⚛️ quantum physics

On substantive Lorentz invariance and quantum theory

This paper investigates the challenge of achieving "substantive Lorentz invariance" in relativistic quantum theory by applying two candidate criteria to various interpretations, revealing that while many models satisfy these formal conditions, they may still lack the deeper intuitive invariance, thereby clarifying which aspects of relativity must be reconsidered in the quantum domain.

Original authors: Ward Struyve

Published 2026-08-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ward Struyve

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

In the universe, the laws of physics are expected to look the same to every observer, regardless of how they are moving. This idea, known as Lorentz invariance, is the bedrock of Einstein's theory of special relativity. It ensures that the speed of light is a constant limit and that time and space stretch and shrink in predictable ways depending on speed. For decades, physicists have successfully combined this relativistic framework with quantum mechanics, the theory that governs the behavior of atoms and subatomic particles. However, a deep tension remains. While the equations describing how particles move are perfectly consistent with relativity, the standard rules for how a quantum system behaves when it is measured are not. When a measurement happens, the standard theory says the system's state changes instantly everywhere, a process that seems to require a specific moment in time that contradicts the flexible nature of relativity.

To resolve this, some physicists have proposed alternative versions of quantum theory that remove the need for this instant change, or "collapse," of the state. These theories, such as Bohmian mechanics and spontaneous collapse models, describe particles as having definite paths or locations at all times. The problem is that making these specific theories work in a relativistic universe is incredibly difficult. Because these theories must allow particles to influence each other instantly across vast distances to match experimental results, they seem to require a hidden, preferred frame of reference—a specific way of slicing time that the universe secretly follows. If such a frame exists, the theory is not truly relativistic, even if its predictions happen to look relativistic to us. The question then becomes: can we build a theory that is truly relativistic in its very structure, or are we forced to accept that relativity is only an approximation of a deeper, non-relativistic reality?

A recent paper by physicist Ward Struyve tackles this head-on by examining several proposed models that attempt to reconcile these conflicting ideas. The author does not simply ask if the math works; he asks what it means for a theory to be "substantively" relativistic. There is a difference between a theory that is mathematically capable of being written in a relativistic form and one that is fundamentally relativistic in its nature. To test this, Struyve applies two different criteria to a variety of models. The first criterion looks for "absolute objects"—fixed elements of the theory that never change and do not react to anything else, acting like a rigid stage upon which the drama of physics plays out. If a theory relies on such a fixed stage, it is not truly relativistic. The second criterion is a principle of isolation: if you take a small, isolated system and move it or change its speed, the laws governing that system should still work perfectly without needing to know about the rest of the universe.

Struyve evaluates six different versions of Bohmian mechanics, a model of spontaneous collapse called rGRWf, and a version of the Many-Worlds theory. The results are nuanced. Most of the models, including the spontaneous collapse theory and the Many-Worlds approach, pass both tests. They appear to be substantively relativistic, meaning they do not rely on a hidden, fixed background and they respect the principle that isolated systems can be transformed without breaking the laws of physics. However, two specific versions of Bohmian mechanics fail these tests. One version relies on a fixed, unchanging grid of time and space that acts as a preferred frame, making it substantively non-relativistic. Another version, while avoiding a fixed grid, still fails the isolation test because the behavior of a small system depends on the state of the entire universe in a way that cannot be transformed independently.

The study suggests that while it is possible to construct quantum theories that are truly relativistic, it is not guaranteed. Some of the most promising alternatives to standard quantum mechanics still struggle to shed the need for a preferred frame of reference. The author notes that even when a model passes these specific tests, it may still feel intuitively non-relativistic, hinting that our current definitions might not fully capture the essence of what it means for a theory to be relativistic. The findings do not prove that one theory is the correct one, but they clarify the landscape. They show that the path to a fully relativistic quantum theory is not blocked by the need for non-locality, but it does require careful construction to ensure that no hidden, absolute structures are left behind. Ultimately, the paper suggests that achieving a truly relativistic quantum theory may require us to give up certain intuitive features of how we think space and time work, moving beyond the classical picture to something more complex.

Problem Statement
Lorentz invariance is a cornerstone of relativistic physics, yet its realization in quantum theory is problematic. While relativistic wave equations (e.g., the Dirac equation) are Lorentz invariant, the standard quantum mechanical collapse postulate is not, as instantaneous collapse requires a preferred time slicing, conflicting with the relativity of simultaneity. Alternative interpretations of quantum mechanics, such as Bohmian mechanics and spontaneous collapse theories (e.g., GRW), resolve the measurement problem by dispensing with the collapse postulate in favor of a "primitive ontology" (particles, flashes, or mass density) defined on space-time. However, extending these theories to the relativistic domain is challenging due to the non-locality implied by Bell's theorem. While non-locality does not strictly preclude Lorentz invariance, reconciling the two is difficult.

A central issue identified in the paper is the distinction between formal and substantive Lorentz invariance. Trivial reformulations of non-relativistic theories (or theories with preferred frames) can be made formally Lorentz invariant by expanding the set of dynamically allowed histories to include all Lorentz-transformed versions. However, such formulations often retain a preferred frame at the dynamical level, failing to capture the physical essence of relativity. The paper addresses the difficulty of articulating a precise definition of "substantive Lorentz invariance" that imposes genuine constraints on theory construction, a problem Bell noted but did not resolve.

