Sidney Coleman

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.

Why There Is Nothing Rather Than Something: A Theory of the Cosmological Constant

Wormholes are topology-changing configurations in euclidean quantum gravity whose importance has recently been advocated by several authors. I argue here that if wormholes exist, they have the effect of making the cosmological constant vanish. The argument involves approximations in dealing with physics at the wormhole energy scale (assumed to be somewhat below the Planck mass) but is exact in all interactions at all lower energies.

More about the massive Schwinger model

Abstract The massive Schwinger model is quantum electrodynamics of a Dirac particle of mass m and charge e in 1 + 1 dimensions. It is known that the physics of the model depends on an arbitrary parameter independent of e and m, an angle, θ,|θ| ⩽ π. I give a physical explanation of this angle, and explain why a corresponding parameter does not appear in (3 + 1)-dimensional electrodynamics. I also compute some quantitative properties of the theory for both weak coupling, e ≪ m, and strong coupling, e ≫ m, and conjecture a qualitative description of the model that interpolates smoothly between weak and strong coupling. A typical quantitative result is that for weak coupling and |θ| ≠ π, the number of stable particles in the theory is 4 m 2 π e 2 1 ( 1 − θ 2 / π 2 ) [ 2 3 − In ( 2 + 3 ] + O ( 1 ) I do similar computations for a generalization of the model with “flavor SU(2), ” i.e., with two fermions of equal charge and mass. For weak coupling the results are very much like those for the massive Schwinger model, but for strong coupling there are some surprising differences.

The Uses of Instantons

In the last two years there have been astonishing developments in quantum field theory. We have obtained control over problems previously believed to be of insuperable difficulty and we have obtained deep and surprising (at least to me) insights into the structure of the leading candidate for the field theory of the strong interactions, quantum chromodynamics. These goodies have come from a family of computational methods that are the subject of these lectures.

Black Holes as Red Herrings: Topological Fluctuations and the Loss of Quantum Coherence

Abstract The topological fluctuations recently suggested by Hawking, by Giddings and Strominger, and by Lavrelashvili, Rubakov, and Tinyakov do not lead to an observable loss of quantum coherence.

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.

Cosmic ray and neutrino tests of special relativity

Abstract Searches for anisotropies due to Earths motion relative to a preferred frame — modern versions of the Michelson-Morley experiment — provide precise verifications of special relativity. We describe other tests, independent of this motion, that are or can become even more sensitive. The existence of high-energy cosmic rays places strong constraints on Lorentz non-invariance. Furthermore, if the maximum attainable speed of a particle depends on its identity, then neutrinos, even if massless, may exhibit flavor oscillations. Velocity differences far smaller than any previously probed can produce characteristic effects at accelerators and solar neutrino experiments.

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.

Classical Lumps and Their Quantum Descendants

A stone thrown into a still body of water makes ripples that spread out and eventually die away. The stone disturbs the water, gives it energy, but, even if we ignore friction, this energy tends in the course of time to spread out over the water. If we imagine the water to be infinite in extent, then, if we wait long enough, at no point is the water appreciably different from its state before the stone was cast. The disturbance dissipates.

No more corrections to the topological mass term in QED3

We consider the theory of a photon interacting with scalar, spinor, and/or vector fields with arbitrary gauge-invariant interactions, in three spacetime dimensions. We show that to all orders in perturbation theory, all corrections to the topological mass term (beyond the known one-loop correction) vanish identically.