The McGucken Nonlocality Principle: All Quantum Nonlocality Begins in Locality, and All Double Slit, Entanglement, Quantum Eraser, and Delayed Choice Experiments Exist in McGucken Spheres

How the Expansion of the Fourth Dimension at the Rate of c Generates Nonlocality from Locality, with a Direct Experimental Test

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

This paper establishes a principle governing the origin and growth of quantum nonlocality — all quantum nonlocality begins in locality — and roots this principle in the more foundational McGucken Principle: the fourth dimension is expanding at the rate of c, dx₄/dt = ic. The expansion of x₄ is the single geometric process that transforms locality into nonlocality: it distributes every local point into a nonlocal expanding wavefront (Huygens’ Principle), generates the nonlocal correlations observed in entanglement experiments, and limits the growth of nonlocality to the velocity of light. Two particles can become entangled only if they have shared a common local origin, or if they have each interacted locally with members of a system that itself originated locally. The geometric object at the center of this framework is the McGucken Sphere — a sphere whose radius expands at c, representing the expansion of x₄. In the simplest case, entangled particles sharing a common origin exist within a single McGucken Sphere: they share a common x₄ wavefront because they originated at the same local event. Entanglement can also be transferred via mediating particles (as in entanglement swapping), and the interaction of those mediating particles represents intersecting McGucken Spheres — each carrying its own wavefront from its own local origin, with entanglement transferred where the spheres intersect. Within a McGucken Sphere, there always exists a frame in which there is no time and no distance between any two events — the frame of the photon, in which proper time and proper distance are zero. This is why entangled particles remain correlated regardless of spatial separation: in the geometry of their shared wavefront, they have never left each other. The paper establishes that the expanding wavefront is a genuine geometric nonlocality in six independent mathematical senses — as a leaf of a foliation, a level set of a distance function, a causal wavefront (Huygens), a Legendrian submanifold in contact geometry, a member of a conformal pencil, and most deeply as a null-hypersurface cross-section, the canonical causal locality of Minkowski geometry. The paper shows that the double-slit experiment, Wheeler’s delayed-choice experiment, and all quantum eraser experiments take place within the confines of McGucken Spheres, and that the apparent paradoxes of these experiments dissolve once the full four-dimensional geometry of the expanding x₄ is recognized. Two formal laws are stated — the First and Second McGucken Laws of Nonlocality — and a concrete experimental test is provided in the form of a challenge to entangle two distant, unentangled electrons without any chain of local contacts. The nonlocality principle connects to the broader McGucken framework, from which not only entanglement but also relativity, the Schrödinger equation, Feynman’s path integral, the Born rule, entropy increase, the canonical commutation relation qp − pq = iℏ, and the constants c and ℏ themselves all emerge — from the single geometric fact that the fourth dimension is expanding at the velocity of light.

1. Introduction

1.1 The foundational principle and the origin of nonlocality

The McGucken Principle — that the fourth dimension is expanding at the rate of c, dx₄/dt = ic — is the geometric foundation from which relativity, quantum mechanics, entropy, and the arrows of time all emerge [4–8]. This paper shows that the same principle also governs the origin and growth of quantum nonlocality: all quantum nonlocality begins in locality, and it does so because the expansion of the fourth dimension is the physical process that transforms the local into the nonlocal.

Quantum entanglement is one of the most striking features of quantum mechanics. Two particles that have interacted can exhibit correlations in their measurement outcomes that persist regardless of the spatial separation between them — correlations that violate Bell inequalities and cannot be explained by any local hidden-variable theory [1, 2]. Einstein famously called this “spooky action at a distance” [3].

The standard treatment of entanglement focuses on the correlations themselves — their mathematical structure, their violation of classical bounds, and their applications in quantum information. But a prior question is rarely asked: where does entanglement come from? Under what physical conditions can two particles become entangled? Is there a universal constraint on the origin of nonlocal correlations? The McGucken Principle answers these questions by identifying the physical process — the expansion of x₄ at c — that generates nonlocality from locality.

Relativity gives us quantum nonlocality and entanglement [22]. As all quantum eraser experiments take place within a McGucken Sphere (or a series of intersecting McGucken Spheres in the case of mediating particles transferring entanglement), all quantum eraser experiments exhibit the same basic physics observed in the double-slit experiment, which also takes place in a McGucken Sphere. Within any McGucken Sphere, relativity tells us there exists a frame — the photon’s frame — in which there is no time and no distance between any two events. What a three-dimensional observer describes as “the past” in a quantum eraser experiment may be considered the present in this frame; what is described as a great spatial separation may be considered no separation at all. The apparent paradoxes of quantum erasure, delayed choice, and double-slit interference all dissolve when the experiments are recognized as taking place within the expanding x₄ geometry of a McGucken Sphere, where the nonlocality that seems mysterious in three dimensions is simply the geometric identity of a shared wavefront in four.

1.2 The McGucken Nonlocality Principle

This paper answers that question with a simple, testable principle:

The McGucken Nonlocality Principle: All quantum nonlocality begins in locality.

No two particles can become entangled unless they have shared a common local origin — unless they were once at the same place at the same time, or unless they have each interacted locally with members of a system that itself shared a common local origin. Nonlocality is not a primitive feature of the universe; it is generated from locality by a specific physical process: the expansion of the fourth dimension at the rate of c, as given by the McGucken Principle dx₄/dt = ic [4–8].

