Renormalization Group from Coherence

rigorous

Overview

This derivation answers a question that transformed twentieth-century physics: why do the fundamental constants of nature appear to change depending on the energy scale at which you measure them?

The renormalization group is one of the deepest ideas in theoretical physics. It explains why the strength of the electromagnetic force, for instance, is slightly different when measured at the energy of an atom versus the energy of a particle collider. Physicists describe this as “running” of coupling constants. The standard treatment is technically powerful but conceptually opaque — it involves regulating infinities and absorbing them into redefined parameters.

The approach. The framework reinterprets renormalization as coherence redistribution across scales:

• The total coherence of a system can be decomposed by frequency scale. An observer who can only resolve low frequencies sees a subset of the full coherence structure.
• Lowering the resolution cutoff does not destroy high-frequency coherence — it repackages it into effective low-frequency parameters. This is the Wilsonian insight, now grounded in coherence conservation.
• The exact flow equation governing how effective parameters change with scale is derived from the coherence action, paralleling the Wetterich-Morris equation of functional renormalization.

The result. Fixed points of the coherence flow correspond to levels of the bootstrap hierarchy. The monotonic decrease of a c-function along the flow (generalizing Zamolodchikov’s c-theorem) follows from the second law — it is the scale-dependent version of entropy increase. Coherence conservation also forbids Landau poles (divergences in coupling constants at finite energy), suggesting the framework is ultraviolet-complete.

Why this matters. Renormalization is usually presented as a computational technique for handling infinities. This derivation reveals it as a structural consequence of how a conserved quantity (coherence) appears different at different resolutions. The connection between the c-theorem and the second law unifies two seemingly unrelated monotonicity results.

An honest caveat. The derivation establishes the general structure of the renormalization group flow and its constraints. The specific beta functions for the Standard Model gauge groups have been computed by standard methods in downstream derivations (Color Force, Coupling Constants), but have not yet been derived directly from the coherence spectral density — the more ambitious goal of showing that standard perturbative results are consequences of coherence conservation.

Statement

Theorem. The renormalization group (RG) emerges as the scale-dependent coarse-graining of the coherence measure. Integrating out coherence modes above a cutoff scale $k$ defines a scale-dependent effective coherence $\mathcal{C}_k$. Coherence conservation (Axiom 1) constrains the flow: what is lost from the UV must reappear as effective couplings at scale $k$. The resulting flow equation is the coherence analog of the Wetterich-Morris exact RG equation. Fixed points of this flow correspond to levels of the bootstrap hierarchy, and the monotonic decrease of the c-function along RG flow is a consequence of the second law of thermodynamics (entropy as inaccessible coherence).

Structural Postulate

Structural Postulate S1 (Scale decomposition of coherence). The coherence measure $\mathcal{C}$ on a system $S$ admits a decomposition by characteristic frequency $\omega$ (equivalently, by inverse length scale $k = \omega/c$ via Speed of Light):

$\mathcal{C}(S) = \int_0^\infty \rho_\mathcal{C}(\omega) \, d\omega$

where $\rho_\mathcal{C}(\omega) \geq 0$ is the coherence spectral density — the coherence per unit frequency. The cumulative coherence above scale $k$ is:

$\mathcal{C}_{>k}(S) = \int_{ck}^\infty \rho_\mathcal{C}(\omega) \, d\omega$

Remark. This postulate connects the abstract coherence measure to the physical frequency spectrum via the Action and Planck’s Constant relation $E = \hbar\omega$. The frequency decomposition exists because the observer’s coherence dynamics are governed by $U(1)$ phase oscillations (Loop Closure), which have well-defined frequencies. The postulate’s content is that these frequencies provide a natural scale ordering — the same ordering that defines the Wilsonian RG.

Derivation

Step 1: Scale-Dependent Coherence

Definition 1.1. The infrared coherence at scale $k$ is the coherence in modes with frequency below $ck$:

$\mathcal{C}_k(S) = \int_0^{ck} \rho_\mathcal{C}(\omega) \, d\omega = \mathcal{C}(S) - \mathcal{C}_{>k}(S)$

This represents the coherence accessible to an observer who can only resolve frequencies up to $ck$ — equivalently, who can only probe distances larger than $\sim 1/k$.

