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Comment on "Relativistic covariance and nonlinear quantum mechanics: Tomonaga-Schwinger analysis''

This paper refutes Hsu's 2026 claim by demonstrating that the Tomonaga-Schwinger equation retains relativistic covariance even with nonlinear modifications, once the original arguments' identified flaws are corrected.

Original authors: Lajos Diósi

Published 2026-02-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Lajos Diósi

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

The Core Conflict: A Misunderstood Rulebook

Imagine the universe as a giant, complex movie being filmed. In standard physics, there is a strict rulebook (called the Tomonaga-Schwinger equation) that tells the camera crew how to film the scene so that the story makes sense no matter which angle or speed the camera moves. This rulebook ensures covariance, which is just a fancy way of saying: "The laws of physics look the same to everyone, regardless of how they are moving or where they are standing."

Recently, a researcher named Hsu published a paper claiming that if you add a specific type of "twist" (a nonlinear modification) to this rulebook, the whole system breaks. He argued that the twist would cause the camera angles to clash, making the movie incoherent and violating the rules of relativity.

Lajos Diósi, the author of this new paper, is here to say: "Not so fast. Hsu made a mistake in his math, and the rulebook actually holds up just fine."

The Analogy: The Chameleon and the Mirror

To understand the disagreement, let's use an analogy involving a Chameleon (the quantum state) and a Mirror (the physical fields).

The Setup: In the standard linear world, the Chameleon changes color based on the background, but the Mirror simply reflects what is there. They don't mess with each other's rules.
The Twist (Nonlinearity): Diósi introduces a scenario where the Chameleon's color depends on what it sees in the Mirror. This is the "nonlinear" part. The Chameleon looks at its own reflection to decide how to change.
Hsu's Argument: Hsu looked at this setup and said, "If the Chameleon changes its color based on its own reflection, the Mirror will start reflecting things that don't exist yet! The timing will get messed up, and the laws of physics will break." He thought the Chameleon and the Mirror were fighting each other.
Diósi's Correction: Diósi points out that Hsu confused who is doing the changing.
- Hsu tried to make the Mirror change its own rules based on the Chameleon's future state. That's impossible and breaks the movie.
- Diósi shows that in the correct version, the Chameleon changes its color, but the Mirror stays perfectly stable and follows the original rules. The Chameleon is just reacting to what is already there.

Because the Mirror (the fundamental fields) doesn't change its rules, the "camera angles" (relativity) remain perfectly synchronized. The movie stays coherent.

The "Entanglement" Misunderstanding

Hsu also brought up a scary idea: that this twist would allow information to travel faster than light (superluminal communication) by instantly entangling distant parts of the universe.

Diósi explains that Hsu is mixing up two different scenarios:

Scenario A (Hsu's view): You start with two strangers who have never met, and the twist instantly makes them best friends (entangled) across the galaxy. Diósi says this doesn't happen in his model. The twist is "local," meaning it only affects what is right in front of it.
Scenario B (The real issue): If two strangers are already best friends (entangled) before the twist starts, the twist might allow them to send secret signals faster than light.

Diósi clarifies: The problem isn't that the twist creates instant connections; the problem is that if connections already exist, the twist might abuse them. But this doesn't mean the fundamental rulebook (covariance) is broken. It just means the universe has a different kind of "causality" issue, one that doesn't destroy the geometry of spacetime.

The Bottom Line

Diósi's paper is a technical "correction note." He argues that:

The Math Works: If you calculate the nonlinear equation correctly (using the "interaction picture," a standard tool in physics), the integrability conditions are met. The "camera angles" don't clash.
The Mistake: Hsu tried to force the physical fields to evolve in a way that depends on the state in a contradictory manner. Diósi shows that if you keep the fields evolving normally and only let the state react, everything stays consistent.
The Conclusion: The nonlinear Tomonaga-Schwinger equation is covariant. It respects the rules of relativity, even though it introduces a nonlinearity that might cause other, different types of causality issues (related to entanglement).

In short: Hsu thought the engine of the car was broken because he was trying to steer with the brakes. Diósi says, "No, the engine is fine; you just need to steer with the wheel." The car (the theory) can still drive straight.

