quantum fields
at i0

Kartik Prabhu
with Gautam Satishchandran
Jim Isenberg PCGM 2021, Virtual!!


Minkowski spacetime outside a light cone

  • \(x \cdot x > 0\): spacelike points relative to some origin.
  • Foliation by hyperboloids: \(\rho^2 = r^2 - t^2\); surfaces of constant \(\rho\) are hyperboloids a.k.a. deSitter spacetime
  • \(x = \rho y\) : \(y \cdot y = 1\) are points on a unit-hyperboloid
  • Similar asymptotic coordinates exist near \(i^0\) in any asymptotically flat spacetime; \(\rho \to \infty\) is the asymptotic Ashtekar-Hansen hyperboloid at \(i^0\)

Massless scalar field: radial modes

Massless scalar in Minkowski: \(\nabla^2 \varphi = 0\). variable separation: \(\varphi(x) = R_q(\rho) \varphi_q(y)\)

  • Radial equation: \(\rho^2 \left( \frac{d^2}{d\rho^2} + \frac{3}{\rho}\frac{d}{d\rho} \right) R_q = - (1+q^2) R_q\)
  • radial mode basis: \(R_q(\rho) \propto 1/\rho^{1+i q}\) where \(q \in \mathbb{R}\) (for completeness)
  • deSitter massive scalar: \(\left[D^2 - (1+q^2)\right] \varphi_q(y) = 0\)

Minkowski field as integral of deSitter fields

  • \(\varphi_q(y) \propto \int_0^\infty d\rho~ \rho \frac{1}{\rho^{1-iq}}~ \varphi(x = \rho y) \) (Mellin transform)
  • \(\varphi(x = \rho y) \propto \int_{-\infty}^\infty dq~ \frac{1}{\rho^{1+iq}}~ \varphi_q(y)\)
\(\varphi_q(y)\) live in the principal series representation of Lorentz

Minkowski QFT as integral of deSitter QFT

Same transform in QFT when \(\hat\varphi(x)\) and \(\hat\varphi_q(y)\) are operator-valued distributions acting on “nice” test functions \(f(x)\) and \(f_q(y)\).
  • \([\hat \varphi[f], \hat\varphi[g]] = \int dq~ \big[\hat \varphi_q[f_q], \hat\varphi_q[\bar g_q] \big] \) (field commutators)
  • \(\langle f | g \rangle = \int dq~ \langle f_q | \bar g_q \rangle \). Minkowski Fock space = integral of Bunch-Davies Fock spaces on deSitter

Note that the radius \(\rho\) disappears on the RHS; Can use the RHS to define an asymptotic QFT at \(i^0\) in any asymptotically flat spacetime!

Scalar charge \(\mathcal{Q}\)

Classical charged solution: \(V(x) \sim \mathcal{Q}(y)/\rho\) as \(\rho \to \infty\)
  • new QFT given by the map: \(\hat\varphi(x) \mapsto \hat\varphi(x) + V(x) \hat 1\)
  • \(V_q(y) \sim \mathcal{Q}(y)/(q+i0^+)\) in the Mellin transformed space

Scalar charges: Inequivalent Hilbert spaces

  • the map \(\hat\varphi(x) \mapsto \hat\varphi(x) + V(x) \hat 1\) can be unitarily-implemented if and only if \(V(x)\) has finite \(1\)-particle norm.
  • but \(V_q(y) \sim \mathcal{Q}(y)/(q+i0^+)\) does not have finite norm
  • So for each classical charged solution \(V(x)\) we get a different Hilbert space representation of the QFT


  • Define a charge operator not just change of charge
  • action of Spi supertranslations (Lorentz is already implemented in Bunch-Davies Hilbert space)
  • Maxwell fields and gravity
  • match to asymptotic quantum states at null infinity constraints on the S-matrix (?)
Look out for an update at APS April meeting!