⚛️ quantum physics

Wave Function Collapse, Lorentz Invariance, and the Third Postulate of Relativity

This paper argues that a specific stochastic, nonlinear modification of the Schrödinger equation, which explains wave function collapse through correlating interactions while preserving conservation laws and avoiding new constants, is Lorentz invariant despite being formulated in a preferred frame, thereby suggesting that local commutativity should be recognized as a third postulate of relativity.

Original authors: Edward J. Gillis

Published 2026-02-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Edward J. Gillis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The Quantum "Snap" vs. The Cosmic Speed Limit

Imagine the universe is a giant, perfectly synchronized dance.

Quantum Mechanics tells us that particles are like dancers who can be in two places at once (a "superposition") until someone watches them. When they are watched, they suddenly "snap" into one specific spot. This snap is random (probabilistic) and happens instantly, even if the dancers are on opposite sides of the galaxy. This is called non-locality.
Relativity (Einstein) tells us that nothing can travel faster than light. If two dancers are far apart, a signal from one to the other takes time.

The Conflict: How can a quantum particle "snap" instantly across the galaxy without breaking Einstein's rule that nothing travels faster than light? If they snap instantly, aren't they sending a secret message faster than light?

For decades, physicists have been stuck in a logical loop. They accept that the snap happens, but they treat it as a weird, unexplainable rule just to keep the math working.

The Paper's Solution: A Hidden "Director" and a Random Walk

Edward Gillis proposes a new way to look at this. He suggests we stop treating the "snap" (wave function collapse) as a magical rule and instead treat it as a physical process driven by interactions between particles.

Here is the breakdown of his idea using a simple analogy:

1. The "Random Walk" Analogy

Imagine a drunk person walking on a tightrope between two poles (Left and Right).

In standard quantum mechanics, the person is a fuzzy cloud covering the whole rope until they fall off one side.
In Gillis's model, the person is taking tiny, random steps left or right.
The Twist: The size of their steps isn't fixed. The closer they get to the edge, the smaller the steps become.
The Result: Eventually, they must fall off one side. The math shows that the chance of falling off the Left side is exactly equal to how much time they spent near the Left side. This naturally creates the famous "Born Rule" (the probability rule of quantum mechanics) without needing to invent it as a separate rule.

2. The "Interaction" Engine

Why does the person start walking? Gillis says it's because of interactions.

When two particles bump into each other (like an electron hitting a detector), they exchange energy and information.
This exchange acts like a "nudge" that starts the random walk.
The stronger the interaction, the bigger the nudge. The paper argues that this "nudge" is what causes the wave function to collapse.

3. The "Hidden Director" (The Preferred Frame)

Here is the controversial part. To make this "nudge" happen instantly across the galaxy, the math requires a preferred reference frame.

Think of the universe as a movie being filmed. In the movie, everything looks relativistic (time slows down, lengths shrink).
But to film the movie, there must be a Director sitting in a control room with a single clock. The Director decides the order of events.
Gillis admits there is a "Director" (a preferred frame of time) that sequences these quantum snaps.
The Magic Trick: Because the "nudge" is random and unpredictable (like a coin flip), no one in the movie can ever see the Director or use the Director's clock to send a secret message. It's like trying to send a telegram using static noise; you can't control the noise to spell out a message. So, even though the Director exists, Einstein's speed limit is never broken in practice.

The "Third Postulate" of Relativity

The paper makes a bold claim about the history of physics.

Postulate 1: The laws of physics are the same for everyone.
Postulate 2: The speed of light is constant.
The New "Third Postulate": We must assume that measurements happening far apart don't interfere with each other (Local Commutativity).

Gillis argues that for a long time, physicists just assumed this third rule to make the math work, without explaining why it was true. He suggests that if we accept his "Hidden Director" model, this third rule isn't just a magic assumption—it's a natural consequence of how the universe is structured. The "Director" ensures the order of events, but the randomness ensures we can't cheat the system.

Why This Matters (The "No Magic Constants" Bonus)

Previous attempts to fix this problem (like the CSL model) had to invent new, made-up numbers (constants) to make the math work. It was like saying, "The universe snaps because of a force we don't know, and its strength is exactly 42."

Gillis's equation is different. It builds the "snap" entirely out of things we already know:

The mass of the particles.
The energy of their interaction.
The speed of light.

It's like building a bridge using only the rocks and wood already on the riverbank, rather than importing special steel beams from another dimension. It also claims that conservation laws (like energy and momentum) are never broken, even in a single event, which solves a major headache in previous theories.

The Bottom Line

This paper suggests that the universe has a hidden "timeline" (a preferred frame) that organizes quantum events. However, because these events are driven by pure randomness, this timeline is invisible to us.

To the Quantum Particle: The collapse is a real, physical process driven by interactions.
To the Human Observer: It looks exactly like Einstein's Relativity. We can't see the hidden timeline, and we can't send faster-than-light messages.

Gillis is essentially saying: "We don't need to choose between Quantum Mechanics and Relativity. We just need to admit that the universe has a hidden structure that makes them play nice together, and that structure is hidden from us by the very nature of randomness."

1. Problem Statement

The paper addresses the fundamental tension between quantum mechanics and special relativity regarding wave function collapse.