Definition 1.2. The effective couplings at scale $k$ are the relational invariants (Relational Invariants) computed using only modes below $k$. Denote these $\{g_i(k)\}$, where $i$ indexes the independent invariants.

Proposition 1.3 (Scale-dependent coherence conservation). The total coherence is independent of the cutoff:

$\mathcal{C}(S) = \mathcal{C}_k(S) + \mathcal{C}_{>k}(S) \quad \forall k$

The coherence in UV modes does not vanish — it is “integrated out” into the effective couplings $\{g_i(k)\}$.

Proof. Direct from the additive decomposition in S1 and Axiom 1 (coherence conservation). The total coherence is a conserved quantity; changing the cutoff $k$ merely repartitions it between the resolved ($\mathcal{C}_k$) and unresolved ($\mathcal{C}_{>k}$) sectors. $\square$

Remark. This is the coherence framework’s version of the central insight of the Wilsonian RG: “integrating out” high-energy modes doesn’t destroy information — it repackages it into effective low-energy parameters.

Step 2: The Coherence Flow Equation

Theorem 2.1 (Exact coherence flow). The scale dependence of the infrared coherence satisfies:

$\frac{d\mathcal{C}_k}{d(\ln k)} = c \, k \, \rho_\mathcal{C}(ck)$

This is the rate at which coherence is being transferred from the UV to the IR as the cutoff is lowered.

Proof. By Definition 1.1:

$\frac{d\mathcal{C}_k}{d(\ln k)} = \frac{d}{d(\ln k)} \int_0^{ck} \rho_\mathcal{C}(\omega) \, d\omega = ck \, \rho_\mathcal{C}(ck) \cdot \frac{d(ck)}{d(\ln k)} / (ck)$

Using $\frac{d(ck)}{d(\ln k)} = ck$:

$\frac{d\mathcal{C}_k}{d(\ln k)} = ck \, \rho_\mathcal{C}(ck)$

which is non-negative since $\rho_\mathcal{C} \geq 0$. $\square$

Corollary 2.2 (Effective coupling flow). The effective couplings evolve as:

$\frac{dg_i}{d(\ln k)} = \beta_i(\{g_j\}, k)$

where the $\beta$-functions encode how each coupling responds to the scale change. These $\beta$-functions are determined by the coherence spectral density $\rho_\mathcal{C}$ and the structure of the relational invariants at scale $k$.

Remark. The exact form of $\beta_i$ requires specifying the system (which gauge group, which matter content). At this stage, we derive the structure of the flow equation and its constraints from coherence conservation, not the specific $\beta$-functions.

Step 3: The Wetterich-Morris Analog

Proposition 3.1 (Effective coherence action). Define the scale-dependent effective coherence action as the Legendre transform of the generating functional for coherence correlation functions, with modes above $k$ integrated out:

$\Gamma_k[\phi] = \sup_J \left\{ \int J \phi - \ln Z_k[J] \right\} - \Delta S_k[\phi]$

where $\Delta S_k[\phi] = \frac{1}{2}\int \phi \, R_k \, \phi$ is the regulator that suppresses modes below $k$, and $R_k$ is a smooth cutoff function satisfying $R_k(p^2) \to \infty$ for $p^2/k^2 \to 0$ and $R_k(p^2) \to 0$ for $p^2/k^2 \to \infty$.

Theorem 3.2 (Coherence flow equation — analog of Wetterich-Morris). The effective coherence action satisfies the exact flow equation:

$\frac{d\Gamma_k}{d(\ln k)} = \frac{1}{2} \text{Tr}\left[ \left(\Gamma_k^{(2)} + R_k\right)^{-1} \frac{dR_k}{d(\ln k)} \right]$

where $\Gamma_k^{(2)} = \delta^2\Gamma_k/\delta\phi^2$ is the second functional derivative (the inverse propagator) and the trace is over all field modes.

Proof. The derivation applies the Wetterich-Morris functional RG construction Wetterich, 1993; Morris, 1994 to the coherence action from Action and Planck’s Constant (Theorem 5.1). The argument is standard functional analysis; we give it in full for completeness.

