Non-Perturbative Renormalization from Informational Coarse-Graining

\labelapp:RG

This appendix develops a non-perturbative renormalization framework derived directly from informational coarse-graining. Renormalization is not introduced as a technical device but emerges as a physical necessity once informational stability, locality, and scale separation are enforced.

The construction applies independently of perturbative expansions and clarifies the origin of universality, running couplings, and fixed points.

Renormalization as Physical Information Loss

Renormalization arises whenever a description discards degrees of freedom below a resolution scale. In the present framework, coarse-graining corresponds to controlled loss of informational detail while preserving structural stability.

Let $I(x)$ denote the informational density field. Define a coarse-graining transformation $\mathcal{C}_\ell$ acting as:

where $K_\ell$ is a smoothing kernel of width $\ell$.

This transformation is not a mathematical convenience but reflects physical inaccessibility of fine-grained informational structure.

Scale-Dependent Effective Dynamics

Under coarse-graining, the effective dynamics of $I_\ell$ changes. The original microscopic equation:

induces a scale-dependent effective equation:

The parameters $(D_\ell,\alpha_\ell,\beta_\ell)$ are not arbitrary. They encode how informational stability is preserved across scales.

Renormalization Group Flow as Stability Flow

Define a logarithmic scale parameter $s = \log \ell$. The induced flow of effective parameters is governed by:

where $\{g\} = \{D,\alpha,\beta,\dots\}$ and $\mathcal{B}_i$ denotes the RG flow functions (to avoid confusion with the cubic saturation coefficient $\beta$).

Crucially, these flow functions are not postulated. They arise from the requirement that the coarse-grained dynamics remains structurally stable under $\mathcal{C}_\ell$.

Unstable directions correspond to irrelevant informational structures that are suppressed dynamically.

Fixed Points and Universality

A fixed point $\{g^\ast\}$ satisfies:

At such points, the coarse-grained dynamics becomes scale invariant. Universality classes correspond to basins of attraction of these fixed points.

Different microscopic informational realizations that share the same stability properties flow toward the same effective QFT description.

This explains why vastly different UV details produce identical IR quantum field theories.

Non-Perturbative Nature of the Flow

The RG flow derived here does not rely on small-coupling expansions. It is defined directly on the space of effective informational dynamics.

In particular:

• strong-coupling regimes are admissible,
• fixed points need not be Gaussian,
• non-perturbative phases are natural outcomes.

This resolves the long-standing reliance of QFT on perturbative control.

Emergence of Relevant and Irrelevant Operators

Operators are classified by their stability under coarse-graining. An operator $\mathcal{O}$ is:

• relevant if it grows under $\mathcal{C}_\ell$,
• irrelevant if it is suppressed,
• marginal if its effect remains scale-invariant.

This classification is purely informational and does not depend on canonical dimensional analysis alone.

Relation to Wilsonian Renormalization

The present construction reproduces the operational content of Wilsonian RG while clarifying its meaning.

Momentum shells are replaced by informational resolution shells. Integrating out high-momentum modes corresponds to discarding fine-grained informational correlations.

The Wilsonian picture is therefore a special case of informational coarse-graining.

Breakdown of RG at Extreme Densities

Renormalization assumes approximate locality and factorization. When informational density exceeds the critical threshold $I_{\rm crit}$:

• scale separation collapses,
• RG flow ceases to be well-defined,
• no effective QFT description survives.

This explains why QFT and RG both fail near black-hole cores and Planckian regimes.

Connection to Emergent Lorentz Symmetry

Lorentz invariance appears when RG flow approaches a relativistic fixed point. Deviations from Lorentz symmetry correspond to irrelevant operators that decay under coarse-graining.

Thus, Lorentz symmetry is dynamically selected rather than imposed.

Summary

Renormalization is an unavoidable consequence of informational coarse-graining. Running couplings, universality, and fixed points arise from stability requirements, not from perturbative artifacts.

This appendix completes the non-perturbative closure of QFT by grounding renormalization group structure in the underlying informational dynamics.