Approaches to Quantum Gravity

152 Questions and answers
Hausdorff dimensions would break Lorentz invariance and translation invariance,
so that much of the beauty and simplicity of dimensional regularization
would get lost. I sometimes tried to speculate that rigorous, finite definitions
of functional integrals can be given in complex dimensions, since all
terms in the perturbation expansion are finite, but did not succeed. I do
know what negative dimensions mean: a negative-dimension coordinate is an
anticommuting coordinate, or equivalently, a negative dimension coordinate
replaces an integration by a differentiation; differentiation is the inverse of
integration.
2. Absolutely. In particular this is an important point for black holes. An ingoing
observer has different pieces of information at his/her disposal than an
observer who stays outside. In view of my conjecture that the quantum states
associated to the primordial basis are informational equivalence classes, this
means that the transformation from a black hole horizon to flat space is not a
transformation in Hilbert space.
3. That is an interesting view, but “measurement” does not play a prominent
role in my proposals. “Measurement” requires a measuring device, which to
my mind is an unnecessary complication when it comes to model building.
• Q - L. Crane - to R. Sorkin:
1. How do you think of the points of a causet? Are they just part of an
approximation to some more subtle spactime structure which has a discrete
aspect, small regions treated as pointlike, or an actual hypothesis about physical
spacetime?
2. If the points in causets are quantum events, isn’t there a superposition
principle? In other words, shouldn’t we be modelling spacetime as a quantum
superposition of an ensemble of causets, rather than just one? Could this
allow symmetry to be restored in the average, and wouldn’t this be an attractive
alternative to the loss of locality?
– A-R.Sorkin:
1. The answer is your third alternative: “an actual hypothesis about physical
spacetime”. The elements of the causet are meant to be constituents
of spacetime that really exist (or better “happen”). Of course, the causet
language would be more fundamental than the geometric, spacetime language,
and as such, it would retain its validity in extreme conditions where
a spacetime description would no longer make sense, inside a black hole for
example.
2. Quantum mechanically, it should indeed be true that spacetime is something
like an ensemble of causets. (Or, as one might express it, reality must
be a quantal causet, not a classical one.) I don’t know whether this could produce
symmetry in the average, but one of the central messages of my article