Information Across Scales and Renormalization

Information measures can diagnose renormalization-group flow only after the physical comparison is fixed. A UV cutoff, a region size, a correlation length, a deformation scale, and a renormalization scale are different variables; changing any one can produce “scale dependence.” This chapter develops the entropic $c$, $F$, and $a$ statements in their proper dimensions, then separates those theorems from subtraction schemes, finite crossovers, channel models, and information geometry on families of theories.

Helpful background. Universal terms and geometry identifies the finite and logarithmic data used at fixed points; data processing supplies the channel inequalities; UV and IR fixed points, flow constraints, and anomaly coefficients supply the RG endpoints. Regulated subregion entropy, relative entropy in QFT, and universal scaling functions provide the continuum and crossover language.

Enter this chapter

Begin with separating scales. It fixes the six quantities most often conflated in an entanglement-flow claim and states a safe order of limits. Information measures along RG flows then asks what is held fixed while a trajectory is varied.

The dimension-specific core follows:

• Two-dimensional entropic monotones derives the interval $c$-function from Lorentz symmetry and strong subadditivity.
• Three-dimensional F monotonicity explains the disk subtraction and its fixed-point sphere-free-energy anchor.
• Four-dimensional a-type flow constraints distinguishes the Euler-anomaly theorem from entropic and dilaton implementations.

The primary results are dimension specific: the interval proof is given by Casini and Huerta 2007, pp. 7032–7035, the disk construction by Casini and Huerta 2012, §§ II–III, and general differential subtractions by Liu and Mezei 2013, § 2. Four-dimensional irreversibility follows through the dilaton argument of Komargodski and Schwimmer 2011, §§ 3–4 or the null-cone entropic argument of Casini, Testé, and Torroba 2017, pp. 2–4.

The remaining pages make those results usable. Renormalized entropy schemes compares differential, counterterm, and mutual-information prescriptions. Relevant deformations and crossovers organizes finite-window scaling. Relative entropy and irreversibility and coarse-graining channels state when data processing has an actual map to act on. Scale-space caveats treats Fourier and wavelet partitions, while information geometry on theory space extracts local response without pretending that all QFTs form one finite-dimensional manifold.

The first diagram is a reading map. Its main point is that fixed-point theorems, crossover functions, and channel recoverability are three branches with different inputs.

Scale dictionary for the chapter. One must first form dimensionless ratios and fix the regional observable, state, regulator, and subtraction prescription. The three lower branches are logically distinct: an endpoint monotonicity theorem, a finite-window crossover, and an operational channel statement. Schematic and not to scale.

What a monotonicity claim must specify

An entropic quantity $\mathcal I(R)$ is not automatically an RG monotone. A complete claim names:

1. the spacetime dimension and theorem hypotheses;
2. the family of regions and the state on each regional algebra;
3. the regulator or algebraic comparison that makes all terms meaningful;
4. the subtraction prescription and its finite local ambiguities;
5. the direction of the scale parameter; and
6. the UV and IR quantities reached in controlled limits.

For example, the two-dimensional result concerns the vacuum interval function $c_E(R)=3R S'(R)$ in a unitary Lorentz-invariant theory. The three-dimensional quantity $\mathcal F(R)=(R\partial_R-1)S(R)$ is a disk construction. In four dimensions the theorem is $a_{\rm UV}>a_{\rm IR}$ under the standard unitary relativistic hypotheses; a proposed interpolating entropic function requires additional care. The statement owner for the fixed-point $c$, $F$, and $a$ theorems is the conformal-field-theory volume. Here they are used as information diagnostics.

Comparison table

The table is the semantic equivalent of the diagrams and should be used when selecting a diagnostic.

RG information diagnostics compared by dimension, region, prescription, theorem hypotheses, endpoint anchor, deformation variable, numerical observable, recovery content, and characteristic failure.

Diagnostic	Region and dimension	Definition or prescription	Hypotheses and fixed-point anchor	Finite-flow observable	Recovery statement	Failure test
Entropic c-function	Vacuum interval, 1+1 dimensions	cE(R) = 3R S′(R)	Unitary Lorentz-invariant QFT, strong subadditivity; equals the CFT central charge at endpoints	Derivative of a cutoff-refined interval entropy	Not required for the theorem	Finite temperature or non-Lorentz-invariant dynamics can spoil the proof
Renormalized F-function	Vacuum disk, 2+1 dimensions	𝓕(R) = (R∂R − 1)S(R)	Relativistic vacuum flow; equals sphere free energy at a CFT	Stable derivative fit after area-law subtraction	Not required for the entropic inequality	Raw disk entropy or a narrow scaling window gives a false plateau
a-type constraint	Spheres or null-cone regions, 3+1 dimensions	Euler-anomaly endpoint difference; logarithmic spherical term at a CFT	Unitary relativistic RG flow between CFTs; aUV > aIR	Endpoint anomaly match or a controlled dilaton/entropic construction	No generic recovery map is implied	Confusing the Weyl-squared coefficient or a scheme-dependent finite term with a
Relative-entropy diagnostic	Any declared common regional algebra	D(ρR ∥ σR) = Δ⟨Kσ⟩ − ΔS	Normal states in one algebra or a controlled common regulator	Region-size dependence and endpoint asymptotics	Equality or small loss can license recovery	Comparing states of different theories without an embedding
Coarse-graining channel	Retained algebra or modes in any dimension	Declared CPTP map 𝒩	Input/output systems and operational task fixed	Distinguishability loss under 𝒩	Approximate sufficiency from a recovery bound	Calling a Wilsonian change of action a channel without a state map
Theory-space metric	Regulated source family at fixed region	Relative-entropy or fidelity Hessian with contact terms specified	Smooth family, fixed operator basis modulo redundant directions	Integrated connected correlators and scaling with R	Contractivity only when an explicit channel acts	Coordinate or counterterm dependence mistaken for an observable distance

Validity gates

The second diagram is a stop rule. If the states do not share an algebra or regulator, relative entropy is not yet defined. If the regional family changes, apparent flow can be kinematic. If neither theorem hypotheses nor a channel are present, the result remains a finite-window diagnostic.

Validity and failure map. Dimension-specific monotonicity theorems require their own relativistic and vacuum hypotheses; recoverability requires a completely positive trace-preserving map and a retained algebra. When a gate fails, cutoff refinement, endpoint matching, and correction-to-scaling fits can still support a limited numerical statement, but not universal irreversibility. Schematic and not to scale.

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.
• Casini, Horacio, Eduardo Testé, and Gonzalo Torroba. “Markov Property of the Conformal Field Theory Vacuum and the a Theorem.” Physical Review Letters 118 (2017): 261602. DOI.
• Komargodski, Zohar, and Adam Schwimmer. “On Renormalization Group Flows in Four Dimensions.” Journal of High Energy Physics 2011, no. 12 (2011): 099. 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.