Approaches to Quantum Gravity

336 Questions and answers
spectrum in the Riemannian theory and a continuous spectrum (for spacelike
intervals) in the Lorentzian case. In four space-time dimensions, the
gauge group of LQG is truly the complexification of SU(2) and the reality
conditions might actually select a non-compact section of the complex
group from which we would then derive a contiuous spectrum. Finally,
these results are only at the kinematical level. They do not use the physical
Hilbert space and inner-product, so we can not be sure of their physical
relevance. Actually, the area operator is itself only defined in the kinematical
Hilbert space (not invariant under diffeomorphism and not in the kernel
of the Hamiltonian constraint) and we have not been able to lift it to a
physical operator acting on physical state. Nevertheless, in three space-time
dimensions, work by Noui & Perez (2004) suggests that we can construct
a physical length operator by introducing particles in the theory and we
then recover the kinematical results i.e a continuous length spectrum for
the Lorentzina theory. The issue is, however, still open in four space-time
dimensions.
• Q - L. Crane - to D. Oriti:
It seems an awful shame to get to the point where each Feynman diagram in a
GFT model is finite, then to describe the final theory as an infinite sum of such
terms. Have you ever thought of the possibility that by specifying the structure
of the observer including its background geometry we limit the number
of simplicial complexes we need to sum over, or at least make most of the
contributions small, thereby rendering the answer to any genuinely physical
question finite?
– A-D.Oriti:
I agree. I would be careful in distinguishing the “definition of the theory”,
given by its partition function (or its transition amplitudes), and the quantities
that, in the theory itself, corresponds to physical observables and are thus
answers to physical questions. The partition function itself may be defined,
in absence of a better way, through its perturbative expansion in Feynman
diagrams, and thus involve an infinite sum that is most likely beyond reach
of practical computability, and most likely divergent. However, I do believe
that, once we understand the theory better, the answer to physical questions
will require only finite calculations. This can happen in three ways, I think.
As you suggest, the very mathematical formulation of the question, involving
maybe the specification of an observer or of a reference frame, or referring
to a finite spacetime volume only, or some other type of physical restriction,
will allow or even force us to limit the sum over graphs to a finite
number of them, thus making the calculation finite. Another possibility is
that, as in ordinary QFT, the answer to a physical question (e.g. the result