Approaches to Quantum Gravity

Questions and answers 573
frame. Just as a frame of reference may have limited validity due to global
geometry, here even if spacetime is flat, its noncommutativity accumulates
the uncertainty the further out one goes from the origin of that frame. Is it a
problem? Only if some other observer with some other origin does not reach
the same conclusion. The other observer would have transformed coordinates
defined via eq. (24.17) which describes a quantum Poincaré transformation,
in particular a shift is allowed. The new variables x μ ′ defined by the RHS
of eq. (24.17) obey the relations eq. (24.1) but are shifted by a μ from the
original. The only thing, which I explain in Section 24.5.1 is that the transformation
parameters such as a μ are themselves operators (its a quantum group
not a classical group) so the new variables are not simply related to the old
ones by a numerical matrix. In short, there is clearly no classical Poincaré
invariance of eq. (24.1) but there is a quantum one. If one takes expectation
values one then has real numbers but the expectation values do not then
transform under a usual Poincaré transformation as the questioner perhaps
assumes. Just because the uncertainty relations are not usual-Poincaré invariant
does not mean an origin is being singled out in the universe. Rather to
actually relate a new observer’s expectations to the old one, one has to know
the expectation value of the a μ and face also that they need not commute with
the x μ . In short, a quantum frame transformation is itself “fuzzy” which is not
surprising since the different observers’ own locations should be fuzzy. To be
sure one has approximated Poincaré invariant to O(λ) but the equations such
as eq. (24.1) are themselves at that level (both sides are zero if λ = 0andwe
have usual commuting x μ ). My goal in Section 24.5.1 is indeed to get physicists
thinking properly about quantum frame rotations as a theory of Quantum
Gravity has to address their expectation values too. However, I don’t see any
inconsistency.
2. The x μ are operators whose expectation values, we suppose, are the physically
observed macroscopic spacetime coordinates at which a particle might
be approximately located. A theory of Quantum Gravity has to provide the
states on which these expectations are computed so the noncommutative
algebra is not the whole of the observed physics. It’s a joint effort between
the (proposed) noncommutative geometry and the effective quantum state in
which the operators are observed.
3. There is no contradiction. The “first predictions” I refer to are order of magnitude
computations for a time-or-arrival experiment that can be done without
solving all problems of interpretation of momentum and their addition. Addition
of momenta would be more relevant in the many particle theory. For a
single photon modelled as a noncommutative plane wave, one does not need
to have solved the many particle theory. One does still need some sort of