A state chaining-based objective collapse model

The Big Problem: Why Don't We See Zombie Cats?

In the quantum world, particles can be in two places at once (superposition). But in our daily life, a cat is either alive or dead, never both. This is the famous "Schrödinger's cat" paradox.

For decades, physicists have argued about why the quantum world turns into the classical world we see.

The Old Idea (Copenhagen): "It collapses when you look at it." (But who counts as a "looker"? A camera? A human? A fly?)
The "Many Worlds" Idea: "It never collapses; the universe just splits into two." (But we only experience one reality.)
The "Gravity" Idea: "Big things collapse because gravity gets too heavy." (This is a popular theory, but hard to prove.)

This paper proposes a new, simpler idea: The universe doesn't care how big an object is. It cares about how many "links" in a chain are holding the object together.

The Core Concept: "Chaining" vs. "Entanglement"

Imagine you are trying to explain how a group of people stay together.

1. Entanglement (The "Handshake"):
Imagine two dancers holding hands. If one spins, the other spins. They are connected, but they are still two separate people. In quantum physics, this is called entanglement. It's like a handshake between particles. It's strong, but it doesn't force them to be the same thing.

2. Chaining (The "Mega-Chain"):
Now, imagine a line of 1,000 people holding hands, but with a twist: They all have to agree on the exact same secret password. If the first person whispers "Red," the second must be "Red," the third must be "Red," and so on. They are no longer just holding hands; they are forced to share a single, unified state.

In this paper, the author calls this "Chaining."

The Rule: Every time a quantum system tries to share a single state across many parts (like a photon hitting a detector and triggering a cascade of electrons), a "chaining" event happens.
The Risk: The author suggests that every time a "chaining" happens, there is a tiny, random chance (like rolling a die) that the chain will snap.
The Snap: When the chain snaps, the quantum superposition collapses. The system is forced to pick one reality (e.g., "Red" or "Blue").

The "Chinese Whispers" Rule

The author calls the mechanism the "Chinese Whispers Rule."

Imagine a game of "Telephone" (Chinese Whispers) where a message is passed down a line of people.

In a normal quantum system, the message stays fuzzy and can be "both A and B."
In this model, every time the message is passed to a new person in the chain, there is a tiny chance (1 in $\Sigma$ ) that the message gets locked in.
If the message gets locked in, the game stops. The superposition is over. The system has "collapsed" into a definite state.

Why does a cat collapse but a mirror doesn't?

The Mirror: When a photon hits a mirror, it bounces off. It's like a single handshake. No long chain is formed. The photon stays in superposition.
The Cat (or a Detector): When a photon hits a detector (or a Geiger counter), it triggers an avalanche. One electron hits another, which hits another, creating a massive chain of "agreement."
- Because there are millions of links in this chain, the odds that at least one link will snap (triggering a collapse) become almost 100%.
- The Cat: The poison doesn't just kill the cat instantly; it spreads through the lungs, blood, and nerves. This creates a massive chain of "chained" particles. The chain is so long that it almost certainly snaps, forcing the cat to be either dead or alive, not both.

The "Magic Number" ( $\Sigma$ )

The theory relies on one universal constant, called $\Sigma$ (Sigma).

Think of $\Sigma$ as the "stability of the universe."
If $\Sigma$ is huge (like a billion), chains rarely snap, and the world stays quantum for a long time.
If $\Sigma$ is small, chains snap easily, and the world becomes classical quickly.

The author looked at real experiments (like bouncing neutrons off slits or using "quantum erasers") to see how often these chains seem to snap.

The Result: The data suggests $\Sigma$ is roughly 1.5.
What this means: The universe is surprisingly "fragile." Every time a complex chain forms, there is a very high chance (about 66%) that it will collapse immediately. This explains why we don't see quantum weirdness in everyday life.

Why This Matters (The "So What?")

It's Not About Size: This theory says a giant atom and a tiny neutron behave the same way if they have the same number of "chaining steps." It's not about being heavy; it's about being connected in a specific way.
Quantum Computers: If you want to build a super-powerful quantum computer, you need to keep your qubits (quantum bits) from "chaining" with the environment. If you can stop the chains from forming, your computer won't collapse, and it will work much longer.
No "Magic" Observer: You don't need a conscious human to collapse the wavefunction. The universe collapses itself automatically whenever a complex chain of events tries to agree on a single state.

Summary Analogy

Imagine the universe is a giant game of Jenga.

Superposition is a tower where the blocks are slightly wobbly and can be in two positions at once.
Chaining is when you try to stack a new block on top that forces the whole tower to lock into a specific shape.
Collapse happens when the tower is so tall and the connections so tight that the structure can't hold the "wobble" anymore. It snaps into one solid shape.

This paper suggests that the universe has a built-in "tipping point" (the number 1.5) where, if you try to link too many things together, the universe forces a decision, turning the fuzzy quantum world into the solid classical world we live in.

1. Problem Statement

The paper addresses the measurement problem in quantum mechanics, specifically the transition from quantum superposition to classical reality.

The Controversy: Standard quantum mechanics (Copenhagen interpretation) relies on an undefined "measurement" to collapse the wavefunction, failing to distinguish between a measuring device and a complex quantum system. Other interpretations (Many-Worlds, Consciousness-causes-collapse) offer philosophical solutions but lack a physical mechanism for the collapse event.
Limitations of Existing Models: Objective collapse theories like Continuous Spontaneous Localization (CSL) and the Diosi-Penrose (DP) model attempt to modify the Schrödinger equation. However, they typically rely on system size or mass (e.g., gravity) to trigger collapse.
The Gap: There is a need for a mechanism that explains why macroscopic objects (like a cat) collapse while microscopic objects (like neutrons) do not, without relying solely on size parameters, and which is consistent with interference experiments showing that different particles (neutrons, atoms, molecules) behave similarly regarding coherence loss.