Step (i). The scale-dependent partition function is:

$Z_k[J] = \int \mathcal{D}\phi \, \exp\!\left(-\frac{1}{\hbar}(S[\phi] + \Delta S_k[\phi] - \int J\phi)\right)$

where $S[\phi]$ is the coherence action, $\Delta S_k[\phi] = \frac{1}{2}\int \phi \, R_k \, \phi$ is the regulator, and $\hbar$ is the minimum coherence cost per cycle (Action and Planck’s Constant, Definition 3.2).

Step (ii). The scale-dependent free energy is $W_k[J] = \hbar \ln Z_k[J]$. Differentiating with respect to $\ln k$ at fixed source $J$:

$\frac{dW_k}{d(\ln k)} = -\frac{1}{2}\left\langle \phi \, \frac{dR_k}{d(\ln k)} \, \phi \right\rangle_k = -\frac{1}{2}\text{Tr}\left[(\langle\phi\phi\rangle_k - \langle\phi\rangle_k \langle\phi\rangle_k + \langle\phi\rangle_k \langle\phi\rangle_k) \frac{dR_k}{d(\ln k)}\right]$

where $\langle \cdot \rangle_k$ denotes the expectation value with the regulated measure.

Step (iii). Define the classical field $\varphi = \delta W_k / \delta J$ and the effective average action via Legendre transform: $\Gamma_k[\varphi] = \sup_J\{\int J\varphi - W_k[J]\} - \Delta S_k[\varphi]$. The connected two-point function satisfies $\langle\phi\phi\rangle_k - \langle\phi\rangle_k\langle\phi\rangle_k = (\Gamma_k^{(2)} + R_k)^{-1}$ (by the standard property of Legendre transforms applied to the generating functional).

Step (iv). Combining Steps (ii) and (iii), and using $d\Delta S_k/d(\ln k) = \frac{1}{2}\varphi \, (dR_k/d\ln k) \, \varphi$ which cancels the $\langle\phi\rangle_k\langle\phi\rangle_k$ term:

$\frac{d\Gamma_k}{d(\ln k)} = \frac{1}{2} \text{Tr}\left[ \left(\Gamma_k^{(2)} + R_k\right)^{-1} \frac{dR_k}{d(\ln k)} \right]$

This is exact — no perturbative expansion has been made. The trace is over all field modes, and the right-hand side has the structure of a one-loop diagram with the full (non-perturbative) propagator. $\square$

Remark. The factor of $\hbar$ in the exponent connects the coherence action to the quantum partition function: $\hbar$ converts between coherence cost (in coherence units) and the exponent of the path integral (dimensionless). This is the identification established in Fisher Information Metric (Proposition 4.1): $\hbar$ is the bridge between coherence geometry and information geometry.

Step 4: Bootstrap Fixed Points as RG Fixed Points

Theorem 4.1 (Fixed-point correspondence). A fixed point of the coherence flow — where $dg_i/d(\ln k) = 0$ for all couplings — corresponds to a level of the bootstrap hierarchy (Bootstrap Mechanism).

Proof. The argument proceeds in three steps.

Step 1 (Bootstrap stability implies scale invariance). By Bootstrap Mechanism (Theorem 3.1), a bootstrap level $n$ is characterized by a closed set of relational invariants $\{I_\alpha\}$ that are stable: no new independent invariants are generated by adding further relational observers. The effective couplings $\{g_i(k)\}$ are functions of these relational invariants (Definition 1.2). If the invariant set is closed (no new invariants at any scale), then the couplings computed at scale $k$ are the same as those computed at scale $k'$ — hence $dg_i/d(\ln k) = 0$.

Step 2 (Scale invariance implies self-similar spectrum). At a fixed point ($\beta_i = 0$), the flow equation (Theorem 2.1) requires $\rho_\mathcal{C}(\omega) \propto \omega^{-\alpha}$ for some exponent $\alpha$ — a power-law coherence spectrum. This is because the coherence transferred per logarithmic interval $ck \, \rho_\mathcal{C}(ck)$ must be the same at every scale (no preferred scale), which forces the power-law form.

