A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
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 H of the harmonic oscillator is paradoxical under the group action induced by SO(2, 1); (3) This paradoxical nature of the Hilbert space 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 H, certain class of quantum gravities naturally emerge from Quantum Mechanics that mediates a self-decoherence of the system.