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arXiv · 2605.23862

Indefinite probabilities in quantum spacetime: A deepening of unpredictability

Vittorio D'Esposito·Giuseppe Fabiano·Domenico Frattulillo
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Abstract

Non-commutative spacetime and quantum groups have been argued to capture non-classical features of spacetime and its symmetries in quantum gravity. In this letter, we show that employing the $SU_q(2)$ quantum group to describe rotational symmetry for spin-$\frac{1}{2}$ systems and Stern-Gerlach apparatuses leads to the description of probabilities of outcomes of spin measurements in terms of non-commuting operators. As a result, we obtain an uncertainty principle between different probability operators, realising a notion of indefinite probabilities. This is then reflected in the non-commutativity of the entries of the rotation matrix relating the reference frames of two observers, hence fundamentally preventing them from sharply measuring their relative orientation.

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Vittorio D'Esposito, Giuseppe Fabiano, Domenico Frattulillo. 2026-05-22. Indefinite probabilities in quantum spacetime: A deepening of unpredictability. https://arxiv.org/abs/2605.23862

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