Alternative dimensional reduction via the density matrix - art. no. 025021

We give graphical rules, based on earlier work for the functional Schroding er equation, for constructing the density matrix for scalar and gauge field s in equilibrium at finite temperature T. More useful is a dimensionally re duced effective action (DREA) constructed from the density matrix by furthe r functional integration over the arguments of the density matrix coupled t o a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expan sions; it encodes all thermal-matrix elements. We term the DREA procedure a lternative dimensional reduction, to distinguish it from the conventional d imensionally reduced field theory (DRFT) which applies at infinite T. The D REA is useful because it gives a dimensionally reduced theory usable at any T including infinity, where it yields the DRFT, and because it does not an d cannot have certain spurious infinities which sometimes occur in the dens ity matrix itself or the conventional DRFT; these come from In T factors at infinite temperature. The DREA can be constructed to all orders (in princi ple) and the only regularizations needed are those which control the ultrav iolet behavior of the zero-T theory. An example of spurious divergences in the DRFT occurs in d=2+1 phi (4) theory dimensionally reduced to d=2. We st udy this theory and show that the rules for the DREA replace these "wrong" divergences in physical parameters by calculable powers of In T; we also co mpute the phase transition temperature of this phi (4) theory in one-loop o rder. Our density-matrix construction is equivalent to a construction of th e Landau-Ginzburg "coarse-grained free energy" from a microscopic Hamiltoni an.