The first week of this series centered around the idea of horizontal fuzziness. This is really about taking quantum mechanics seriously. Quantum fluctuations do not only apply to measurements inside a detector (vertical fuzziness), but also to the measurement of the detector's location (horizontal fuzziness). Ignoring horizontal fuzziness amounts to a hidden assumption about an infinitely massive detector, which does not work in the presence of gravity.
The second week I described the multi-dimensional Virasoro algebra, i.e. the Virasoro-like extensions of the diffeomorphism algebra in \(d\) dimensions. The classical representations act on tensor fields, but that is not a good start for quantization, because normal ordering gives rise to infinite extensions due to unrestricted sums over transverse directions. Instead we must start from \(p\)-jets, i.e. Taylor series truncated at order \(p\). Since a Taylor series depends not only on the field being expanded, but also on the choice of expansion point, we naturally have a horizontal observable that displays fuzziness.
The multi-dimensional Virasoro algebra is not really physics, since it only involves kinematics but not dynamics. It is more like a quantum version of tensor calculus rather than a quantum version of gravity. Nevertheless, already at this level there are some consequences which are very much in disagreement with conventional wisdom. This is what we will discuss during this third and final week.
Friday, December 5, 2025
Third Week