1.3 The McGucken Sphere

Definition 1.2 (The McGucken Sphere). The McGucken Sphere is a sphere whose radius expands at the velocity of light c, representing the expansion of the fourth dimension x₄ as given by dx₄/dt = ic. Centered on a spacetime event O = (x₀, t₀), the McGucken Sphere at time t is the set of all spatial points satisfying |x − x₀| = c(t − t₀). This is the forward light cone of O intersected with the spatial slice at time t. No matter how large the sphere grows, a photon on its surface remains at the same position in the fourth dimension x₄ — the photon is stationary in x₄, demonstrating that it is x₄ that is moving and expanding [4, 22].

The surface of the McGucken Sphere is not merely a set of points that happen to be equidistant from the origin — it is a geometric nonlocality in the strongest possible sense. As established in Section 4 of this paper, the wavefront’s spatially separated points share a common geometric identity in six independent mathematical frameworks: foliation theory (same leaf), metric geometry (same level set), wave optics (same Huygens caustic), contact geometry (same Legendrian submanifold), conformal geometry (same pencil member), and most fundamentally, Lorentzian geometry (same null-hypersurface cross-section). All six frameworks yield the same conclusion: what appears from a three-dimensional perspective as a collection of widely separated points is, in the full four-dimensional geometry, a single unified object traceable to a single local origin.

The McGucken Sphere has a remarkable property that connects relativity directly to quantum nonlocality. Relativity tells us that the photon experiences no proper time in its frame — and no proper distance. To a photon, the spatial distance between any two points is zero, and the time between any two events is zero. Therefore, within a McGucken Sphere, there always exists a frame (the photon’s frame) in which there is no time and no distance between any two events on the sphere’s surface. What is considered “the past” in a quantum eraser or double-slit experiment may be considered the present in the photon’s frame. What is considered a great spatial separation may be considered no separation at all [22].

This is why entangled particles remain correlated regardless of spatial separation: in the simplest case, they share a common McGucken Sphere, and in the geometry of their shared wavefront, they have never left each other. In the more general case — where entanglement is transferred via mediating particles, as in entanglement swapping — each mediating particle carries its own McGucken Sphere from its own local creation event, and entanglement is transferred at the intersection of these spheres. Even in this general case, every link in the chain traces back to a local origin: the intersecting McGucken Spheres all began as local events. Entanglement is not a mysterious connection across space — it is the shared geometric identity of particles on the same x₄ wavefront, or the transferred geometric identity across intersecting wavefronts, where in the photon’s frame there is neither time nor distance.

1.4 The mechanism: from locality to nonlocality via the expanding fourth dimension

The McGucken Principle provides the physical mechanism by which locality becomes nonlocality. 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. This is Huygens’ Principle [9]: every local point becomes a nonlocal wave. Every point on the expanding wavefront shares a common origin — the original local event — and therefore shares a common x₄ identity, a common position on the expanding McGucken Sphere [4].

Two particles created at the same local event are on the same expanding McGucken Sphere. As the sphere expands, the particles separate in three-dimensional space, but they remain on the same x₄ wavefront. Their entanglement — their nonlocal correlation — is the geometric consequence of their shared wavefront identity. The nonlocality grew from locality through the expansion of x₄ at c.

2. The Two Laws of Nonlocality

2.1 First Law: All nonlocality begins as locality

First McGucken Law of Nonlocality. Two quantum systems can exhibit nonlocal correlations (entanglement) only if they have shared a common local origin, or if each has interacted locally with members of a system that itself shared a common local origin.

Equivalently: only systems of particles with intersecting light spheres — with each light sphere having originated from each respective particle — can ever be entangled [10]. The property of entanglement between particles is limited by the velocity of light, even though the nonlocal influences found in entanglement are instantaneous.

This law does not contradict the instantaneous character of entanglement correlations. It constrains the creation of entanglement, not the manifestation of it. Once two particles are entangled (by having shared a local origin, or by each having interacted with particles sharing a local origin), their correlations are instantaneous and persist regardless of separation. But the entanglement itself could not have been established without a prior local interaction.

2.2 Second Law: Nonlocality grows over time, limited by c

Second McGucken Law of Nonlocality. Nonlocality grows over time, in a manner limited by the velocity of light c.

As the fourth dimension expands at c, the McGucken Sphere grows. At time t after a local event, the sphere of nonlocality has radius r = ct. Particles within this sphere may be entangled with the original event; particles outside it cannot be, because the expansion of x₄ has not yet reached them. The boundary of entanglement possibility is exactly the light cone — the causal boundary of relativity.

This law connects quantum nonlocality directly to the causal structure of spacetime. The expansion of x₄ at c simultaneously generates: the light cone (the boundary of causal influence in relativity), the expanding wavefront (Huygens’ Principle in wave optics), and the sphere of potential entanglement (the boundary of nonlocality in quantum mechanics). All three are the same geometric object — the McGucken Sphere — viewed from different physical perspectives [4].

3. The Experimental Test: The New York–Los Angeles Challenge

3.1 The setup

Consider two electrons: electron A in a laboratory in New York, and electron B in a laboratory in Los Angeles. Both electrons’ positions, spins, and momenta are being continuously measured. The experimentalists in New York and Los Angeles communicate by phone and determine that there is no correlation between the measurements on the two electrons. They conclude that electrons A and B are not entangled.

