The Quantum Structure of Space and Time (293 pages)

(^212) The Quantum Structure of Space and Time
why do you think that is not true? Well, obviously the reason, the most obvious
reason is that gravity has problems that you get into, problems of ultraviolet

divergences. That is really the wrong way to look at the problem, because if

you allow all possible terms in the Lagrangian, with arbitrary powers of the

curvature, you can cancel the divergences the same way you do in quantum

electrodynamics. But then you say: oh, but the problem is that you have an

infinite number of free constants, and the theory loses all predictive power when

you go to sufficiently high energy. Well, that is not necessarily true, although

it might be true. It might be that there is a fixed point in the theory, that

is that there is a point in the infinite-dimensional space of all these coupling

constants where the beta function vanishes. And furthermore, when that hap-

pens, you actually expect that the surface of trajectories which are attracted

into that fixed point as you go to high energy is finite-dimensional. So the

theory would in fact have precisely as much predictive power as an ordinary

renormalizable field theory, although much more difficult to calculate since the

fixed point would not be near the origin. Even so, you would then say: oh, but

even so, this theory has a lot of free parameters, maybe a finite number, but

where do they come from? What we were hoping for was that string theory

would tell us how to calculate everything. And even there, that might not be

true, it might be that the surface of trajectories that are attracted to the fixed

point is one-dimensional. And we know that there are examples of extremely

complicated field theories in which in fact there is a non-Gaussian fixed point

with a one-dimensional attractive surface -just a line of trajectories that are

ultravioletly attracted to the fixed point and therefore avoid problems when
you go to large energy. That is shown by the existence of second order phase
transitions. Basically, the condition is that the matrix of partial derivatives of
all the various beta functions with respect to all the various coupling parame-
ters should have a finite number of negative eigenvalues, and in the case of a
second order phase transitions, in fact, there is just one negative eigenvalue:
that is the condition. So it is possible that that is the answer. I suspect it is
not. I suspect that these really revolutionary ideas are going to turn out to
be necessary. But I think we should not altogether forget the possibility that
there is no revolution that’s needed, and that good old quantum field theory,
although with a non-Gaussian fixed point, is the answer.
I. Klebanov I just had a comment on Brian’s question. Of course in AdS/CFT,
not all of spacetime is emergent: 3 + 1, say, dimensions are put in from the be-
ginning, but the other six emerge. And for example in the story I discussed, you
can see the emergent Calabi-Yau with the complex structure epsilon emerging
from the infrared effects of the field theory. And then I have a brief unrelated
comment about locality. I think there are some speculations that you can for-
mulate string theory just on a lattice, and string theory completely erases this

lattice, as long as the lattice spacing is small enough. Namely, it’s not just an