Universal Emergence of Bosonic and Fermionic Algebras in a Deterministic Proper-Time Framework

The gist This work develops a deterministic, discrete proper-time framework where quantum field theory emerges as an effective infrared description characterized by a running Planck constant, providing both bosonic and fermionic canonical algebras with a common normalization derived from microscopic dynamics.

How it works

The construction begins with an ordered set of fundamental events, defined by the causal chains that generate the microscopic structure <ref:2608.14816#pg3>. The microscopic evolution proceeds through elementary updates separated by a minimal proper-time interval, denoted as τmin <ref:2608.14816#pg5>. This evolution is described by an information-preserving discrete map F, where each application of F represents the finite change of the complete microscopic configuration during one elementary proper-time interval <ref:2608.14816#pg6>.

The emergence of continuous quantum field theory occurs only after coarse-graining over these deterministic histories <ref:2608.14816#pg3>. The effective quantization scale, ħeff(µ), is determined by the fraction of microscopic information that remains unresolved within the coarse-grained equivalence classes, specifically ħeff(µ) = ħQ(nµ) = ħ (1 - 1/nµ) <ref:2608.14816#pg7>. This links microscopic evolution to the emergent canonical algebra by representing the reversible microscopic dynamics through finite-difference translations in field-configuration space <ref:2608.14816#pg4>.

Algebraic Realization of Dynamics

The local representation of an elementary reversible update is achieved by a pair of finite translations, Tϵ and T−ϵ, which define a symmetric finite-difference operator Dϵψ(x) = ψ(x + ϵ) − ψ(x − ϵ) <ref:2608.14816#pg8>. This operator leads to the exact deformed commutation relation [x, ˆ pˆϵ] = iħCϵ, where Cϵ is the symmetric averaging operator <ref:2608.14816#pg8>. In the smooth regime, this reduces to the ordinary canonical commutation relation as ϵ approaches zero <ref:2608.14816#pg8>.

The resulting operator-valued deformation is projected onto the coarse-grained macroscopic state using a projector Pˆµ = ϕµ⟩⟨ϕµ <ref:2608.14816#pg8>. For the minimal boundary-crossing realization, this yields PˆµCˆϵPˆµ = Q(nµ)Pˆµ = 1 - 1/nµ Pˆµ <ref:2608.14816#pg10>. This demonstrates that the effective canonical algebra is defined by the scalar relation hϕ j, π ki eff = iħeff(µ)δjk, where ħeff(µ) = ħ (1 - 1/nµ) <ref:2608.14816#pg10>.

Extension to Fermionic Degrees of Freedom

The construction is extended to fermionic degrees of freedom by representing the universal update within an effective graded algebra, where the Grassmann coordinates and Berezin derivatives determine the anticommuting canonical structure <ref:2608.14816#pg22>. The minimal effective representation combines the symmetric finite-update structure with the Berezin derivative via Dθ = Cˆ(F) ∂/∂θ <ref:2608.14816#pg24>.

The resulting fermionic canonical anticommutator is given by the relation PˆµCˆ(F)Pˆµ = Q(nµ)Pˆµ <ref:2608.14816#pg31>. This shows that both bosonic and fermionic sectors inherit a common effective Planck constant, ħeff(µ) = ħ (1 - 1/nµ) <ref:2608.14816#pg33>.

Implications for High-Energy Physics

The deterministic crossover manifests in two possible phenomenological signatures <ref:2608.14816#pg35>. First, radiative corrections are expected to be suppressed when the characteristic proper-time resolution of a high-energy process approaches τmin, with the relative effect schematically written as R eff rad(E) = Srad [Q(E)] R QFT rad (E) <ref:2608.14816#pg34>. Second, Hawking radiation is suppressed as the effective quantization scale decreases, leading to T eff H = Q(nµ) T H <ref:2608.14816#pg36>.

The framework provides a microscopic rationale for energy-dependent suppression factors in quantum tunneling and suggests that the vanishing of Hawking temperature at nµ → 1 is a direct dynamical consequence of the underlying deterministic update rate <ref:2608.14816#pg36>.

Summary

The present work establishes that the effective quantization scale is determined by the unresolved microscopic multiplicity, and this common factor Q(nµ) governs both bosonic and fermionic canonical algebras <ref:2608.14816#pg38>. This framework provides a microscopic interpretation of renormalization group structure, an operatorial realization of the coarse-grained quantization scale, and a common canonical normalization for bosonic and fermionic degrees of freedom <ref:2608.14816#pg38>.

1. Processing of Scale-Dependent Renormalization Group Flows: The improved AI can distinguish between standard RG flow and running Planck constant crossover by analyzing how changes in macroscopic resolution affect both the coupling renormalization group structure and the effective quantization scale, as discussed in The standard renormalization-group flow and the running of the effective Planck constant therefore refer to two different regimes of the same change in resolution.

2. Framework for Deformed Canonical Algebras: The improved AI can derive operator-valued canonical relations by explicitly representing elementary reversible updates through finite translations in field-configuration space, allowing it to construct a deformed Leibniz rule and an operator-valued field-momentum commutator that governs the effective quantization scale, as shown in Equation (58) and (110).

3. Consistency Check for Bosonic and Fermionic Sectors: The improved AI can verify the consistency of its framework by demonstrating that the bosonic and fermionic sectors inherit a common effective Planck constant without introducing an independent fermionic update or quantization scale, as established in Section 5.4 where both sectors share the same common coarse-graining factor Q(nµ) = 1 − 1/nµ.

4. Model for High-Energy Suppression: The improved AI can identify potential signatures of the deterministic crossover by calculating the relative radiative contribution to a cross section, Rrad(E) = σfull(E) − σtree(E), predicting that the loop-induced channel should be suppressed relative to the tree-level reference as Q(nµ) approaches zero at high energies.

5. Model for Effective Hawking Radiation Temperature: The improved AI can calculate a modified effective Hawking temperature, T eff H = Q(nµ) TH, predicting that the effective temperature is suppressed as the deterministic regime is approached, nµ → 1, providing a dynamical reason for the suppression of quantum radiation.