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.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.

quant-pharXiv:2605.23862May 25, 20260 citations

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Rajesh R. Parwani

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Springer eBooksApr 16, 20070 citations

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.