Classical dynamics from spinors

Ben Maybeeb.maybee@ed.ac.uk

Higgs Centre for Theoretical Physics,

The University of Edinburgh QCD meets Gravity VI, 30/11/20

With Alfredo Guevara, Alex Ochirov, Donal O’Connell and Justin Vines

Outline

1. The worldline NJ shift

2. From worldsheets to spinors

▪ Electrodynamics

▪ Gravity

3. Electric-magnetic duality

▪ Kerr-Taub-NUT

4. Discussion

Worldline NJ shift

Take sheet coordinates 𝓏 𝜏, 𝜆 = 𝑟 𝜏 + 𝑖𝜆𝑎 𝜏 .

Worldsheet action → chiral Newman-Janis shift to worldline

Kerr

worldsheet:

Spinors in classical physics

NJ shifts natural for spinors:

Weyl:

Maxwell:

Exploit chiral splitting: spinor-helicity.

Newman & Janis, 1965

From worldsheets to spinors

Common structure:

Common basis:

Dynamical spinorsLittle group tensor

Arkani-Hamed,

Huang &

Huang, 2017

Particle dynamics encoded by spinors!

Spinor equations

𝑝 and 𝑎 form distinct little group irreps:

SymmetricSinglet state

Set 𝑎 = 0; must recover Lorentz force!

Example

Kerr scattering

Static Kerr background:

LO probe trajectory 𝑟1 𝜏 = 𝑏 + 𝑢1𝜏 :

Kerr probe:

Worldline NJ shift

Background NJ shift

Arkani-Hamed, Huang &

O’Connell, 2019Spin-spin scattering as easy as spin-static!

Impulse:

Δ𝑎𝜇 free.

Gravity

Linearised Kerr worldsheet:

→ spinor equations in gravity are

Worldsheet provides covariant meaning for trajectory’s NJ shift.

Connection spinor:

Porto, Rothstein; Levi & Steinhoff

EM duality for fields

Turn on magnetic charge for background field:

Huang, Kol, & O’Connell, 2019

+ Emond & Moynihan, 2020

→ simple maps between spinors:

Talbot, 1969 Kim & Shim, 2020

Einstein-Maxwell:

Maxwell:

EM duality for particles

∴ include phase rotation on spinors for particle:

Spinor equations for Kerr-Taub-NUT

Phase rotation for on-shell amplitudes:

→ EM duality for interactions!

Kerr-Taub-NUT scattering

Scatter off static Kerr-Taub-NUT background:

Background

NJ

from amplitudes: Emond, Huang, Kol,

Moynihan & O’Connell, 2020

Angular impulse easy. Difficult calculations from KTN metric!

Worldline dualityBackground duality

Worldline NJ

𝑤 = rapidity

KTN probe:

Impulse:

Discussion

• NJ shifts, EM duality natural for spinors → exploit to provide compact

description of dynamics of spinning + charged black holes.

• What are the spinor equations at higher orders?

• Angular impulse just as easy as impulse.

• Worldsheet actions = chiral splitting of spin couplings.

Spinor equations = chiral splitting of dynamics.

• Do the worldsheet symmetries constrain tidal 𝒪 𝐹2 or 𝒪 𝑅2 terms?

• Do the worldsheet action and spinor equations constrain the classical

Compton amplitude?

Kim & Shim, 2020

see Jung-Wook’s talk

see Michele’s talk

Next steps:

Results known for Taub-NUT at NLO

Also easier than amplitudes:

BM, O’Connell & Vines, 2019