In critical phenomena, we can enlarge the block size(momenta fluctuation) by Kadanoff transformation, say(adsbygoogle = window.adsbygoogle || []).push({});

[tex]k \rightarrow bk (b<=1) [/tex], and scale the new Hamiltonian by [tex]k' = k/b, x'=bx[/tex] to recover to the original block size.

In QFT, similarly integrating out the high momenta produces the effective Langrangian,

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

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

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# Question about RG and scaling in qft

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