⚛️ quantum physics

Towards relativistic generalization of collapse models

This paper proposes a relativistic generalization of spontaneous collapse models using a local field collapse operator and Lorentz-invariant non-Markovian noise, successfully overcoming previous obstacles such as microcausality violations and infinite energy rates while maintaining consistency with non-relativistic limits.

Original authors: Anirudh Gundhi, Lajos Diósi, Matteo Carlesso

Published 2026-07-10

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CC BY 4.0

Original authors: Anirudh Gundhi, Lajos Diósi, Matteo Carlesso

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

Imagine the universe as a giant, humming orchestra. In the standard rules of quantum mechanics, this orchestra plays a perfect, continuous melody where every instrument (every particle) exists in a superposition of playing all notes at once. But when we look at the world, we only ever hear one clear note. This is the "measurement problem": how does the fuzzy, many-note reality snap into a single, definite note?

For decades, physicists have proposed a "collapse model" to explain this snap. Think of it as a cosmic DJ who occasionally hits a button to force the music to pick a single track. This paper, by Gundhi, Diósi, and Carlesso, tries to upgrade this DJ from a non-relativistic, local version to a fully relativistic one that respects the speed of light and the structure of spacetime.

The Problem with Previous Attempts

Previous attempts to make this "cosmic DJ" work in a relativistic universe (where time and space are woven together) hit some serious roadblocks. Imagine trying to run a relay race where the runners are allowed to run faster than light or create energy out of thin air.

The "Ghost" Problem: Some models allowed the collapse to happen instantly across the universe, violating the rule that nothing travels faster than light (microcausality).
The "Infinite Energy" Problem: Other models suggested the collapse process would pump infinite energy into the universe, heating everything up instantly.
The "Particle Factory" Problem: Some versions implied the vacuum (empty space) would spontaneously burst into a shower of new particles, which we don't see happening.

The authors argue that these previous attempts failed because they tried to make the "collapse noise" too simple or too local.

The New Proposal: A "Fuzzy" and "Memory-Keeping" DJ

The authors propose a new way to run the show. Instead of a sharp, instant "snap," their model uses a non-Markovian noise.

The Analogy: Imagine a standard radio signal that is just static (Markovian). It has no memory; the noise at this second is totally unrelated to the noise a second ago. This leads to the "infinite energy" and "particle creation" problems.
The New DJ: The authors suggest the noise should be like a fuzzy, echoing memory. The static at this moment is connected to the static from a moment ago, but in a way that respects the speed of light (Lorentz invariant). This "memory" (non-Markovianity) smooths out the rough edges that caused the infinite energy problems.

They also introduce a specific rule called normal ordering.

The Analogy: Think of the vacuum as a room full of invisible, jittery dust motes. In previous models, the collapse process counted these motes as real energy, causing the "infinite energy" explosion. The authors say, "Let's agree to ignore the baseline jitter of the empty room." By using normal ordering, they subtract this background noise, ensuring the collapse doesn't accidentally create particles from nothing.

What They Actually Found

The paper suggests and demonstrates mathematically that this new setup works:

It respects the speed of light: Because the "collapse operator" (the thing being collapsed) is local (happening at a specific point in space) and the noise correlations are carefully designed, the model respects microcausality. This means a measurement here cannot instantly mess up a measurement there if they are too far apart for light to travel between them.
It stops the energy explosion: By using this "memory-keeping" noise and the "ignore the background jitter" rule, the rate at which energy is added to the system becomes finite. It doesn't blow up to infinity.
It looks like the old model when things are slow: When they slow everything down to non-relativistic speeds (like our everyday world), the math reduces to the familiar Continuous Spontaneous Localization (CSL) model, which is the standard version used in current experiments.

The "Magic" Numbers and Limits

The authors don't claim to have measured this in a lab yet. Instead, they propose a specific mathematical shape for the noise correlation function, $G(q)$ , to make the math work.

They suggest a function that looks like $G(q) \to \exp(-(q^2)^2/\beta^4)$ .
This choice is suggested to be simple and to ensure the energy rate stays finite.
They calculate that in the non-relativistic limit, the energy rate is proportional to the total number of particles ( $N$ ).
They note that current experimental bounds on energy production are roughly $2 \times 10^{-11}$ Watts per kilogram. Their model's parameters ( $\alpha, \beta, \gamma$ ) would need to fit within these existing limits.

The Big "Maybe"

While the math looks consistent, the authors are very careful about what this means for reality.

The Caveat: They point out that while the average behavior (the density matrix) works perfectly, there is still a deep conceptual puzzle. If you look at a single specific realization of the collapse (one specific "take" of the universe), different observers moving at different speeds might disagree on what the state of the universe is at a specific moment.
The Conclusion: The paper does not claim to have solved the measurement problem or proven that this is how the universe works. It proposes a mathematically consistent framework that avoids the fatal flaws of previous attempts. It offers a "possible" relativistic generalization that is free of infinite energy and particle production, but the question of whether this can be interpreted as a true, objective collapse of reality for every observer remains open.

