# Question about RG and scaling in qft

1. Feb 4, 2004

### beacon

In critical phenomena, we can enlarge the block size(momenta fluctuation) by Kadanoff transformation, say
$$k \rightarrow bk (b<=1)$$, and scale the new Hamiltonian by $$k' = k/b, x'=bx$$ to recover to the original block size.
In QFT, similarly integrating out the high momenta produces the effective Langrangian,

$$\int_{k<=b\Lambda} [D\phi] exp(iS_{eff}) = \int_{b\Lambda <k < \Lambda} [D\phi] exp(iS)$$.

The parameters $$y$$ in the effective langrangian $$S_{eff}$$ should depend on $$b$$. We can also do a scaling $$k' = k/b, x'=bx$$ in $$S_{eff}$$ to get $$S'_{eff}$$ whose path integral is now $$\int_{k' <= \Lambda}$$. The parameters $$y'$$ also depend on $$b$$. My puzzle is that which are the so-called beta fuctions, $$dy \over db$$ or $$dy' \over db$$

Last edited: Feb 4, 2004