parameter field . Furthermore, it is also shown that CS p-branes are found splitting or merging at the degenerate point of field function but the total topological charges of the CS p-branes are still unchanged.     

The geometry of generalized quantum logics

Let II be a quantum logic; by this we mean an orthocomplemented, orthomodular, partially ordered set. We assume that II carries a sufficiently large collection of states (probability measures). Then, is embedded as a base for the cone of a partially ordered normed spaceL and II is also embedded in the dual order-unit Banach spaceL ^*. We consider conditions on the pairs (, II) and (L,L ^*) that guarantee that II is a dense subset of the extreme points of the positive part of the unit ball ofL ^*. We demonstrate a connection of these conditions in noncommutative measure theory. The assumptions made here are far weaker than the assumptions of the traditional quantum mechanical formalisms and also apply to situations quite different from quantum mechanics. Finally, we show the connections of this theory to the well-known models of quantum mechanics and classical measure theory.     

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.     

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.     

Enhancedt −3/2 long-time tail for the stress-stress time correlation function

Nonequilibrium molecular dynamics is used to calculate the spectrum of shear viscosity for a Lennard-Jones fluid. The calculated zero-frequency shear viscosity agrees well with experimental argon results for the two state points considered. The low-frequency behavior of shear viscosity is dominated by an ^1/2 cusp. Analysis of the form of this cusp reveals that the stress-stress time correlation function exhibits at ^–3/2 long-time tail. It is shown that for the state points studied, the amplitude of this long-time tail is between 12 and 150 times larger than what has been predicted theoretically. If the low-frequency results are truly asymptotic, they imply that the cross and potential contributions to the Kubo-Green integrand for shear viscosity exhibit at ^–3/2 long-time tail. This result contradicts the established theory of such processes.     

Analyticity of effective coupling and propagators in massless models of quantum field theory

For massless models of quantum field theory, some general theorems are proved concerning the analytic continuation of the renormalization group functions as well as the effective coupling and the propagators. Starting points are analytic properties of the effective coupling and the propagators in the momentum variablek ², which can be converted into analyticity of - and -functions in the coupling parameter . It is shown that the -function can have branch point singularities related to stationary points of the effective coupling as a function ofk ². The type of these singularities of () can be determined explicitly. Examples of possible physical interest are extremal values of the effective coupling at space-like points in the momentum variable, as well as complex conjugate stationary points close to the realk ²-axis. The latter may be related to the sudden transition between weak and strong coupling regimes in quantum chromodynamics. Finally, for the effective coupling and for the propagators, the analytic continuation in both variablesk ² and is discussed.On leave from the Max-Planck-Institut für Physik und Astrophysik, D-8000 München, Federal Republic of Germany     

Generalized Apollonian packings

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,) as generators:R andT that are hyperbolic rotations of 2/3 and 2/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique     

Fixed points of Feigenbaum's type for the equationf p (λx)≡λf(x)

Existence and hyperbolicity of fixed points for the mapN _p :f(x) ^–1 f ^p (x), withf ^p p-fold iteration and =f ^p (0) are given forp large. These fixed points come close to being quadratic functions, and our proof consists in controlling perturbation theory about quadratic functions.Supported in part by the Swiss National Science Foundation     

Mathematical and physical aspects of gauge theories

Conclusion It is now a broadly-admitted fact that gauge theories provide intense research stimulation for both mathematicians and physicists. We tried to show that, as far as bundles and connections are the adequate minimal concepts to formalize the very act of beings which produce their own space in a consistent way, or, which is the same, the very act of informing a basis world B through a parametrized space of operations E, gauge theories realize a true translation between geometrical and physical information.In Section I.A, we tried to show how the concept of communication via experimentation finds this equivalence and makes a necessary collision between mathematics and physics.We gave some striking theoretical and practical facts: the obtaining of interaction Lagrangians, Higgs phenomenon and good energy behaviour of graphs arising from broken symmetries, the miraculous encounter between connection and path integral formalism. We could have added the relation with twistor theory, the theory of strong interactions, super gravity theories where the basis world B is not even a manifold. Some authors even speak of geometrodynamics and pregeometry [21] and would like to understand physics as a cosmogony.In a way, this makes us understand Einstein's unsatisfied mind when he wrote down his famous equation. The equivalence between a curvature tensor and an impulsion-energy tensor requires a careful analysis of the act of setting consistent spaces (via experimentation) when one doesn't reduce material beings (monads) to material points ruled by the universal Newton's observer.So, as far as gauge theories pretend to found and explain the process of translation between Geometry and Physics, they provide an extremely solid epistemological basement for the so-called mathematical Physics.     

