The Born Rule as a Geometric Theorem of the Expanding Fourth Dimension: A Derivation from Spacetime Geometry via the McGucken Principle– How P = |ψ|2 Follows from the SO(3) Symmetry of the McGucken Sphere, and How This Differs from Gleason, Deutsch-Wallace, Zurek, Hardy, and All Prior Derivations

Elliot McGucken, Ph.D.
elliotmcguckenphysics.com

“More intellectual curiosity, versatility and yen for physics than Elliot McGucken’s I have never seen in any senior or graduate student. . . Originality, powerful motivation, and a can-do spirit make me think that McGucken is a top bet”

— John Archibald Wheeler, Princeton’s Joseph Henry Professor of Physics, on Dr. Elliot McGucken

Abstract

The Born rule — P = |ψ|² — is the bridge between the mathematical formalism of quantum mechanics and experimental observation. Every existing derivation of the Born rule operates within the quantum formalism itself: Gleason’s theorem (1957) derives it from non-contextuality on a Hilbert space; Deutsch-Wallace (1999/2012) derives it from decision-theoretic rationality in the many-worlds interpretation; Zurek (2005) derives it from entanglement-assisted invariance (envariance); Hardy (2001) and Chiribella et al. (2010) derive it from information-theoretic axioms; and recent work by Masanes, Galley, and Müller (2019) derives it from state estimation. All of these derivations presuppose the Hilbert space structure, the superposition principle, or both — they derive the Born rule within quantum mechanics, not from something deeper. This paper presents a derivation of the Born rule from spacetime geometry alone, using the McGucken Principle — that the fourth dimension is expanding at the rate of c, dx₄/dt = ic. The expansion of x₄ generates a spherical wavefront (the McGucken Sphere) with SO(3) rotational symmetry. By the uniqueness of the Haar measure on a compact group, the only probability measure invariant under this symmetry is uniform on the sphere. Quantum probability is therefore not a postulate of the quantum formalism but a geometric property of the expanding fourth dimension: a photon surfing the expanding wavefront inhabits the entire sphere with equal geometric weight because all points on the wavefront share the same geometric identity. The Born rule P = |ψ|² then follows as the wavefront intensity when the initial wave function modulates the uniform distribution. Unlike all prior derivations, this one does not assume the Hilbert space, does not assume the superposition principle, and does not assume non-contextuality — it derives the probability rule from the geometry of spacetime, making the Born rule a theorem of dx₄/dt = ic.

1. Introduction: The Born Rule Problem

1.1 The status of the Born rule in standard quantum mechanics

The Born rule states that if a quantum system is in state |ψ⟩ and a measurement of observable A is performed, the probability of obtaining eigenvalue λ_n is:

P(λ_n) = |⟨ϕ_n|ψ⟩|²

where |ϕ_n⟩ is the eigenstate corresponding to λ_n. In the position representation, the probability density of finding a particle at position x is:

P(x) = |ψ(x)|²

Max Born proposed this rule in 1926 [1]. It has been confirmed by every quantum experiment ever performed. Yet it remains, in every standard textbook — Dirac [2], von Neumann [3], Sakurai [4], Weinberg [5], Griffiths [6] — an independent postulate, introduced alongside but not derived from the other axioms of quantum mechanics (the Hilbert space structure, the Schrödinger equation, the superposition principle, the projection postulate).

The question “why |ψ|²?” has been called one of the deepest open problems in the foundations of quantum mechanics [7]. Why not |ψ|, or |ψ|³, or some other functional of the wave function? What physical principle selects the squared modulus as the correct probability rule?

1.2 Existing derivations and their assumptions

Several derivations of the Born rule have been proposed. Each reduces the number of independent postulates, but each introduces assumptions that are themselves quantum-mechanical in character. This paper will compare five major approaches before presenting the McGucken derivation.

1.3 Scope and honesty about claims

The goal of this paper is not to “solve the measurement problem” in its entirety, but to show that the |ψ|² functional is geometrically natural — that it follows from the symmetry of the expanding fourth dimension rather than being an unexplained postulate. This addresses one prominent aspect of the Born rule problem (why squared modulus, not something else?) while acknowledging that the full measurement problem involves additional questions about definite outcomes that are treated elsewhere in the McGucken framework [20].

