1991/12/18 by Olaf Lechtenfeld · 1 citation
Physics and Astronomy · #hep-th
paper · pdf · doi:10.1142/s0217751x92003264
published as Int.J.Mod.Phys.A7:7097-7118,1992 · 27 pages / 3 figures (ps file fixed)
arxiv created 1991/12/18 · arxiv updated 2010/11/01
We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~N. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated \it exactly\/. The semiclassical loop expansion turns out \it not\/ to coincide with the (topological) 1\over N~expansion, because the classical background has a non-trivial N-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around N = ∞. This leads to IR singularities in the large-N expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results.