Nature abhors macroscopic superpositions

1. Problem Statement

The paper addresses two fundamental questions at the intersection of quantum mechanics and general relativity:

1. Spacetime Entanglement: Why do we never observe macroscopic objects in superpositions of different mass distributions, which would theoretically cause the object to become entangled with the geometry of spacetime (as different mass locations curve spacetime differently)?
2. The Measurement Problem: How does a quantum measurement device, which theoretically evolves into a superposition of pointer states (e.g., "0" and "1"), collapse into a single definite outcome?

Existing proposals often suggest instability mechanisms (e.g., Penrose-Diósi models) where superpositions decay over time. Roux proposes a different mechanism: a variational principle that prevents macroscopic superpositions from forming in the first place, rather than destroying them after formation.

2. Methodology

The author employs a functional phase space formalism to model quantum states and their evolution without modifying standard quantum dynamics.

• Model System: The superposition is modeled as a Schrödinger cat state consisting of a "fiducial state" (a single-mode thermal state representing a massive scalar field) that undergoes spatial translation.
• Translation Mechanism: Instead of using displacement operators (which affect particle number), the author uses phase modulations in the Fourier domain to implement spatial translations. A translation vector $x_0$ is applied via a phase factor $e^{ix_0 \cdot k}$ to the spectral function $\zeta(k)$.
• Variational Quantity: The total energy of the system is treated as the variational quantity. The author calculates the expectation value of the energy, $\langle E \rangle$, for the cat state as a function of the separation distance ($z$) between the superposed terms.
• Mathematical Framework:
  • The calculation utilizes Wigner functionals and characteristic functionals ($\chi[\eta]$).
  • The energy is derived using functional derivatives of the characteristic functional (Eq. 5).
  • The cat state characteristic functional is constructed using the star product of the fiducial thermal state and the translation operator.
  • The mass distribution is modeled using a normalized Gaussian function to simplify the convolution integrals.

3. Key Results

The core finding is the existence of an energy dip in the total energy of the Schrödinger cat state as a function of the separation distance.

• Energy Landscape: The total energy $\langle E \rangle_{cat}$ exhibits a minimum (a "dip") very close to the origin (zero separation).
• Dependence on Particle Number ($\langle n \rangle$):
  • As the number of particles in the state increases (macroscopic limit), the energy dip becomes narrower and moves closer to the origin.
  • The depth of the dip relative to the baseline remains significant, but the slope of the energy curve near the dip increases drastically with $\langle n \rangle$.
• Opposing Force: The energy curve acts as a potential function. The steep slope near the origin generates a massive opposing force that resists any increase in the separation distance between the superposed terms.
• Macroscopic Implication: For macroscopic objects (large $\langle n \rangle$), the energy minimum is so close to zero separation that the "superposition" is indistinguishable from a single localized state. The force required to separate the terms into a discernible macroscopic superposition is prohibitively large, effectively preventing the formation of such states.
• Generality: The phenomenon was tested with different states (coherent states) and different interaction kernels (four-point interactions). In all cases, the energy (or similar variational quantities) curves exhibited dips near the origin that sharpened with increasing particle numbers.

4. Significance and Implications

• Resolution of Spacetime Entanglement: The results suggest that nature "abhors" macroscopic superpositions not because they decay, but because the energy cost to create a spatially separated superposition of massive objects is too high. This explains why we do not observe entanglement between mass distributions and spacetime geometry.
• Solution to the Measurement Problem: The paper proposes a mechanism for wavefunction collapse. During a measurement, the system evolves via a continuous parameter (e.g., time or interaction strength). The energy landscape creates a "dip" that forces the system to evolve deterministically toward one outcome (e.g., "0" or "1") while the other outcome is "dragged along" at the bottom of the dip. The system does not settle into a stable superposition of macroscopic outcomes; instead, the variational force selects a single trajectory.
• No New Physics Required: Unlike other collapse theories that introduce new constants or modify the Schrödinger equation, this mechanism arises naturally from the total energy of the existing theory. It requires no new fundamental principles, only a specific interpretation of the energy landscape in the context of superpositions.
• Experimental Outlook: The author suggests that while macroscopic dips are too narrow to observe, experiments with mesoscopic systems (e.g., optical Schrödinger cat states with high photon numbers) might detect these energy dips, providing empirical verification of the variational principle.

Conclusion

Roux demonstrates that the total energy of a quantum state containing a superposition of mass distributions creates a potential barrier that prevents macroscopic separation. This "reluctance" of nature to form macroscopic superpositions offers a unified, energy-based explanation for the absence of spacetime entanglement in massive objects and the emergence of single outcomes in quantum measurements, all without altering the fundamental dynamics of quantum mechanics.