Algebraic approach to Quantum Gravity II: noncommutative spacetime
Shahn Majid2009, Toward a New Understanding of Space, Time and Matter
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Abstract
We provide a self-contained introduction to the quantum group approach to noncommutative geometry as the next-to-classical effective geometry that might be expected from any successful quantum gravity theory. We focus particularly on a thorough account of the bicrossproduct model noncommutative spacetimes of the form [t, xi] = ıλxi and the correct formulation of predictions for it including a variable speed of light. We also study global issues in the Poincaré group in the model with the 2D case as illustration. We show that any off-shell momentum can be boosted to infinite negative energy by a finite Lorentz transformaton.
Figures (3)
Key takeaways
- We provide a full introduction to our theory of 'bicrossproduct quantum groups', which is one of the two main classes of quantum group to come out of physics (the other class, the q-deformation quantum groups, came out integrable systems rather than quantum gravity).
- In summary, the bicrossproduct theory constructs both the deformed Poincaré enveloping algebra and coordinate algebra at the same time and provides their canonical action and coaction respectively on another copy of U (m) as noncommutative spacetime.
- Now, consider θ infinitesimal, i.e. we differentiate all expressions (20)-(21) by ∂ ∂θ | 0 which is all we need for the algebraic part of the bicrossproduct Hopf algebra (the full operator algebra structure needs the full global data).
- The correct way to address the first issue according to current understanding is to treat the noncommutative algebra as an operator algebra, construct representations or 'states' of this 'prequantum system' and consider that what would be observed macroscopically are expectation values x µ , ψ p (x) etc. in this state.
- This operator is a 2-cocycle in any Hopf algebra containing ∂ ∂X 0 , X i ∂ ∂X i which means it also fits into a 'twist functor approach to quantisation' [18,22] leading to a different NCG on the same algebra than the one from the bicrossproduct picture.
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