Peter G. O. Freund

Dynamics of dimensional reduction

In d-dimensional unified theories that, along with gravity, contain an antisymmetric tensor field of rank s-1, preferential compactification of d-s or of s space-like dimensions is found to occur. This is the case in 11-dimensional supergravity where s = 4.

Adelic string amplitudes

Abstract We show that the Veneziano and Virasoro-Shapiro four-particle scattering amplitudes can be factored in terms of an infinite product of non-archimedean string amplitudes. This factorization is equivalent to the functional equation for the Riemann zeta function.

Topology of Higgs fields

It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ? a = φ a (φ b φ b )−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.

p-Adic numbers in physics

Abstract The boundary of the ordinary open string world sheet is the real line. Along with the usual open string, one can consider p -adic open strings whose world sheet has as boundary the p -adic line instead (the points on this boundary are labelled by p -adic numbers rather than real numbers). The world sheet itself is then no longer a continous manifold but becomes a discrete homogeneous Bethe lattice, or Bruhat-Tits tree of incidence number p + 1. The p -adic strings thus correspond to specific discretizations of the world sheet. Studying all these discretizations (one for each prime p ) together, as suggested by number theory, sheds new light on the ordinary string, via adelic product formulae. One is also led this way to a new type of string, the adelic string. These issues and their generalization to closed strings are discussed after a self-contained mathematical introduction. Other physical systems with natural p -adic counterparts are considered and in their context the connection between p -adic theories and quantum groups is explored.

Kaluza-Klein Cosmologies

Abstract In generalized Kaluza-Klein theories the scale set by the size of the extra space-dimensions is close to the grand unification scale of supersymmetric GUTs with minimal number of Higgs supermultiplets. In view of this observation, we explore cosmologies in which the “effective” dimensionality of space depends on time. Such cosmologies are studied in higher-dimensional Jordan-Brans-Dicke theories, and in 10- and 11-dimensional supergravity. The preferential expansion of three space-like dimensions is noted in the latter theory. Cosmology in pure higher-dimensional Einstein theory, where there is no preferential expansion of three space-like dimensions, has been discussed by Chodos and Detweiler.

Non-archimedean string dynamics

Abstract Explicit formulas for the N-point tree amplitudes of the non-archimedean open string are derived. These amplitudes can be generated from a simple non-local lagrangian involving a single scalar field (the tachyon) in ambient space-time. This lagrangian is studied and is found to possess a tachyon free vacuum with no “particles” but with soliton solutions. The question of generalizing the adelic product formular to N-point amplitudes is taken up. The infinite product of 5-point amplitudes is shown to converge in a suitably chosen kinematic region whence it can be analytically continued. Though the precise form of the product formula for the 5-point (and N-point)amplitudes is not found, it is shown that the product is not equal to one as it is for the 4-point amplitudes but rather involves the famous zeros of the Riemann zeta function. Chan-Paton rules for non-archimedean open strings are given. A string over the (global) field of rational numbers is constructed. Other problems that are addressed are the introduction of supersymmetry, the nature of a p-adic string lagrangian, and the possibility of strings over other locally compact fields.

Non-archimedean strings

Abstract A full set of factorized, dual, crossing-symmetric tree-level N -point amplitudes is constructed for non-archimedean closed strings. Momentum components and space-time coordinates are still valued in the field of real numbers, quantum amplitudes in that of complex numbers. It is the world-sheet parameters, which one integrates over, that become p -adic. For the closed string the parameters are valued in quadratic extensions of the fields Q p of p -adic numbers (p = prime).

DISCRETE SCALE INVARIANCE IN STOCK MARKETS BEFORE CRASHES

We propose a picture of stock market crashes as critical points in a system with discrete scale invariance. The critical exponent is then complex, leading to log-periodic fluctuations in stock market indexes. We present “experimental” evidence in favor of this prediction. This picture is in the spirit of the known earthquake-stock market analogy and of recent work on log-periodic fluctuations associated with earthquakes.

S-matrices for perturbations of certain conformal field theories

Abstract We present a family of factorizable S -matrix theories in 1+1 dimensions with an arbitrary number N of particles of distinct masses, and find the conservation laws of these theories. An analysis of the conservation laws of the family of nonunitary CFTs with central charge c = c 2,2 N +3 = −2 N (6 N +5)/(2 N +3) perturbed by the φ (1,3) operator, leads us to conjecture the identification of these perturbed CFTs with the S -matrix theories we found. The case N =1 was treated by Cardy and Mussardo. We also present the S -matrix of an E 7 -related unitary model.

Local scale invarlance and gravitation

Abstract A gauge field theory of (local) scale invariance is constructed. When coupled to gravity it yields the old theory of Weyl. A Brans-Dicke scalar field is introduced a la Dirac. With a suitable choice of parameters the theory then reduces in a special gauge to Einsteins theory even in the presence of matter. If the gauge field of scaling (even under charge conjugation) is identified with the gauge field of an internal symmetry, a (spontaneous) breakdown of charge conjugation invariance (and of CP ) can follow. This effect has a reasonable order of magnitude if gravity is replaced by strong gravity and the internal charge identified with lepton number of baryon number, or some neutral weak charge. Certain difficulties arise, however, if one insists on the Weyl identification of the gauge field of scaling with the electromagnetic field. It is shown that the observed tensor mesons ( f , f ′, etc.), while basic in tensor meson dominance, are not the quanta of a gauge field of strong gravity.