Approaches to Quantum Gravity

Questions and answers 573frame. Just as a frame of reference may have limited validity due to globalgeometry, here even if spacetime is flat, its noncommutativity accumulatesthe uncertainty the further out one goes from the origin of that frame. Is it aproblem? Only if some other observer with some other origin does not reachthe same conclusion. The other observer would have transformed coordinatesdefined via eq. (24.17) which describes a quantum Poincaré transformation,in particular a shift is allowed. The new variables x μ ′ defined by the RHSof eq. (24.17) obey the relations eq. (24.1) but are shifted by a μ from theoriginal. The only thing, which I explain in Section 24.5.1 is that the transformationparameters such as a μ are themselves operators (its a quantum groupnot a classical group) so the new variables are not simply related to the oldones by a numerical matrix. In short, there is clearly no classical Poincaréinvariance of eq. (24.1) but there is a quantum one. If one takes expectationvalues one then has real numbers but the expectation values do not thentransform under a usual Poincaré transformation as the questioner perhapsassumes. Just because the uncertainty relations are not usual-Poincaré invariantdoes not mean an origin is being singled out in the universe. Rather toactually relate a new observer’s expectations to the old one, one has to knowthe expectation value of the a μ and face also that they need not commute withthe x μ . In short, a quantum frame transformation is itself “fuzzy” which is notsurprising since the different observers’ own locations should be fuzzy. To besure one has approximated Poincaré invariant to O(λ) but the equations suchas eq. (24.1) are themselves at that level (both sides are zero if λ = 0andwehave usual commuting x μ ). My goal in Section 24.5.1 is indeed to get physiciststhinking properly about quantum frame rotations as a theory of QuantumGravity has to address their expectation values too. However, I don’t see anyinconsistency.2. The x μ are operators whose expectation values, we suppose, are the physicallyobserved macroscopic spacetime coordinates at which a particle mightbe approximately located. A theory of Quantum Gravity has to provide thestates on which these expectations are computed so the noncommutativealgebra is not the whole of the observed physics. It’s a joint effort betweenthe (proposed) noncommutative geometry and the effective quantum state inwhich the operators are observed.3. There is no contradiction. The “first predictions” I refer to are order of magnitudecomputations for a time-or-arrival experiment that can be done withoutsolving all problems of interpretation of momentum and their addition. Additionof momenta would be more relevant in the many particle theory. For asingle photon modelled as a noncommutative plane wave, one does not needto have solved the many particle theory. One does still need some sort of