Twistors from Killing Spinors alias Radiation from Pair Annihilation I: Theoretical Considerations

Abstract

This paper is intended to be a further step through our Killing spinor programme started with Class. Quantum Grav. 32, 175007 (2015), and we will advance our programme in accordance with the road map recently given in arXiv:1611.04424v2. In the latter reference many open problems were declared, one of which contained the uncovered relations between specific spinors in spacetime represented by an arrow diagram built upon them. This work deals with one of the arrows with almost all of its details and ends up with an important physical interpretation of this setup in terms of the quantum electrodynamical pair annihilation process. This method will shed light on the classification of pseudo-Riemannian manifolds admitting twistors in connection with the classification problem related to Killing spinors. Many physical interpretations are given during the text some of which include dynamics of brane immersions, quantum field theoretical considerations and black hole evaporation.

Spin Geometry and Some Applications

“…We will call S 1 = ΣM as the spinor bundle and S 2 = ker(c) as the twistor bundle. Now, we will see that we can define two first-order differential operators on ΣM by considering two projections of ∇ on S 1 and S 2 and their composition with the Clifford action c [5,38];…”

Field equations from Killing spinors

“…Applying Hodge-de Rham operator d = d − d † to this sum gives d( V ψ + B ψ ) = d V ψ + d B ψ , the right hand side of which, because of the first equation of ( 20) and the second equation of ( 21), reduces to d V ψ + d † B ψ . Then using the remaining equations in (20) and ( 21) we find…”

“…then we define the Killing reversal ψ ς of ψ by [7,20]. To every Killing spinor pair ψ, ψ ς there corresponds a twistor pair Ψ + , Ψ − such that…”

Gauged twistor spinors and symmetry operators

“…For the coordinates t, x, r, ρ and a relevant coframe basis e a with a = 0, ..., 3, the spinor κ = e it (cosh ρ + i sinh ρ) 1 − e it (sinh ρ + i cosh ρ) e 2 is a geometric Killing spinor. One can construct more general twistor spinors from geometric Killing spinors by using the Killing reversal method [29,30]. So, the following spinor φ = κ + z.κ is a twistor spinor in AdS 4 where z is the volume form.…”