On blocking Dispersion of Matter by Energy conservation

This paper revisits a nonlinear energy-conservation mechanism proposed to suppress macroscopic superpositions, deriving necessary commutation relations to validate spatial confinement terms while demonstrating that analogous terms for non-pure spin models fail these constraints, and concludes by presenting a toy model of the resulting energy barrier alongside a comparison with collapse models.

The Gist

The Big Question: Why Don't We See "Quantum Cats"?

Imagine you have a coin. In the quantum world, a coin can be spinning so fast that it is both Heads and Tails at the exact same time. This is called a "superposition." If you have a single atom, this is normal. But if you have a giant object—like a cat, a car, or a baseball—quantum mechanics says it should also be able to be in two places at once (a "Schrödinger's Cat" state).

Yet, in our daily life, we never see a cat that is simultaneously on the sofa and in the kitchen. It's always in one place.

The Problem: Why does the quantum world (where things can be in two places) stop working when things get big? Is there a "magic line" where quantum rules turn off?

The Author's Idea: The "Energy Tax"

Most scientists think the answer is "decoherence" (the environment messes up the quantum state) or "collapse models" (random noise forces the object to pick a spot).

Leonardo De Carlo proposes a different idea. He suggests that nature has a strict Energy Budget.

Think of a quantum superposition like a high-speed race car.

• Small objects (atoms): They are like tiny go-karts. They can zip around, split into two paths, and rejoin without needing much fuel. The "energy cost" of being in two places is tiny.
• Big objects (cats): To make a giant object be in two places at once, the universe would have to pay a massive Energy Tax.

De Carlo argues that the laws of physics have been tweaked to add a "surcharge" for large objects trying to be in two places. If the object is too big, the energy required to maintain that "split" state is so huge that it becomes impossible to create, unless you have an infinite power source.

The Analogy: Imagine trying to inflate a balloon.

• A small balloon (an atom) is easy to blow up.
• A giant balloon (a macroscopic object) requires so much air pressure that the rubber would snap before you could finish. The "rubber snapping" is the Energy Conservation law blocking the superposition.

The "Wavefunction Energy" (WFE)

To make this math work, the author introduces a new term called Wavefunction Energy (WFE).

• The Rule: The more spread out an object is in space (or momentum), the more energy it costs.
• The Catch: For a single atom, this cost is negligible. But for a billion atoms moving together, the cost grows exponentially (like squaring the number of atoms).
• The Result: The universe naturally "traps" large objects in one location because spreading out is too expensive. It's like a hill: small marbles can roll up the hill easily, but a boulder cannot. The boulder stays at the bottom (in one place).

The "Measurement" Mystery

This theory also tries to solve the Measurement Problem.

• Old View: When you measure a quantum particle, it magically "collapses" from a wave to a particle.
• New View: The particle doesn't magically collapse. Instead, when it interacts with a measuring device (like a needle on a gauge), the system tries to split into two outcomes (needle left / needle right).
• The Barrier: Because the needle is macroscopic, the "Energy Tax" for it to be in two places is too high. The system hits an Energy Barrier. It can't afford to split. So, it is forced to "fall" into one state or the other.
• Chaos: The author suggests that the tiny difference that decides which side it falls to comes from Chaos (extreme sensitivity to tiny starting conditions), not random magic.

Testing the Theory: The "Toy Model"

The author builds a simple computer model (a "toy") to see if this works.

• Imagine a ball in a double-well potential (a valley with two dips, separated by a hill).
• In normal quantum physics, the ball can be in both dips at once.
• In De Carlo's model, if the ball gets too heavy (too many atoms), the "Energy Tax" makes the hill between the dips infinitely high. The ball gets stuck in one dip.
• The Prediction: If you try to create a "Quantum Cat" (a big object in two places), you will hit a Critical Size. Below this size, you can make a superposition. Above it, the energy barrier is too high, and the object will stay in one place.

How is this different from other theories?

