[Paper Review] Unitarity issues in higher derivative field theories

M. Asorey, Lesław Rachwał+1 more|arXiv (Cornell University)|Feb 3, 2018

Quantum Electrodynamics and Casimir Effect3 references3 citations

TL;DR

This paper investigates unitarity in higher derivative quantum field theories, particularly non-local, ghost-free models with exponential form-factors. Despite being free of ghosts and UV-finite, the authors demonstrate that these theories violate reflection positivity and thus fail to be unitary due to negative-norm states, showing that ghost absence is insufficient for consistency in relativistic quantum field theories of gravity.

ABSTRACT

We analyze the unitarity properties of higher derivative quantum field theories which are free of ghosts and ultraviolet singularities. We point out that in spite of the absence of ghosts most of these theories are not unitary. This result confirms the difficulties of finding a consistent quantum field theory of quantum gravity.

Motivation & Objective

• To assess whether non-local, higher derivative field theories—free of ghosts and UV divergences—can be unitary.
• To investigate the consistency of non-local theories with fundamental principles like unitarity, causality, and relativistic invariance.
• To determine whether the absence of ghosts in such models guarantees a consistent quantum field theory.
• To analyze the Schwinger functions and 2-point functions for reflection positivity violations in scalar and spin-2 theories.
• To clarify the role of Lorentz covariance and higher time derivatives in generating unitarity issues.

Proposed method

• Constructs non-local, super-renormalizable scalar field theories using an exponential form-factor $ e^{(oxslash / ilde{\Lambda}^2)^s} $ to regulate UV divergences.
• Analyzes the Euclidean 2-point Schwinger function and checks for reflection positivity via the positivity of $ S^{(2)}_{ijij} $, which must be non-negative for unitarity.
• Applies the Källén-Lehmann spectral representation to the 2-point function and checks for positive-definite spectral densities.
• Evaluates the propagator $ \Delta(p) = \frac{e^{-p^{2s}/\Lambda^{2s}}}{p^2 + m^2} $ in momentum space to assess UV finiteness and analytic structure.
• Extends the analysis to spin-2 theories using projection operators $ \mathcal{P}^{(2)} $ and $ \mathcal{P}^{(0)} $ to decompose the graviton 2-point function.
• Demonstrates that the Schwinger function fails reflection positivity due to non-positive definite integrands involving $ e^{p^{2s}/\Lambda^{2s}} $, even with real coefficients.

Experimental results

Research questions

• RQ1 Can higher derivative field theories that are ghost-free and UV-finite still violate unitarity?
• RQ2 Does the absence of ghosts in non-local higher derivative theories guarantee a consistent quantum field theory?
• RQ3 Why do theories with exponential form-factors $ e^{(\Box/\Lambda^2)^s} $ fail to satisfy reflection positivity despite being non-local and UV-finite?
• RQ4 What is the role of Lorentz covariance in generating unitarity violations in higher derivative theories?
• RQ5 Can non-local theories with entire function form-factors (e.g., exponential) be viable candidates for quantum gravity?

Key findings

• The 2-point Schwinger function of the non-local scalar theory fails reflection positivity, indicating the presence of negative-norm states.
• The expression $ S^{(2)}_{ijij} $ for the scalar theory is not always positive definite, violating a necessary condition for unitarity.
• For spin-2 theories, the 2-point function does not satisfy reflection positivity due to the structure of the exponential form-factor in the denominator.
• Even though the theory is UV-finite and ghost-free, the analytic structure of the propagator leads to non-positive spectral functions.
• The violation of unitarity arises specifically from the combination of Lorentz invariance and higher-order time derivatives, not from ghost modes.
• The results imply that such non-local, exponential-form-factor theories cannot be considered consistent fundamental theories of quantum gravity.