True Quantum Dirac Equation vs Emergent Effective Dirac Equation

Quantum Dirac Equation vs Emergent Effective Dirac Equation

True Quantum Dirac Equation (Fundamental)

Define the fundamental relativistic wave equation:

$$ (i \gamma^\mu \partial_\mu - m)\,\psi(x) = 0 $$

Key properties:

\(\psi(x)\) is a quantum operator-valued spinor field (in QFT), or a complex probability-amplitude spinor (in single-particle quantum mechanics).

Hilbert space:

$$ \psi \in L^2(\mathbb{R}^3,\mathbb{C}^4) $$

Inner product:

$$ \langle \phi | \psi \rangle = \int \phi^\dagger(x)\,\psi(x)\,d^3x $$

Unitary evolution:

$$ \frac{\partial}{\partial t} \langle \psi | \psi \rangle = 0 $$

Linearity is fundamental. If \(\psi_1\) and \(\psi_2\) are solutions, then any linear combination is also a solution:

$$ \psi(x) = a\,\psi_1(x) + b\,\psi_2(x) $$

Anticommutation relations (QFT):

$$ \{\psi_\alpha(x),\psi_\beta^\dagger(y)\} = \delta_{\alpha\beta}\,\delta^3(x-y) $$

Probability interpretation:

$$ j^0 = \psi^\dagger \psi \ge 0 $$ $$ \int j^0\,d^3x = 1 $$

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From My Derivation (Effective / Emergent)

Define an effective Dirac-like evolution equation:

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

Where the emergent spinor is defined as:

$$ \Psi(x,t) = \begin{pmatrix} \psi_+(x,t) \\ \psi_-(x,t) \end{pmatrix} $$

With phase-amplitude components:

$$ \psi_\pm(x,t) = A_\pm\,e^{i\phi_\pm} $$

Critical Defining Relation

The emergent fields are defined as coarse-grained limits of discrete substrate variables:

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

Interpretation

\(\Psi(x,t)\) is not a fundamental quantum field.

It is a classical, coarse-grained observable constructed from many underlying discrete contributions — analogous to temperature, pressure, or fluid velocity.

Quantum structure is not assumed at the substrate level; it appears only after statistical averaging.

Lack of Operator Structure and Fundamental Linearity

The effective equation resembles the Dirac equation formally, but it lacks the operator algebra and intrinsic linearity that make Dirac’s equation genuinely quantum.

Initial Value Problem Comparison

Dirac (Quantum Mechanics)

Given an initial state:

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

The future state is uniquely and coherently determined by linear, unitary evolution.

Superposition is physical and persists at the level of individual systems.

TFP (Temporal Flow Physics)

An initial condition of the form

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

cannot be prepared as a single substrate state.

Each concrete realization of the underlying micro-variables \(F_i(t)\) produces exactly one definite \(\Psi(x,t)\).

There is no physical superposition at the level of an individual realization.

Ensemble-Level Superposition Only

What appears as “superposition” exists only after averaging across many realizations.

Linearity emerges statistically, not dynamically.

Individual runs are always definite; coherence exists only at the ensemble level.