Saturday, 10 September 2016

mathematical physics - functional equation in renormalization group theory

In the Renormalization Group Theory, a key step is the derivation of the so called scaling equation, which in general is in the form of a functional equation of the kind:
g(μ(λ)x,ν(λ)y)=λg(x,y)

where x, y are real numbers and λ>0, moreover μ(λ=1)=ν(λ=1)=1.



We can assume g differentiable both on x and y and μ(λ),ν(λ) analytic function of λ.



The (hopefully unique) solution of the functional equation should correspond to μ(λ) and ν(λ) being of the form of power law with two independent exponents.



Is it necessary to add other conditions on the functions g,μ and ν to prove this result and its uniqueness? Or is it possible to weaken the hypotheses?



Edit:




Assuming μ(λ) being an invertible function (this is an additional hypothesis), setting y=0 we get
g(λx,0)=μ1(λ)g(x,0)
It easy to get
g(λ1λ2x,0)=μ1(λ1)μ1(λ2)g(x,0)=μ1(λ1λ2)g(x,0)
from which follows (at least for differentiable μ1(λ):

μ1(λ)=λs and then μ(λ)=λ1s.



My question is if it is necessary to add an additional hypothesis like that of an invertible μ(λ), if there are alternatives to the invertibility or if is possible to proof the result even under weaker conditions.

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...