Doubly Quantum Mechanics

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-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:math> measurements by promoting the group of spatial rotations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> to the quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> case.