QUANTUM EVOLUTION OF INHOMOGENEITIES IN CURVED SPACE

We obtain the renormalized equations of motion for matter and semiclassical gravity in an inhomogeneous space–time. We use the functional Schrödinger picture and a simple Gaussian approximation to analyze the time evolution of the λϕ4 model, and we establish the renormalizability of this nonperturbative approximation. We also show that the energy–momentum tensor in this approximation is finite once we consider the usual mass and coupling constant renormalizations, without the need of further geometrical counterterms.

THE EXACT RENORMALIZATION GROUP AND APPROXIMATE SOLUTIONS

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff Green’s functions are given. A promising nonperturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in “irrelevancy” of operators. We illustrate with two simple models of four-dimensional λφ4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

LEADING INFRARED LOGARITHMS AND VACUUM STRUCTURE OF QCD3

QCD 3 is a superrenormalizable massless theory; therefore off-mass-shell infrared divergences appear in the loop expansion. We show how certain infrared divergences can be subtracted by changing the boundary conditions in the functional integral, letting the vector potentials approach non-zero constant values at infinity. Infrared divergences, in the Green's functions, come together with powers of logarithms of the external momenta, and among the infrared divergences we deal with, there are those that give rise to the leading and first subleading logarithms. We show how for two-point functions it is possible to sum the leading and first subleading logarithms to all orders. This procedure defines a nonperturbative approximation for QCD 3. We find that in the ultraviolet region these summations are well defined, while in the infrared region, some additional prescription is needed to make sense out of them. From the construction developed to cancel infrared divergences, we derive the presence of arbitrarily large field strengths in the vacuum. We respect Lorentz invariance, because in this construction, one averages all directions and strengths of these fields. Possible applications to the infrared problem of QCD 4 are indicated.