Qft Renormalization Group Rs

The Renormalization Group in RS

The renormalization group is a standard tool in quantum field theory. It tracks how coupling constants, masses, and other parameters flow as the energy scale of an experiment changes. In quantum chromodynamics, the one-loop beta function predicts asymptotic freedom: the strong force weakens at high energies. The coefficient b_0 = 11N_c/3 - 2N_f/3, where N_c is the number of colors and N_f the number of flavors, determines this flow.

In Recognition Science, the framework models this same structure through a recognition cost, a measure of the price reality pays to distinguish two quantities. The module defines domainCost(m, e) = Jcost(m/e), where Jcost(x) = (x + 1/x)/2 - 1. This cost vanishes when the mass m equals the energy e, is nonnegative for positive inputs, and the canonical threshold phi - 3/2 is positive. These three facts are proved in the machine-checked library of formal theorems.

The research note attached to the module records an intended identification: colors N_c = 3 as the configuration dimension, flavors N_f = 6 as twice that, giving b_0 = 7 = 2^3 - 1. The note calls this exact, but the formal proofs establish nothing specific to quantum chromodynamics. The definition of domainCost uses only the ratio m/e, without reference to colors, flavors, or any QFT quantity.

What the module actually establishes is a template. The three proved facts about domainCost hold for any positive m and e. The structure RGFlowCert packages these properties, and cert_inhabited shows such a certificate exists. This is a general statement about the cost function, not a theorem about the renormalization group. The bridge from the cost template to QFT's beta function remains open.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove b_0 = 7 for QCD. The research note's color and flavor identification is not a formal result. No claim is made that the renormalization group flow in nature follows Jcost.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/QFT/Renormalization_Group_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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