Causality as an emergent macroscopic phenomenon: The Lee-Wick $O(N)$ model

In quantum mechanics the deterministic property of classical physics is an emergent phenomenon appropriate only on macroscopic scales. Lee and Wick introduced Lorentz invariant quantum theories where causality is an emergent phenomenon appropriate for macroscopic time scales. In this paper we analyze a Lee-Wick version of the $O(N)$ model. We argue that in the large-$N$ limit this theory has a unitary and Lorentz invariant $S$ matrix and is therefore free of paradoxes in scattering experiments. We discuss some of its acausal properties.

Figure 1

Contour given by the Lee-Wick prescription for integration in the complex $p^0$ plane. The crosses denote the poles at $p^0=\pm \sqrt{{\overrightarrow{p}}^2+M_c^2}$ and at $p^0=\pm \sqrt{{\overrightarrow{p}}^2+M_c^{*2}}$ and the circles those at $p^0=-q^0\pm \sqrt{(\overrightarrow{p}+\overrightarrow{q}{)}^2+m^2}$. The heavy line denotes the cuts on the real axis starting at $\pm 3m$. The contour of integration is deformed as the interactions are turned on and the LW poles move into the complex plane so that the complex poles do not cross the contour.

Figure 2

Feynman diagram for the two-to-four scattering amplitude proceeding through the decay of a Lee-Wick particle. The solid line denotes a “normal” particle, the zigzag line a “Lee-Wick” particle, and the dashed line with the shaded blob denotes the dressed $\sigma$ auxiliary field propagator.