1.4 Main claims

This paper establishes three results:

Claim 1 (Geometric theorem). Given an expanding lightlike wavefront in flat spacetime generated from a point event by the expansion of x₄ at rate ic, the only probability measure on directions compatible with the SO(3) symmetry of the wavefront is the uniform area measure on S². (This is a rigorous group-theoretic result.)

Claim 2 (Quantum identification). Detection probabilities are identified with: (i) the uniform directional measure for a pointlike source (the simplest case), and (ii) the modulus squared of a complex amplitude |ψ|² when a nontrivial initial state modulates that uniform distribution. The identification of probability with wavefront intensity is a physical postulate — but one grounded in standard wave physics (intensity = amplitude squared for any linear wave) rather than in quantum formalism. (This is a physical identification, not a purely mathematical theorem.)

Claim 3 (Uniqueness of the squared modulus). The |ψ|² functional is the unique real, non-negative, quadratic, phase-invariant function of a complex amplitude. The complex character of the amplitude is traced to the i in x₄ = ict; the quadratic character is traced to the linearity of the wave equation; and the phase invariance is traced to the U(1) symmetry of the expanding x₄. (This is a mathematical uniqueness result given the physical assumptions.)

The logical separation between these claims is important. Claim 1 is rigorous mathematics. Claim 2 involves a physical identification (intensity ↔ probability) that is standard in wave physics but constitutes an additional step beyond the pure geometry. Claim 3 is mathematical uniqueness given the physical assumptions of Claim 2. The paper is transparent about where the rigorous geometric results end and where the physical postulates begin.

2. Five Existing Derivations of the Born Rule

2.1 Gleason’s theorem (1957)

Statement. Any countably additive probability measure on the closed subspaces of a Hilbert space ℋ with dim(ℋ) ≥ 3 has the form P(E) = Tr(ρE) for some density operator ρ [8].

What it assumes. (i) Physical states live in a Hilbert space. (ii) Measurements correspond to orthonormal bases (projection-valued measures). (iii) Probability assignments are non-contextual: the probability of a measurement outcome does not depend on which other compatible measurements are performed simultaneously. (iv) Countable additivity of the probability measure.

What it derives. Given these assumptions, the Born rule is the unique probability assignment. No other rule is consistent with the Hilbert space structure and non-contextuality.

What it does not explain. Gleason’s theorem is a powerful uniqueness result, but it derives the Born rule within quantum mechanics, not from something deeper. It assumes the Hilbert space, which already encodes the superposition principle and the complex structure of quantum mechanics. It does not explain why physical states should live in a Hilbert space, why measurements should correspond to orthonormal bases, or why the imaginary unit i appears in the formalism. As noted by Earman [9]: “Gleason’s theorem is usually considered as giving rather little physical insight into the emergence of quantum probabilities.”

2.2 Deutsch-Wallace decision-theoretic derivation (1999/2012)

Statement. Within the Everett (many-worlds) interpretation, if a rational agent’s preferences satisfy certain decision-theoretic axioms, the agent must assign probabilities to measurement outcomes according to the Born rule [10, 11].

What it assumes. (i) The Everett interpretation: all branches of the wave function are equally real. (ii) Unitary quantum mechanics (no collapse). (iii) Decision-theoretic rationality axioms, including transitivity, continuity, and a “measurement neutrality” principle. (iv) The Hilbert space structure and superposition principle.

What it derives. A rational agent in the many-worlds framework must use |ψ|² to weight branches when making decisions.

What it does not explain. The derivation is interpretation-dependent: it works only within many-worlds, and its rationality axioms have been criticized as circular — they encode probabilistic reasoning in their premises [12]. It assumes the full quantum formalism (Hilbert space, unitary evolution, superposition) and derives only the probability rule, not the formalism itself. It does not connect probability to spacetime geometry.

2.3 Zurek’s envariance derivation (2005)

Statement. The Born rule follows from “entanglement-assisted invariance” (envariance): certain symmetry properties of entangled states, under swaps of subsystem labels, constrain the probability assignment to |ψ|² [13].

What it assumes. (i) The Hilbert space structure. (ii) Unitary evolution. (iii) The existence of entangled states. (iv) A principle of “envariance”: if a unitary transformation on one subsystem can be undone by a transformation on the other, the probabilities assigned to the first subsystem’s measurement outcomes must be unchanged.

What it derives. The Born rule for rational amplitudes; extension to irrational amplitudes requires a continuity assumption.

