# space at the Planck scale

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All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 11 dimensional manifold in order to model particles. Do they do this because there is no mathematical alternatives or because the nature (mathematical essence) of space at the Planck scale is still not yet discovered?

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user Q Q
retagged Feb 16, 2015

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There are approaches to quantum gravity where spacetime is described as a quantum superposition of labelled piecewise-linear CW complexes or other related combinatorial/algebraic entities. See for example:

However, your question feels more like a physics question than a math question to me.

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user John Baez
answered Feb 11, 2015 by (365 points)
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No, wait. In perturbative string theory (as well as, in fact, in spectral triples) the input datum is not in general a geometry modeled on manifolds, but is the purely algebraic datum of a 2d SCFT . From this one recovers an "emergent" effective target spacetime at low energies by computing the string perturbation series given by summing up correlators of this abstractly, algebraically defined 2d SCFT and then asking for an ordinary manifold with an ordinary field theory on it which produces the same S-matrix elements at low energy. This may or may not exist! There are plenty of completely non-geometric 2d SCFTs, the famous examples being the Gepner models which model completely non-geometric "phases" of spacetime such as famously the flop transition.

It just so happens that for phenomenological reasons, since one is trying to match to the observed physics well below the Planck scale, there is so much focus on those 2d SCFTs which are constructed geometric as sigma-models from differential geometric input data. But this is human prejudice, not string theory's preference. The moduli space of all 2d SCFTs (the true landscape) has no reason to be dominated by geometric sigma-models. That these are at the focus of attention is because it is easier for us humans to deal with them. One fine day the time will come that mathematical tools have advanced to the point that a genuine analysis of the space of all 2d SCFTs is possible and then we'll be speaking much more about non-geometric backgrounds.

Indeed, for the comparatively simple case of rational 2d CFTs a full mathematical description of the moduli space ("landscape") of all these is already known, by the FRS theorem. And indeed, this classification turns out to be completely algebraic, no differential geometry anywhere. The theorem says that rational 2d CFTs are given essentially by certain Frobenius algebra objects internal to the modular representation categories of rational vertex operator algebras. This does include geometric models such as WZW models, but the general point in this space of rational CFTs has no reason at all to have any relation to smooth manifolds.

answered Feb 19, 2015 by (5,865 points)
edited Feb 19, 2015

Excellent answer!  Maybe this is a different question in its own right, but we know there are tons of beautiful results in math by taking the geometric target space interpretation.  To name a few, all the highly non-trivial symmetries in various partition functions coming from string dualities.  This seems related to the appearance of modular objects appearing all over the place in enumerative geometry like Gromov-Witten, Donaldson-Thomas, and Gopakumar-Vafa theory.  Is it conjectured that a lot of this nice structure arising from the geometric point of view is actually something more fitting in the general 2D SCFT picture?

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This paper by Carlip http://arxiv.org/abs/gr-qc/0108040 is a good, relatively nontechnical explanation of why it's hard to reconcile quantum mechanics (QM) with general relativity (GR).

GR says that spacetime is a real manifold with a semi-Riemannian metric. QM says that the possible states of a system form a complex vector space.

If you naively try to combine these two ideas, it's hard to make sense of the result. Given one manifold-with-metric $M_1$ and another one $M_2$, what would it even mean to talk about the linear combination $c_1M_1+c_2M_2$, where $c_1$ and $c_2$ are complex numbers? The spacetimes $M_1$ and $M_2$ do not have any built-in way of matching up points in one with points in the other. The two spacetimes don't even need to have the same topology. In quantum mechanics, we would also have the Born rule, which says that $|c_1|^2$ and $|c_2|^2$ have interpretations as the probabilities of outcomes of measurements. It's not clear what these probabilities would mean in this context.

So should spacetime be described at the Planck scale as a real manifold, or if not, then what? Straightforward application of the fundamental principles of the two theories seems to lead to nonsense answers. We really don't know.

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user Ben Crowell
answered Feb 12, 2015 by (1,070 points)
That naive attempt at combining the two ideas is way too naive!

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user Mariano Suárez-Alvarez
I essentially agree with the general message of this answer, but I disagree with the objection given to the naive proposal. The linear combination $c_1M_1+c_2M_2$ can trivially be taken in the vector space generated by the $M_i$'s and it is usually this kind of thing one has to do in quantum mechanics. For example, in gauge theory, we have classically bundles-with-connections and if $E_1$ and $E_2$ are two such objects then $c_1M_1+c_2M_2$ is a well-defined element in the Hilbert space of the theory. More generally, the "linearity" of quantum mechanics is something which has nothing to do ..

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user user25309
with the linearity of the classical objects. I agree that the naive proposal does not work but one has to give better reasons.

This post imported from StackExchange MathOverflow at 2015-02-16 11:45 (UTC), posted by SE-user user25309

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