# Renormalization group in time and space, finite size systems

I have few questions about RG. For completeness and clearness I'll start more or less from the beginning.

Initially I have bare two-point propagator: $$G_0(r)=\frac{1}{r}$$ Where $r=\sqrt{x^2+V^2t^2}$,$V$--some velocity.

Now I want to renormalize my theory, but not within Wilson's scheme, but in real space and time (my natural cutoff is connected with finite lattice constant, I'll call it $r_0$). Now I want to apply Callan-Symanzik equation: $$[\partial/\partial(\log r) +\beta(\lambda)\partial/\partial\lambda+2\gamma(\lambda)]G(r,\lambda)=0$$ Where my beta-function and anomalous dimensionality have the form ( $\lambda>0$--coupling constant): $$\beta(\lambda)=-\lambda^2 \\ \gamma(\lambda)=\frac{1}{2}-\lambda$$ We have here asymptotic freedom: $$\lambda(r) =\frac{\lambda_0}{1+\lambda_0\log(r/r_0)}$$ The general solution of Callan-Symanzik equation can be written as follows (skipping here one derivation step): $$G(r)=\frac{1}{r}\sqrt{\frac{\lambda_0}{\lambda(r)}} \sum_m F_m(r)\lambda^m(r)$$ Functions $F_m(r)$ here can be found by direct calculation of correlator in perturbation theory, say, in zero order we have $F_0=1$ (since we should restore free theory without RG flow in zero order) .

Further, if we have conformal field theory on a ring in $x$-space then we know, that our bare propagator is (by mapping plane to a cylinder): $$G(r)=\frac{\pi}{L\sin(\pi r/L)}$$ Correspondingly: $$F_0 \sim \frac{\pi r/L}{\sin(\pi r/L)}$$ The following questions appear after that:

1) Generally speaking, I have two variables: space $x$ and time $t$. So that bare propagator not necessarily “isotropic”, the theory currently I am dealing with is also on a circle, and my Green's function is something like $\frac{1}{\sin(x+Vt) \sin(x-Vt)}$ already in zero order. So, how should I perform RG-step in that case? Should I vary coordinate and time separately?

And how to rewrite Callan-Symanzik equation then? Will logarithmic derivative with respect to $r$ remain or I need to write something like $\partial/\partial(\log x)+\partial/\partial(\log Vt)$?

In other words I don't understand how to define small scales now (if I want to average over them): is it small time, small coordinate $x$ or still small $r$?

2) About second question. Actually I am interested in finite-size system, when initially fields satisfy periodic boundary conditions (when I place them on a cylinder, as it was mentioned before), or (what is more interesting for me) open boundary conditions (when fields are zero on the boundary). In both cases a new scale $L$-system's length appears. Then the question : should I also renormalize it? What is the physical meaning of it then? In articles I've seen so far, people keep $r/L$ constant in RG procedure, is it done just for convenience?

Or I'll ask naive question. What changes in RG-corrections to Green's function can I expect for the system, where fields satisfy some boundary conditions? Will beta-fuction remain the same?

I have the feeling (sorry for this word “feel”) that prefactor $\frac{1}{r}\sqrt{\frac{\lambda_0}{\lambda(r)}}$ (that is literally calculated through beta function and anomalous dimensionality) is universal, like it does not depend on boundary conditions, system size and it doesn't know that I have some nonisotropic $(x,t)$ dependence in free theory… Am I correct?