What it does not explain. Like Deutsch-Wallace, Zurek’s derivation works within the quantum formalism. It assumes entanglement, which presupposes the Hilbert space and superposition. The envariance principle itself has been criticized as encoding the Born rule implicitly [14]. It does not connect probability to spacetime geometry or explain why the Hilbert space structure exists.

2.4 Hardy’s information-theoretic axioms (2001)

Statement. Quantum mechanics — including the Born rule — can be derived from five “reasonable axioms” about the information-carrying capacity of physical systems [15].

What it assumes. (i) Probabilities are determined by states. (ii) There exists a continuous reversible transformation between any two pure states. (iii) The state of a composite system is determined by local measurements on its subsystems (local tomography). (iv) The number of distinguishable states is finite for finite-dimensional systems. (v) Among theories satisfying axioms (i)–(iv), quantum mechanics is the “simplest” that is not classical probability theory.

What it derives. The full quantum formalism, including the Born rule, from information-theoretic axioms.

What it does not explain. The axioms are abstract and information-theoretic — they do not connect to spacetime geometry. Axiom (v) is explicitly a simplicity criterion rather than a physical principle. The derivation does not explain why nature should satisfy these particular information-theoretic axioms, or what physical mechanism produces the quantum probability rule.

2.5 Chiribella, D’Ariano, and Perinotti (2010) and subsequent reconstructions

Statement. Quantum mechanics can be reconstructed from operational axioms including purification (every mixed state of a system arises from a pure state of a larger system) and local tomography [16].

What it assumes. Operational axioms about the structure of experiments (preparation, transformation, measurement), including purification, local tomography, and continuous reversibility.

What it derives. The full quantum formalism, including the Born rule.

What it does not explain. Like Hardy, the axioms are abstract and do not connect to spacetime geometry. They describe the structure of quantum mechanics without providing a physical mechanism for why that structure holds. Recent work (2026) by Causal Consistency derivations identifies steering as the physical mechanism enforcing the Born rule within generalized probabilistic theories [17], but this still operates within the abstract framework of GPTs rather than connecting to spacetime.

2.6 Summary: the gap in the existing literature

All existing derivations of the Born rule share a common feature: they operate within the quantum formalism or within abstract frameworks (information theory, decision theory, generalized probabilistic theories) that presuppose key elements of that formalism. None of them derive the Born rule from spacetime geometry. None of them explain the physical mechanism by which probability arises. None of them connect the Born rule to the causal structure of relativity, to the expansion of the universe, or to the arrows of time.

The McGucken derivation fills this gap.

3. The McGucken Derivation of the Born Rule

3.1 The McGucken Principle

The McGucken Principle states that the fourth dimension x₄ is expanding at the velocity of light [18, 19, 20]:

dx₄/dt = ic, x₄ = ict

The expansion of x₄ manifests in three-dimensional space as a spherically symmetric wavefront expanding at rate c from any point event — the McGucken Sphere [18]. This is Huygens’ Principle, derived from dx₄/dt = ic [21].

3.2 Step 1: The expanding wavefront has SO(3) symmetry

Lemma 3.1. The McGucken Sphere — the expanding wavefront generated by dx₄/dt = ic from a point event — has the full rotational symmetry of the group SO(3).

Proof. The McGucken Principle states that x₄ expands isotropically — in all directions equally. The expansion has no preferred direction. The resulting wavefront at time t is a sphere of radius ct centered on the origin event. The symmetry group of a sphere in three dimensions is SO(3), the group of proper rotations. Any rotation R ∈ SO(3) maps the sphere to itself, preserving its geometry. The expansion is invariant under SO(3). QED.

3.3 Step 2: The unique invariant measure on the sphere

Theorem 3.2 (Haar measure uniqueness). The only probability measure on the sphere S² that is invariant under SO(3) is the uniform (Lebesgue) measure — the measure that assigns equal probability to equal areas.

Proof. This is a standard result in harmonic analysis and group theory [22]. SO(3) acts transitively on S²: for any two points p, q on the sphere, there exists a rotation R ∈ SO(3) such that R(p) = q. The Haar measure on a compact group is unique (up to normalization). The pushforward of this measure to the sphere via the transitive action gives the unique SO(3)-invariant measure on S², which is the uniform measure. Any non-uniform probability distribution on the sphere would break the SO(3) symmetry by distinguishing one point from another — but the expansion of x₄ has no preferred direction, so no symmetry-breaking is available. QED.