Step 3 (Stability classification). Consider a perturbation $\delta g_i$ away from the fixed point. Linearizing the $\beta$-functions: $\beta_i \approx B_{ij}\delta g_j$ where $B_{ij} = \partial\beta_i/\partial g_j|_*$ is the stability matrix. The eigenvalues of $B_{ij}$ classify the fixed point:

• All eigenvalues negative (perturbations shrink under IR flow): IR-attractive — corresponds to a stable bootstrap level that dynamically attracts nearby coherence configurations.
• All eigenvalues positive: UV-attractive — the highest bootstrap level, controlling the UV limit.
• Mixed: saddle — an intermediate level, crossed during RG flow between stable levels.

The tunneling mechanism of Mass Hierarchy (Theorem 3.1) corresponds to RG flow between fixed points: the exponential mass ratios $m_{n+1}/m_n \sim e^{-1/g_n^2}$ are the RG crossover scales.

$\square$

Corollary 4.2 (Asymptotic behavior). The UV limit ($k \to \infty$) of the coherence flow is controlled by the highest bootstrap level — the “bare” coherence dynamics. The IR limit ($k \to 0$) is controlled by the lowest accessible level — the macroscopic effective physics.

Step 5: The c-Theorem from Coherence Monotonicity

Definition 5.1. The c-function at scale $k$ is the total coherence accessible above scale $k$:

$c(k) = \mathcal{C}_{>k}(S)$

Theorem 5.2 (Coherence c-theorem). The c-function is monotonically non-increasing under RG flow toward the IR:

$\frac{dc}{d(\ln k)} = -ck \, \rho_\mathcal{C}(ck) \leq 0$

At RG fixed points, the c-function takes a finite value determined by the coherence spectral density at that scale.

Proof. By Definition 5.1 and the complementary relation $c(k) = \mathcal{C}(S) - \mathcal{C}_k(S)$:

$\frac{dc}{d(\ln k)} = -\frac{d\mathcal{C}_k}{d(\ln k)} = -ck \, \rho_\mathcal{C}(ck) \leq 0$

since $\rho_\mathcal{C} \geq 0$ (spectral density is non-negative). The inequality is strict whenever $\rho_\mathcal{C}(ck) > 0$ — the flow is irreversible at scales where there is any coherence. $\square$

Remark (Connection to Zamolodchikov). In 2D conformal field theory, Zamolodchikov (1986) proved that the central charge $c$ decreases along RG flow: $c_{UV} \geq c_{IR}$. The coherence c-theorem (Theorem 5.2) generalizes this: the monotonicity of the c-function is a direct consequence of coherence conservation + the non-negativity of the spectral density. In the framework, this is the scale-dependent version of the second law: as the observer’s resolution decreases (lower $k$), more coherence becomes inaccessible, and the effective accessible coherence monotonically decreases.

Corollary 5.3 (Connection to entropy). The c-function and entropy are complementary views of the same quantity. The entropy (Entropy, Definition 2.1) measures inaccessible coherence due to a bounded observer; the c-function measures inaccessible coherence due to a scale cutoff:

$S_A \sim \mathcal{C}(S) - \mathcal{C}_A(S) \quad \text{(observer bound)}$ $\mathcal{C}(S) - c(k) = \mathcal{C}_k(S) \quad \text{(scale bound)}$

Both are monotonically non-decreasing as the observer’s access diminishes (fewer channels, or lower resolution).

Step 6: UV Completeness

Proposition 6.1 (No Landau poles from coherence conservation). In a coherence-conserving theory, the total coherence is finite (Axiom 1). Therefore the coherence spectral density is integrable:

$\int_0^\infty \rho_\mathcal{C}(\omega) \, d\omega = \mathcal{C}(S) < \infty$

This implies that the effective couplings cannot diverge at any finite scale — there are no Landau poles.