The McGucken Nonlocality Principle makes a specific, falsifiable prediction: it is impossible to entangle electrons A and B without some form of local contact, either directly or through a locally-originated intermediary.

3.2 The challenge

To falsify the McGucken Nonlocality Principle, one would need to demonstrate a method for entangling the two distant, unentangled electrons in New York and Los Angeles that satisfies all of the following conditions:

The electrons are never brought into direct local contact.
No intermediary particle or system that has shared a local origin with itself is used to mediate the entanglement.
The entanglement is established faster than the velocity of light — i.e., without waiting for any signal or system to travel from one laboratory to the other.

If such a method can be demonstrated, the McGucken Nonlocality Principle is falsified. If no such method can be found — either experimentally or even in a thought experiment consistent with the known laws of physics — then the principle stands.

3.3 Why the challenge cannot be met

Every known method for entangling distant particles requires a locally-originated intermediary. The standard procedure is entanglement swapping [11]:

Prepare two entangled pairs: particles C and D are entangled (having been created locally together), and particles E and F are entangled (also created locally together).
Transport particle C to New York, where it interacts locally with electron A. Transport particle F to Los Angeles, where it interacts locally with electron B.
Perform a Bell-state measurement on particles D and E at some intermediate location.
After the Bell-state measurement, electrons A and B become entangled — but only because the chain of local interactions (A↔C, C↔D at creation, D↔E at measurement, E↔F at creation, F↔B) connects them through a sequence of locally-originated contacts.

Every step in this chain involves local contact. The nonlocality of the final A–B entanglement traces back, through the chain, to the local creation events of the C–D and E–F pairs. All nonlocality begins in locality.

No alternative method has ever been proposed — in any interpretation of quantum mechanics, in any extension of the Standard Model, in any thought experiment — that creates entanglement between distant particles without some chain of local contacts originating in a common locality. The McGucken Nonlocality Principle is not merely unfalsified; it appears to be unfalsifiable within the known laws of physics, because those laws themselves enforce it.

4. The Expanding Wavefront as a Geometric Nonlocality: Six Independent Proofs

The McGucken Nonlocality Principle — that all nonlocality begins in locality — rests on a specific geometric claim: the expanding wavefront generated by dx₄/dt = ic is a genuine nonlocal entity whose points all share a common identity traceable to a local origin. This section establishes this claim rigorously by showing that the wavefront is a geometric locality (and therefore a geometric nonlocality, since its spatially separated points share a common identity) in six independent mathematical senses [16a].

4.1 Foliation theory

The expanding McGucken Sphere defines a foliation of three-dimensional space: a family of nested 2-spheres S²(t) parameterized by time. Each sphere is a leaf of the foliation, and the entire family carries a well-defined transverse geometry. The wavefront at any moment is a leaf — a single geometric object that separates space into inside/outside regions with sharp topological meaning. All points on the leaf share a common identity as members of the same leaf. This is the first sense in which spatially separated points on the wavefront constitute a single nonlocal entity traceable to a local origin: they are all on the same leaf of the foliation generated by the expansion of x₄ from a single local event.

4.2 Level sets of a distance function

The wavefront is the level set d(x) = ct of the distance function from the origin of the expansion. Every point on the wavefront is equidistant from the origin — a metric locality that is geometrically canonical. In any metric space, the level sets of the distance function from a point define the “spheres” around that point. All points on a level set share the same metric relationship to the origin. This is the second sense: spatially separated points on the wavefront are the same “distance” from the local origin in the induced metric, and therefore share a common metric identity.

4.3 Caustics and wavefronts (Huygens)

The wavefront is a caustic in the sense of geometric optics — the envelope of secondary wavelets emanating from every point on the previous wavefront. This makes the wavefront a causal locality: it is the boundary between the region that has received the disturbance and the region that has not. All points on the wavefront have the same causal status — they are all on the boundary of the causal future of the origin event. This is the third sense: causal equivalence traceable to a common local origin.

4.4 Contact geometry

In the jet space with coordinates (x, y, z, t), the growing wavefront traces a cone that is a Legendrian submanifold of the contact structure. The wavefront at each t is a contact locality — defined by the contact distribution rather than by position alone. All points on the Legendrian submanifold share a common contact-geometric identity. This is the fourth sense.

4.5 Conformal and inversive geometry

Growing spheres under inversion map to other spheres or to planes. The family of expanding wavefronts belongs to a pencil in the inversive/Möbius geometry of space — a conformal locality invariant under the conformal group. All members of the pencil share a common conformal identity. This is the fifth sense.

4.6 Null-hypersurface locality: the deepest answer

The five frameworks above identify the wavefront as a locality in progressively deeper senses: topological, metric, causal, contact-geometric, and conformal. But the deepest identification is Lorentzian.

Consider the Lorentzian line element ds² = dx² + dy² + dz² − c²dt². The growing wavefront (radius = ct) is precisely a null-hypersurface cross-section — the intersection of the light cone with a spacelike slice. This is the most fundamental geometric locality possible: it is causal, metric, and topological simultaneously. It is the boundary of the causal future of the origin point.

Null hypersurfaces have a special status in Lorentzian geometry: they are neither spacelike nor timelike but causally extremal, and they are the only surfaces on which signals propagate at the invariant speed c. Every point on the wavefront has the same causal relationship to the source — they are all on the light cone. This is the sixth and deepest sense: null-hypersurface equivalence.

