Symmetry And Group

2-Generator golod p-groups by Timofeenko A.V.

By Timofeenko A.V.

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Again, formal transformations on microscopic Hamiltonians do not give a much better definition of this cutoff. Fortunately, as in the case of the other parameters, critical exponents and other universal quantities are independent of the cutoff, as a result of the long-range nature of fluctuations near the critical point. 2-4 THE PARTITION FUNCTION, OR THE GENERATING FUNCTIONAL Once the Lagrangian has been chosen the probability distribution is written as in Eq. (2-l), namely 16 FIELD THEORY. THE RENORMALlZATlON GROUP, A N D CRITICAL PHENOMENA In addition we introduce a source h, or an external field, which couples linearly to all the relevant fields.

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. 47) (3-48) with zo the generating functional for the non-interacting theory. Z can be written also as: Z { a = X-' ! 9 @ exp ( i 5 [U(@) + J @ ] d4x] (3-49) with @ a c-number field. , destruction operators to the right of the creation operators.

40 FIELD THEORY. THE RENORMALIZATION GROUP, A N D CRITICAL PHENOMENA One should keep in mind that it is only outside the interval (f, t‘) that we continue to the imaginary time axis. Because there J = 0, and hence we know the time-dependence explicitly (Eq. (3-28); see Fig. 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 .

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