beacon
Feb4-04, 12:58 AM
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
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