Proposition 4.1. The expanding McGucken Sphere in (x₁, x₂, x₃, x₄) space, with x₄ advancing at rate ic, intersects any three-dimensional spatial slice in a growing sphere whose radius expands at c. This sphere is the intersection of a null hypersurface with a spatial slice — the canonical geometric locality in Minkowski geometry [16a].

The six framework analyses are mutually reinforcing: each frames the same physical object (the expanding wavefront) in the language of a different mathematical discipline, and each yields the same conclusion that the wavefront’s spatially separated points share a common geometric identity traceable to a single local origin. What appears from a three-dimensional perspective as a collection of causally disconnected points is, in the full four-dimensional geometry, a single unified object.

4.6a Proof of the nonlocality of the light cone’s surface

The following constructive proof demonstrates that the surface of a light cone — the McGucken Sphere — is a nonlocal entity [22a].

Theorem 4.2 (Nonlocality of the light cone’s surface). The surface of the forward light cone emanating from any spacetime event is a geometric nonlocality: its spatially separated points share a common quantum identity.

Proof.

Step 1. Consider a pair of entangled photons created at the origin and traveling in opposite directions in the (x₁, x₂) plane. No matter how far apart they travel, they remain entangled — their two positions define a nonlocality.
Step 2. Now consider numerous entangled pairs of photons, where the two photons in each pair travel away from the origin in opposite directions, with different pairs oriented along different directions in the plane. Together, the positions of the photons in all the entangled pairs define, as the number of pairs approaches infinity, a circle of nonlocality.
Step 3. Introduce a third axis — time. As each pair of entangled photons travels outward at c, their positions trace the circle at successive times. Together, the positions of the photons in all the entangled pairs define a conical surface — the light cone — whose every point is connected by entanglement to the antipodal point on the same time-slice.
Step 4. Extending to three spatial dimensions, the circle at each time-slice becomes a sphere of radius ct — the McGucken Sphere. Every point on this sphere is entangled with the antipodal point through the origin event, and every point shares the same null-hypersurface identity established in Section 4.6.

Therefore the surface of the light cone is a nonlocality: it is a geometric surface every point of which is connected to the origin and to other points on the surface by entanglement traceable to the common local origin. QED. [22a]

4.6b The McGucken Proof: the fourth dimension is expanding at the velocity of light

The nonlocality established in Theorem 4.2 rests on the McGucken Principle itself. This subsection reproduces the formal proof that the fourth dimension is expanding at c [4d].

Theorem 4.3 (The McGucken Proof). The fourth dimension x₄ is expanding at the velocity of light relative to the three spatial dimensions: dx₄/dt = ic.

Proof.

The magnitude of the four-velocity of every physical system through the four dimensions of spacetime is c: u_μu^μ = −c².
The faster a system moves through the three spatial dimensions, the slower it moves through the fourth dimension (the four-speed budget: |v|² + |dx₄/dt|² = c²).
As a system’s spatial velocity approaches c, its velocity through the fourth dimension approaches zero.
Therefore photons — which travel at c through the spatial dimensions — are stationary in the fourth dimension x₄.
Photons thus track and trace the movement and character of x₄: since photons do not advance through x₄, any observed expansion of a photon’s wavefront in three-dimensional space must reflect the expansion of x₄ itself.
Light is observed to propagate as a spherically symmetric wavefront expanding at c. Since photons are stationary in x₄, this spherical expansion is the geometric signature of x₄ expanding at the rate of c in a spherically symmetric manner, distributing locality into nonlocality.

Therefore dx₄/dt = ic. The fourth dimension is expanding at the velocity of light. QED. [4d]

This proof parallels Einstein’s conceptual step with Planck’s relation E = hf: where Planck treated E = hf as a calculational device, Einstein promoted it to a physical postulate about quantized energy. The McGucken Principle takes the analogous step with Minkowski’s x₄ = ict: it promotes a coordinate convenience to an ontological statement that reality advances through a fourth geometric dimension at speed c, and that photons — stationary in x₄ — trace the geometry of its expansion [4d].

4.7 From geometric locality to quantum nonlocality

The six-fold geometric identity of the expanding wavefront is the formal foundation of the McGucken Nonlocality Principle. Two particles created at the same local event are on the same wavefront — the same leaf, the same level set, the same caustic, the same Legendrian, the same conformal pencil member, the same null-hypersurface cross-section. They share a common geometric identity in all six senses. As the wavefront expands at c, the particles separate in three-dimensional space, but they remain on the same geometric object. Their entanglement — their nonlocal correlation — is the direct consequence of this shared identity.

The nonlocality did not exist before the local event that created both particles on the same wavefront. It grew from locality through the expansion of x₄ at c. All nonlocality begins in locality because all shared wavefront identity begins in a common local origin.

4.8 From geometric nonlocality to quantum probability

The same geometric identity that produces nonlocality also produces quantum probability [16a]. Because the expansion of x₄ is spherically symmetric, the rotation group SO(3) acts transitively on the expanding sphere. By the uniqueness of the Haar measure on a compact group, the only probability measure on the sphere that is invariant under SO(3) is the uniform measure. A photon surfing the expanding wavefront inhabits the entire sphere of nonlocality with equal geometric weight — there is no geometric structure that distinguishes one point from another — until a measurement event localizes it in three spatial dimensions.

