Doubly Quantum Mechanics

TL;DR

This paper introduces 'Doubly Quantum Mechanics', a framework where geometrical configurations and reference frames are quantized, leading to new non-classical probability features and intrinsic uncertainties in measurements.

Contribution

It develops a formalism for a covariant quantum theory with quantized geometries using quantum groups, extending standard quantum mechanics.

Findings

Probability becomes a self-adjoint operator on geometry states.

Intrinsic uncertainties affect measurement outcomes in semi-classical geometries.

Unavoidable fuzziness persists in reference frame alignment, even with infinite exchanged qubits.

Abstract

Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin-$\frac{1}{2}$ measurements by promoting the group of spatial rotations $SU(2)$ to the quantum group $SU_q(2)$ and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of $SU_q(2)$. This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard $SU(2)$ case.