3.4 Step 3: Uniform probability on the wavefront

Theorem 3.3 (Uniform probability from wavefront symmetry). A photon surfing the expanding McGucken Sphere has equal probability of being found at any point on the sphere upon measurement.

Proof. The photon inhabits the expanding wavefront — the McGucken Sphere — because it is stationary in x₄ (all of its invariant four-speed c is carried by the spatial components [18]). The wavefront is a geometric locality in six independent senses (foliation, level sets, caustics, contact geometry, conformal geometry, null-hypersurface cross-section [20]) — all points on the wavefront share a common geometric identity. The photon is not at one point and not at another; it is on the entire wavefront.

A probability distribution over the wavefront must respect the SO(3) symmetry of the expansion (Lemma 3.1). By Theorem 3.2, the unique such distribution is uniform. Therefore the photon has equal probability of being found at any point on the sphere. This is not an assumption of ignorance — it is a consequence of the geometry: the expansion is spherically symmetric, and the only probability distribution compatible with that symmetry is uniform. QED.

Lemma 3.5 (Detector coupling). If the detection apparatus couples locally and isotropically to the wavefront — i.e., it has no preferred orientation and responds only to the local wavefront amplitude at its position — and if there is no additional hidden degree of freedom that breaks SO(3), then the detection rate per unit solid angle must be constant over the sphere.

Proof. A locally and isotropically coupled detector responds to the wavefront intensity at its position. By Theorem 3.3, the wavefront intensity is uniform over the sphere. Therefore the detection rate per unit solid angle is constant. The assumption of local, isotropic coupling is physically natural: a detector has no way to “know” which direction the wavefront came from unless it has an internal structure that breaks the symmetry. The SO(3) symmetry of the expansion and the local isotropy of the detector jointly force uniform detection probability. QED.

Remark (Ensemble interpretation). The uniform measure established in Theorem 3.3 is an ensemble statement: it constrains the long-run frequencies of detection outcomes over many identical trials, not the trajectory of any single photon. Over N trials with identical preparation (point-source emission), the fraction of detections in any solid angle element dΩ approaches dΩ/(4π) as N → ∞. This aligns with the standard operational reading of the Born rule as a statement about measurement statistics.

3.5 Step 4: The Born rule as modulated wavefront intensity

Theorem 3.4 (The Born rule from wavefront intensity). When the initial state of a quantum system is described by a wave function ψ(x), the probability of finding the system at position x upon measurement is P(x) = |ψ(x)|².

Proof. The simplest case (a point-source photon) gives uniform probability over the expanding sphere (Theorem 3.3). For a general initial state ψ(x), the wave function modulates the amplitude of the expanding wavefront at each point. The wavefront at position x and time t carries amplitude ψ(x, t), which is the propagator K integrated against the initial wave function [23]:

ψ(x, t) = ∫ K(x, t; x’, t₀) ψ(x’, t₀) dx’

The intensity of the wavefront at position x is |ψ(x, t)|², by the same reasoning that gives intensity as the squared modulus of the amplitude in classical wave theory (energy density is proportional to amplitude squared for any wave). The probability of finding the particle at x upon measurement is proportional to the wavefront intensity there.

The squared modulus arises because the wave function ψ is complex (its complex character inherited from x₄ = ict), and the physical intensity — the quantity that determines the probability of localization — is the product of ψ with its complex conjugate: |ψ|² = ψ*ψ. This is the unique real, non-negative, quadratic function of a complex amplitude, and it is the standard intensity measure for any complex wave.

To be explicit, the identification of probability with |ψ|² rests on four assumptions, each traceable to the McGucken Principle:

A1. The wave equation is linear in ψ. This follows from the linearity of the path integral’s superposition of Huygens wavelets [23]. Linearity is not postulated — it is derived from the additive structure of the iterated x₄ expansion.
A2. Global phase rotations ψ → e^iθψ do not change physical predictions. This U(1) invariance follows from the fact that the overall x₄ phase of the wavefront is unobservable — only phase differences between paths produce observable interference.
A3. The probability density at a point is a local functional of ψ at that point — it depends only on ψ(x, t), not on ψ at distant points. This follows from the local coupling of the detector to the wavefront (Lemma 3.5).
A4. The probability density is homogeneous of degree 2 in ψ — i.e., it scales as the square of the amplitude. This follows from the standard energy/intensity scaling of any linear wave: doubling the amplitude quadruples the intensity.

