Information Measures Along RG Flows

There is no single “information along the RG flow.” Entropy, mutual information, relative entropy, entanglement negativity, and information metrics depend on different regional and operational choices. A useful flow observable is a family with fixed conventions, controlled endpoint limits, and either a theorem or an explicit channel that fixes the monotonicity direction.

Required background. UV and IR fixed points supplies the endpoint theory data; separating scales fixes the independent dimensionless ratios.

Helpful background. Universal scaling functions explains crossover collapse; measure selection matches an information quantity to a physical task.

A flow is a family of comparisons

Let $g$ be a relevant coupling and $R$ a regional length. A typical dimensionless variable is

$$x=gR^{d-\Delta}\sim (R/\xi)^{d-\Delta}.$$

The family $I(R,g)$ becomes an RG diagnostic only after the state, shape, cutoff removal, and finite counterterms are fixed. One may then compare:

• the same theory at different $R$, which changes the regional algebra;
• different deformed states on a common regulated algebra at fixed $R$;
• fixed-point limits $x\to0$ and $x\to\infty$;
• or states related by a declared coarse-graining channel.

These comparisons need not have the same monotonicity. In particular, Wilsonian integration of a momentum shell is not automatically the partial trace associated with a spatial subregion.

The chapter diagram keeps the branches distinct.

Information-flow dictionary. Fixed-point monotones use dimension-specific theorems, crossover functions use controlled finite ratios, and recovery statements require an explicit channel. Schematic and not to scale.

Selecting the measure

For a single sharp region, the raw von Neumann entropy is UV divergent. Differential renormalization can remove local terms for symmetric families, but the answer may retain scheme dependence. Mutual information

$$I(A:B)=S(A)+S(B)-S(AB)$$

is finite for separated regions in ordinary continuum QFTs and probes correlations at separation $L$, yet it is not a universal count of local degrees of freedom. Relative entropy

$$D(\rho_A\Vert\sigma_A) =\Delta\langle K_\sigma\rangle-\Delta S_A$$

is finite in many common-algebra comparisons and has data processing, but it depends on a reference state and modular Hamiltonian. Negativity is adapted to mixed-state entanglement rather than total correlations, and theory-space metrics describe infinitesimal distinguishability rather than a global count.

Thus the measure is selected by the question: fixed-point degree count, correlation range, deformation distinguishability, or recoverable operational information.

Endpoint matching and monotonicity candidates

A candidate $C(R)$ should satisfy more than endpoint agreement. A strong checklist is

$$C(R\to0)=C_{\rm UV},\qquad C(R\to\infty)=C_{\rm IR},\qquad R\frac{dC}{dR}\le0,$$

with the last inequality derived under stated hypotheses. Endpoint matching alone does not imply monotonicity: a crossover can overshoot. Conversely, a numerically decreasing function can depend on a subtraction prescription and fail to equal a universal fixed-point coefficient.

In two dimensions the interval $c_E(R)=3RS'(R)$ passes this test for a unitary Lorentz-invariant vacuum. In three dimensions the disk function $\mathcal F(R)=(R\partial_R-1)S(R)$ has the correct CFT anchor and entropic monotonicity under its hypotheses. Four-dimensional irreversibility is the $a$ theorem; no arbitrary finite spherical entropy is automatically its interpolating function.

For the precise interval inequality see Casini and Huerta 2007, pp. 7032–7035; for the disk proof see Casini and Huerta 2012, §§ II–III. The systematic differential construction and its fixed-point limits are developed in Liu and Mezei 2013, §§ 2–3.

Massive deformation example

For a massive relativistic vacuum, compare two diagnostics. A renormalized interval or disk entropy probes correlations crossing the boundary as $mR$ grows. Mutual information between separated regions probes the decay with both $mR$ and $mL$. They can display different crossover widths because the geometric kernels differ.

A credible numerical flow plot therefore includes:

1. several values of $R/\epsilon$ at fixed $mR$;
2. the expected UV and IR constants or asymptotic powers;
3. finite-volume and derivative-discretization errors; and
4. at least one deliberately too-narrow window showing how a false plateau arises.

Validity boundaries

The map below separates theorem, channel, and finite-window claims.

Validity map for information-flow claims. A monotone requires either the hypotheses of a dimension-specific theorem or a declared information channel. A trend across a finite range is useful evidence but is not by itself an RG irreversibility theorem. Schematic and not to scale.

Common pitfalls

Equating spatial trace with Wilsonian integration. They act on different structures unless a specific state map identifies them. State the input and output algebras before invoking data processing.

Assuming every measure is monotone. Mutual information, negativity, and finite renormalized entropies can have nonmonotone crossover behavior outside special geometries or hypotheses.

References

• Casini, Horacio, and Marina Huerta. “A c-Theorem for the Entanglement Entropy.” Journal of Physics A: Mathematical and Theoretical 40 (2007): 7031–7036. DOI.
• Casini, Horacio, and Marina Huerta. “On the RG Running of the Entanglement Entropy of a Circle.” Physical Review D 85 (2012): 125016. DOI.
• Liu, Hong, and Mark Mezei. “A Refinement of Entanglement Entropy and the Number of Degrees of Freedom.” Journal of High Energy Physics 2013, no. 4 (2013): 162. DOI.