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Banach-Tarski Paradoxes in Quantum Mechanics

Abhishek Mehta

Published Feb 11, 2026• 0 citations
Publisher

Abstract

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 U ≤ U(H) × U(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.

DOI
10.20944/preprints202602.0669.v1

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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 U ≤ U(H) × U(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.
Schrodinger's cat and Wigner's friend paradoxes are analyzed using the `wave-particle non-dualistic interpretation of quantum mechanics at a single-quantum level' and are shown to be non-paradoxes within the quantum formalism. Then, the extended version of Wigner's friend thought experiment, proposed in a recent article titled, ``Quantum theory cannot consistently describe the use of itself '', Nature Communications {\bf 9}, 3711 (2018), by Frauchiger and Renner (FR) is considered. In quantum mechanics, it's well-known that, statistically observing a large number of identical quantum systems at some particular quantum state, which results in Born's probability, and merely inferring its presence in the same quantum state with the same probability yield distinct physical phenomena. If this fact is not taken care while interpreting any experimental outcomes, then FR type paradoxes pop up. ``What an astonishingly self-consistent the Quantum Theory is!' - is explicitly worked out in the case of FR gedankenexperiment. The present work shows the importance of single-quantum phenomenon for the non-paradoxical interpretation of statistically observed experimental outcomes.
By assuming (i) a universal, observer-independent quantum state, by considering (ii) measurement processes with wave function collapse as part of quantum mechanical time evolution (also within isolated systems), and by (iii) clearly distinguishing the state of a quantum system from the knowledge of conscious observers about this state, we suggest a modified Copenhagen interpretation of quantum mechanics that resolves Wigner's Friend and extended Wigner's Friend paradoxes. The suggested interpretation leads to differences in some outcome probabilities compared to previous analyses which, in principle, make it possible to falsify the suggested modifications or the previous analyses (or both). The fundamental problem of the incompleteness of quantum mechanics in the Copenhagen interpretation regarding the lack of a precise definition and a mechanistic description of the measurement process (including the collapse of the wave function) is not addressed in this study. However, the argumentation regarding the resolution of Wigner's Friend paradoxes also applies to attempts for such a mechanistic description using collapse models with stochastic extensions to the Schrödinger equation or trying to model wave function collapse with unitary time evolution and irreversibility of the measurement process resulting from quantum statistical mechanics.
This paper offers a comprehensive analysis of Wigner’s Friend paradox (1961), a foundational thought experiment in quantum mechanics that highlights the tension between the internal perspective of a measuring observer (who experiences a definite outcome) and the external perspective of a super-observer who treats the entire laboratory as a quantum system in superposition. It traces the historical lineage from Schrödinger’s cat and von Neumann’s measurement theory, provides a formal mathematical description of the state evolution and logical structure of the paradox, examines implications for major interpretations of quantum mechanics (Copenhagen, Everettian many-worlds, QBism, relational quantum mechanics, and objective collapse theories), and discusses modern extensions such as the Frauchiger–Renner paradox. The work explores profound philosophical and epistemological consequences for the nature of observation, objectivity, intersubjectivity, and the role of consciousness in quantum theory (2026).
The purpose of this paper is to resolve two paradoxes, which occur in quantum theory, by using the discussion of the theory of measurement presented in two earlier papers by the author [3], [4], [5]. The two paradoxes discussed will be the Schrödinger cat paradox and the Einstein, Podolski, Rosen paradox [2]. An introductory section will be included which summarizes the relevant results from the author's previous papers. Also a discussion will be made regarding the author's interpretation of the density operator.
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.