The Measurement Problem: Standard quantum theory posits that measurement induces a probabilistic, non-local collapse of the wave function. This process is currently treated as an ad hoc postulate (the "measurement postulate") rather than a dynamic consequence of fundamental physical laws.
Conflict with Relativity: Non-local collapse appears to conflict with the principle of relativity, which prohibits superluminal signaling and implies no preferred reference frame. While standard Quantum Field Theory (QFT) preserves Lorentz invariance by assuming spacelike-separated operators commute, this is often viewed as a mathematical constraint rather than an explanation of the underlying physical mechanism.
Limitations of Existing Models: Previous attempts to model collapse dynamically (Stochastic Nonlinear Schrödinger equations, e.g., GRW, CSL) often introduce ad hoc physical constants, violate conservation laws (specifically energy) in individual instances, or fail to maintain strict Lorentz invariance.

2. Methodology

The author proposes a specific modification to the Schrödinger equation based on the premise that wave function collapse is induced by the elementary interactions that establish correlations between physical systems.

Stochastic Nonlinear Modification: The paper utilizes a stochastic, nonlinear modification of the Schrödinger equation. This equation includes a stochastic term driven by a Wiener process (white noise) to ensure the probabilistic nature of collapse and prevent superluminal signaling (per Gisin's theorem).
Interaction-Induced Collapse Operator: Unlike previous models that use constant collapse rates, this model defines the collapse operator ( $V$ $V$ ) and the rate parameter ( $\gamma$ $γ$ ) dynamically based on the physical interaction between systems ( $j$ $j$ and $k$ $k$ ):
- Collapse Operator: Proportional to the interaction potential energy ( $V_{jk}$ ) minus its expectation value, normalized by the total relativistic energy (rest mass + kinetic) of the systems:
  $V_{jk} \equiv \frac{V_{jk} - \langle \psi | V_{jk} | \psi \rangle}{(m_j + m_k)c^2}$
- Rate Parameter ( $\gamma_{jk}$ ): Defined by the rate of change of the interaction potential energy relative to the maximum potential energy during the interaction. This ensures the stochastic term integrates to a value of order 1 over the duration of the interaction.
Preferred Frame Assumption: The equation is formulated in a preferred reference frame (or a foliation of spacetime) to sequence the interactions. The paper argues this frame is "hidden" due to the probabilistic nature of the collapse.

3. Key Contributions

The paper makes three primary technical contributions:

Derivation of the Born Rule from Dynamics:
The author demonstrates that the stochastic nonlinear equation naturally drives the system to one of the eigenstates of the interaction operator. By modeling the evolution as a random walk in Hilbert space, the probability of ending in a specific state is shown to be exactly $|\alpha|^2$ or $|\beta|^2$ , thereby deriving the Born rule as a dynamic consequence rather than an imposed postulate.
Strict Conservation Laws in Individual Instances:
A major critique of stochastic collapse models is that they often violate conservation laws (e.g., energy) in single events. This paper proves that the proposed interaction-induced model strictly conserves momentum, angular momentum, and energy in individual measurement instances.
- Mechanism: Because the collapse is driven by the exchange of conserved quantities (interaction potentials) and the operator is constructed from the interaction itself, the total conserved quantities are redistributed among branches without net creation or destruction.
- No Ad Hoc Constants: The model eliminates the need for arbitrary constants (like the collapse rate $\lambda$ in CSL) by deriving all parameters from fundamental physical quantities (mass, interaction potential, $c$ ).
Proof of Lorentz Invariance:
The most significant contribution is the demonstration that the proposed stochastic modification is Lorentz invariant, despite being formulated in a preferred frame.
- The ratio of interaction energy to total relativistic energy is shown to be invariant across frames.
- The integration of the rate parameter $\gamma_{jk}$ over the interaction duration yields a frame-independent value (order 1).
- The "ping-pong" nature of causal influences in entangled systems ensures that the stochastic changes at spacelike-separated points remain consistent, preserving the non-signaling condition.

4. Results

Mathematical Consistency: The modified equation (Eq. 10 in the text) successfully reproduces standard quantum mechanical predictions for ensembles while providing a mechanism for individual outcomes.
Lorentz Invariance: The amplitude shifts induced by the collapse are shown to be independent of the reference frame. The preferred foliation required for the stochastic process remains undetectable, satisfying the empirical requirements of relativity.
Conservation: The model resolves the "tails problem" (where small amplitudes persist) and conservation violations found in other models (like CSL) by tying the collapse strength directly to the interaction energy and relativistic mass.
Predictions: The model predicts minute deviations from standard quantum linearity (proportional to the square of the amplitude shift, roughly $10^{-9}$ for electron interactions at the Bohr radius), offering a potential experimental test.

5. Significance and Implications

The "Third Postulate" of Relativity: The paper argues that the assumption of local commutativity (spacelike-separated operators commute) is not merely a mathematical convenience but constitutes a third postulate of relativity. This postulate is necessary to regulate non-local quantum effects and ensure no superluminal signaling.
Spacetime Ontology: The work suggests that relativistic spacetime requires additional structure (a foliation or preferred frame) to explain how non-local correlations propagate and are regulated. However, due to the fundamental probabilistic nature of the collapse, this structure remains hidden, preserving the empirical success of standard relativity.
Unification: By deriving measurement postulates and conservation laws from a single fundamental equation, the paper offers a more coherent logical framework for physics, bridging the gap between the unitary evolution of the Schrödinger equation and the probabilistic nature of measurement outcomes.
Future Work: The author acknowledges the current formulation is non-relativistic but argues that the demonstrated Lorentz invariance removes conceptual barriers to a full relativistic quantum field theory extension, potentially resolving issues like the "tails problem" and energy conservation discrepancies in high-energy regimes.

In summary, Gillis proposes a physically motivated, Lorentz-invariant stochastic collapse model that eliminates ad hoc parameters, enforces strict conservation laws in individual events, and reinterprets the structure of spacetime to accommodate non-local quantum effects.