Gerald S. Guralnik

Dimensionally reduced gravitational Chern-Simons term and its kink

Abstract When the gravitational Chern–Simons term is reduced from 3 to 2 dimensions, the lower dimensional theory supports a symmetry breaking solution and an associated kink. Kinks in general relativity bear a close relation to flat-space kinks, governed by identical potentials.

Path integral formulation of mean-field perturbation theory

Abstract We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting φ 4 scalar field theory and a J μ J μ four-Fermion field theory. To solve the φ 4 theory we introduce an auxiliary field χ and rewrite the Lagrangian so that the interaction term has the form χφ 2 . The vacuum generating functional is then expressed as a path integral over the fields χ and φ. Since the χ field is introduced to make the action no more than quadratic in φ, we do the φ integral exactly. Then we use Laplaces method to expand the remaining χ integral in an asymptotic series about the mean field χ 0 . We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field φ and the mean-field bound-state propagator for the composite field χ. The φ and χ propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a χφ 2 field theory in which χ and φ are both elementary fields. We therefore argue that by renormalizing these theories so that the mean-field propagators are equivalent, the two theories are described by the same renormalized Greens functions containing the same three parameters, μ 2 , m 2 , and g . The quartic theory is completely specified by the renormalized masses μ 2 and m 2 of the χ and φ fields. These two masses determine the coupling constant g = g ( μ 2 , m 2 ). The cubic theory depends on μ 2 and m 2 and a third parameter g 0 , g = g ( μ 2 , m 2 , g 0 ), where g 0 is the bare coupling constant. We indicate that g ( μ 2 , m 2 , g 0 ) ⩽ g ( μ 2 , m 2 ) with equality obtained only in the limit g 0 → ∞. When g 0 → ∞ the wave function renormalization constant for the χ field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same graphical topology in the mean-field approximation. To illustrate this we show that the mean-field expansion of the four-Fermion current-current interaction theory is renormalizable and reproduces the results of the usual vector meson theory. A coupling-constant eigenvalue condition is derived which could serve to distinguish current-current interactions from normal electrodynamics.

Complexified path integrals and the phases of quantum field theory

Abstract The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee–Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee–Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.

Complex Langevin equations and Schwinger–Dyson equations

Abstract Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger–Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. The relevance to the study of quantum field theory solution space is discussed.

THE HISTORY OF THE GURALNIK, HAGEN AND KIBBLE DEVELOPMENT OF THE THEORY OF SPONTANEOUS SYMMETRY BREAKING AND GAUGE PARTICLES

I discuss historical material about the beginning of the ideas of spontaneous symmetry breaking and particularly the role of the paper by Guralnik, Hagen and Kibble in this development. I do so adding a touch of some more modern ideas about the extended solution-space of quantum field theory resulting from the intrinsic nonlinearity of nontrivial interactions.

ϵ′/ϵ from the lattice☆

Abstract Results from a lattice calculation of the matrix elements of the weak hamiltonian which determine ϵ′ are presented. They suggest a smaller value of ϵ′/ϵ than previously estimated.

Effective potential for complex Langevin equations

Abstract We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space–time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger–Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.

Programming a Parallel Computer: The Ersatz Brain Project

There is a complex relationship between the architecture of a computer, the software it needs to run, and the tasks it performs. The most difficult aspect of building a brain-like computer may not be in its construction, but in its use: How can it be programmed? What can it do well? What does it do poorly? In the history of computers, software development has proved far more difficult and far slower than straightforward hardware development. There is no reason to expect a brain like computer to be any different. This chapter speculates about its basic design, provides examples of “programming” and suggests how intermediate level structures could arise in a sparsely connected massively parallel, brain like computer using sparse data representations.

On the finite-temperature transition of QCD

Abstract Using an exact algorithm to incorporate dynamical quarks, we present evidence from numerical simulations that the finite-temperature transition in QCD is first order for realistic quark masses. For two light flavors of quarks we see a two state signal, with flip-flops between these states. For four flavors we show that the first-order transition extends to quark masses heavier than T c .

Instantons and vortices in two dimensions

Abstract We consider two-dimensional non-linear sigma models with XY-like and Ising-like behavior. For the XY-like case, instanton effects reproduce the Kosterlitz-Thouless vortex gas. In the Ising-like case, instanton effects can be used to obtain two different expansions. One is similar to an Ising model low-temperature expansion; the other is a vortex plus spin-wave expansion. The explicit relation between instantons and disorder variables is given.