The Quantum Structure of Space and Time (293 pages)

120 The Quantum Structure of Space and Time

N. Seiberg Do you have in mind time being an operator which does not commute

with something?

R. Dijkgraaf I am completely ignorant about that. I think you know more than

I.

H. Ooguri Talking about a new direction to go, I think stringy Lorentzian geom-

etry is completely uncharted territory that we need to explore. We have gained

lots of insights into quantum geometry but those are all mostly static geometries

and the important question about how Lorentzian geometry can be quantized

needs to be understood.

A. Ashtekar Partly going back to what Horowitz and Seiberg were saying: in

loop quantum gravity we do have a quantum geometry. The coordinates are

commuting. There is no problem with that, the manifold is as it is. It is the

Riemannian structures which are not commuting. So for example, areas of

surfaces which intersect with each other are not commuting and therefore you

cannot measure the areas arbitrarily accurately, for example. So there is this

other possibility also. Namely that observable quantities such as areas, etc,

are not commuting, but there is commutativity for the manifold itself. The

manifold itself does not go away.

R. Dijkgraaf I think that is important. One thing I did not mention, but also

a good open question, is just to go to three dimensional gravity because there

are many ways in which all these approaches connect. Of course, from the loop

quantum gravity point of view, three dimensional gravity, written as a Chern-

Simon theory, is very interesting. In fact many of these topological theories,

when we reduce them down to three dimensions, you get also some Chern-

Simon theories. But again, there are many open issues: “Are these Chern-

Simons theories really well defined? Do they really correspond to semi-classical

quantum gravity theories?” If you want to think about more precise areas, I

feel that that point should be developed.

E. Rabinovici I would like to make two comments. One is that when we use

strings as probes, it seems that all mathematical concepts we are used to some-

how become ambiguous. When you describe T-duality, geometry becomes am-

biguous or symmetric. You have two totally different representations of the

same geometry. The same goes for topology and the number of dimensions. It

also applies to the questions: “Is some manifold singular or not?” and “Is a

manifold commutative or not commutative?” So I was wondering: “Is there

anything which remains non-ambiguous when we study it with strings?” That

was the first comment.

The second comment, which also relates to a discussion we had yesterday, re-

lates to what you said about algebra and geometry. When we use the relation

between affine Lie algebras and their semi-classical geometrical description, we

can sometimes treat systems which have curvature singularities and large R2

corrections. Even if we do not know what the Einstein equations are nor what