Quantum probability is therefore not a separate postulate but the direct geometric consequence of the wavefront’s nonlocal identity: the photon has equal chance of being found at any point because all points on the wavefront share the same geometric identity in all six senses established above. Nonlocality and probability arise from the same source — the expanding fourth dimension — and the McGucken Nonlocality Principle and the Born rule are two faces of one geometric fact [16a].

5. Connection to the Broader McGucken Framework

4.1 Huygens’ Principle: every local point becomes a nonlocal wave

The McGucken Nonlocality Principle is a direct consequence of the McGucken Principle dx₄/dt = ic. The expansion of the fourth dimension distributes every local point across a spherically symmetric wavefront — this is Huygens’ Principle [9], derived from dx₄/dt = ic [8, 9]. Every local point becomes a nonlocal wavefront. The nonlocality is generated by the expansion; it did not exist before the expansion began.

Huygens’ Principle is therefore the bridge between locality and nonlocality. It is the physical process by which a point (local) becomes a sphere (nonlocal). And because the sphere expands at c, the growth of nonlocality is limited by the velocity of light — the Second Law of Nonlocality.

4.2 Entanglement as shared x₄ geometry

In the McGucken framework, entangled particles share a common position on the expanding McGucken Sphere — they are on the same x₄ wavefront because they originated at the same local event [4, 10]. Their correlations are not transmitted between them; they are inherited from their shared geometric origin. Measurement of one particle localizes its x₄ phase, which constrains the other’s because they share the same wavefront. No signal is sent; no action at a distance occurs. The correlation is geometric, and it was established at the local creation event.

This is why all nonlocality begins in locality: entanglement is shared x₄ geometry. In the simplest case, shared x₄ geometry requires a common origin on the same McGucken Sphere, which requires a common local event. In the general case, entanglement can be transferred via mediating particles whose own McGucken Spheres intersect — but every intersection traces back, through the chain, to local creation events. Whether the entanglement is direct (one McGucken Sphere) or transferred (intersecting McGucken Spheres), all nonlocality begins in locality.

4.3 Time’s arrows and the growth of nonlocality

The Second Law of Nonlocality — that nonlocality grows over time, limited by c — connects directly to the arrows of time derived from the McGucken Principle [5, 12]. As the fourth dimension expands:

Entropy increases (because the expansion grows the accessible phase-space volume) [12].
Radiation propagates outward as expanding spheres (the radiative arrow) [4].
The universe expands (the cosmological arrow) [4].
Nonlocality grows (the nonlocality arrow).

The growth of nonlocality is a new arrow of time, joining the thermodynamic, radiative, cosmological, causal, and psychological arrows as manifestations of the one-way expansion of the fourth dimension. All arrows point in the same direction because they are all driven by the same geometric process: dx₄/dt = ic.

4.4 Relativity and quantum mechanics unified in nonlocality

The McGucken Nonlocality Principle reveals a deep unity between relativity and quantum mechanics. Relativity says that causal influence cannot propagate faster than c — the light cone is the boundary of causal contact. The First Law of Nonlocality says that entanglement cannot be created faster than c — the light cone is also the boundary of entanglement creation. The light cone, the McGucken Sphere, and the boundary of potential entanglement are the same geometric object.

Consider two photons sharing a common origin and traveling in opposite directions. Relativity teaches us that photons experience no proper time and traverse no proper distance — they have never left each other in the geometry of their own worldlines. Quantum mechanics teaches us that entangled photons from a common source remain correlated regardless of spatial separation. These are the same statement: the photons share a common x₄ wavefront, and their correlation is the geometric consequence of that shared identity. Both relativity and quantum mechanics arise from dx₄/dt = ic, and the nonlocality principle makes this unity visible.

6. The Double-Slit Experiment, Wheeler’s Delayed Choice, and Quantum Erasers Within McGucken Spheres

The McGucken Nonlocality Principle — that all nonlocality begins in locality and grows via the expansion of x₄ at c — provides a unified geometric account of three of the most celebrated and apparently paradoxical experiments in quantum mechanics. All three take place within the confines of McGucken Spheres, and all three are resolved by recognizing the full four-dimensional geometry of the expanding x₄ [22].

6.1 The double-slit experiment

In the double-slit experiment, a particle is emitted from a source, passes through one of two slits in a barrier, and is detected on a screen. When no which-path information is available, an interference pattern appears — the particle seems to have passed through both slits simultaneously.

In the McGucken framework, the entire experiment takes place within a single McGucken Sphere centered on the emission event. At the moment of emission, the expansion of x₄ distributes the particle across an expanding wavefront (Huygens’ Principle). This wavefront encounters the barrier and passes through both slits — not because the particle “chooses” both paths, but because the expanding x₄ physically distributes the particle’s position across the wavefront, which intersects both slits. Beyond the barrier, the wavefront from each slit generates new Huygens wavelets, and these wavelets overlap and interfere at the detection screen.

The interference pattern is the visible manifestation of x₄-phase histories through both slits. Each path carries a complex phase e^iS/ℏ from the expansion of x₄ = ict. At some screen positions, paths through both slits arrive in phase and reinforce; at others, they cancel. The particle “takes all paths” because the expanding fourth dimension opens all Huygens histories between source and detector.

