Quantum mechanics of 4-derivative theories

TL;DR

This paper develops a quantum theory for 4-derivative gravity that achieves renormalizability and positive energy states by employing a novel representation in a negative-norm space, enabling unitary evolution.

Contribution

It introduces a formalism for quantizing 4-derivative theories with unusual parity, ensuring positive energies and unitarity in a negative-norm framework.

Findings

Achieves a renormalizable quantum gravity model with 4-derivative terms.

Demonstrates positive energy eigenvalues and normalisable wave functions.

Provides a formalism for quantum mechanics in a negative-norm space.

Abstract

A renormalizable theory of gravity is obtained if the dimension-less 4-derivative kinetic term of the graviton, which classically suffers from negative unbounded energy, admits a sensible quantisation. We find that a 4-derivative degree of freedom involves a canonical coordinate with unusual time-inversion parity, and that a correspondingly unusual representation must be employed for the relative quantum operator. The resulting theory has positive energy eigenvalues, normalisable wave functions, unitary evolution in a negative-norm configuration space. We present a formalism for quantum mechanics with a generic norm.