Conformal invariance of a Brownian chain

We may wonder whether there is a direct relation between the structure of polymers and this conformal invariance which proved so useful for the study of the properties of correlation functions in field theory. Thus, in order to cast some light on this question, we shall show here directly how the property of conformal invariance applies to a two-dimensional Brownian chain. 

The term / (z) z is a local dilatation factor. Just as an infinite chain is statistically invariant for dilatations, a local dilatation amounts to a local increase of matter, i.e, of Brownian area. Consequently, we have to set 

we feel that Kuhnian chains must have very similar invariance properties for conformal transformations. 

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