[Paper Review] Fourth Order Theories Without Ghosts

Philip D. Mannheim, Aharon Davidson|ArXiv.org|Jan 19, 2000

Cosmology and Gravitation Theories2 references19 citations

TL;DR

This paper demonstrates that fourth-order gravitational theories, such as conformal gravity, can be unitary and ghost-free when the limit of equal frequencies is taken in the Pais-Uhlenbeck oscillator model. Using the Dirac constraint method, it shows that while ghost states appear in the unequal-frequency case, they decouple in the equal-frequency limit, leaving only positive-norm, energy-eigenstate composite states, thus making fully unitary, renormalizable quantum gravity in 4D feasible.

ABSTRACT

Using the Dirac constraint method we show that the pure fourth-order Pais-Uhlenbeck oscillator model is free of observable negative norm states. Even though such ghosts do appear when the fourth order theory is coupled to a second order one, the limit in which the second order action is switched off is found to be a highly singular one in which these states move off shell. Given this result, construction of a fully unitary, renormalizable, gravitational theory based on a purely fourth order action in 4 dimensions now appears feasible.

Motivation & Objective

• To resolve the longstanding ghost problem in fourth-order gravitational theories, particularly in conformal gravity.
• To determine whether the presence of negative-norm states in the unequal-frequency Pais-Uhlenbeck model implies non-unitarity in the equal-frequency limit.
• To establish the feasibility of a fully unitary, renormalizable quantum gravitational theory based on a purely fourth-order action in four spacetime dimensions.
• To analyze the structure of the Hilbert space and S-matrix in the singular equal-frequency limit using canonical quantization.

Proposed method

• Applying the Dirac constraint method to quantize the higher-derivative Pais-Uhlenbeck oscillator Lagrangian with unequal frequencies.
• Introducing auxiliary variables and Lagrange multipliers to reformulate the fourth-order equation into a first-order form suitable for canonical quantization.
• Constructing a Fock space basis for the $ε \neq 0$ case using creation and annihilation operators with non-orthogonal overlaps.
• Analyzing the $ε \to 0$ limit by holding states fixed and studying the behavior of norm and matrix elements of creation/annihilation operators.
• Using the algebraic structure of the $a$ and $b$ operators to identify surviving states in the limit, particularly composite states like $a^\dagger b^\dagger |\Omega\rangle$.
• Demonstrating that only composite, positive-norm states survive the limit and become energy eigenstates, while single-particle states become null vectors.

Experimental results

Research questions

• RQ1 Does the presence of ghost states in the unequal-frequency limit of the fourth-order Pais-Uhlenbeck model imply non-unitarity in the equal-frequency limit?
• RQ2 Can a purely fourth-order gravitational theory be unitary if ghost states decouple in the singular $M^2 \to 0$ limit?
• RQ3 What is the structure of the Hilbert space and S-matrix in the equal-frequency limit of a higher-derivative oscillator model?
• RQ4 How do the dynamics of creation and annihilation operators behave in the singular limit, and can they still define a consistent quantum theory?
• RQ5 Can a unitary, renormalizable quantum gravity theory be constructed from a purely fourth-order action in four spacetime dimensions?

Key findings

• In the unequal-frequency case, the theory has a complete, positive-norm Fock space basis with non-orthogonal overlaps, and all states have positive norm.
• In the $\epsilon \to 0$ limit, the single-particle states $a^\dagger|\Omega\rangle$ and $b^\dagger|\Omega\rangle$ become null vectors, but not identically annihilating the vacuum due to non-vanishing commutator matrix elements.
• Only the composite two-particle state $a^\dagger b^\dagger |\Omega\rangle / \mu$ survives the limit and remains a positive-norm, energy-eigenstate.
• Higher multiparticle states of the form $(a^\dagger b^\dagger)^n |\Omega\rangle$ are the only surviving observable states in the limit, forming a one-dimensional harmonic oscillator-like sector.
• The S-matrix remains unitary in the equal-frequency limit, as only positive-norm, energy-eigenstate composite states are observable.
• The result implies that a fully unitary, renormalizable quantum gravitational theory based on a purely fourth-order action in four spacetime dimensions is now feasible.