Spectral Density of the Causal Propagator

The causal propagator (or Pauli-Jordan function), which multiplied by $i$ is the spacetime commutator of the field $[\phi (x),\phi (x^{\mathrm{\prime }})]$, plays an essential role in scalar quantum field theory. We discuss the role of the causal propagator and its spectrum in recent developments in defining quantum field theory in a more explicitly covariant manner, as well as in causal set theory. We then present a conjecture for its asymptotic spectral density in a free theory, and give examples that lend evidence to the conjectured scaling. Our work has implications for Lorentzian spectral geometry in much the same way as Weyl’s asymptotic law has for Riemannian spectral geometry.