Alternative dimensional reduction via the density matrix

Abstract

We give graphical rules, based on earlier work for the functional Schr\"odinger equation, for constructing the density matrix for scalar and gauge fields in equilibrium at finite temperature T. More useful is a dimensionally reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to 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 expansions; it encodes all thermal matrix elements. We term the DREA procedure alternative dimensional reduction, to distinguish it from the conventional dimensionally reduced field theory (DRFT) which applies at infinite T. The DREA 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 and cannot have certain spurious infinities which sometimes occur in the density matrix itself or the conventional DRFT; these come from $\mathrm{ln}T$ factors at infinite temperature. The DREA can be constructed to all orders (in principle) and the only regularizations needed are those which control the ultraviolet behavior of the zero-T theory. An example of spurious divergences in the DRFT occurs in $d=2+1{\ensuremath{\varphi}}^{4}$ theory dimensionally reduced to $d=2.$ We study this theory and show that the rules for the DREA replace these ``wrong'' divergences in physical parameters by calculable powers of $\mathrm{ln}T;$ we also compute the phase transition temperature of this ${\ensuremath{\varphi}}^{4}$ theory in one-loop order. Our density-matrix construction is equivalent to a construction of the Landau-Ginzburg ``coarse-grained free energy'' from a microscopic Hamiltonian.