Crucially, in the photon’s frame — the frame defined by the McGucken Sphere — there is no time and no distance between the emission event and the detection event. The entire experiment, which a three-dimensional observer describes as “emission, travel, slit passage, interference, detection,” is a single geometric fact in the four-dimensional expansion of x₄.

6.2 Wheeler’s delayed-choice experiment

In Wheeler’s delayed-choice thought experiment [23a], the decision to measure “which path” or “interference” is made after the particle has already passed through the slits. Wheeler argued that this means the particle’s past behavior (wave-like or particle-like) is determined by a future measurement choice — an apparent retroactive influence.

In the McGucken framework, no retroactive influence occurs. The expanding x₄ wavefront carries all paths through both slits at all times. The past behavior of the particle is not changed by the future measurement choice. The measurement choice determines which aspect of the x₄ geometry is revealed at the point of detection:

If the detector is configured to observe interference, both paths contribute coherently to the propagator at the detection point, and fringes appear.
If the detector is configured to observe which-path information, the paths through different slits carry distinguishable x₄ phases that cannot interfere, and no fringes appear.

The entire experiment — emission, slit passage, delayed choice, detection — takes place within a single McGucken Sphere. In the photon’s frame, there is no temporal ordering between “passing through the slits” and “choosing the measurement.” These events are at the same x₄ location. The apparent paradox of delayed choice arises from imposing a three-dimensional temporal ordering on a four-dimensional geometric process that has no such ordering in the frame of the photon [22].

6.3 Quantum eraser experiments

In a quantum eraser experiment [24a], which-path information is first obtained (destroying the interference pattern) and then “erased” (restoring it in a subset of the data, conditioned on a measurement on the entangled partner). This seems to imply that a future choice — to erase or not erase — can retroactively change whether interference occurred.

All quantum eraser experiments take place within McGucken Spheres [22]. The entangled photon pairs share a common McGucken Sphere because they were created at the same local event (the First Law of Nonlocality). The “signal” photon and the “idler” photon are both on the same x₄ wavefront. In the photon’s frame, there is no time and no distance between the creation event, the signal photon’s detection at the screen, and the idler photon’s measurement at the eraser.

The “erasure” does not change the past. It changes which paths on the shared McGucken Sphere are allowed to interfere at the detection point. When which-path information is available (via the idler), the paths through different slits carry distinguishable x₄ phases — they are on different branches of the entangled state — and cannot interfere. When the which-path information is erased (by measuring the idler in a complementary basis), the phases become indistinguishable and interference is restored in the conditioned subset.

The key insight is that the signal and idler photons share the same McGucken Sphere — the same x₄ wavefront — and therefore the measurement on the idler is not a “distant” or “later” event in the four-dimensional geometry. It is an event on the same wavefront, at the same x₄ location as the signal photon’s passage through the slits. The quantum eraser is not paradoxical; it is the expected behavior of entangled particles sharing a common McGucken Sphere.

In more complex quantum eraser configurations involving mediator particles — where entanglement is transferred via intermediaries rather than shared directly from a single creation event — each mediator particle exists on its own McGucken Sphere traceable to its own local creation event. The entanglement is transferred at the local intersections of these McGucken Spheres, all of which originated in a respective locality represented by their point origin. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality.

6.4 The unifying principle

The double-slit experiment, Wheeler’s delayed-choice experiment, and all quantum eraser experiments exhibit the same basic physics: the expansion of x₄ at c distributes particles across wavefronts, assigns complex phases to all paths, and produces interference when path information is unavailable. All three experiments take place within McGucken Spheres — within the expanding x₄ wavefront generated by the source event. The apparent paradoxes (simultaneous passage through both slits, retroactive determination of particle behavior, delayed erasure of which-path information) all dissolve once the full four-dimensional geometry is recognized: in the frame of the photon, within the McGucken Sphere, there is neither time nor distance between any of the events in these experiments. The nonlocality is real — it is the geometric identity of the expanding wavefront — but it is not mysterious, because it began in locality and grew through the expansion of the fourth dimension [22].

7. Addressing Potential Objections

7.1 “What about entanglement swapping and teleportation?”

Entanglement swapping [11] and quantum teleportation [13] are sometimes cited as examples of entanglement being “created at a distance.” But as analyzed in Section 3.3, every step in these protocols involves local interactions mediated by locally-originated intermediaries. The final nonlocal correlation between distant particles traces back, through the chain of interactions, to local creation events. Entanglement swapping does not create nonlocality from nothing; it transfers nonlocality that was already present in locally-created entangled pairs. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality.

7.2 “What about the vacuum and virtual particles?”

In quantum field theory, the vacuum contains virtual particle-antiparticle pairs that are created and annihilated at the same spacetime point. These pairs are local by construction — they originate at a single point. Any entanglement associated with vacuum fluctuations (such as the Unruh effect or Hawking radiation) originates in the local vacuum state and is then spread by the expansion of x₄ at c. The McGucken Nonlocality Principle is consistent with vacuum entanglement because the vacuum itself is local — it is the ground state of the expanding fourth dimension at every point. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality.

7.3 “Does this violate Bell’s theorem?”