C=1 conformal field theories on Riemann surfaces

We study the theory ofc=1 torus and ₂-orbifold models on general Riemann surfaces. The operator content and occurrence of multi-critical points in this class of theories is discussed. The partition functions and correlation functions of vertex operators and twist fields are calculated using the theory of double covered Riemann surfaces. It is shown that orbifold partition functions are sensitive to the Torelli group. We give an algebraic construction of the operator formulation of these nonchiral theories on higher genus surfaces. Modular transformations are naturally incorporated as canonical transformations in the Hilbert space.     

Can atomic processes be described by non-linear wave mechanics in space-time?

Generalizing from a classical application of the paradigm of Elementary Measurement discussed elsewhere (Leiter, 1969), we consider a non-linear, spinor wave-mechanical field theory of Elementary Measurement. In this theory, charged particles are represented by complex spinorc-number fields interacting through their associated electromagnetic fields in space-time. The paradigm of Elementary Measurement implies that the particle fields, and their associatedc-number electromagnetic fields, are interdependent degrees of freedom in an action principle associated with the measurement interaction, and are not elementary in themselves. Making the action stationary with respect to the interacting field degrees of freedom gives the equations of motion of the measurement.The application of this model theory to atomic hydrogen yields the result that the inherent limit cycle solutions (LCS) of the non-linear measurement equations correspond to the quantum levels of conventional relativistic Dirac quantum mechanics of hydrogen, in the approximation that the nucleus has infinite mass.Superpositions of these Dirac-LCS solutions have the property of collapsing (reduction of the wave packet) into one of the LCS in the superposition,in a characteristic time which is identical to the lifetime of the associated atomic levels as calculated from conventional quantum mechanics. Hence, in thisc-number electromagnetic theory,both spontaneous and induced transitions can be accounted for. Photons, in this theory, are not elementary particles, but instead are associated with the secondary dynamics related to the inherent nonlinear structure in the elementary measurement equations of motion. The hidden variable characteristics of this measurement theory (as seen from the point of view of ordinary quantum mechanics), in describing a universe made up of such hydrogen atoms, is discussed. Within this context, a consistent derivation of the Planck blackbody radiation formula is given, in which the associated electromagnetic fields arec-numbers and arenot second quantized. Finally, a generalization of this prototype model theory, to a more consistent form which can account for the presence of vacuum interaction processes and negative energy states, is suggested.Preliminary study for this work was done at the Albert Einstein Institute of Physics, Technion, in Haifa, Israel, during the summer of 1965.     

Are there critical points on the boundaries of singular domains?

Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree 2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z ⁿ +a, withn,n2 anda. We also show that the singular disks, associated to periodic elliptic points off(z)=e ^az that satisfy a diophantine condition, are unbounded in . In the end of the paper, we give a survey of the theory of iteration of entire functions of .     

The quantum pentacle

Quantum set theory permits the formulation of a quantum simplicial topology suitable for a quantum theory of time space and gravity without prior time space structure. The quantum simplex differs strikingly from the classical: It is isotropic (points in all directions) and all quantum simplexes of the same signature are congruent. Quantum simplexes and complexes are described byS numbers, elements of the Clifford algebra of quantum sets. The isotropy groups of noncontiguous simplexes commute, like local invariance groups in a gaugeinvariant theory.     

