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How Closed Space Breaks the Quantum Rules of Rotation ✛ ✎

Original: "Orbital angular momentum can take non-integer values in a closed universe"
· Daniel Burgarth, Paolo Facchi
arXiv:2506.03254v1 · 2025-06-03 · arXiv non-exclusive · independent analysis · 1 min
Quantum Physics General Relativity Math Physics
If space loops back on itself, orbital angular momentum ceases to be quantized and can take any value.

Orbital angular momentum, the measure of an object's rotation around a point, usually appears discrete in quantum physics: like climbing a staircase where each step is a whole number. This rule was considered unshakable. But if space has periodic boundaries, looping back on itself like the surface of a donut, the staircase turns into a smooth ramp. On it, rotation can take any value — from fractions to irrational numbers like √2.

Such a continuous spectrum challenges the long-held belief that orbital momentum must be an integer. Half-integer values, once thought to be the exclusive privilege of spin, now turn out to be possible here as well.

Interestingly, the effect doesn't depend on size: even on the scale of the entire Universe, rotation can lose its quantum strictness. It's not yet detectable in the lab, but traces could show up in the cosmic microwave background — the ancient light after the Big Bang. Finding them would confirm the nontrivial structure of our spacetime.

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For a long time, physicists believed that orbital angular momentum could only be integer, and half-integer values were characteristic only of spin — a particle's intrinsic rotation. The new discovery shows that the boundaries of space can mix these properties.

L_z = m\hbar, \quad m \in \mathbb{Z}
In infinite space, the orbital angular momentum along the z-axis is quantized: it takes only integer values m times Planck's constant ħ.
|L_z| \geq \hbar
In a closed universe, a continuous band of angular momentum values appears outside the interval from -ħ to ħ. In this band, the momentum can be anything, including half-integers and irrationals.
Scientists
Johannes KeplerIsaac NewtonEmmy NoetherWerner HeisenbergKip ThorneHendrik Lorentz
Concepts
angular momentum orbital angular momentum quantum orbital motion spin axis commutator continuum probability organic molecules in space vacuum nuclear spin orbit Hilbert space rotation period wavenumber vacuum energy uncertainty principle ring quantum number spin