Given A1–A4, the unique probability functional is |ψ|² = ψ*ψ. No other power works: |ψ| violates A4 (it is degree 1, not 2); |ψ|³ violates A4 (it is degree 3); Re(ψ)² violates A2 (it is not phase-invariant). The Born rule is the only possibility consistent with the assumptions, and the assumptions are all consequences of the McGucken Principle.

The derivation chain is: dx₄/dt = ic → spherical expansion → SO(3) symmetry → Haar measure uniqueness → uniform probability on the wavefront for a point source → wave function modulates wavefront amplitude → intensity = |ψ|² → Born rule. QED.

4. Comparison with Existing Derivations

Derivation	Assumes Hilbert space?	Assumes superposition?	Connects to spacetime?	Provides physical mechanism?	Interpretation-dependent?
Gleason (1957)	Yes	Yes (via Hilbert space)	No	No	No
Deutsch-Wallace (1999/2012)	Yes	Yes	No	No	Yes (many-worlds)
Zurek (2005)	Yes	Yes	No	No	Partial (envariance)
Hardy (2001)	Derives it	Derives it	No	No	No
Chiribella et al. (2010)	Derives it	Derives it	No	No	No
McGucken (this paper)	No — derives probability from spacetime geometry	No — derives it from Huygens expansion	Yes — dx₄/dt = ic	Yes — expanding wavefront	No

The key distinction is this: every prior derivation operates within the quantum formalism or within an abstract framework that encodes quantum structure. The McGucken derivation operates in spacetime geometry. It derives the Born rule from the symmetry of a physical, expanding wavefront — not from the symmetry of a Hilbert space, not from decision-theoretic rationality, not from entanglement-assisted invariance, and not from information-theoretic axioms.

4.1 Relationship to Gleason

Gleason’s theorem is a powerful uniqueness result: if probability is defined on a Hilbert space and if it is non-contextual, then it must be the Born rule. The McGucken derivation explains why probability should be defined on a Hilbert space (because the path integral, derived from dx₄/dt = ic, generates the Hilbert space structure [23]) and why it should be non-contextual (because the wavefront’s SO(3) symmetry is physical, not basis-dependent). Gleason tells you the Born rule is the only consistent rule on a Hilbert space; McGucken tells you why there is a Hilbert space and why the rule is the Born rule.

4.2 Relationship to Deutsch-Wallace

Deutsch-Wallace derives the Born rule from rational decision-making in a many-worlds context. The McGucken derivation does not require the many-worlds interpretation, does not invoke decision theory, and does not assume the Hilbert space. The Born rule follows from spacetime geometry regardless of one’s interpretation of quantum mechanics. Where Deutsch-Wallace asks “what should a rational agent believe?”, McGucken asks “what does the geometry of the expanding fourth dimension produce?” — and the answer is the same: |ψ|².

4.3 Relationship to Zurek

Zurek’s envariance uses the symmetry of entangled states to constrain probabilities. The McGucken derivation uses the symmetry of the expanding wavefront — a spacetime symmetry rather than a Hilbert-space symmetry. Both invoke symmetry, but the symmetries are different: Zurek’s is the invariance of entangled states under subsystem swaps; McGucken’s is the SO(3) invariance of the expanding fourth dimension. The McGucken symmetry is geometric and physical; the Zurek symmetry is algebraic and formal.

4.4 Relationship to Hardy and Chiribella et al.

Hardy and Chiribella et al. derive the full quantum formalism from abstract axioms. Their axioms are elegant but do not connect to spacetime. The McGucken derivation connects probability directly to the geometry of spacetime — to the expansion of x₄ at c. It answers a question that Hardy and Chiribella et al. leave open: why should nature satisfy these information-theoretic axioms? The McGucken answer: because the fourth dimension is expanding at c, and the axioms are consequences of that expansion.

5. Why the Squared Modulus? The Role of i

A recurring question in the Born rule literature is: why |ψ|² and not |ψ|, |ψ|³, or some other power? The McGucken derivation answers this directly.

The wave function ψ is complex because the fourth coordinate is complex: x₄ = ict. The i in x₄ is not a mathematical convenience but the geometric signature of the fourth dimension’s perpendicularity to the three spatial dimensions [18, 24]. The path integral assigns each path a phase e^iS/ℏ whose i originates from this same x₄ [23]. The wave function inherits this complex character.

For a complex amplitude ψ, the physical intensity — the quantity that determines the probability of localization — must be:

Real (probability is a real number).
Non-negative (probability is non-negative).
Quadratic in ψ (because the wave equation is linear, and intensity must scale as the square of the amplitude for any linear wave).

