Differential Renormalization of Massive Quantum Field Theories

We extend the method of differential renormalization to massive quantum field theories treating in particular \ph4-theory and QED. As in the massless case, the method proves to be simple and powerful, and we are able to find, in particular, compact explicit coordinate space expressions for the finite parts of two notably complicated diagrams, namely, the 2-loop 2-point function in \ph4 and the 1-loop vertex in QED.Comment: 8 pages(LaTex, no figures

Modified Weak Energy Condition for the Energy Momentum Tensor in Quantum Field Theory

The weak energy condition is known to fail in general when applied to expectation values of the the energy momentum tensor in flat space quantum field theory. It is shown how the usual counter arguments against its validity are no longer applicable if the states |\psi \r for which the expectation value is considered are restricted to a suitably defined subspace. A possible natural restriction on |\psi \r is suggested and illustrated by two quantum mechanical examples based on a simple perturbed harmonic oscillator Hamiltonian. The proposed alternative quantum weak energy condition is applied to states formed by the action of scalar, vector and the energy momentum tensor operators on the vacuum. We assume conformal invariance in order to determine almost uniquely three-point functions involving the energy momentum tensor in terms of a few parameters. The positivity conditions lead to non trivial inequalities for these parameters. They are satisfied in free field theories, except in one case for dimensions close to two. Further restrictions on |\psi \r are suggested which remove this problem. The inequalities which follow from considering the state formed by applying the energy momentum tensor to the vacuum are shown to imply that the coefficient of the topological term in the expectation value of the trace of the energy momentum tensor in an arbitrary curved space background is positive, in accord with calculations in free field theories.Comment: 27 pages, 1 figure, uses harvmac, epsf and boldmath (included). Change of title and some text changes, form to be publishe

Gradient Flows from an Approximation to the Exact Renormalization Group

Through appropriate projections of an exact renormalization group equation, we study fixed points, critical exponents and nontrivial renormalization group flows in scalar field theories in $2<d<4$. The standard upper critical dimensions $d_k={2k\over k-1}$, $k=2,3,4,\ldots$ appear naturally encoded in our formalism, and for dimensions smaller but very close to $d_k$ our results match the \ee-expansion. Within the coupling constant subspace of mass and quartic couplings and for any $d$, we find a gradient flow with two fixed points determined by a positive-definite metric and a $c$-function which is monotonically decreasing along the flow.Comment: 10 pages, TeX, 3 postscript figures available upon request, UB-ECM-PF-93/2

Scheme Independence and the Exact Renormalization Group

We compute critical exponents in a $Z_2$ symmetric scalar field theory in three dimensions, using Wilson's exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved.Comment: 15 pages, TeX with harvmac, 2 figures in compressed postscript; presentation of first section revised, several minor errors corrected, two references adde