In short, the authors have built a new, sturdy bridge across a canyon that previous bridges fell into. They haven't crossed it yet to see what's on the other side, but they've shown the bridge won't collapse under its own weight.

Technical Summary: Towards Relativistic Generalization of Collapse Models

Problem Statement
Spontaneous collapse models, such as Continuous Spontaneous Localization (CSL) and the Diosi-Penrose (DP) model, offer a potential solution to the quantum measurement problem by modifying the Schrödinger equation with non-linear and stochastic terms. However, a fully consistent relativistic generalization of these models remains elusive. Previous attempts to extend collapse dynamics to Quantum Field Theory (QFT) have encountered significant theoretical obstacles, including the violation of microcausality, infinite energy production rates, particle creation from the vacuum, and tachyonic dynamics. Furthermore, there is a fundamental conceptual tension between the objective collapse of the wavefunction and the principles of relativity.

Methodology
The authors propose a relativistic generalization of the continuous collapse master equation (ME) rather than a specific non-linear unraveling of the statevector. The construction relies on three primary components:

Local Field Collapse Operator: The collapse operator $\hat{Q}(x)$ is chosen to be a local field operator, specifically $\hat{Q}(x) = \frac{1}{2\alpha}\hat{\phi}^2(x)$ , which is a Lorentz scalar. This choice ensures that in the non-relativistic (NR) limit, the operator becomes proportional to the mass density operator.
Non-Markovian Noise: The model employs a stochastic noise field with a Lorentz-invariant correlation function $G(x, y) = G((x-y)^2)$ . Unlike Markovian noise (which corresponds to a delta function in spacetime), this non-Markovian approach allows for a broader class of correlation functions.
Normal Ordering: To address energy divergences, the authors apply a normal-ordering prescription to all observables of interest. This removes the vacuum contribution (the $1/2$ term in the occupation number) that typically leads to infinite particle production and energy rates in previous models.

The dynamics are analyzed using a unitary unraveling in the Heisenberg picture, which allows for the calculation of operator evolution to any order in the coupling constant $\sqrt{\gamma}$ using standard perturbation theory. The authors focus on the resulting master equation for the density matrix, as experimental access is limited to density matrices rather than specific statevector realizations.

Key Contributions and Results

Preservation of Microcausality: The paper demonstrates that the Microcausality Condition (MCC)—the requirement that commutators of observables vanish for spacelike separations—is respected to all orders in $\sqrt{\gamma}$ . This is achieved because the collapse operator $\hat{Q}(x)$ is local. The analysis shows that for the MCC to hold, the noise correlations do not need to be ultra-local (i.e., $G(x,y) \neq \delta^4(x-y)$ is permissible); rather, the combination of a local operator and non-divergent correlations is sufficient.
Finite Energy Rate: The authors derive an expression for the energy rate $dE/dz^0$ . They show that while Markovian noise leads to a divergent energy rate (proportional to $\delta(0)$ ), the proposed non-Markovian noise combined with normal ordering yields a finite energy rate. The divergence associated with vacuum particle production is eliminated by the normal-ordering prescription.
Lorentz Covariance: The model is constructed to be manifestly Lorentz covariant. By ensuring $\hat{Q}(x)$ is a Lorentz scalar and the noise correlation $G(x,y)$ depends only on the invariant interval, the collapse dynamics transform correctly under reference frame changes.
Non-Relativistic Limit: In the NR limit, the model recovers behavior consistent with standard CSL models, where the energy rate is proportional to the total number of particles $N$ . The specific choice of the noise correlation function in momentum space, $G(q^2) = \exp(-(q^2)^2/\beta^4)$ , ensures convergence in both the NR and fully relativistic regimes.

Significance and Claims
The paper claims to provide a mathematically consistent relativistic generalization of the collapse master equation that overcomes the specific pathologies (infinite energy, vacuum particle production, and microcausality violation) found in prior literature. The authors emphasize that their construction successfully decouples the need for ultra-local noise from the preservation of causality.

However, the authors remain modest regarding the conceptual interpretation of the model. They acknowledge that while the master equation (the ensemble average) is consistent, the interpretation of the statevector for a single realization of the noise remains an open conceptual issue. Specifically, they note that for a single realization, quantum expectation values computed along different spacelike hypersurfaces by relatively moving observers may differ, posing a challenge to assigning an objective meaning to the wavefunction at all times.

The work positions itself as a pragmatic step forward, offering a consistent relativistic non-Markovian dynamics via the master equation. The authors suggest that future work could explore connections with relativistic open quantum system dynamics, though they note that their approach is less restrictive regarding the initial state of the "environment" (substituted here by noise fields) compared to standard open system frameworks. The free parameters of the model ( $\alpha, \beta, \gamma$ ) are noted to be subject to future experimental bounds, similar to the non-relativistic CSL model.