Reply to 'Comment on "Ideal clocks -- a convenient fiction'' '

This paper rederives a previously questioned de-excitation probability formula for a quantum scalar field in an accelerated cavity using a perturbation theory calculation formulated entirely within the relevant Rindler wedge, thereby addressing concerns about the use of causally disconnected modes in the original work.

The Gist

Imagine you are trying to listen to a specific radio station (the "cavity") that is speeding through space at a constant, intense acceleration. This radio is trying to tune into a signal from the vast, empty background of the universe (the "ambient field").

In 2015, a team of physicists (Lorek, Louko, and Dragan) published a paper calculating exactly how likely this speeding radio is to "drop a beat" or lose energy (de-excite) while listening to the background noise. They used a specific mathematical recipe to get the answer.

Recently, a critic (Vladimir Toussaint) wrote a letter saying, "Wait a minute! Your recipe is cheating."

The Critic's Complaint:
The critic argued that the original recipe was "unfair" because, in the middle of the calculation, it peeked into a part of the universe that is causally disconnected from the radio.

• The Analogy: Imagine you are in a sealed room on a speeding train. To calculate how the train moves, the original recipe briefly looked at a parallel universe on the other side of a wall that you can never see or touch. The critic said, "You can't use information from a place you can't reach to calculate what happens in your room! That breaks the rules of cause and effect."

The Authors' Reply (The "Simple" Explanation):
The authors of the original paper have now written a reply to say, "We hear you, but we aren't breaking any rules. Here is why, and here is a new way to prove we were right."

1. The "Sealed Room" Proof

Instead of peeking over the wall, the authors went back and re-did the entire calculation using only the tools available inside the sealed room (the "Rindler wedge").

• The Metaphor: Think of the accelerated cavity as a self-contained bubble. The authors showed that this bubble is a complete, logical universe all by itself. You don't need to look outside the bubble to understand what happens inside it. They re-derived their original formula using only the physics inside the bubble, and guess what? They got the exact same answer.
• The Takeaway: The result is solid. It doesn't matter if you peek outside or stay inside; the math for the radio's energy loss is the same.

2. Why the "Peeking" Wasn't Cheating

The authors also explained why the original method (which did look at the "other side") wasn't actually a violation of physics.

• The Analogy: Imagine you are playing a game of billiards in a small room. To calculate the path of the balls, you might use a giant, invisible grid that covers the entire city, including the park across the street.
  • The critic says: "You're using the park to calculate the billiards! That's wrong!"
  • The authors reply: "No, the grid is just a mathematical map. The actual collision only happens inside the room. The fact that our map extends to the park doesn't mean the balls are hitting the park. The 'interaction' (the collision) is strictly limited to the room."
• The Physics: In the original paper, they used "Minkowski modes" (waves that stretch across the whole universe) as a temporary stepping stone. Even though these waves technically exist in the "other side" of the universe, the interaction (the part where the radio actually loses energy) only happens inside the cavity. Because the interaction is strictly local, the "ghostly" presence of the waves in the disconnected region doesn't break the rules of cause and effect.

The Bottom Line

• The Result: The formula for how much energy the accelerated clock (or radio) loses is correct.
• The Method: You can prove it by staying strictly inside the "bubble" (which they did in this new paper), or you can use the "whole universe" map (which they did in the old paper). Both roads lead to the same destination.
• The Lesson: Just because a mathematical tool uses a concept that seems to reach into a "forbidden" zone, it doesn't mean the physical result violates the laws of physics, as long as the actual physical event stays within its own boundaries.

In short: The math works, the physics is safe, and the "cheating" accusation was just a misunderstanding of how mathematical maps work.

Technical Summary

1. Problem Statement

The paper addresses a critique raised by V. Toussaint (Comment [2]) regarding a previous calculation by the authors (Reference [1]) concerning the de-excitation probability of a quantum scalar field confined in a uniformly linearly accelerated cavity.

• The Original Calculation ([1]): Derived a formula for the probability that a cavity field (initially in an excited state) de-excites due to interaction with an ambient scalar field (initially in the Minkowski vacuum state restricted to the Rindler wedge).
• The Critique ([2]): Toussaint argued that the derivation in [1] was flawed because the intermediate steps invoked Rindler modes from two distinct regions:
  1. The right Rindler wedge (where the accelerated cavity resides).
  2. The left Rindler wedge (causally disconnected from the cavity).
     The critique suggested that using modes from the causally disconnected wedge violated causality or physical consistency.

2. Methodology

The authors respond by re-deriving the original de-excitation probability formula using a strictly causal, single-wedge framework.

