R. Jackiw

A new improved energy-momentum tensor

Abstract We show that the matrix elements of the conventional symmetric energy-momentum tensor are cut-off dependent in renormalized perturbation theory for most renormalizable field theories. However, we argue that, for any renormalizable field theory, it is possible to construct a new energy-momentum tensor, such that the new tensor defines the same four-momentum and Lorentz generators as the conventional tensor, and, further, has finite matrix elements in every order of renormalized perturbation theory. (“Finite” means independent of the cut-off in the limit of large cut-off.) We explicitly construct this tensor in the most general case. The new tensor is an improvement over the old for another reason: the currents associated with scale transformations and conformal transformations have very simple expressions in terms of the new tensor, rather than the very complicated ones they have in terms of the old. We also show how to alter general relativity in such a way that the new tensor becomes the source of the gravitational field, and demonstrate that the new gravitation theory obtained in this way meets all the epxerimental tests that have been applied to general relativity.

Three-dimensional Einstein gravity: dynamics of flat space

In three spacetime dimensions, the Einstein equations imply that source-free regions are flat. Localized sources can therefore only affect geometry globally rather than locally. Some of these effects, especially those generated by mass and angular momentum are discussed.

Current algebra and anomalies

Current algebra remains our most successful analysis of fundamental particle interactions. This collection of surveys on current algebra and anomalies is a successor volume to Lectures on Current Algebra and Its Applications (Princeton Series in Physics, 1972). Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Three-Dimensional Cosmological Gravity: Dynamics of Constant Curvature

In three space-time dimensions, Einsteins equations with cosmological constant Λ imply that the curvature is constant outside sources. When particles are present, they alter the global properties of the exterior geometry. In the De Sitter case, space is a two-sphere and static many-body solutions are quite different from their Λ = 0 counterparts; in particular particles lie at antipodal points, with the great-circle wedge between them excised. These configurations are analyzed in terms of the general static solution to the exterior field equations.

Charge shielding and quark confinement in the massive schwinger model

The Schwinger model is quantum electrodynamics with massless fermions in two dimensions. It is known that the asymptotic states of the theory contain no states corresponding to free fermions (“quark trapping”) and that local charge conservation is spontaneously broken (“Higgs phenomenon”). We investigate to what extent these phenomena persist when the fermion is given a bare mass. We find quark trapping but no Higgs phenomenon. The second of these results is dependent on mass perturbation theory; the first is not.

Lower Dimensional Gravity

Abstract Gravity theory on a line and in the plane is reviewed. The triviality of the planar Einstein model is avoided by adding sources and a topological mass term. A constant curvature model for two dimensional space-time, analogous to the theory in three dimensional space-time, is proposed.

Why dilatation generators do not generate dilatations

We show that the Ward identities associated with broken scale invariance contain anomalies in renormalized perturbation theory. In low orders, these anomalies can be absorbed into a redefinition of the scale dimensions of the fields in the theory, but in higher orders this is not possible. Also, these anomalies cannot be removed by studying the Greens functions for objects other than canonical fields, e.g., currents. These results are established to first nontrivial order in perturbation theory by explicit Feynman calculations (which give us information at all momentum transfers), and in higher orders by the method of Callan and Symanzik (which gives information only at zero momentum transfer). The two approaches are consistent within their common domain of validity. Two appendices contain self-contained treatments of the formal canonical theory of scale and conformal transformations and of the derivation of Ward identities. In another appendix, we derive the Callan-Symanzik equations for Greens functions of currents, and show that no redefinition of scale dimension is necessary for these objects, although the other anomalies remain.

Chern-Simons modification of general relativity

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate v_\mu -- an external quantity, which we fix to be a non-vanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension.

Radiatively induced Lorentz and CPT violation in electrodynamics

In a nonperturbative formulation, radiative corrections arising from Lorentz and CPT violation in the fermion sector induce a definite and nonzero Chern-Simons addition to the electromagnetic action. If instead a perturbative formulation is used, an infinite class of theories characterized by the value of the Chern-Simons coefficient emerges at the quantum level. {copyright} {ital 1999} {ital The American Physical Society}

Zero modes of the vortex-fermion system

Abstract Electric charge of a fermion that is coupled only to the gauge field of an abelian vortex need not be quantized, in contrast the situation for a magnetic monopole field. Correspondingly, zero-energy modes of the Dirac equation are not determined topologically. On the basis of mathematical and physical arguments we suggest that a complete description of the fermion-vortex system should include a fermion-scalar field interaction. This quantizes the charge and leads to a non-linear Dirac equation which possesses | n | zero-eigenvalue modes in the n -vortex background field. The result suggests that an index theorem exists for this Dirac equation on the non-compact space R 2 . When regularity requirements on the wave function are relaxed, additional normalizable zero-energy modes are present, even in the vacuum sector, n = 0, and fermions are converted to bosons.