• Collapse Models (The "Random Noise" Theory): These say the universe is like a foggy room where random bumps knock the object into one spot. This adds randomness and creates heat (energy is lost).
• De Carlo's Model (The "Energy Budget" Theory): This says the universe is like a strict accountant. It doesn't use random noise; it strictly enforces energy conservation. The system is deterministic (no randomness), but it looks random because of Chaos. It also doesn't create extra heat.

The "Spin" Twist (What Didn't Work)

The author also tried to apply this idea to Spin (the magnetic direction of atoms, like a tiny compass).

• He tried to combine the position of the atom with its spin direction.
• The Result: It failed. The math showed that if you try to do this, the "Center of Mass" (the average position of the object) would start moving strangely, violating basic laws of motion.
• The Conclusion: This "Energy Tax" likely applies to position (where things are) and momentum (how fast they move), but not directly to the internal "spin" of atoms. It suggests the "cat" we are trying to stop is a spatial cat (being in two places), not a spin cat.

Summary in One Sentence

The universe might not have a "magic switch" that turns off quantum mechanics for big things; instead, it might have a strict energy bill that makes it impossible for large objects to afford the luxury of being in two places at once.

What's Next?

The author suggests we should look for experiments where we try to create "Quantum Cats" with larger and larger objects. If his theory is right, there will be a specific size where, no matter how hard we try, the object refuses to split because the energy barrier is too high. This would be a new way to find the boundary between the quantum world and our everyday world.

Technical Summary

1. Problem Statement

The paper addresses the Measurement Problem and the Quantum-Classical Boundary. Specifically, it investigates why macroscopic objects do not exhibit spatial superpositions (Schrödinger's cat states) despite the linearity of the Schrödinger equation.

• Context: While experimental efforts aim to create macroscopic superpositions, standard quantum mechanics predicts they should exist. Current solutions often rely on Collapse Models (e.g., CSL, GRW), which introduce stochastic, non-unitary dynamics that violate energy conservation.
• The Gap: The author seeks an alternative framework where macroscopic superpositions are suppressed not by stochastic collapse, but by energy conservation. The goal is to modify the Schrödinger equation with deterministic, non-linear terms that penalize "macroscopic dispersion" (spatial delocalization) without violating fundamental conservation laws (norm and energy).

2. Methodology

The author proposes a modification to the Schrödinger equation by adding a "Wavefunction-Energy" (WFE) term to the Hamiltonian. The evolution is governed by:
$$i\hbar \frac{\partial \psi}{\partial t} = \frac{\partial}{\partial \psi^*} E(\psi)$$
where $E(\psi) = E_{QM}(\psi) + E_{WFE}(\psi)$.

Key Principles:

1. Negligible Microscopically, Macroscopic Effect: The modification must be tiny for single particles but scale as $N^2$ for $N$ particles to block macroscopic dispersion.
2. Conservation: The norm $\|\psi\|$ and total energy $E(\psi)$ must be conserved.
3. Center of Mass (COM) Invariance: The modification must not alter the classical equation of motion for the center of mass.
4. Chaos: The dynamics of a system coupled to a "measuring" apparatus must exhibit chaos to explain the emergence of definite outcomes.

The WFE Term:
The proposed energy term is:
$$E_{WFE}(\psi) = w N^2 D(\psi)$$
where $w$ is a small positive constant and $D(\psi)$ represents the spatial dispersion (variance) of an observable $O_i$:
$$D(\psi) = \left\langle \psi \left| \left( \frac{1}{N}\sum O_i - \langle \frac{1}{N}\sum O_i \rangle \right)^2 \right| \psi \right\rangle$$

Analytical Approach:

• Commutation Relations: The author derives necessary commutation relations between the position/momentum operators ($X_k, P_k$) and the operators $O_i$ to ensure the COM equation remains unchanged.
• Spin Ensembles: The paper revisits previous work on spin models (Curie-Weiss/Ising) to check if the WFE term induces spontaneous magnetization in wavefunction ensembles.
• Toy Models: A discrete spin toy model is used to demonstrate the transition from linear dispersion to chaotic collapse when the non-linearity exceeds a critical threshold (using a "Determinant Criterion of Instability").
• Comparison: A rigorous comparison is made with Continuous Spontaneous Localization (CSL) models regarding energy conservation, reversibility, and phenomenological predictions.