No. Bell’s theorem [1] constrains the correlations between measurement outcomes on entangled particles; it does not address the origin of entanglement. Bell’s theorem says that no local hidden-variable theory can reproduce the quantum correlations. The McGucken Nonlocality Principle says that the entanglement itself must originate locally, even though the resulting correlations are nonlocal. These are compatible statements: the creation of entanglement is local (constrained by the light cone), while the correlations of entanglement are nonlocal (violating Bell inequalities). The McGucken Principle provides the geometric mechanism for both: shared x₄ wavefront identity, established locally, producing nonlocal correlations.

Consider the general case of entanglement mediated by intermediary particles. Two distant particles A (in New York) and B (in Los Angeles) are initially unentangled. To entangle them, one prepares two entangled pairs: particles C and D (created locally together at event E₁), and particles E and F (created locally together at event E₂). Particle C is transported to New York and interacts locally with A; particle F is transported to Los Angeles and interacts locally with B; particles D and E are brought together for a Bell-state measurement.

In the McGucken framework, each locally-created pair exists on its own McGucken Sphere: the C–D pair shares the McGucken Sphere centered on E₁, and the E–F pair shares the McGucken Sphere centered on E₂. When D and E are brought together for the Bell-state measurement, their respective McGucken Spheres intersect at the measurement event. It is at this intersection — a local event — that the entanglement is transferred from the C–D and E–F pairs to the A–B pair. The chain is: A ↔ C (local interaction in NY) ↔ D (shared McGucken Sphere from E₁) ↔ E (intersecting McGucken Spheres at Bell measurement) ↔ F (shared McGucken Sphere from E₂) ↔ B (local interaction in LA).

Every link in this chain is either a shared McGucken Sphere (from a common local creation event) or an intersection of McGucken Spheres (at a local measurement event). At no point does entanglement arise without a local origin. The mediator particles C, D, E, F each exist on McGucken Spheres traceable to local creation events, and the entanglement is transferred — never created from nothing — at the local intersections of these spheres. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality.

7.4 “Is this just the no-signaling theorem?”

The no-signaling theorem states that entanglement cannot be used to transmit information faster than light. The McGucken Nonlocality Principle is stronger: it states that entanglement cannot be created faster than light. No-signaling constrains the use of existing entanglement; the McGucken principle constrains the origin of entanglement itself. No-signaling is a consequence of the Second Law of Nonlocality, but the Second Law says more: it says that the sphere of potential entanglement grows at c, not just that signals cannot exceed c. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality, and its growth is limited by c.

7.5 “Shared light cone doesn’t mean entangled — many systems share a light cone without being entangled”

This is correct, and the distinction is important. Being on the same McGucken Sphere is a necessary condition for entanglement, not a sufficient one. Two classical light pulses from the same source share a light cone but are not entangled. Two photons emitted at different times share overlapping light cones but are not entangled. The McGucken Nonlocality Principle does not claim that every pair of systems on the same wavefront is entangled — it claims that every entangled pair must share a wavefront (or intersecting wavefronts) traceable to a local origin.

The additional ingredient required for entanglement — beyond shared wavefront identity — is quantum coherence: the particles must be in a coherent superposition of joint states, with their phases correlated through the shared x₄ geometry. Classical systems on the same light cone have decohered — their phases are randomized by interaction with macroscopic degrees of freedom. Entangled systems are those that maintain quantum coherence on the shared wavefront. The McGucken framework provides the geometric substrate for entanglement (the shared wavefront); quantum coherence provides the condition under which that substrate produces nonlocal correlations. The McGucken Nonlocality Principle is upheld: shared wavefront identity is necessary for entanglement, and that identity always traces to a local origin.

7.6 “Isn’t this just restating what relativistic QFT already enforces?”

There is overlap, and this should be acknowledged honestly. Relativistic quantum field theory respects microcausality — spacelike-separated field operators commute — and this effectively prevents entanglement creation outside the light cone. In that sense, the First Law of Nonlocality is consistent with, and in part implied by, the structure of QFT.

However, the McGucken Nonlocality Principle goes beyond what QFT states in three respects. First, QFT’s microcausality is a formal axiom imposed on the field algebra; the McGucken Principle provides a geometric mechanism for why microcausality holds — because entanglement is shared wavefront identity, and wavefronts expand at c. Second, the McGucken framework unifies the entanglement constraint with the causal structure of relativity, the wavefront structure of optics, and the arrows of time under a single geometric principle — while also providing a common, deeper foundation for special and general relativity [4, 7], the Second Law of Thermodynamics and entropy increase [12], the Schrödinger equation and Feynman’s path integral [17], the Born rule and quantum probability [18], the canonical commutation relation qp − pq = iℏ [16a], the uncertainty principle [12a], Huygens’ Principle and the Principle of Least Action [8], Newton’s law of gravitation [4a], the Milgrom acceleration scale and the Tully-Fisher relation [4b], the cosmological constant and vacuum energy [4c], and the physical constants c and ℏ themselves [10a]. QFT treats all of these as separate structures requiring separate postulates; the McGucken framework derives them all from dx₄/dt = ic. Third, the McGucken framework identifies a new arrow of time — the growth of nonlocality — that is not explicitly recognized in standard QFT.

The relationship between the McGucken Nonlocality Principle and QFT is analogous to the relationship between thermodynamics and statistical mechanics: thermodynamics states the laws (entropy increases), and statistical mechanics provides the mechanism (phase-space growth). QFT states the constraint (microcausality), and the McGucken Principle provides the mechanism (expanding x₄). The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality, and the McGucken Principle provides the deeper geometric reason why.

