Moshe

Quantum field theory in the large N limit: A Review

Abstract We review the solutions of O ( N ) and U ( N ) quantum field theories in the large N limit and as 1/ N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large N , the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar U ( φ 2 ) field theory for N large and discuss various non-perturbative physical issues such as critical behaviour. We show how large N results can also be obtained from variational calculations. We illustrate these ideas by showing that the large N expansion allows to relate the ( φ 2 ) 2 theory and the non-linear σ -model, models which are renormalizable in different dimensions. Similarly, a relation between CP ( N −1) and abelian Higgs models is exhibited. Large N techniques also allow solving self-interacting fermion models. A relation between the Gross–Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows. We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry. This also serves as an introduction to the study of the 3D supersymmetric quantum field theory. Large N methods are useful in problems that involve a crossover between different dimensions. We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories. We also use large N methods to investigate the weakly interacting Bose gas. The solution of the general scalar U ( φ 2 ) field theory is then applied to other issues like tricritical behaviour and double scaling limit.

Critical Exponents for the Reggeon Quantum Spin Model

Abstract The Reggeon quantum spin (RQS) model on the transverse lattice in D dimensional impact parameter space has been conjectured to have the same critical behaviour as the Reggeon field theory (RFT). Thus from a high “temperature” series of ten ( D = 2) and twenty ( D = 1) terms for the RQS model we extrapolate to the critical temperature T = T c by Pade approximants to obtain the exponents η =0.238±0.008, z =1.16±0.01, v =1.271±0.007 for D =2 and η =0.317±0.002, z =1.272±0.007, v =1.736±0.001, λ =0.57±0.03 for D =1. These exponents naturally interpolate between the D =0 and D =4− e results for RFT as expected on the basis of the universality conjecture.

Recent developments in Reggeon field theory

Abstract In the last few years Reggeon Field Theory (RFT) has been developed into a very powerful tool for analyzing the complex angular momentum structure of high energy scattering amplitudes. In a previous review of this subject (Abarbanel et al. [8]) RFT was motivated, formulated, and its early results were discussed. These issues will be mentioned only briefly in the introduction to the present review in order to make it self-contained. It will then be followed by a discussion of the recent developments and progress in the field. A major part of this review is devoted to the discussion of various efforts made to elucidate the s -channel content of Reggeon Field Theory. The formulation of RFT on a lattice and the analysis of production amplitudes are emphasized as the main approaches towards understanding the implication of RFT on s -channel unitarity issues. The path integral and the Hamiltonian formulations of RFT on a lattice, along with their results, are treated in detail. The work done on production amplitudes, in particular cut Reggeon field theory, its derivation and applications, are presented. The problem of the approach to scaling, the evaluation of the transition energy and scaling functions are discussed as they emerge from the representations of Reggeon Greens functions. Practical problems such as the confrontation of the RFT results with the experimental data and the relevance of different approximations are also analyzed.

Nonperturbative physics from interpolating actions

Abstract We study the expansion in an artificial parameter δ which interpolates between a solvable theory at δ = 0 and the desired theory at δ = 1. The interpolating actions are form δS + (1− δ ) S 0 ; and augmented by an optimization procedure which introduces nonperturbative features into our results. This procedure relies on the freedom in choosing the best S 0 without affecting the convergent results at δ =1. Our linear interpolation is similar in spirit but differs in detail from the novel δ expansion that was recently formulated for scalar theories where the parameter 2(1+δ) was the power of the field in the interaction lagrangian. Here we use interpolating actions for the first time in fermionic and gauge theories.

String theory, misaligned supersymmetry, and the supertrace constraints.

We demonstrate that string consistency in four spacetime dimensions leads to a spectrum of string states which satisfies the supertrace constraints Str{bold 1}=0 and Str {ital M}{sup 2}{proportional_to}{Lambda} at tree level, where {Lambda} is the one-loop string cosmological constant. This result holds for a large class of string theories, including critical heterotic strings. For strings lacking spacetime supersymmetry, these supertrace constraints will be satisfied as a consequence of a hidden ``misaligned supersymmetry`` in the string spectrum. These results suggest a new intrinsically stringy mechanism whereby such supertrace constraints may be satisfied without phenomenologically unacceptable consequences.

t/s scaling in strong interactions

Abstract We discuss the phenomena of t / s scaling in proton-proton scattering. After a theoretical introduction based on a parton approach we analyze the data and show the scaling properties. The parton model leads us to similar expressions for the inclusive productions of protons.

Generalized ensemble of random matrices.

A random matrix ensemble incorporating both Gaussian unitary ensemble and Poisson level statistics while respecting [ital U]([ital N]) invariance is proposed and shown to be equivalent to a system of noninteracting, confined, one dimensional fermions at finite temperature.

Spontaneous Breaking of Scale Invariance in a Supersymmetric Model

Abstract The phase structure of a large N, O(N) supersymmetric model in three dimensions is studied. Of special interest is the spontaneous breaking of scale invariance which occurs at a fixed value of the coupling constant, λ0=λc=4π. In this phase the bosons and fermions acquire a mass while a Goldstone boson (dilaton) and Goldstone fermion (“dilatino”) are dynamically generated as massless bound states. The absence of renormalization of the dimensionless coupling constant λ0 leaves these Goldstone particles massless.

A field-theoretic approach to non-equilibrium work identities

We study non-equilibrium work relations for a space-dependent field with stochastic dynamics (model A). Jarzynskis equality is obtained through symmetries of the dynamical action in the path-integral representation. We derive a set of exact identities that generalize the fluctuation–dissipation relations to non-stationary and far-from-equilibrium situations. These identities are prone to experimental verification. Furthermore, we show that a well-studied invariance of the Langevin equation under supersymmetry, which is known to be broken when the external potential is time dependent, can be partially restored by adding to the action a term which is precisely Jarzynskis work. The work identities can then be retrieved as consequences of the associated Ward–Takahashi identities.

Semiclassical analysis of quasiexact solvability

Higher-order WKB methods are used to investigate the border between the solvable and insolvable portions of the spectrum of quasiexactly solvable quantum-mechanical potentials. The analysis reveals scaling and factorization properties that are central to quasiexact solvability. These two properties define a new class of semiclassically quasiexactly solvable potentials.