Monday, October 31, 2005

Doubly special relativity in 3D

Nima who just returned from the Perimeter Institute was excited about very reasonable discussions with Laurent Freidel who is a loop quantum gravity person.

Laurent has shown that the so-called DSR (double or deformed special relativity) may arise naturally in 3 dimensions.

The Harvard interpretation is that 3 dimensions are special; there are no gravitons; and moreover, there is an invariant mass scale - the maximal mass you can have to avoid a closure of space (deficit angles exceeding 2 pi) - and these things won't hold in 4 dimensions or above four.

Nevertheless, the basic story of Laurent is quite interesting. Take 3D gravity coupled to a scalar field PHI with a cubic coupling, and integrate out the gravitational field. What you obtain is an action for PHI only; it differs from the original PHI-part of the action by having a new kind of a "star product" instead of the original one. However, it is not a Moyal product but rather a new kind of product relevant for addition of momenta in DSR.

The rule is
  • exp(iPx) * exp (iQx) = exp(iRx)
  • R = P sqrt(1-L^2.P^2) + Q sqrt(1-L^2.Q^2) + L P x Q
where "x" is the cross product in three dimensions, involving an epsilon, and L is the Planck length, more or less. As you can see, the last term in "R" makes it very noncommutative but differently than in noncommutative geometry as we know it.




I am puzzled how the "epsilon" terms can suddently arise when one integrates out gravity in 3D. Using the Chern-Simons interpretation of 3D gravity (which is apparently used in their derivations), it is easier to envision how this spontaneous "violation of parity" may arise. But has the result anything to do with gravity? It is not the first time in which the CS theory behaves differently under parity than the "true" gravitational description.

Also, it is not clear to me how the expressions above are defined if the arguments of the square roots become negative. A loop quantum gravity person may want to explain us these issues.

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