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  1. While the idea of extra spatial dimensions has been around for quite a while now, is there a possibility or any theory which proposes extra time dimensions?

  2. If so, what could be theoretical implications of multidimensional time on existing theories like relativity or QM?

  3. Would they work just fine?

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Such a spacetime is called an ultrahyperbolic spacetime, so called because it produces ultrahyperbolic equations (equations with more than one negative eigenvalue). Those spacetimes are not overly nice to work with.

  • They pretty trivially include closed timelike curves, since a closed curve in any plane of two time directions will be timelike.
  • They permit particles to decay without any restriction regarding masses, including single photon decay. [1]
  • A hypersurface of dimension $n-1$ is required to solve matter fields equations on a spacetime, but obviously such a hypersurface will not be purely spacelike in the case of an ultrahyperbolic spacetime. Hence you cannot predict events given informations on the current time.

There are still a few theories regarding it but overall it is not really that studied, for those reasons.

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    $\begingroup$ By compactifying the 2nd time direction, does this not solve the decay issue (for light enough particles)? I may be missing a sign so glad to be corrected. $\endgroup$ Commented Jan 20, 2016 at 16:00

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