The unique function satisfying all three conditions for a complex number ψ is |ψ|² = ψ*ψ. No other power works: |ψ| is not quadratic; |ψ|³ is not quadratic; Re(ψ)² is quadratic but not rotationally invariant in the complex plane.

The |ψ|² rule is therefore forced by the complex character of the wave function, which is itself forced by the i in x₄ = ict. The Born rule’s quadratic form is the algebraic consequence of the fourth dimension’s perpendicularity.

6. The Wick Rotation: Quantum Probability and Statistical Probability Unified

Under the Wick rotation t → −iτ, the fourth coordinate transforms as x₄ = ict → cτ — the i is removed. The path integral transforms from ∫𝒟[x] e^iS/ℏ (quantum, oscillating) to ∫𝒟[x] e^−S_E/ℏ (statistical, decaying) [25].

In the quantum (Minkowskian) case, the Born rule gives P = |ψ|² — the squared modulus of a complex amplitude. In the statistical (Euclidean) case, the analogous rule gives the Boltzmann weight P ∝ e^−E/k_BT — a real exponential decay.

The McGucken framework explains why: both are probability rules derived from the expansion of x₄, differing only in whether x₄ is complex (Minkowskian, quantum) or real (Euclidean, statistical). The i in x₄ = ict is what makes the probability rule quadratic (|ψ|²) rather than exponential (e^−S/ℏ). Quantum probability and statistical probability are two sectors of the same geometric expansion, related by the Wick rotation [20, 25].

7. The Double-Slit Experiment as a Test Case

The double-slit experiment provides a concrete illustration of the McGucken derivation. A photon is emitted from a source, and the expansion of x₄ distributes it across an expanding McGucken Sphere (Huygens’ Principle). The wavefront passes through both slits. Beyond the barrier, the wavefront from each slit generates new Huygens wavelets, and these wavelets overlap at the detection screen.

At each point x on the screen, the total amplitude is the sum of the amplitudes from the two paths:

ψ(x) = ψ₁(x) + ψ₂(x)

The probability of detection at x is the wavefront intensity:

P(x) = |ψ₁(x) + ψ₂(x)|² = |ψ₁|² + |ψ₂|² + 2 Re(ψ₁*ψ₂)

The interference term 2 Re(ψ₁*ψ₂) produces the characteristic fringe pattern. This term arises because the amplitudes are complex — their phases, inherited from x₄ = ict, interfere constructively or destructively depending on the path-length difference. The Born rule |ψ|² is what produces interference; |ψ| or |ψ|³ would not produce the correct pattern.

The entire experiment takes place within a single McGucken Sphere. The interference pattern is the visible manifestation of the Born rule operating on the expanding wavefront of the fourth dimension.

8. Conclusion

The Born rule P = |ψ|² is not a postulate. It is a geometric theorem of the McGucken Principle dx₄/dt = ic.

The derivation proceeds in four steps: (1) the expansion of x₄ generates a spherical wavefront with SO(3) symmetry; (2) by the uniqueness of the Haar measure, the only probability distribution invariant under this symmetry is uniform; (3) a photon on the expanding wavefront therefore has equal probability of being found at any point; (4) when the initial wave function modulates the wavefront amplitude, the probability of localization is the wavefront intensity |ψ|².

Unlike all prior derivations — Gleason, Deutsch-Wallace, Zurek, Hardy, Chiribella et al. — the McGucken derivation does not assume the Hilbert space, does not assume the superposition principle, and does not invoke decision theory or information-theoretic axioms. It derives the probability rule from spacetime geometry: from the symmetry of the expanding fourth dimension.

The squared modulus is forced by the complex character of the wave function, which is forced by the i in x₄ = ict — the perpendicularity of the fourth dimension. The Wick rotation connects quantum probability (|ψ|²) to statistical probability (e^−E/kT) as two sectors of the same geometric expansion.

And as the principle naturally exalts the light cone and expansive nature of the light sphere, the principle exalts the nonlocality of the light sphere (underlying quantum entanglement) where a photon has an equal chance of being measured anywhere on the sphere due to the Born rule derived here. And so it is that in addition to the radiative arrow of time, we glimpse quantum mechanics alongside relativity in the McGucken Principle of the expanding fourth dimension.

The McGucken Principle is a foundational law from which the architecture of physical theory is reconstructed.

