Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

Abstract

The integrals of the motion associated with conformal Killing vectors of a curved space-time with an additional electromagnetic background are studied for massive particles. They involve a new term which might be non-local. The difficulty disappears for pp-waves, for which explicit, local conserved charges are found. Alternatively, the mass can be taken into account by "distorting" the conformal Killing vectors. The relation of these non-point symmetries to the charges is analysed both in the Lagrangian and Hamiltonian approaches, as well as in the framework of Eisenhart-Duval lift.

Hidden Symmetries in Deformed Very Special Relativity

“…They are converted into the higher order symmetries, X uv , X ui , which reduce on mass shell to expressions given by the original Killing vectors distorted appropriately by b, the ξ uv and the ξ ui respectively. Similar distortions of broken symmetries have emerged in a different setting involving proper conformal Killing vectors in the case of the motion of a massive particle in Riemannian pp-wave space-times [31]: For example, it is well known that in the case of null geodesics the proper Conformal Killing vectors (CKVs) generate integrals of motion. However this property is lost when one considers a massive particle (in a sense the presence of the mass m breaks these symmetries).…”

“…However this property is lost when one considers a massive particle (in a sense the presence of the mass m breaks these symmetries). In [31] it was shown that the proper CKVs still contribute by producing conservation laws under a similar mass dependent distortion. Here, in a Finsler geometry, we see it happening at the level of Killing vectors whose symmetry property is broken by the introduction of the nonzero parameter b.…”

Global propagation of massive quantum fields in the plane gravitational waves and electromagnetic backgrounds

“…Finally, it is worth to notice that these two families of fields are distinguished due to the conformal symmetry. Namely, these gravitational metrics exhibit the 7-dimensional conformal symmetry maximal among all vacuum solutions to the Einstein equations (see [52,53]); the electromagnetic fields are invariant under the action of a special conformal generator [54,55]. As we noted above the classical motion in gravitational (electromagnetic) fields (4.1) and (4.2) is explicitly solvable.…”

Generation of new symmetries from explicit symmetry breaking

“…Studies for symmetries of geodesic motion in generic manifolds can be found in [15,16]. There are also particular geometries which are of special physical interest, like plane gravitational waves, see [17,18,19].…”

“…Interestingly enough the additional space-time vector N b of (19), that appears as an extra symmetry in the b = 0 case, is given by the linear combination…”