Comparison to our previous results

Let us now embed the renormalization group. Constructed in Chap. 8, iAto this general framework. As mentioned above, relation (8.5) shows that the RG we are searching for must be a nonlinear representation of the group of dilatations in the space of parameters These are the microscopic 

the explicit form (8.1), (8.38), (8.42) of the mapping the Ward identities (10.35) read 

These equations are known as the renormalization gmup equations. They are clearly equivalent to the integrated form 

This representation shows two fixed points. We first consider the trivial, or Gaussian , fixed point l/n — 0,/3e = 0. The associated linear representation is of the form (10.19) with resulting 

The nontrivial fixed point 1/u = 0, = /5 induces another linear rep- 

Let us now embed the renormalization group, Constructed in Chap. 8, iftto this general framework. As mentioned above, relation (8.5) shows that the RG we are searching for must be a nonlinear representation of the group of dilatations in the space of parameters. , n,/ e). These are the microscopic parameters of the model, and the representation shall leave macroscopic observables invariant. Furthermore we want the representation to show a nontrivial fixed point. In Sect. 8.2 we have constructed such a representation based on first order perturbation theory. The invariance constraint is obeyed within deviations of order 1+e 2, no = n(A = 1). Equations (8.38), (8.42) give the parameter flow under this nonlinear representation in the standard form (10.28), 

This representation shows two fixed points. We first consider the trivial, or Gaussian5, fixed point 1 jn — 0,/3e = 0, The associated linear representation is of the form (10.19) with = —2tD = —e, resulting immediately from Eq. (8.29). These are the canonical dimensions1 of 1/n or ft., respectively. (Dg = — 1 holds by definition of A.) Showing negative scaling dimensions, both 1/n or j3e for A — 0 are driven away from the Gaussian fixed point l/n(A) = A 2/tio /3e(A) = A-e/ Clo. It should be noted that the 

The nontrivial fixed point 1/n = 0, Qe = J3 induces another linear representation. To find the corresponding scaling dimensions, we write, following Eq. (10.30) 

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