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Can the electroweak and strong forces be written as geometric theories? - Why and why not?

Can quantum mechanics in general?

For example, the Kaluza-Klein theory explains the electromagnetic field as "twists" that include an extra dimension of space. (As written by Lubos Motl in a previous question).

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Well, I get the impression from reading your previous question that you would not call (the standard formulation of) the standard model geometric. What is your definition of a geometrical theory? That it can be viewed as a higher-dimensional Kaluza-Klein theory? – Qmechanic Jun 8 '11 at 21:15
@Qmechanic I suppose I mean any kind of geometric theory - something like the Kaluza-Klein theory in that the electromagnetic field is treated in a geometric way, but not necessarily that, and for more than just the electromagnetic field. – Calvin Jun 9 '11 at 15:03
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From Qmechanic's comment, maybe something like the noncommutative geometry approach would appeal to you. See e.g., Noncommutative standard model on Wikipedia and Alain Conne's homepage.

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that looks interesting! Thanks, I'll take a closer look. – Calvin Jun 9 '11 at 15:13

By "geometric" I'm going to assume that you mean "having to do with the geometry of the usual 3+1 dimensions", that is, geometric in the sense that electricity and magnetism is geometric.

This is a question that was researched deeply in the 1950s especially by Coleman and Mandula after which the "Coleman-Mandula" theorem is named. As wikipedia puts it, "It states that "space-time and internal symmetries cannot be combined in any but a trivial way". Thus the internal symmetries are not related to "geometry" in the sense of our usual world.

The Coleman-Mandula theorem depends on a lot of complicated mathematics. One can imagine a bunch of ways that one could get around it. If you google Coleman-Mandula on, you can find papers on this subject, i.e. extensions and consequences of the Coleman-Mandula theorem, as well as papers proposing how one might get around it:

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It would be nice to list the possible ways around it. E.g. Supersymmetry, noncommutative geometry, quantum groups, fields with no mass gap, ... – Simon Jun 9 '11 at 6:22

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