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By H.S.M. and W.O.J. Moser Coxeter

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Also, Eq. (2-79), together with the definition of p 2 , Eq. (2-64), gives y = 1. If the correlation length is defined by: it can be expressed in terms of C(k) as: Inserting Go, we find andu=;. Notice that at this level none of the quantities required any knowledge of the cutoff. No momentum integrals enter. They make their first appearance in the computation of the specific heat. For this calculation the Gaussian integral 28 FIELD THEORY. THE RESORMALIZATIOK GROUP. A N D CRITICAL P H E N O M E S A in (2-77) has to be carried out.

3-1). , by performing a Wick rotation. All we have to do is to define time-ordering for imaginary times, and this we do by defining 1. I (3-33) so that the T ~ ’ can S be ordered along the imaginary axis - to later ti’s correspond “later” T ~ The . only change that will occur in the previous discussion is that I t - t n+l €=-+€ , , 7-5- =-=ie n+l (3-34) and we have to add a boundary condition insuring that the solutions of the Schrodinger equation for large imaginary times will remain finite. ECfectively we are converting the Schrodinger equation into a diffusion equation.

The complications involved in resurrecting such things as field equations, interpolating fields, etc. have been discussed at length by Zimmermann (Brandeis Lecture Notes). Perhaps the most rational attitude is that of t'Hooft and Veltman (Diagrammar) in which the theory is postulated in terms of its regularized perturbation expansion. So there seems little point in giving a lengthy exposition. Here we will follow the presentation of Fried4 which in turn follows Symanzik. The logic is as follows: (1) For an interacting bose field @(x)we can define the operator where J ( x ) is a c-number source, and T is the time-ordering operator.