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

Benjamin Grinstein, Donal O'Connell, Mark B. Wise

TL;DR

This paper investigates whether causality-violating ideas can be reconciled with unitarity by studying a Lee–Wick extension of the $O(N)$ model in the large-$N$ limit. Using the CLOP prescription and an auxiliary sigma-field formulation, the authors show the $S$-matrix remains unitary and Lorentz invariant, even in the presence of acausal effects, and they explicitly compute the self-energy and time-dependent scattering amplitudes. They reveal that acausal behavior emerges as late-time tails in certain LW-loop processes, including power-law contributions that persist alongside standard exponential decays. The work establishes a controlled framework to analyze acausal, yet unitary, quantum field theories and discusses potential implications for hierarchy puzzles and quantum gravity.

Abstract

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.

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

TL;DR

This paper investigates whether causality-violating ideas can be reconciled with unitarity by studying a Lee–Wick extension of the model in the large- limit. Using the CLOP prescription and an auxiliary sigma-field formulation, the authors show the -matrix remains unitary and Lorentz invariant, even in the presence of acausal effects, and they explicitly compute the self-energy and time-dependent scattering amplitudes. They reveal that acausal behavior emerges as late-time tails in certain LW-loop processes, including power-law contributions that persist alongside standard exponential decays. The work establishes a controlled framework to analyze acausal, yet unitary, quantum field theories and discusses potential implications for hierarchy puzzles and quantum gravity.

Abstract

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.

Paper Structure

This paper contains 15 sections, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Time Dependence in a Scattering Experiment
  3. Kinematics
  4. Standard resonant behavior
  5. Lee-Wick resonant behavior
  6. Review of the $O(N)$ Model
  7. Unitarity of the Two Particle Scattering Amplitudes
  8. Time dependence of Two Particle Scattering Processes
  9. The Lee-Wick $O(N)$ Model
  10. The Lagrangian
  11. Loops
  12. Computation of the Self-Energy
  13. Unitarity
  14. Time Dependence: Acausality
  15. Concluding Remarks

Figures (2)

  • Figure 1: Contour given by the Lee-Wick prescription for integration in the complex $p^0$ plane. The crosses denote the poles at $p^0=\pm\sqrt{\vec{p}^2+M_c^2}$ and at $p^0=\pm\sqrt{\vec{p}^2+M_c^{*2}}$ and the circles those at $p^0=-q^0\pm\sqrt{(\vec{p}+\vec{q})^2+m^2}$. The heavy line denotes the cuts on the real axis starting at $\pm3m$. 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 zig-zag line a "Lee-Wick" particle, and the dashed line with the shaded blob denotes the dressed $\sigma$ auxiliary field propagator.