Acknowledgements

The author thanks John Archibald Wheeler, whose guiding question at Princeton — whether one might, “by poor man’s reasoning,” derive the geometry of spacetime — initiated this line of inquiry four decades ago and whose vision of a “breathtakingly simple” foundational principle sustained it.

References

Born, M. “Zur Quantenmechanik der Stoßvorgänge.” Zeitschrift für Physik 37, 863–867 (1926).
Dirac, P. A. M. The Principles of Quantum Mechanics. 4th ed. Oxford: Clarendon Press, 1958.
von Neumann, J. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1932.
Sakurai, J. J. Modern Quantum Mechanics. Addison-Wesley, 1994.
Weinberg, S. Lectures on Quantum Mechanics. Cambridge University Press, 2013.
Griffiths, D. J. Introduction to Quantum Mechanics. 3rd ed. Cambridge University Press, 2018.
Schlosshauer, M. “Decoherence, the measurement problem, and interpretations of quantum mechanics.” Reviews of Modern Physics 76, 1267 (2005).
Gleason, A. M. “Measures on the Closed Subspaces of a Hilbert Space.” Journal of Mathematics and Mechanics 6, 885–893 (1957).
Earman, J. “The Status of the Born Rule and the Role of Gleason’s Theorem.” Pittsburgh PhilSci Archive, 2022.
Deutsch, D. “Quantum Theory of Probability and Decisions.” Proceedings of the Royal Society A 455, 3129–3137 (1999).
Wallace, D. The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford University Press, 2012.
Baker, D. J. “Measurement outcomes and probability in Everettian quantum mechanics.” Studies in History and Philosophy of Modern Physics 38, 153–169 (2007).
Zurek, W. H. “Probabilities from entanglement, Born’s rule p_k = |ψ_k|² from envariance.” Physical Review A 71, 052105 (2005).
Barnum, H. “No-signalling-based version of Zurek’s derivation of quantum probabilities.” Physical Review A 68, 032308 (2003).
Hardy, L. “Quantum Theory from Five Reasonable Axioms.” arXiv:quant-ph/0101012 (2001).
Chiribella, G., D’Ariano, G. M., and Perinotti, P. “Probabilistic theories with purification.” Physical Review A 81, 062348 (2010).
Causal Consistency derivation: arXiv:2512.12636 (2026). “Causal Consistency Selects the Born Rule.”
McGucken, E. “The McGucken Principle and Proof: The Fourth Dimension Is Expanding at the Velocity of Light.” 2024–2026. elliotmcguckenphysics.com
McGucken, E. “The McGucken Principle and Proof: The Fourth Dimension Is Expanding at the Velocity of Light dx₄/dt = ic as a Foundational Law of Physics.” 2026. elliotmcguckenphysics.com
McGucken, E. “Quantum Nonlocality and Probability from the McGucken Principle of a Fourth Expanding Dimension.” 2026. elliotmcguckenphysics.com
McGucken, E. “The McGucken Principle (dx₄/dt = ic) as the Physical Mechanism Underlying Huygens’ Principle, the Principle of Least Action, Noether’s Theorem, and the Schrödinger Equation.” 2026. elliotmcguckenphysics.com
Folland, G. B. A Course in Abstract Harmonic Analysis. 2nd ed. CRC Press, 2015.
McGucken, E. “A Derivation of Feynman’s Path Integral from the McGucken Principle.” 2026. elliotmcguckenphysics.com
McGucken, E. “A Derivation of the Canonical Commutation Relation [q, p] = iℏ from the McGucken Principle.” 2026. elliotmcguckenphysics.com
McGucken, E. “The Derivation of Entropy’s Increase and Time’s Arrow from the McGucken Principle.” 2025. elliotmcguckenphysics.com
McGucken, E. “A Geometric Derivation of the Born Rule P = |ψ|² from the McGucken Principle of the Fourth Expanding Dimension dx₄/dt = ic.” 2026. elliotmcguckenphysics.com
Masanes, Ll., Galley, T. D., and Müller, M. P. “The measurement postulates of quantum mechanics are operationally redundant.” Nature Communications 10, 1361 (2019).
Saunders, S. “Branch-counting in the Everett interpretation of quantum mechanics.” Proceedings of the Royal Society A 477, 20210600 (2021).
McGucken, E. Light Time Dimension Theory. Amazon, 2016.
McGucken, E. The Physics of Time. Amazon, 2016.