Showing papers by "Roman Jackiw published in 1974"

Symmetry Behavior at Finite Temperature

Abstract: Spontaneous symmetry breaking at finite temperature is studied. We show that for the class of theories discussed, symmetry is restored above a critical temperature ${{\ensuremath{\beta}}_{c}}^{\ensuremath{-}1}$. We determine ${\ensuremath{\beta}}_{c}$ by a functional-diagrammatic evaluation of the effective potential and the effective mass. A formula for ${\ensuremath{\beta}}_{c}$ is obtained in terms of the renormalized parameters of the theory. By examining a large subset of graphs, we show that the formula is accurate for weak coupling. An approximate gap equation is derived whose solutions describe the theory near the critical point. For gauge theories, special attention is given to ensure gauge invariance of physical quantities. When symmetry is violated dynamically, it is argued that no critical point exists.

Effective Action for Composite Operators

Abstract: An effective action and potential for composite operators is obtained. The formalism is used to analyze, by variational techniques, dynamical symmetry breaking and coherent solutions to field theory. A Rayleigh-Ritz procedure is introduced which replaces arbitrary variations with parametric variations. Previously unsolved nonlinear equations become, in the Rayleigh-Ritz approximation, solvable algebraic equations.

Functional evaluation of the effective potential

Abstract: By use of the path-integral formulation of quantum mechanics, a series expansion for the effective potential is derived Each order of the series corresponds to an infinite set of conventional Feynman diagrams, with a fixed number of loops As an application of the formalism, three calculations are performed For a set of $n$ self-interacting scalar fields, the effective potential is computed to the two-loop approximation Also, all loops are summed in the leading-logarithmic approximation when $n$ gets large Finally, the effective potential for scalar, massless electrodynamics is derived in an arbitrary gauge It is found that the potential is gauge-dependent, and a specific gauge is exhibited in which all one-loop effects disappear

Spontaneous Symmetry Breaking in the O(N) Model for Large N

Abstract: We study the O(N) generalization of the σ model in the limit of large N, for four, three, two, and one space-time dimensions. We compute the effective potential and some momentum-dependent Green's functions. In one and two dimensions, spontaneous symmetry breakdown is impossible; any asymmetric minimum inserted in the tree-approximation potential is immediately filled in by the effects of radiative corrections. This is in agreement with general theorems. In four dimensions, the model is inconsistent; it possesses a tachyon. In three dimensions, the model seems to be consistent, and offers an interesting example of some nonlinear effects associated with spontaneous symmetry breakdown that are not present in the usual (tree-approximation) models.

Gauge-invariant signal for gauge-symmetry breaking

Abstract: The effective potential is computed to order $\ensuremath{\hbar}$ in an Abelian gauge theory---scalar electrodynamics. The calculation is performed first in the ghost-requiring ${R}_{\ensuremath{\xi}}$ gauges. The corresponding expression is also derived from the unitary Lagrangian. We discuss the gauge dependence of the effective potential and its minima in connection with spontaneous symmetry breakdown; and we interpret the unitary computation to be the physically relevant one.