How can one recover/obtain information from the renormalization group procedure?

Question

I know the basic idea behind the renormalization group approach as it is used in mathematical physics to study both QFT and statistical mechanics. However, I have trouble understanding how can one recover the information one was trying to obtain using this technique. Let me elaborate.

Although there is no such thing as a 'general approach' to RG, I want to try to sketch the ideas from a generic model. As stated in Brydges & Kennedy's article, one starts with integrals of the form: \begin{eqnarray} Z(\varphi) = \int d\mu_{C}(\psi) e^{-V_{0}(\psi+\varphi)} = (\mu_{C}*e^{-V_{0}})(\varphi) \tag{1}\label{1} \end{eqnarray} where $\varphi = (\varphi_{x})_{x\in \Lambda}$ is a Gaussian process with joint distribution $\mu_{C}$, mean zero and covariance $C$. Suppose we can write $C$ as a sum $C=C_{1}+C_{2}$. Then: \begin{eqnarray} Z(\varphi) = \int d\mu_{C_{1}+C_{2}}(\psi)e^{-V_{0}(\psi+\varphi)} = \int d\mu_{C_{2}}(\zeta)\int d\mu_{C_{1}}(\psi) e^{-V_{0}(\psi+\varphi+\zeta)} = \int d\mu_{C_{2}}(\zeta) (\mu_{C_{1}}*e^{-V_{0}})(\varphi+\zeta) = \int d\mu_{C_{2}}(\zeta)e^{-V_{1}(\varphi+\zeta)} \tag{2}\label{2} \end{eqnarray} where: \begin{eqnarray} V_{1} = -\ln \mu_{C_{1}}*e^{-V_{0}} \tag{3}\label{3} \end{eqnarray} Thus, we can define a map on the (informal) space of actions, called renormalization group map and denoted by $RG$, such that $RG: V_{0} \to V_{1}$. Analogously, if $C=C_{1}+\cdots +C_{n}$, $n \ge 2$, then sucessive applications of (\ref{2}) lead to: \begin{eqnarray} Z(\varphi) = \int d\mu_{C_{n}}(\zeta_{n})(\mu_{C_{n-1}}*e^{-V_{n-1}})(\varphi+\zeta_{n}) = \int d\mu_{C_{n}}(\zeta_{n})e^{-V_{n}(\varphi+\zeta_{n})} \tag{4}\label{4} \end{eqnarray} where $V_{n}= RG(V_{n-1})=\cdots = RG^{n-1}(V_{0})$. We thus defined a 'trajectory' $V_{0}\to V_{1}\to V_{2}\to \cdots \to V_{n}$.

All these being said, I believe the main idea of the process is to (luckily) prove that the above trajectory ends up in a fixed point. In other words, luckily we have $RG^{n}(V_{0}) = V^{*}$ for every $n$ suficiently large.

The above scenario, although very generic, appears in some discussions on the topic. As an example, see Salmhofer's book. Now comes my questions.

(1) How can one recover the information about $Z(\varphi)$ once we was lucky and obtained $V^{*}$? See, $Z(\varphi)$ was our object of study in the first place, right? But I don't see how to get back and obtain it.

(2) One situation in which the covariance splits into a sum of covariances is when one is trying to approach the continuum limit from a scaled lattice. This can be done in either QFT and statistical mechanics, but I believe it is more common for QFT models. But when we think about statistical mechanics, can one obtain critical temperatures, critical exponents and other thermodynamics entities from the above process? Is it possible to ilustrate how it could be done considering this very generic model?

Answer 1

1. The limiting function $V^\ast$ is such that any further convolutions of $e^{-V^\ast}$ with $\mu$ return $e^{-V^\ast}$, so $Z^\ast(\phi)=e^{-V^\ast(\phi)}$.

2. To obtain critical properties, you need the correlator $K(x,x')=\langle\phi(x)\phi(x')\rangle$. The decay length of the correlator diverges at the critical temperature $T_c$ as a power law $(T-T_c)^{-\alpha}$ and the power $\alpha$ is the critical exponent. The correlator is obtained by adding a source term $\lambda\psi(x)\psi(x')$ to the exponent in the definition of $Z(\phi)$ and then evaluating $dZ/d\lambda$ at $\phi=0$.

In reference to the title of the post: "How can one recover/obtain information from the renormalization group procedure?" Information that depends on features that appear at small distances cannot be recovered, it is lost in the renormalization flow (which is not reversible). The information that remains refers to features that persist at large distances, such as a diverging correlation length and the critical exponents associated with it.