Quantum mechanics: the finite path integral has modulus at most the number of configurations
ProvedEveryday.amplitude_boundThe source equation's left-hand side W is a transition amplitude: the integral of the phases exp(iS) over all field configurations below the ultraviolet cutoff Λ. The item is the path integral's coherence, in the arena's exact finite form: on a finite configuration space, the amplitude — the sum of the phase of each configuration — has modulus at most the number of configurations. Each summand is a unit phase; the bound is the triangle inequality; the interest of the everyday equation's dynamics lies in the interference of these phases, never in their magnitude.
Formalization Note The finite configuration space is the exact discrete stand-in for the continuum of fields below the cutoff Λ; no measure-theoretic or renormalization content is claimed. The proof: the norm of the sum is at most the sum of the norms — the triangle inequality — each summand's norm is one by the phase milestone, and the sum of a constant one over the space is its cardinality.
import Mathlib import Definitions.Def_Everyday_spacetime import Definitions.Def_Everyday_sectors
namespace Everyday
/-- Quantum mechanics: the path integral is a coherent sum. On a finite
configuration space, the amplitude W = ∑_c exp(i S c) has modulus at
most the number of configurations — a finite sum of unit phases. Each
summand is a phase; the bound is the triangle inequality. -/
theorem amplitude_bound (C : Type) [Fintype C] (S : C → ℝ) :
‖amplitude S‖ ≤ (Fintype.card C : ℝ) := by sorry
end Everyday
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What the Lean code literally says, in plain math · glm-5-latest-short
The statement says: for every type C equipped with the structure of a finite type and every function S assigning a real number to each element of C, the complex norm of amplitude S is at most the coercion to the real numbers of the cardinality of C. Unfolding, amplitude S is the sum over all elements c of C of the complex exponential of the imaginary unit times the coercion of S c, and the norm of that sum is compared with a cardinality coerced to the reals. The conclusion is an inequality between real numbers.