Methodology
The paper investigates two candidate criteria for substantive Lorentz invariance, applying them to various relativistic quantum models:

Anderson's Criterion (Absolute Objects): Based on the work of Anderson (1967, 1971), this criterion distinguishes between dynamical objects (which vary across solutions) and absolute objects (which remain the same across all dynamically possible histories, modulo symmetry transformations). A theory is substantively invariant if its symmetry group (the transformations leaving the set of absolute objects invariant) includes the Lorentz group. If a theory relies on a fixed structure (like a preferred foliation) that acts as an absolute object, it fails this criterion.
Relativity Principle for Isolated Subsystems: Inspired by Brown and Sypel (1995) and others, this criterion posits that a theory satisfies a relativity principle if the equations of motion allow for (approximately) isolated subsystems to be transformed (e.g., via a Lorentz boost) while leaving the environment unchanged, resulting in a new, valid solution of the dynamical equations with consistent probabilities. This tests whether the theory's laws are local enough to permit independent subsystem dynamics.

The author applies these criteria to:

Six distinct formulations of Lorentz-invariant Bohmian mechanics for a fixed number of non-interacting Dirac particles.
The relativistic flash ontology model (rGRWf).
The relativistic version of the Many-Worlds "Sm" theory (based on mass/energy density).

The analysis is restricted to non-interacting particles to maintain tractability, though entanglement (and thus non-locality) is permitted.

Key Contributions and Results

Analysis of Bohmian Models: The paper evaluates six models (labeled i–vi) differing in how the space-like hypersurfaces $\Sigma_{k,x}$ (which determine the non-local guidance of particles) are defined:
- Models (i)–(iii): These rely on a foliation of space-time.
  - Model (i) uses a foliation of hyperplanes defined by a constant vector field $n^\mu$ .
  - Models (ii) and (iii) allow for more general foliations (curved surfaces) determined by specific differential laws.
- Models (iv)–(vi): These do not introduce an independent foliation; the surfaces are determined by the wave function or particle worldlines.
  - Model (iv) determines the foliation covariantly from the total wave function (e.g., via the total energy-momentum vector).
  - Models (v) and (vi) determine surfaces based on individual particle momenta or tangent vectors, potentially resulting in non-foliated structures.
Evaluation Results:
- Anderson's Criterion: Model (i) fails because the constant vector field $n^\mu$ is an absolute object. Models (ii) and (iii) pass because their laws allow for a broader class of foliations, making the specific foliation dynamical rather than absolute. Models (iv)–(vi) pass as they lack extra space-time structure.
- Relativity Principle: Model (i) fails because a boost of an isolated subsystem cannot be accommodated without altering the global constant vector field $n^\mu$ , which is not allowed locally. Model (iv) also fails because the foliation is determined by the total wave function; thus, transforming a subsystem changes the global foliation, preventing the subsystem from being truly isolated. Models (ii), (iii), (v), and (vi) satisfy the principle.
Spontaneous Collapse (rGRWf): The rGRWf model satisfies both criteria. It possesses no absolute objects beyond the Minkowski metric, and its probability distribution for flashes factorizes for separable wave functions, allowing for the transformation of isolated subsystems.
Many-Worlds (Sm Theory): The relativistic Sm theory, which posits an energy-momentum tensor as the primitive ontology, satisfies both criteria. The ontology transforms as a tensor, and the theory allows for isolated subsystems.
Limitations of the Criteria: The paper notes a counterintuitive result: Model (i) (with a hyperplane foliation) is judged as not substantively invariant by Anderson's criterion, while Models (ii) and (iii) (with more complex foliations) are judged as invariant. This suggests that Anderson's criterion might be sensitive to the specific mathematical formulation of the laws rather than the physical intuition of "action without reaction." Furthermore, Model (iv) fails the subsystem criterion despite being covariant, suggesting the criteria may not fully capture the nuances of substantive invariance in all cases.

Significance and Claims
The paper claims that while the proposed criteria (Anderson's and the subsystem principle) are useful tools, they may not fully capture the concept of substantive Lorentz invariance. Specifically, the fact that some Bohmian models satisfying these criteria still appear intuitively non-substantive (due to their dependence on a global wave function or specific foliation structures) suggests that a more stringent definition may be necessary.

The author concludes that these criteria help clarify which aspects of relativity theory might need to be modified or abandoned when transitioning from classical to quantum theory. They suggest that Lorentz invariance might hold only at the statistical level (empirically) rather than the fundamental dynamical level, particularly in theories with preferred foliations. The paper does not claim to definitively solve the problem of substantive invariance but rather to test existing proposals against concrete quantum models, highlighting the tension between non-locality and the requirements of a relativistic space-time ontology. The work emphasizes that achieving a "fundamentally relativistic" quantum theory requires navigating a trade-off between non-locality, the existence of a primitive ontology, and the nature of space-time symmetries.

More like this