Understanding the True Quantum Dirac Equation vs. TFP Emergent Version

By John Gavel

I want to share a detailed comparison between the fundamental Dirac equation and an emergent version I derived within my framework of Temporal Flow Physics (TFP). This post is intended to be as complete and informative as possible, showing all the key mathematical structures and interpretations.

1. The True Quantum Dirac Equation (Fundamental)

The standard Dirac equation is written as:

\[ (i \gamma^\mu \partial_\mu - m) \psi(x) = 0 \]

Key properties of this fundamental equation:

• Quantum field: \(\psi(x)\) is a quantum operator-valued spinor field in quantum field theory (QFT), or a complex probability-amplitude spinor in single-particle quantum mechanics (QM).
• Hilbert space: \[ \psi \in L^2(\mathbb{R}^3, \mathbb{C}^4) \] with inner product \[ \langle \phi | \psi \rangle = \int \phi^\dagger(x) \psi(x) \, d^3x \]
• Unitary evolution: \[ \frac{\partial}{\partial t} \langle \psi | \psi \rangle = 0 \] which guarantees probability conservation.
• Linear superposition: If \(\psi_1\) and \(\psi_2\) are solutions, any linear combination \[ a \psi_1 + b \psi_2 \] is also a solution.
• Anticommutation relations (QFT): \[ \{ \psi_\alpha(x), \psi_\beta^\dagger(y) \} = \delta_{\alpha\beta} \delta^3(x-y) \]
• Probability interpretation: The probability density \[ j^0 = \psi^\dagger \psi \ge 0, \quad \int j^0 \, d^3x = 1 \] ensures a consistent probabilistic framework.

This is truly quantum: superposition exists at the level of individual systems, and the operator structure ensures coherent, unitary evolution.

2. My Emergent / Effective Dirac Equation

From my derivation in TFP, the emergent equation takes the form:

\[ (i \hbar_\text{eff} \gamma^\mu \partial_\mu - m_\text{eff} c_\text{eff}) \Psi(x,t) = 0 \]

where

\[ \Psi(x,t) = \begin{pmatrix} \psi_+(x,t) \\ \psi_-(x,t) \end{pmatrix}, \quad \psi_\pm(x,t) = A_\pm e^{i \phi_\pm} \]

The critical defining relation connecting \(\Psi\) to the underlying micro-variables is:

\[ \psi_\pm(x,t) = \lim_{R \to x} \frac{1}{|R|} \sum_{i \in R} F_i^\pm(t) e^{i \phi_i^\pm(t)} \]

2.1 Interpretation

\(\Psi(x,t)\) is not a fundamental quantum field. Instead, it is a classical, coarse-grained observable, analogous to macroscopic quantities like temperature, pressure, or fluid velocity. It emerges from many underlying discrete contributions \(F_i(t)\).

2.2 Discrete Update Rule

The microscopic variables evolve according to a discrete update rule:

\[ F_i(t+1) - 2 F_i(t) + F_i(t-1) = \alpha \sum_j (F_j - F_i) - \mu_\text{eff}^2 F_i \]

This rule shows how each micro-variable \(F_i\) interacts with its neighbors, its own history, and a nonlinear self-term, producing a substrate from which the emergent \(\Psi(x,t)\) is constructed.

2.3 Lack of Quantum Operator Structure and Linearity

The effective equation lacks the operator structure and linearity that make the Dirac equation genuinely quantum. In particular, superposition is not physical at the level of a single realization.

3. Comparison of Initial Value Problems

3.1 Quantum Dirac (QM)

Given an initial state

\[ \psi(x,0) = \alpha \psi_1(x) + \beta \psi_2(x) \]

The future state is uniquely determined by linear, unitary evolution. Superposition is real and persists at the level of individual systems.

3.2 Temporal Flow Physics (TFP)

In TFP, you cannot prepare an initial condition of the form

\[ \Psi(x,0) = \alpha \Psi_1 + \beta \Psi_2 \]

Each realization of the micro-variables \(F_i(t)\) produces exactly one \(\Psi(x,t)\). There is no superposition at the single-substrate level. What appears as “superposition” arises only after averaging across many realizations — that is, ensemble-level superposition only. Linearity emerges statistically, not dynamically.

In other words, individual runs are always definite; coherence exists only at the ensemble level.

4. Why This Matters

This distinction is crucial. The standard Dirac equation provides true quantum behavior for individual particles. My emergent version shows how classical-like, coarse-grained observables can mimic a Dirac equation at the ensemble level. Quantum-like properties such as superposition and coherence emerge only statistically, not dynamically. This highlights how complex macroscopic phenomena can arise from deterministic microscopic dynamics, reshaping how we think about “quantum” versus “emergent” physics.