Spontaneous wave function collapse from non-local gravitational self-energy

This paper proposes a model-independent mechanism for spontaneous wave function collapse driven by non-local gravitational self-energy, which arises from the fundamental tension between the equivalence principle and quantum superposition in a semiclassical spacetime, resulting in a collapse time inversely proportional to the system's mass.

The Gist

The Big Picture: Why Don't We See "Schrödinger's Cats" in Real Life?

In the weird world of quantum mechanics, tiny particles (like electrons) can exist in two places at once. This is called superposition. It's like a coin spinning in the air—it's technically both "heads" and "tails" simultaneously until it lands.

However, in our everyday world, we never see a cat that is both dead and alive, or a rock that is in two different rooms at the same time. Something must be stopping these "superpositions" from happening when things get big (macroscopic). This is the famous "Measurement Problem."

This paper proposes a new reason why the "coin" stops spinning: Gravity.

The authors argue that gravity acts like a cosmic "reality check." When an object gets heavy enough, its own gravity becomes so strong that it forces the object to pick a single location, collapsing the quantum "blur" into a definite reality.

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The Core Idea: Gravity vs. The "Blur"

Imagine you are trying to draw a map of a city.

• Quantum Mechanics says the city is a fuzzy cloud of possibilities. You can be in the park and the library at the same time.
• General Relativity (Gravity) says the city has a specific shape. If you are in the park, the ground is flat; if you are in the library, the ground might be sloped.

The Conflict:
If you are in a superposition (both park and library), you are creating two different shapes of the ground at the same time. The paper argues that the universe hates this contradiction. It's like trying to stretch a rubber sheet in two different directions simultaneously; eventually, the sheet snaps.

The authors suggest that this "snapping" is the wave function collapse. The universe forces the object to choose one spot so the "rubber sheet" (spacetime) can settle down.

The New Twist: String Theory and "Fuzzy" Gravity

Previous theories (like those by Roger Penrose) suggested this happens, but they had a mathematical problem: when you calculate the gravity of a tiny particle, the numbers blow up to infinity (a singularity).

The Authors' Solution:
They use a concept from String Theory called T-duality.

• The Analogy: Imagine looking at a string. If you zoom in really close, you see it's a string. But if you zoom out, it looks like a point. T-duality suggests there is a minimum size to the universe. You can't zoom in forever; there is a "pixel" size (called the zero-point length).
• The Result: Because space has these "pixels," gravity doesn't get infinitely strong at tiny distances. It gets "smeared out" or "fuzzy." This fixes the math and makes the theory work without breaking.

How the Collapse Happens (The "Tug-of-War")

The paper describes a tug-of-war between two fundamental rules of physics:

1. The Superposition Principle: "I can be everywhere at once."
2. The Equivalence Principle (Gravity): "Gravity pulls everything down the same way."

The Scenario:
Imagine a heavy rock is in a superposition of being in two places.

• In Place A, the rock creates a specific gravitational pull.
• In Place B, the rock creates a different gravitational pull.

The authors argue that if the rock is heavy enough, the difference in gravity between Place A and Place B becomes so confusing that spacetime itself gets "ill-defined." It's like trying to drive a car where the road is simultaneously flat and a steep cliff. The car (the quantum state) can't handle the confusion and crashes (collapses) into one reality.

The "Collapse Time" Formula

The paper calculates exactly how long it takes for this to happen.

• Tiny Particles (Electrons): They have very little mass, so their gravity is weak. The "tug-of-war" is weak. They can stay in a superposition for a very, very long time (longer than the age of the universe).
• Big Objects (Rocks, Cats): They have lots of mass. Their gravity is strong. The "tug-of-war" is intense. They collapse almost instantly.

The Rule: The heavier the object, the faster it collapses. The time it takes to collapse is inversely proportional to the square of its mass.

• Analogy: Think of a feather (light) floating in the wind (staying in superposition) vs. a bowling ball (heavy) dropping straight to the ground (collapsing immediately).

