Banach-Tarski Paradoxes in Quantum Mechanics

We show that Quantum Mechanical Hilbert space can be paradoxical under some group action and explore its physical consequences. (1) Is there a more natural way of resolving the paradox of Wigner's friend without invoking the Heisenberg's cut?; (2) We notice the qualitative similarities between the paradox and paradoxical sets and use it as a motivation to rigorously prove that the Hilbert space $\mathcal{H}$ of the harmonic oscillator is paradoxical under the group action $\mathcal{U} \leq U(\mathcal{H})\times U(1)$; (3) This paradoxical nature of the Hilbert space $\mathcal{H}$ provides the natural resolution for the paradox by using the Axiom of Choice instead of the Heisenberg's cut; (4) Finally, we show that due to the very same paradoxical nature of $\mathcal{H}$, certain class of quantum gravities naturally emerge from Quantum Mechanics that mediates a self-decoherence of the system.