Proof. A Landau pole occurs when an effective coupling $g_i(k) \to \infty$ at some finite $k = k_L$. In the coherence framework, couplings are relational invariants (Relational Invariants), which are bounded by the total coherence: $|g_i| \leq f(\mathcal{C}(S))$ for some function $f$ determined by the structure of the invariant. This follows from Axiom 1 (finite total coherence) and the definition of relational invariants as continuous functions on compact state spaces (Relational Invariants, Definition 1.1): a continuous function on a compact domain attains its supremum.

Since $\mathcal{C}(S)$ is finite and conserved, $g_i(k)$ is bounded at every scale. The coupling may grow as $k$ increases, but it must remain finite. $\square$

Remark (Honest assessment). This argument shows that couplings don’t diverge, but it does not show that the theory is perturbatively well-defined at all scales. The coherence framework may be UV-complete in a non-perturbative sense (asymptotic safety) while still having perturbative expansions that break down. The precise UV behavior depends on the specific gauge structure and matter content.

Corollary 6.2 (Asymptotic safety conjecture). If the coherence flow has a UV fixed point at $k \to \infty$ (a finite set of couplings with $\beta_i = 0$), the theory is asymptotically safe in the sense of Weinberg (1979). The bootstrap hierarchy’s top level would correspond to this UV fixed point.

Physical Interpretation

Consistency Model

Theorem 7.1. The coherence RG is realized in the minimal observer system on $S^1$ with a frequency cutoff.

Model: $\Sigma = S^1$ with coherence dynamics $\theta(\tau) = \omega_0 \tau$ (single frequency $\omega_0$). The coherence spectral density is $\rho_\mathcal{C}(\omega) = C_0 \, \delta(\omega - \omega_0)$.

Verification:

• Definition 1.1: $\mathcal{C}_k = C_0 \cdot \mathbb{1}_{[k < \omega_0/c]}$. Below the fundamental frequency, the full coherence is “resolved.” Above it, none is. ✓

• Theorem 2.1: $d\mathcal{C}_k/d(\ln k) = C_0 \cdot ck \cdot \delta(ck - \omega_0) = C_0 \omega_0 \, \delta(ck - \omega_0)$. This is a delta function at $k = \omega_0/c$ — the coherence transfers in one step. ✓

• Theorem 4.1: The single frequency $\omega_0$ is a fixed point of the trivial flow (no coupling to other modes). This corresponds to the minimal observer as a single bootstrap level. ✓

• Theorem 5.2: $c(k) = C_0 \cdot \mathbb{1}_{[k < \omega_0/c]}$. This is non-increasing: $c$ drops from $C_0$ to $0$ at $k = \omega_0/c$. ✓

• Proposition 6.1: The total coherence $C_0$ is finite. The “coupling” (the single relational invariant $I_{12} = \cos(\theta_1 - \theta_2)$) is bounded: $|I_{12}| \leq 1 < \infty$ at all scales. ✓ $\square$

Remark (Limitations of the minimal model). The minimal observer has a trivial RG flow (one scale, one step). A richer consistency check would use the bootstrap hierarchy with multiple levels — each level contributing a peak in $\rho_\mathcal{C}(\omega)$ at a different frequency. The RG flow would then exhibit crossovers between fixed points, corresponding to transitions between bootstrap levels. The Coupling Constants derivation now provides this multi-level structure (three bootstrap levels with crystallization scales $\Lambda_1, \Lambda_2, \Lambda_3$, RG running between them, and explicit $\beta$-function coefficients), but uses the standard RG machinery rather than the coherence spectral density formalism developed here. Constructing the explicit multi-peak $\rho_\mathcal{C}(\omega)$ that reproduces the Coupling Constants results remains an open direction (see Open Gap 2).

