Nuclear Physics B230 [FS10] (1984) 317-335 © North-Holland Publishing Company MASTER VARIABLES AND SPECTRUM EQUATIONS IN LARGE-N THEORIES Antal JEVICKI ~ Laboratoire de Physique Thdorique de I'Ecole Normale Supdrieure, Paris, France and Department oj: Physics, Brown University, Providence, RI 02912, USA Joao P. RODRIGUES2 Department of Physics, Brown University, Providence, RI 02912, USA Received 15 September 1983 Based on the loop-space effective action approach to the large-N limit, we consider the issue of constrained minimization and the problem of small fluctuations. The master variables are introduced to allow for an unconstrained minimization; the second derivative of the effective loop-space action with respect to these variables leads to master equations for the spectrum of particle states. We show that even though these variables contain redundant degrees of freedom, the only additional eigenvalues are of zero energy. We also establish a simplification through use of a reduced ansatz. 1. Introduction Even though Wilson loop variables lead in the large-N limit to a simplified set of Schwinger-Dyson equations the situation for weak coupling is still rather cumber- some. Specifically, in our recent numerical approach [1] to the hamiltonian loop- space problem [2] it was emphasized that in the weak-coupling phase one has a minimization problem with non-linear constraints. Consequently, a direct derivative of the effective potential does not lead to a correct result. In the euclidean case also it was realized sometime ago [3] that the MigdaI-Makeenko equation [4] is not a derivative of the effective action: rather it represents a projection of the naive saddle-point equation. With the above situation in mind one can ask what happens to the subsequent important problem of small fluctuations. It was suggested before to derive spectrum eigenvalue equations simply by linearization in terms of loop-space variables; because of the non-trivial nature of the loop-space minimization, we believe that this is neither a correct nor a complete procedure. To make it work one would need Alfred P. Sloan Fellow. 2 Address after August 1, 1983: CEN-Saclay, Division de Physique Th6orique, F-91191 Gif-sur- Yvettc, France. 317