• Physical Setup:
  • Spacetime: (1+1)-dimensional Minkowski spacetime.
  • Coordinates: Right Rindler wedge ($x > |t|$) using Rindler coordinates $(\tau, \xi)$, where $\tau$ is the proper time for the accelerated observer.
  • Fields:
    • Cavity Field ($\phi$): A massless real scalar field confined to a cavity of proper length $l$ within the wedge, subject to Dirichlet boundary conditions.
    • Ambient Field ($\Phi$): A massive real scalar field ($M > 0$) filling the entire Rindler wedge.
  • Initial States:
    • Cavity field: First excited state of the lowest mode ($n=1$).
    • Ambient field: The state induced by the Minkowski vacuum restricted to the Rindler wedge (a thermal mixed state with temperature $T = \alpha/2\pi$).
• Interaction:
  • A linear coupling $H_{int}(\tau) = \lambda \int d\xi \, \phi(\tau, \xi)\Phi(\tau, \xi)$ is introduced, active only within the spatial bounds of the cavity.
• Calculation Framework:
  • Perturbation Theory: The evolution is treated to first order in the coupling constant $\lambda$ (resulting in a probability of order $\lambda^2$).
  • Causal Evolution: The authors explicitly formulate the time evolution entirely within the right Rindler wedge. They utilize the fact that the Rindler wedge is a globally hyperbolic spacetime on its own, allowing for a Cauchy foliation using constant $\tau$ hypersurfaces.
  • Density Matrix Formalism: The system evolves from an initial density matrix $\rho_i$ to a final state $\rho_f$ using the interaction picture. The de-excitation probability is calculated by tracing over the ambient field and projecting the cavity field onto its ground state.

3. Key Contributions

1. Independent Re-derivation: The authors successfully re-derive the de-excitation probability formula from [1] without ever referencing the left (causally disconnected) Rindler wedge. This proves that the original result is robust and does not depend on the specific choice of mode expansion in the unphysical region.
2. Clarification of Causality: The paper provides a rigorous explanation of why the original method (using modes from both wedges) did not violate causality.
   • The authors draw an analogy to an inertial cavity in Minkowski space: intermediate calculations often use plane waves that extend over all spacetime (including spacelike separated regions), yet causality is preserved because the interaction Hamiltonian has compact support (it is zero outside the cavity).
   • Similarly, for the accelerated cavity, while the mathematical representation of Minkowski vacuum states in terms of Rindler modes involves both wedges, the physical evolution is strictly confined to the cavity's worldline. The "support of the interaction" enforces causality, rendering the inclusion of the second wedge's modes in intermediate steps a mathematical artifact rather than a physical violation.

4. Results

The re-derived de-excitation probability $P_{\downarrow}$ matches the original result in [1] exactly.

The final formula (Equation 11) is:
$$P_{\downarrow} = \lambda^2 \int_0^\infty d\Omega \left( \cosh^2 r_\Omega |\gamma_{\Omega 1}|^2 + \sinh^2 r_\Omega |\tilde{\gamma}_{\Omega 1}|^2 \right)$$

Where:

• $\Omega$ is the frequency of the ambient field modes.
• $r_\Omega$ is the squeezing parameter defined by $\tanh r_\Omega = e^{-\pi\Omega/\alpha}$ (related to the Unruh temperature).
• $\gamma_{\Omega n}$ and $\tilde{\gamma}_{\Omega n}$ are overlap integrals between the cavity modes and the ambient field modes over the interaction time and spatial volume.
• The term $\cosh^2 r_\Omega$ corresponds to spontaneous emission (stimulated by the thermal bath), while $\sinh^2 r_\Omega$ relates to absorption processes.

Verification: The authors confirm that this result is identical to Equation (19) in their 2015 paper, validating the original calculation despite the critique.

5. Significance

• Validation of "Ideal Clocks" Model: The paper reinforces the validity of the "Ideal clocks" model proposed in [1], which is relevant for understanding timekeeping in non-inertial frames and the interplay between acceleration, quantum fields, and thermodynamics (Unruh effect).
• Resolution of Conceptual Dispute: It settles a technical dispute regarding the necessity of the "left wedge" in calculations involving accelerated observers. It demonstrates that while the full Minkowski vacuum is entangled across the Rindler horizons, physical processes confined to one wedge can be calculated entirely within that wedge's causal structure.
• Methodological Rigor: By providing a derivation that stays strictly within the globally hyperbolic right wedge, the authors remove any ambiguity regarding the physical interpretation of the intermediate mathematical steps, strengthening the theoretical foundation of quantum field theory in accelerated frames.