Fuzzy Probabilities in a Blurry Spacetime express
The quantum world is known for uncertainties: for example, you can't measure a particle's position and its speed at the same time. But new research shows that in a 'fuzzy' spacetime, even probability itself — the measure of chance — loses sharpness. A foggy compass doesn't give an exact direction: its needle wavers. Two people with such compasses can never perfectly align their devices. Similarly, in a world churned by quantum foam, observers can't precisely match the orientations of their detectors. Because of the shakiness of space itself, the probabilities of different spin measurement outcomes become interdependent — pinning one down blurs another, like one compass needle jiggling the other. These findings echo [scientist:John Archibald Wheeler]'s ideas about 'quantum foam' spacetime. And the mathematical tool for this description — quantum groups — was created in the 1980s for an abstract problem about deforming multiplication, only later becoming useful for physicists. It turns out that probability fuzziness might be key to uniting quantum mechanics with gravity: the more we pin down one probability, the more disorder (entropy) floods into the others.
Original abstract
as written by the authors arXiv:2605.23862
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.
The tool that captured probability’s fuzziness — quantum groups — was created in the 1980s for a completely different purpose: mathematicians were exploring how to smoothly deform ordinary multiplication operations. Only later did it turn out that this abstraction fits perfectly for describing quantum foam.