Theory of the diffusive glide of dislocations with narrow kinks

A theory of glide velocity of a dislocation with narrow kinks which encounter spatially periodic, relatively high energy barriers is developed. The thermally activated generation of double kinks is considered as the mechanism of the dislocation movement. It is assumed that the saddle point is determined by the elastic interaction between the two kinks and that diffusion of the kinks is rate controlling. According to this theory velocities of screw dislocations in -iron are calculated in dependence on temperature and the applied stress with 2E _k =0·68 eV andE=0·07 eV (E _k is the energy of an isolated kink,E is the second-order Peierls energy). Relations to three other theories, which may be considsred for calculation of velocities of screw dislocations in b.c.c. metals are discussed and demonstrated by numerical calculations for iron. It appears that there are no serious objections suggested by experiments which might be raised against the screw dislocation velocities in iron calculated according to the presented theory.     

Estimates of general Mayer graphs. II. Long-range behavior of graphs with two root points occurring in the theory of ionized systems

We find the asymptotic behavior of general Mayer 2-graphs (Mayer graphs with two root points), which occur in the theory of ionized systems. This problem arises when one wants to compute corrections to the Debye length for large values of the plasma parameter. For a given 2-graph (r) with Debye-Hückel linese ^– /r, we prove the inequalitiesC _m r ^– e ^– (r) (r ₀)C_Mr^3k–l e ^– , for anyrr ₀, and whereC _m andC _M are positive and finite constants which depend only on . These bounds are finite whenever (r) is not infinite everywhere. The integersl, k, and denote, respectively, the number of lines of the graph , its number of field points, and its local line connectivity (the maximum number of chains linking the root points, which have no line in common). From this result, we deduce that the simple irreducible 2-graphs dominant at large distances decay exponentially likee ^– and have an isthmus between the root points (an isthmus is a line whose deletion separates the graph into two disjoint components, each one containing a root point). We prove also that 2-graphs that have a number of linesl > 3k+ are infinite. We exhibit simple, irreducible prototypes satisfying this condition, for anyk 6. This implies that the Abe-Meeron theory of ionized gases as applied to a classical plasma is not free from divergences. Finally, we extend the preceding results to 2-graphs with lines F_L=(e ^– /r)^k L, withk _L real positive. We prove that they still decay exponentially likee ^– , where is now the maximal flow in a network associated to by assigning the capacityk _L to each lineL.     

Analytic fields on Riemann surfaces. II

The properties of analytic fields on a Riemann surface represented by a branch covering of ¹ are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of operators forZ _n -symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.     

Macrocausality,unitary and physical-region structure of the multiparticleS matrix

The physical-region analytic structure of theS-matrix is established on the basis of refined macrocausality, unitary and a weak supplementary analyticity assumption for a 33 process below the four-particle threshold, and away from points where two initial, or two final, four-momenta are colinear: the connectedS matrix is decomposed as a sum of contributions that generalize Feynman graphs and are associated with the various +-Landau surfaces encountered (=surfaces of graphs with one internal line and triangle graphs). The property of separation of singularities in unitarity equations, which was used as a crucial ad hoc assumption in traditional derivations of related results inS-matrix theory, is obtained as a byproduct in the region considered.     

Critical Points and Supersymmetric Vacua, III: String/M Models

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates.Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle over the moduli space of complex structures on X × T ² with respect to the Weil-Petersson connection. Flux superpotentials form a lattice of full rank in a 2 b ₃(X)-dimensional real subspace . We show that the density of critical points in for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1,DSZ2,AD,DD1].Research partially supported by DOE grant DE-FG02-96ER40959 (first author) and NSF grants DMS-0100474 (second author) and DMS-0302518 (third author).     

Origin of rate dependence in frictional sliding

Experiments indicate that frictional resistance to sliding between macroscopic, clean, dry surfaces depends on the average rateV at which the surfaces are translated relative to each other. Using a new lattice automaton, we obtain results suggesting that rate-dependent macroscopic dynamics may arise from microscopic interactions between contact points which decay from a metastable state with a finite lifetime. Sliding is accommodated by clusters, or avalanches, of failed lattice contact points, and corresponds to successive quenches into the metastable state by an electromechanical loading system with a finite response time. Under the quasistatic assumption , rate dependence is a consequence of the increase in correlation length _d of clusters of failed lattice points as quench rate increases. Special cases of the model are isomorphic to the selforganized criticality model for sandpiles, and to block-spring models of the type first studied by Burridge and Knopoff for earthquakes.