Abstract
In the first part of this book, we argued that if one takes into account self-gravity effects, a particle becomes dressed with topological degrees of freedom of its own gravitational field so that its kinematics becomes deformed. This results in the energy and momentum of the particle being described by elements of a non-abelian Lie group. In this chapter, we turn to mathematics describing this deformation. The first step will be the introduction of some key notions in the theory of Hopf algebras and their relevance in physical applications.
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Notes
- 1.
For massless particles denoting their states with \(|\mathbf {p}, h\rangle \) one has
$$\begin{aligned} W^{\mu } W_{\mu } |\mathbf {p}, h\rangle = W^{\mu } P_{\mu } |\mathbf {p}, h\rangle = P^{\mu } P_{\mu } |\mathbf {p}, h\rangle = 0\,. \end{aligned}$$(5.52)Since \(W^{\mu }\) is null on physical states and orthogonal to \(P^{\mu }\) it must be proportional to the latter, i.e.
$$\begin{aligned} W^{\mu } |\mathbf {p}, h\rangle = h\, P^{\mu } |\mathbf {p}, h\rangle \,, \end{aligned}$$(5.53)and the proportionality factor h is just the helicity of the particle with the helicity operator given by
$$\begin{aligned} \hat{h} = - \frac{\mathbf {P}\cdot \mathbf {J}}{|\mathbf {P}|}\,. \end{aligned}$$(5.54)
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Arzano, M., Kowalski-Glikman, J. (2021). Hopf Algebra Relativistic Symmetries: The \(\kappa \)-Poincaré Algebra. In: Deformations of Spacetime Symmetries. Lecture Notes in Physics, vol 986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63097-6_5
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