Why This Paper is Different (and Safe from Recent Experiments)

Recently, scientists did an experiment (Donadi et al.) that ruled out some theories about gravity causing collapse. They looked for "spontaneous radiation" (like static noise) that these theories predicted. They found nothing, so those theories were discarded.

Why this paper survives:

• The Old Theory: Was like a noisy radio. It said gravity causes collapse by adding random "static" (noise) to the universe. The experiment found no static, so the theory failed.
• This New Theory: Is like a silent, deterministic movie. It says gravity causes collapse because of a smooth, mathematical tension in spacetime, not because of random noise.
• The Takeaway: Because this model doesn't predict "static noise," the recent experiment didn't rule it out. It's a different kind of mechanism entirely.

Summary in a Metaphor

Imagine the universe is a giant trampoline.

• Quantum Superposition is like a child bouncing on the trampoline, creating a wavy, blurry shape.
• Gravity is the weight of the child.
• The Paper's Argument: If the child is light (an electron), the trampoline can handle the blur. But if the child is a giant (a rock), the trampoline gets so distorted that it can't hold the blur anymore. The fabric of the trampoline forces the child to sit still in one spot.

The authors have built a new mathematical model using "string theory pixels" to show exactly how this happens, proving that gravity is the invisible hand that turns the quantum world into the solid, definite world we see every day.

Technical Summary

1. Problem Statement

The paper addresses the measurement problem in quantum mechanics, specifically the mechanism behind wave function collapse. While standard quantum mechanics relies on the linear superposition principle, it fails to explain why macroscopic objects do not exist in superpositions.

• The Conflict: There is a fundamental tension between the Equivalence Principle (general relativity) and the Superposition Principle (quantum mechanics). When a massive object is in a superposition of two locations, it creates a superposition of two different spacetime geometries (gravitational fields).
• Limitations of Existing Models: The Diósi-Penrose (DP) model proposes that this gravitational self-energy causes instability and collapse. However, the standard DP formulation relies on phenomenological inputs, suffers from ultraviolet (UV) divergences (singularities at short distances), and often employs stochastic (random) noise terms that have been constrained by recent experiments (e.g., Donadi et al., 2021).
• Goal: The authors aim to derive a deterministic, regularized (singularity-free), and model-independent mechanism for wave function collapse rooted in fundamental spacetime non-locality, avoiding the constraints of recent experimental bounds on stochastic collapse models.

2. Methodology

The authors employ a framework based on String Theory T-duality to modify the gravitational interaction at short scales.

• T-Duality and Minimal Length: They utilize T-duality, which relates string theories with compactification radius $R$ to those with radius $R_*^2/R$. This implies the existence of a minimal length scale (zero-point length), denoted as $l_0 \approx 2\sqrt{\alpha'}$.
• Regularized Propagator: Instead of the standard Newtonian potential ($1/r$), they use a quantum-corrected propagator derived from path-integral duality. This leads to a regularized gravitational potential:
  $$V_G(r) = -\frac{GM}{\sqrt{r^2 + l_0^2}}$$
  This potential is finite everywhere, removing the $r=0$ singularity.
• Non-Local Gravitational Self-Energy: They incorporate the gravitational self-energy ($E_{GSE}$) arising from this non-local potential into the Schrödinger equation. The mass density is treated as a "smeared" quantum probability density $\rho = M|\Psi|^2$.
• Modified Schrödinger-Newton Equation: By adding the self-energy Hamiltonian term to the Schrödinger equation, they derive a non-linear evolution equation:
  $$i\hbar\frac{\partial\Psi}{\partial t} = \left[ -\frac{\hbar^2}{2M}\nabla^2 - \frac{1}{2} \int V_G(x) \rho(x) d^3x \right] \Psi$$
  Crucially, this equation is deterministic and contains no stochastic noise terms.

