A fine-tuning measure is defined from the eigenvalues of a rescaled Fisher information matrix on parameter space, with a geometric interpretation as the pullback of the Euclidean metric from observable space.
Relative entropy and the RG flow
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abstract
We consider the relative entropy between vacuum states of two different theories: a conformal field theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the difference of entanglement entropies. As a result, this difference has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d=2 space-time dimensions and, for d>2, the proof that the coefficient of the area term in the entanglement entropy decreases along the renormalization group (RG) flow between fixed points. We comment on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension $\Delta$ of the perturbation that triggers the RG flow.
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Naturalness and Fisher Information
A fine-tuning measure is defined from the eigenvalues of a rescaled Fisher information matrix on parameter space, with a geometric interpretation as the pullback of the Euclidean metric from observable space.