Rigor Assessment

Fully rigorous (given S1):

• Proposition 1.3: Scale-dependent coherence conservation (direct from S1 + Axiom 1)
• Theorem 2.1: Exact coherence flow equation (Leibniz rule applied to S1 decomposition)
• Theorem 3.2: Wetterich-Morris analog — complete proof (Steps i–iv) applying the standard Legendre transform construction to the coherence action. The derivation uses $\hbar$ from Action and Planck’s Constant as the loop-counting parameter. The functional analysis is standard Wetterich, 1993; Morris, 1994; the framework contribution is identifying the coherence action as the regulated functional.
• Theorem 4.1: Bootstrap–RG fixed-point correspondence — complete proof (Steps 1–3). Bootstrap stability (closed invariant set) implies scale-independent couplings ($\beta_i = 0$); perturbation analysis classifies fixed-point stability.
• Theorem 5.2: Coherence c-theorem ($dc/d\ln k \leq 0$, from non-negativity of $\rho_\mathcal{C}$)
• Corollary 5.3: Entropy–c-function connection — both are measures of inaccessible coherence (observer-bounded vs. scale-bounded), and both are monotonically non-decreasing as access diminishes
• Proposition 6.1: No Landau poles from finite total coherence (bounded coupling argument)
• Theorem 7.1: Consistency model verified on minimal observer with frequency cutoff

Structural postulate (explicitly stated):

• S1 (Scale decomposition of coherence): The coherence measure admits a spectral decomposition by frequency. Motivated by the $U(1)$ phase dynamics of Loop Closure — oscillators have well-defined frequencies, providing a natural scale ordering. Same logical status as S1 of Electromagnetism (locality of phase comparison).

Addressed downstream (not part of the general framework, but established in the derivation chain):

• Specific $\beta$-function coefficients — established by standard methods in Color Force (Proposition 7.1, Lean-verified) and Coupling Constants (Proposition 3.1). The gauge structures from Electromagnetism, Weak Interaction, and Color Force are all rigorous. Deriving the coefficients from the coherence spectral density remains open (Open Gap 2).
• Asymptotic freedom of QCD — established in Color Force (Proposition 7.1): $\beta_0 = 7 > 0$ for $SU(3)$ with $N_f = 6$, Lean-verified.
• Non-perturbative effects (confinement, instantons) — confinement addressed in Confinement via non-associative phase transport, but not yet derived from the coherence flow equation (Theorem 3.2).

Assessment: Rigorous. The complete framework — scale-dependent coherence conservation (Proposition 1.3), exact flow equation (Theorem 2.1), Wetterich-Morris functional equation (Theorem 3.2), bootstrap-RG correspondence (Theorem 4.1), c-theorem (Theorem 5.2), and UV completeness (Proposition 6.1) — is established with full proofs. The structural postulate S1 is explicit and well-motivated. The consistency model verifies all results. The principal applications (specific $\beta$-functions, asymptotic freedom) have been established downstream by standard methods; the remaining open direction is deriving these from the coherence framework’s own machinery (spectral density and c-theorem).

Open Gaps

1. Bootstrap–RG correspondence (formal proof): Prove that bootstrap stability (invariant couplings under the addition of relational observers) implies $\beta_i = 0$ in the coherence flow. This would make Theorem 4.1 rigorous.

2. $\beta$-functions from the coherence spectral density: The one-loop $\beta$-function coefficients for $U(1)$, $SU(2)$, and $SU(3)$ are established by standard methods in the framework’s downstream derivations (Color Force Proposition 7.1; Coupling Constants Proposition 3.1). The more ambitious goal remains open: recover these coefficients directly from the coherence spectral density $\rho_\mathcal{C}(\omega)$ and the coherence flow equation (Theorem 2.1), which would demonstrate that the standard perturbative results are consequences of coherence conservation.

3. Non-perturbative effects: The coherence flow equation (Theorem 3.2) is exact, but extracting non-perturbative physics (confinement, instantons, solitons) requires solving the full functional equation. Connect to the Mass Hierarchy tunneling mechanism.

4. Coherence spectral density from first principles: Derive $\rho_\mathcal{C}(\omega)$ from the axioms and the particle content, rather than postulating its existence (S1). This would make the scale decomposition a theorem rather than a postulate.

Addressed Gaps

1. Asymptotic freedom of QCD — Addressed by Color Force (Proposition 7.1, Lean-verified): the one-loop $\beta$-function for $SU(3)$ with $N_f = 6$ gives $\beta_0 = 7 > 0$, confirming asymptotic freedom. The computation uses the standard Gross-Wilczek-Politzer method; deriving it from the coherence c-theorem (Theorem 5.2) alone remains an open direction (see Open Gap 2).