We know that space-time dimensions are 3+1 macroscopically, but what if 2+2? Obviously it is tough to imagine two time dimensions, but mathematically we can always imagine as either having two parameters $t_1$ and $t_2$ or else in Lorentz matrix $$\eta_{00} = \eta_{11} = -1$$ and, $$\eta_{22} = \eta_{33} = 1.$$
Is there any physical reason for not taking this, like the norms become negative or something else?
Answer
As Cumrun Vafa explains in the video linked to below the picture of him in this article, F-theory works in a total of $10+2$ dimensions. The signature of the last two infinitesimal dimensions is ambiguous, so that they can indeed both be timelike. Since these are only infinitesimal dimensions, any causality issues etc are not a problem in this case.
And as Cumrun Vafa nicely explains in his talk, F-theory gives quite a nice phenomenology with an astonishingly realistic CKM-Matrix, coupling constants, etc; so it is NOT true that theories that operate in more than one time dimension are completely off base, as some people claim. There is no reason to dogmatically dismiss every theory that has more than one time dimension.
BTW, the talk is very accessible and enjoyable.
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