Abstract
In Part I of these notes, we showed how gravity in 2+1 dimensions deforms particle kinematics. It turned out that the original dynamical degrees of freedom of the particle merge with the ‘would be gauge’ degrees of freedom of the gravitational field making the particle’s kinematics effectively deformed. One could say that the deformed particle Lagrangian describes the particle along with the back reaction of its own gravitational field. Another way of thinking about the deformation is that, effectively, the originally flat momentum space becomes, as a result of incorporating this back reaction, a curved manifold, which, in the cases considered above, became a group manifold of the group \(\mathsf {SO}(2,1)\) or \(\mathsf {AN}(2)\).
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Notes
- 1.
The property is easily verified by taking the derivative in the direction of X of \(w^R(e) = 0\).
- 2.
Sometimes, the r-matrix is written directly as the skew-symmetric part \(r_- = \frac{1}{2}\left( P_\mu \otimes X^\mu - X^\mu \otimes P_\mu \right) \equiv P_\mu \wedge X^\mu \), which obviously yields the same Lie bi-algebra structure.
- 3.
It is possible, however, to foliate the group manifold in a set of symplectic leaves with the Poisson structure of the manifold restricted to each leaf [5].
- 4.
It is worth to mention that using the adjoint representation \(\mathscr {R}\) for \(\mathfrak {an}(n)\), we can describe the right decomposition of \(d = g\ t \in T \times AN(n)\). The Poisson brackets are again obtained from (4.78) by just changing the sign of the r-matrix, \(r \rightarrow -r\) and are given by (4.80), (4.81), and
$$\begin{aligned} \{x^\mu , g\} = g\ \tilde{X}^\mu . \end{aligned}$$(4.83) - 5.
The generalization to the case of both momentum space and spacetime being curved is not-trivial. The reader can find a detailed description of this construction in the paper [12].
- 6.
For example, the statement I see the table means really that the worldline of the table crossed the worldlines of photons that hit my retina.
- 7.
- 8.
If we choose another point as \({\mathscr {O}}\), we would get different realizations of the group AN(3). For example, if \({\mathscr {O}}=(\kappa ,\ldots , 0)\), we would get the Euclidean realization and for \({\mathscr {O}}=(\kappa ,0,0,0,\kappa )\), the light-cone one. See [7] for more details of this construction.
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Arzano, M., Kowalski-Glikman, J. (2021). Deformed Classical Particles: Phase Space and Kinematics. In: Deformations of Spacetime Symmetries. Lecture Notes in Physics, vol 986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63097-6_4
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