# Asymptotic quantum fieldsati0

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

# Why?

• Classical GR, asymptotic fields at $$\mathscr{I}^\pm$$ and $$i^0$$: can match these fields at $$i^0$$
• charges of BMS symmetries at $$i^0$$ are conserved in classical scattering
• Can quantize fields at $$\mathscr{I}^\pm$$: asymptotic QFT for radiative modes in/out states for scattering
• Is there an asymptotic QFT at $$i^0$$ which “interpolates” between in/out states?

# 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

# To-do

• 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!