3. Key Contributions and Results

A. Derivation of Admissibility Conditions

The author derives strict conditions on the operators $O_i$ used in the WFE term.

• Valid Cases: The conditions are satisfied if $O_i = X_i$ (Position) or $O_i = P_i$ (Momentum). These terms successfully block spatial or momentum cats while preserving COM dynamics and energy conservation.
• Invalid Case (Main Result): The author proves that the case $O_i = L_i + s_i$ (Total Angular Momentum + Spin), which was used in previous spin-ensemble models, fails the COM invariance condition.
  • Implication: The non-linear terms added to pure spin models in previous literature cannot be directly interpreted as acting on the spin system itself without violating kinematics. The author suggests these terms effectively act on the measurement apparatus (e.g., a magnetometer) rather than the spins, or require a lattice model with spatial coordinates to be physically consistent.

B. The Energy Barrier Mechanism

The paper establishes that forming a macroscopic superposition (a "cat state") requires overcoming an energy barrier proportional to $N^2$.

• Scaling: For a product state, $E_{WFE} \sim O(N)$. For a cat state (delocalized over distance $R$), $E_{WFE} \sim O(N^2 R^2)$.
• Consequence: Unless an external potential supplies this massive energy, the system cannot evolve into a cat state. This creates a "self-trapping" mechanism where the wavefunction remains localized.

C. Chaos and Measurement

Using a toy model of a spin system coupled to an apparatus, the author demonstrates that:

• When the non-linearity parameter $w$ exceeds a critical threshold ($w > w^*$), the system exhibits deterministic chaos (positive Lyapunov exponents).
• This chaos amplifies microscopic instabilities (initial asymmetries in the spin state) into macroscopic outcomes (the system "falls" into one well or the other), effectively solving the measurement problem without stochastic collapse.

D. Experimental Estimates

The author provides rough estimates for the coupling constant $w$ based on potential experiments (e.g., graphene resonators or cold atom clouds):

• Estimated range: $w \sim 10^{-25} \text{ to } 10^{-21} \, \text{J/m}^2$.
• Testable Prediction: Unlike collapse models which predict a sharp "step function" in dispersion based on size, WFE predicts a smooth transition dependent on the energy barrier height. If an external potential is strong enough to overcome the $N^2$ barrier, cat states should be observable even for large $N$.

E. Comparison with Collapse Models (CSL)

• Energy: CSL models continuously pump energy into the system (violating conservation) and cause spontaneous heating/radiation. WFE models conserve energy exactly and are time-reversible.
• Mechanism: CSL uses stochastic noise; WFE uses deterministic non-linearity and chaos.
• Phenomenology: CSL predicts spontaneous X-ray emission from free electrons. WFE predicts no spontaneous radiation but a specific suppression of superposition formation unless sufficient external energy is supplied.

4. Significance

• Theoretical Consistency: The paper resolves a potential inconsistency in previous work regarding spin models, clarifying that the WFE mechanism likely acts on the spatial degrees of freedom of the measuring apparatus, not the internal spin states.
• Alternative Paradigm: It offers a compelling alternative to stochastic collapse theories. By grounding the suppression of macroscopic superpositions in energy conservation and deterministic chaos, it aligns better with standard physical principles (unitarity, reversibility).
• Experimental Roadmap: It provides a concrete, testable hypothesis: the existence of an energy barrier for cat states. This suggests that experiments should focus not just on increasing mass ($N$), but on the energy landscape (barrier height) to see if superpositions can be "unlocked" by supplying sufficient energy.
• Chaos Connection: It strengthens the link between the quantum-to-classical transition and deterministic chaos, suggesting that the "randomness" of measurement outcomes arises from sensitivity to initial conditions in a non-linear deterministic system.

In summary, De Carlo refines the "Wavefunction Energy" theory, proving its mathematical consistency for position and momentum operators, rejecting its application to pure spin models without spatial context, and proposing a new experimental signature based on energy barriers rather than stochastic collapse.