Technical Summary: "Relativistic covariance and nonlinear quantum mechanics: Tomonaga-Schwinger analysis"

Problem Statement
The paper addresses a recent claim by Hsu (2026) published in Physics Letters B, which argues that the nonlinear modification of the Tomonaga-Schwinger (TS) equation violates relativistic covariance. Hsu's argument posits that the state-dependent Hamiltonian term in the nonlinear TS equation breaks microcausality, thereby failing the integrability conditions required for foliation independence and, consequently, covariance. The author, Lajos Diósi, contends that Hsu's conclusion is incorrect due to fundamental misunderstandings regarding the mechanism of causality violation in nonlinear dynamics and errors in the mathematical treatment of field and state evolution.

Methodology
Diósi revisits Hsu's analysis by:

Re-evaluating the Integrability Condition: The paper examines the commutator of functional derivatives with respect to spacelike hypersurfaces, $[\frac{\delta}{\delta\sigma(x)}, \frac{\delta}{\delta\sigma(y)}]|\Psi, \Sigma\rangle = 0$ , which is the condition for foliation independence.
Critiquing Hsu's Transformation Logic: The author analyzes Hsu's assumption that the transformation operator $\hat{U}$ , which evolves the state, fails to preserve the commutators of field operators because it is "not unitary in the usual sense." Diósi demonstrates that while the state evolution is non-unitary in a specific sense, the transformation of the field operators (in the interaction picture) remains unitary, preserving microcausality.
Identifying Inconsistencies in Hsu's Model: The paper points out that Hsu's specific example leads to a degenerate structure where the nonlinear Hamiltonian density depends on a time-independent initial expectation value, rendering the proposed combination of field and state evolutions incorrect.
Applying the Standard Interaction Picture: Diósi reformulates the problem using the standard interaction picture of field theory. In this framework, the quantized local fields evolve with the free Hamiltonian, while the interaction Hamiltonian density depends on the local expectation value of the field operator.
Direct Calculation of the Integrability Term: The author explicitly calculates the variation of the expectation value term $\langle\Psi_\Sigma|\hat{O}(x)|\Psi_\Sigma\rangle$ with respect to the hypersurface $\sigma(y)$ . By applying the TS equation and the microcausality condition (commutation of operators at spacelike separations), the author shows that the extra term arising from nonlinearity vanishes.
Clarifying Causality vs. Covariance: The paper distinguishes between the generation of entanglement and the violation of microcausality, referencing Gisin's work to correct Hsu's interpretation of how nonlinearity leads to acausality.

Key Contributions and Results

Proof of Covariance: The paper provides a direct proof that the nonlinear TS equation with operator-expectation nonlinearity (specifically $\hat{N}_x[\Psi] = \lambda \hat{O}(x)\langle\Psi, \Sigma|\hat{O}(x)|\Psi, \Sigma\rangle$ ) satisfies the integrability conditions. Consequently, the equation remains covariant despite the nonlinearity.
Resolution of Hsu's Loopholes: The author identifies that Hsu's failure to prove covariance stems from:
- A misunderstanding of how the transformation operator acts on field operators versus state vectors.
- An incorrect mixing of state and field dynamics that leads to a degenerate nonlinearity.
- A misinterpretation of the relationship between nonlinearity, entanglement, and superluminal communication.
Clarification of Acausality Mechanisms: The paper clarifies that while nonlinear quantum dynamics can lead to acausality (superluminal signaling) in the presence of initial entanglement and local measurements, this does not imply a violation of microcausality (commutation of spacelike operators) or a loss of relativistic covariance. The nonlinearity in the TS equation is local and does not generate entanglement between initially unentangled spacelike separated systems.
Consistency with Semiclassical Gravity: The author notes that the integrability and covariance of the nonlinear TS equation are essential for the consistency of semiclassical gravity theories, which encompass such equations. If Hsu's claim were correct, these theories would be mathematically impossible.

Significance
The paper asserts that the central claim of Hsu (2026)—that nonlinear modifications to the Tomonaga-Schwinger equation violate relativistic covariance—is incorrect. By correcting the mathematical treatment of the integrability conditions and clarifying the distinct mechanisms of causality violation, the paper reaffirms that the nonlinear TS equation is integrable and covariant. This result supports the viability of local field theories with operator-expectation nonlinearity, particularly within the context of semiclassical gravity, and corrects the theoretical record regarding the compatibility of such nonlinearities with relativistic principles.

The Core Conflict: A Misunderstood Rulebook

The Analogy: The Chameleon and the Mirror

The "Entanglement" Misunderstanding

The Bottom Line

Technical Summary: "Relativistic covariance and nonlinear quantum mechanics: Tomonaga-Schwinger analysis"

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