7.7 “What about entanglement in condensed matter — momentum-space entanglement, topological order, ground-state entanglement?”

In condensed-matter physics, long-range entanglement can exist in the ground state of many-body systems (e.g., topologically ordered states, spin liquids). This entanglement does not obviously map to simple light-cone structures. However, these systems do not violate the McGucken Nonlocality Principle, because the entanglement in their ground states was established through local interactions — the Hamiltonian of a condensed-matter system involves only local (nearest-neighbor or short-range) interactions, and the ground-state entanglement is built up through these local interactions over the system’s history. The ground state is the result of the system’s evolution under a local Hamiltonian, and the entanglement in it traces back, through the evolution, to local interactions at every step.

Momentum-space entanglement is a Fourier-transform restatement of position-space correlations that were themselves established locally. The McGucken Nonlocality Principle is a statement about the origin of correlations in spacetime, not about the basis in which they are described. Changing the basis (from position to momentum) does not change the causal history of the correlations. The McGucken Nonlocality Principle is upheld: even in condensed matter, all quantum nonlocality begins in locality.

7.8 “What prediction does this make that differs from standard QM + relativity?”

The McGucken Nonlocality Principle, as a constraint on the origin of entanglement, is consistent with and largely implied by standard relativistic quantum theory. Its primary value is not as a source of new predictions at the level of standard quantum experiments, but as a foundational clarification and a unifying principle that connects the causal structure of relativity, the wavefront structure of optics, and the origin of quantum entanglement under a single geometric postulate.

That said, the broader McGucken framework from which the Nonlocality Principle derives does make specific predictions that differ from standard physics — including the redshift evolution of the MOND acceleration scale a₀(z) = cH(z)/(2π), the dark energy equation of state w(z) = −1 + Ω_m(z)/(6π), and a proposed sidereal modulation of Bell correlations (the McGucken-Bell experiment). These predictions arise from the same geometric postulate dx₄/dt = ic and are testable with current and forthcoming observational programs. The Nonlocality Principle is one theorem of a framework that makes many testable claims; it stands or falls with the framework as a whole.

Additionally, the Nonlocality Principle makes a specific negative prediction: it predicts that proposals involving indefinite causal order, closed timelike curves, or post-selected teleportation that claim to create entanglement without any local-origin chain will, upon careful analysis, always be found to involve a hidden chain of local contacts. This is a testable claim about the structure of future theoretical proposals, not just about experiments. The McGucken Nonlocality Principle is upheld: all quantum nonlocality begins in locality, and no theoretical framework has yet produced a counterexample.

8. The Two Laws Stated Formally

First McGucken Law of Nonlocality. Two quantum systems A and B can be in an entangled state only if there exists a chain of local interactions (A ↔ C₁ ↔ C₂ ↔ ⋯ ↔ C_n ↔ B) such that each interaction in the chain is local (the interacting systems are at the same spacetime point or within each other’s light cones) and each adjacent pair in the chain has shared a common local origin at some point in its causal past. Equivalently: only systems of particles with intersecting light spheres, with each light sphere centered about each respective particle, can ever be entangled.

Second McGucken Law of Nonlocality. The sphere of potential entanglement emanating from any local event grows at the velocity of light c. No entanglement can be established between two systems whose causal pasts do not overlap — i.e., between systems outside each other’s light cones. Nonlocality grows over time, limited by c.

Corollary. The growth of nonlocality constitutes a sixth arrow of time — the nonlocality arrow — joining the thermodynamic, radiative, cosmological, causal, and psychological arrows as manifestations of the one-way expansion of the fourth dimension dx₄/dt = ic.

9. Conclusion

The McGucken Principle — that the fourth dimension is expanding at the rate of c, dx₄/dt = ic — is the geometric foundation from which the origin and growth of quantum nonlocality are derived. All quantum nonlocality begins in locality, because the expansion of the fourth dimension is the physical process that transforms the local into the nonlocal. Every local point becomes a nonlocal expanding wavefront (Huygens’ Principle). Every entangled pair shares a common McGucken Sphere, or intersecting McGucken Spheres, traceable to local origins. The growth of nonlocality is limited to the velocity of light because the expansion of x₄ proceeds at c.

The same principle that governs nonlocality also governs much else. From dx₄/dt = ic emerge special and general relativity, the Schrödinger equation, Feynman’s path integral, the Born rule, the canonical commutation relation qp − pq = iℏ, entropy increase and the Second Law of Thermodynamics, the uncertainty principle, the physical constants c and ℏ, the Milgrom acceleration scale, the cosmological constant, and now the two laws of nonlocality. The nonlocality principle is one theorem of a framework that unifies physics under a single geometric postulate.

The New York–Los Angeles challenge provides a direct, falsifiable test: demonstrate a method for entangling two distant, unentangled electrons without any chain of local contacts, or accept that all nonlocality begins in locality. No such method has been proposed in any interpretation of quantum mechanics, in any extension of the Standard Model, or in any thought experiment. The principle stands.

Locality becomes nonlocality via the expansion of the fourth dimension at the rate of c. This process gives rise to Huygens’ Principle, time and all its arrows, all of relativity via the spacetime metric x₄ = ict, and quantum nonlocality, entanglement, and probability. The local and the nonlocal are not separate categories of physics; they are the beginning and the present 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 due to quantum mechanics. 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.

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