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  1. The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be though of as a flux in a space of possible Hamiltonians for a system. For example, in the Kadanoff picture, if I understand correctly, every step of "scaling up" by averaging across a spin block sends me to a new point of my parameter space, with (possibly) different values of the field and interaction strength. Critical points and certain other points are fixed points of this transformation. So if my family of Hamiltonians is indicated by $H_\theta(X)$, where $X$ is the state of my system and $\theta$ is a vector of parameters, the renormalisation process is a flux in the $\theta$ space. See for example Fisher, M. E. (1998), Rev. Mod. Phys., in particular figure 4.

  2. On the other hand, I have learned in a completely different context that this space of parameters has an interesting metric, the Fisher-Rao metric, which is defined as $$g_{ij}=-\left\langle\frac{\partial \log p(x,\theta)}{\partial \theta_{i}}\frac{\partial \log p(x,\theta)}{\partial \theta_{j}}\right\rangle,$$ where $p$ is a probability distribution, and the average is taken with respect of $p$ itself. In the canonical ensemble formalism, take $$ p(x, \theta) = \frac{e^{-H_\theta(x)}}{Z_\theta}.$$ This connects it to what was said above.

Now, my question is the following: does this metric say anything useful about the renormalisation flux? Maybe its geodesics have anything to do with it?

Why do I think there is something in common? Because $g_{ij}$ diverges at critical points, for reasons that would be long to explain here. A general idea is, from the physical point of view, that the specific heat, the magnetic susceptibility (for example in an Ising-like case) are entries of the F.I. tensor. From the statistical point of view, critical points are points from which even an infinitesimal deviation in the parameter space leads to a finite change in the parametrised probability distribution.

If this is an interesting idea, please don't steal it. Thanks.

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