Symmetries in Entangled States under Bohm’s Quantum Potential (FTL Assumption)

Assuming faster-than-light (FTL) communication is possible via Bohm’s Quantum Potential, you’re entering a
non-local hidden variable interpretation of quantum mechanics. In that framework—Bohmian Mechanics—the
quantum potential acts instantaneously across spatial distances, coordinating entangled particles.

In such a universe, if FTL influence via the quantum potential exists, Lorentz invariance is broken or modified,
and a preferred foliation of spacetime (absolute simultaneity) must be assumed. Given that, we can explore what
symmetries might still be applicable to entangled states:

1. Permutation Symmetry of Entangled States

• Entangled states of identical particles are symmetric (bosons) or antisymmetric (fermions) under exchange.
• This symmetry is preserved even across large spatial separations.

Example:

|Ψ⟩ = (1/√2)(|↑⟩ₐ |↓⟩ᵦ − |↓⟩ₐ |↑⟩ᵦ)

2. Gauge Symmetry

• The global phase invariance of quantum states (a U(1) symmetry) still holds.
• The Bohmian wavefunction evolves under Schrödinger dynamics and remains gauge invariant.

3. Non-Relativistic Galilean Symmetry

• Bohmian mechanics typically violates Lorentz symmetry, but preserves Galilean invariance in the non-relativistic limit.
• The quantum potential is invariant under Galilean transformations.

4. Time-Reversal Symmetry

• Bohmian trajectories are deterministic and time-reversible.
• The Schrödinger equation is time-reversal symmetric (modulo complex conjugation), and so is Bohmian evolution.

5. Configuration Space Symmetry

• In Bohmian mechanics, the wave function lives in configuration space, not 3D physical space.
• Entangled states are non-factorizable and inherently nonlocal in this space.

Caveat: Breaking Lorentz Symmetry

• Allowing FTL interaction via the quantum potential violates Lorentz symmetry, unless a preferred reference frame is introduced.
• Spatial rotation symmetry (SO(3)) may be preserved, but boost symmetry is broken.

Summary Table

Symmetry	Preserved?	Notes
Permutation (Exchange)	✅	Important for identical entangled particles
Gauge (U(1))	✅	Global phase invariance
Galilean	✅	Non-relativistic limit; Bohmian-compatible
Lorentz	❌	Violated by non-local influences
Time-Reversal	✅	Bohmian mechanics is deterministic and reversible
Configuration Space Symmetry	✅	Fundamental in Bohmian interpretation

In Conclusion

If FTL communication via Bohm’s quantum potential exists, the entangled states still obey permutation, gauge, time-reversal, and
configuration-space symmetries, but Lorentz invariance must be relinquished. In its place, a hidden preferred frame
or foliation of spacetime is assumed. The overall coherence of entangled states is preserved through the nonlocal structure of the
wavefunction in configuration space, guided by a universal quantum potential.