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 quantization. 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, normalizable wavefunctions, unitary evolution in a negative-norm configuration space. We present a formalism for quantum mechanics with a generic norm.
Highlights
This classical arbitrarily negative energy is avoided by quantization with anti-commutators, if the vacuum state is appropriately chosen
We here study the concrete system that lies at the basis of perturbative Quantum Field Theory: the harmonic oscillator
Starting from the harmonic oscillator, we describe a more general representation of a pair of canonical coordinate variables q, p
Summary
Let us introduce the main issues in the simplest relevant case. Our final goal will be 4-derivative gravity; the graviton components can be Fourier expanded into modes with given momentum and four time derivatives, and at leading order in the perturbative expansion one has decoupled harmonic oscillators. Ostrogradski introduced an auxiliary coordinate q2 that allows one to describe the 4-derivative oscillator in canonical Hamiltonian form Using the Poisson parentheses { , } one computes the Hamiltonian equations of motion:. For any λ they imply the classical Lagrangian equation of motion. The correspondin..g. classical solution, for given initial conditions q0, q0, q0, q 0 at t = 0, is q (t ). This is a well-behaved oscillator without run-away issues because the positive-energy and negative-energy components are decoupled.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
