From Dirac Solutions to Physical Reality

The Gap in the Dirac Framework

The Dirac equation is a cornerstone of relativistic quantum mechanics. It predicts electron spin, the existence of antiparticles, and the fine-structure spectrum of hydrogen. Combined with quantum field theory, it underpins the Standard Model's treatment of fermionic matter.

The equation defines a precise mathematical space of admissible states — solutions satisfying boundary conditions, symmetry requirements, and normalisability constraints. This solution space contains particle and antiparticle branches, multiple spin orientations, and a continuum of momentum modes. All are admissible at the level of the equation.

Yet the Dirac equation does not answer a fundamental question: given the full space of admissible states, what determines which configuration is realised in a concrete physical context?

---

Existing Approaches

Several interpretations address the gap between admissible states and realised outcomes, each with a different strategy:

• Copenhagen — provides operational closure through axiomatic collapse but does not specify an underlying mechanism.
• Many-worlds — avoids single-outcome collapse by positing universal branching; selection becomes observer-relative.
• GRW / objective collapse — introduces explicit stochastic modifications to the dynamics.
• Pilot wave — provides a deterministic guidance equation but requires additional ontology (the pilot wave itself).

None of these formulates state selection as optimisation over a constraint functional on the Dirac solution space. That is what this paper proposes.

---

Crystallisation as Selection Architecture

The paper introduces crystallisation: a deterministic, constraint-based selection architecture acting over a restricted admissible subset of the Dirac solution space. The core idea is that state selection is not random but is the result of asymptotic convergence under a projected relaxation flow toward states that extremise a constraint functional.

The admissible domain

Crystallisation does not act over the full (infinite-dimensional) Dirac solution space. It acts over a restricted admissible domain S_adm — normalised, finite-energy states satisfying the boundary conditions and symmetry constraints of a concrete physical context. In measurement settings, S_adm is further conditioned by the interaction structure and macroscopic boundary conditions of the experimental arrangement.

The constraint functional

The constraint functional F encodes the physical requirements for a realised state. At this stage of development, F is a schematic template — a structural proposal whose specific form is not uniquely derived:

• C(ψ) — Coherence: internal phase alignment and structural consistency
• H(ψ) — Entropy: degree of internal decoherence (penalised)
• R(ψ) — Resonance: alignment with the constraint geometry
• V(ψ) — Variance: fluctuations and structural fragility (penalised)

The dynamical flow

The selected state is not chosen instantaneously. The system evolves via continuous gradient flow on the constraint landscape:

The projection Π ensures the flow remains within the admissible domain. Crystallisation is defined as the asymptotic convergence of this flow:

---

Physical Interpretation

Particle stability

Within this framework, candidate stable realised state classes — with particle states as the primary examples — correspond to local maxima of F over S_adm. Distinct attractor basins of the flow correspond to distinct stable state classes. The discreteness of the observed particle spectrum is associated with a discrete set of relevant local maxima.

Antiparticles

The Dirac solution space contains both particle (E > 0) and antiparticle (E < 0, reinterpreted) branches. In the crystallisation framework, antiparticles correspond to symmetry-related extrema of F on the conjugate solution branch. If the constraint functional respects CPT-compatible symmetries, particle and antiparticle extrema are related by conjugation: F(ψ) = F(C_CPTψ).

Annihilation

In a heuristic field-theoretic extension (not fully developed in this paper), annihilation is represented as a particle-antiparticle pair transitioning to a configuration of higher global stability — typically radiation modes. The full QFT formulation is deferred to future work.

---

How It Compares

Feature	Standard QM	Crystallisation
Collapse	Axiom	Emergent convergence
Probability	Fundamental (operational)	Potentially epistemic
Measurement	Primitive	Constraint interaction
State selection	Unspecified	Extremisation of F
Indeterminism	Irreducible (standard treatment)	Deterministic given F

Feature	Copenhagen	GRW	Crystallisation
Selection principle	Axiomatic collapse	Stochastic collapse	Constraint extremisation
Randomness	Operational	Fundamental	Epistemic (conjectured)
Collapse dynamics	Unspecified	Finite rate	Relaxation flow on F
Preferred basis	Unspecified	Position	Extrema of F
Extends Dirac	Adds axiom	Adds noise	Adds constraint layer

---

Testable Implications

The crystallisation architecture generates several structural implications that distinguish it from axiomatic and stochastic collapse models:

• Preferred states. Realised states correspond to extrema of a constraint functional. If effective proxies for F can be operationally characterised, preferred states should exhibit higher coherence measures than generic superpositions.
• Stability spectra. Candidate stable state classes correspond to local maxima of F. The stability spectrum is linked to constraint geometry rather than arbitrary parameter choices.
• Deterministic selection signatures. If the full constraint state were exactly replicated, the framework predicts reproducible selection. Apparent statistical spread may reflect uncontrolled variation in the effective constraint environment.
• Transition-time structure. Dynamical crystallisation via the projected flow naturally introduces a finite convergence timescale governed by the gradient structure of F.
• Connection to mass structure. If F is related to the spectral geometry of the underlying space, mass ratios of stable particles should be computable from the constraint landscape — a direction explored in companion papers.

---

Geometric Structure

The constraint functional F defines a landscape over S_adm whose topology determines the structure of physically realised states:

• Constraint manifold — the subspace where all constraints are exactly satisfied. Physically realised states lie on or near this manifold.
• Attractor basins — regions of S_adm that flow toward a specific local maximum of F. Distinct basins correspond to distinct stable state classes.
• Basin boundaries — separatrices between attractor basins. Transitions between basins correspond to state changes in the physical system.

---

Stated Limitations

The paper is explicit about what it does not yet accomplish:

• The functional form of F is schematic, not derived from first principles
• The coefficients a₁, a₂, a₃, a₄ are structural choices, not determined
• Lorentz covariance of F has not been established
• Compatibility with full QFT (operator algebras, renormalisation, gauge invariance) remains open
• Born rule recovery is conjectured, not proven
• Existence and uniqueness of the flow limit are not proven for general F
• No operational protocol for estimating F from experimental data is yet given
• This is not a replacement for the Standard Model — it addresses the selection mechanism only

---

What This Paper Is

This is a bridge paper: it introduces the formal architecture and physical interpretation of crystallisation in the context of relativistic quantum mechanics. It connects the general crystallisation programme — developed in earlier work on closure functionals, gradient flows, and deterministic state selection — to the specific mathematical structure of the Dirac equation.

The paper works primarily at the level of a Dirac-inspired selection framework. Interpretations involving annihilation and particle creation are heuristic extensions to field-theoretic settings; a full QFT formulation is deferred. The constraint functional is proposed, not uniquely determined.

---