2. Methodology: The Qil-Formalism

The author introduces a new diagrammatic framework called "qils" (quantum illustrations) to describe quantum states based on their degrees of freedom (DoF) rather than their internal structure.

Qil Notation: A qil is represented as $(q)_{\beta_1...\beta_m}^{\alpha_1...\alpha_n}$ $(q)_{β_{1} ... β_{m}}^{α_{1} ... α_{n}}$ , where:
- Upper indices ( $\alpha$ ) represent defined DoFs (specific values).
- Lower indices ( $\beta$ ) represent undefined DoFs (superpositions of values).
Two Fundamental Operations:
1. Entanglement (Square Bracket $[\dots]_p$ ): Constrains DoFs of different subsystems via a rule (e.g., conservation of momentum). It preserves the total number of DoFs but links them.
2. Chaining (Curly Bracket $\{\dots\}_p$ ): Identifies multiple DoFs as a single, shared value. This forces multiple subsystems to share one specific parameter value (e.g., "all electrons are triggered or none are").
The Collapse Mechanism:
- An Event is defined as the transition of an undefined DoF to a defined state.
- The "Chinese Whispers" Rule: The core postulate states that whenever a chaining occurs (merging DoFs), there is a fixed, universal probability $1/\Sigma$ that an objective collapse event is triggered.
- Key Distinction: Collapse depends on the number of chaining steps required to propagate a superposition through a system, not the system's physical size.

3. Key Contributions

A. Theoretical Framework

Scale Invariance: The model posits that all qils are equivalent regardless of internal complexity. A composite system is treated as a single entity defined by its external DoFs.
Mechanism for Classicality: Classicality emerges not because an object is "large," but because macroscopic systems involve a vast number of chaining steps when interacting with a quantum trigger.
- Example (Schrödinger's Cat): The poison triggering the cat involves a cascade of ionization events (Geiger counter $\to$ lungs $\to$ blood $\to$ heart). Each step adds a chaining. With $N$ steps, the probability of no collapse is $(1 - 1/\Sigma)^N$ . For large $N$ , collapse is near-certain.
- Example (Mirror): Reflection involves only entanglement of momentum, not chaining. Thus, mirrors do not trigger collapse, preserving superposition.

B. Application to Scenarios

The paper applies the formalism to:

Delayed-Choice Quantum Eraser: The model explains the statistics without retrocausality. The "erasure" or "which-path" information corresponds to whether the idler photon's path involves a chaining event that triggers a collapse.
Photomultiplier Tubes (PMT): The avalanche effect in a PMT creates a massive chain of entangled/triggered electrons. The model predicts a near-100% probability of collapse, consistent with detector efficiency.
Atomic Relaxation: Spontaneous decay is modeled as a chaining of the atom's world-line with the emitted photon's world-line, allowing for non-unitary collapse events.

C. Experimental Estimation of $\Sigma$

The author derives a method to estimate the fundamental constant $\Sigma$ (the inverse of the collapse probability per chaining step) using interference experiments with compound particles.

Logic: In an interference experiment (e.g., double-slit or Mach-Zehnder) with a compound particle, the splitting of the particle creates a chaining of its internal components. If a collapse occurs (probability $P = 1/\Sigma$ ), the interference pattern is destroyed, creating a "bias" (non-interfering background).
Formula: $\Sigma = I / B$ , where $I$ is the total integral and $B$ is the bias integral.

4. Results

Consistency with Data: The model was tested against existing experimental data for:
- Neutrons (beam splitter & double-slit)
- Helium atoms (double-slit)
- Neon atoms (double-slit)
- Delayed-Choice Quantum Eraser
Estimated Value: The analysis yields a lower bound for the fundamental constant: $\Sigma \geq 1.5$ .
- Specific estimates ranged from $\Sigma \geq 1.21$ (Helium) to $\Sigma \geq 1.54$ (Neutrons).
- The author suggests the true value is likely around 1.5, assuming experimental decoherence is minimized.
Universality: The model successfully explains why neutrons, atoms, and molecules show similar levels of "non-coherent bias" in interference experiments, a feature difficult for size-dependent models (like CSL) to explain without fine-tuning.

5. Significance and Implications

Unified Objective Collapse: The theory provides a parameter-sparse, unified mechanism for wavefunction collapse that does not rely on subjective observation or arbitrary size thresholds.
Distinction from Decoherence: It offers a distinct physical mechanism (chaining-induced collapse) that is mathematically separable from environmental decoherence, though they may appear similar in data.
Quantum Computing Implications: The paper suggests a surprising implication for quantum computing. If qubits are composed of elementary particles (which have no internal components to chain), the "chaining-induced collapse" mechanism would be absent. This implies that elementary particle qubits could theoretically possess significantly longer coherence times than composite particle qubits, as they lack the internal chaining steps that trigger collapse.
New Formalism: The introduction of "qils" offers a novel diagrammatic tool for visualizing and calculating quantum events based on the topology of degrees of freedom rather than wavefunction amplitudes.

Conclusion

Roman V. Li proposes that the quantum-to-classical transition is driven by the probabilistic collapse of chained degrees of freedom. By formalizing this through "qils" and postulating a universal collapse probability per chaining step ( $1/\Sigma$ ), the model resolves the measurement problem without invoking consciousness or size-dependent parameters. The theory is consistent with current interference data, yielding an estimate of $\Sigma \approx 1.5$ , and offers new perspectives on the stability of quantum states in elementary versus composite systems.