By Simon A.W.
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E. the 8's, do not feel the boundary of the cubes. e. a(x}d3 x in the cubes in which they are produced and the fermion propagator 8 by the fermion propagator 8 p with periodic boundary conditions on the cube of production. We are then in a situation where it is possible to estimate exactly log(det3(1 + K))N. We get ± for it ;::: for it ~ v'NMi v'NMi (41) 51 The quadratic bound we get for large a comes from the fact that in log (det3 (1 + K)) N it is the quadratic term -(N/2)TrK2 which dominates.
Rivasseau and R. SencOl, A Renormalizable Field Theory: The Massive Gross-Neveu Model in Two Dimensions, Comm. Math. , 103:67 (1986) 11. C. de Calan, P. Faria de Veiga, J. Magnen and R. Seneor, to be published 12. P. Faria de Veiga, thesis 13. J. Magnen and R. Seneor, Yukawa quatum field theory in three dimensions (Y3), in "Third International Conference on Collective Phenomena", J. Lebowitz, J. Langer and W. , The New-York Academy of Sciences, New-York (1980) 53 CONSTRUCTIVE GAUGE THEORY II T.
1 The N = +00 limit The limit N --+ 00 will be more properly defined if we remove the >'p dependance of the interaction. This is obtained by a scaling of a: A a --+ a. The a-covariance is now (>'p)-l + '1I"p(p) and in order to have a well defined limit when N --+ 00, it has to be positive and finite when p --+ +00. Let us introduce the decomposition p T/p(p) = I>i(p) (16) i=O in (9) and associated to it a splitting of Sp as a sum of Si, s. ) and of course of'1l"p is controlled by it O-momentum value.