3. Key Contributions

1. Derivation of a Regularized Schrödinger-Newton Equation: The authors derive a new form of the Schrödinger-Newton equation where the gravitational potential is regularized by the zero-point length $l_0$. This eliminates the UV divergences present in the standard Diósi-Penrose formulation.
2. Demonstration of Non-Linearity: They explicitly show that the inclusion of gravitational self-energy makes the evolution equation non-linear. The Hamiltonian becomes a functional of the wavefunction itself ($H[\Psi]$), violating the linear superposition principle.
3. Phase Shift Analysis: They analyze the wave function in two frames: an inertial frame and a freely falling (Einsteinian) frame. They demonstrate that these frames differ by a gravitationally induced phase shift containing:
   • A constant global term (from the regularized self-energy).
   • A linear time-dependent term.
   • A cubic time-dependent term ($t^3$).
4. Collapse Mechanism via Metric Indefiniteness: The authors argue that collapse occurs when the uncertainty in gravitational acceleration ($\Delta g$) becomes comparable to the acceleration itself ($|g|$). This condition ($\Delta g \sim |g|$) corresponds to a regime where the spacetime metric becomes ill-defined due to quantum fluctuations, forcing the superposition to collapse.
5. Distinction from Stochastic Models: They clarify that their model is deterministic, whereas the models constrained by recent underground experiments (Donadi et al.) are stochastic. Therefore, their model is not subject to the experimental bounds that ruled out specific stochastic implementations of the DP model.

4. Key Results

• Collapse Time Formula: The paper derives a spontaneous collapse time ($\tau_{collapse}$) that is inversely proportional to the square of the mass ($M$) and the spatial separation ($\Delta r$) of the superposition:
  $$\tau_{collapse} \sim \frac{\hbar M_{Pl} \Delta r}{M^2 l_{Pl} c^2} \propto \frac{1}{M^2}$$
  • Microscopic Systems: For small masses (e.g., electrons), $\tau_{collapse}$ is extremely large, allowing stable superpositions.
  • Macroscopic Systems: For large masses, $\tau_{collapse}$ is vanishingly small, explaining why macroscopic objects do not exhibit quantum superposition.
• Numerical Estimates: Table I in the paper provides numerical values showing that for masses approaching the Planck mass, the collapse time is instantaneous, while for atomic masses, it exceeds the age of the universe.
• ER = EPR Connection: The authors extend their analysis to entangled states, suggesting that the entangled state corresponds to an Einstein-Rosen (ER) wormhole geometry. They show that the energy of this wormhole matches the gravitational self-energy of the quantum system, and the instability of the wormhole geometry corresponds to the collapse of the wave function.
• Experimental Viability: The model predicts no spontaneous photon emission or momentum diffusion (unlike stochastic DP models), making it consistent with current null-result experiments. Future tests could involve atom interferometry to detect the specific $t^3$ phase shift.

5. Significance

• Resolution of Singularities: By introducing the zero-point length $l_0$ via T-duality, the theory naturally resolves the short-distance divergences that plague standard gravitational collapse models.
• Deterministic Collapse: The paper offers a compelling alternative to stochastic collapse theories. It suggests that collapse is an inevitable, deterministic consequence of the incompatibility between quantum superposition and the non-local structure of spacetime, rather than a random noise process.
• Bridging QM and GR: The work provides a concrete mechanism linking quantum mechanics and general relativity without requiring a full theory of quantum gravity. It posits that the "classicality" of the macroscopic world emerges naturally from the non-local gravitational self-energy of quantum states.
• Experimental Guidance: By identifying the specific $t^3$ phase shift and the lack of radiation, the paper guides future experimental efforts (e.g., in optomechanics or atom interferometry) to test gravity-induced collapse without the false positives associated with stochastic noise models.

In summary, Jusufi, Singleton, and Lobo propose that spontaneous wave function collapse is a deterministic phenomenon arising from the non-local gravitational self-energy inherent in a T-duality inspired spacetime. This mechanism naturally suppresses macroscopic superpositions while preserving quantum behavior for microscopic particles, all while avoiding the singularities and experimental constraints of previous stochastic models.