Quantization of (1+1)-dimensional Ho\v{r}ava-Lifshitz theory of gravity

TL;DR

This paper explores the quantization of (1+1)-dimensional Hořava-Lifshitz gravity, demonstrating that it can be quantized using Dirac methods, with wavefunctions that are normalizable and well-defined even in classically singular spacetimes.

Contribution

It provides a detailed canonical quantization of (1+1)-dimensional HL gravity, including coupling to scalar fields, and highlights the importance of operator ordering in obtaining solutions.

Findings

Wavefunctions are normalizable for certain operator orderings.

The Hamilton can be expressed as harmonic oscillators, with one having negative energy.

Quantization remains consistent even in classically singular spacetimes.

Abstract

In this paper, we study the quantization of the (1+1)-dimensional projectable Ho\v{r}ava-Lifshitz (HL) gravity, and find that, when only gravity is present, the system can be quantized by following the canonical Dirac quantization, and the corresponding wavefunction is normalizable for some orderings of the operators. The corresponding Hamilton can also be written in terms of a simple harmonic oscillator, whereby the quantization can be carried out quantum mechanically in the standard way. When the HL gravity minimally couples to a scalar field, the momentum constraint is solved explicitly in the case where the fundamental variables are functions of time only. In this case, the coupled system can also be quantized by following the Dirac process, and the corresponding wavefunction is also normalizable for some particular orderings of the operators. The Hamilton can be also written in terms of two interacting harmonic oscillators. But, when the interaction is turned off, one of the harmonic oscillators has positive energy, while the other has negative energy. A remarkable feature is that orderings of the operators from a classical Hamilton to a quantum mechanical one play a fundamental role in order for the Wheeler-DeWitt equation to have nontrivial solutions. In addition, the space-time is well quantized, even when it is classically singular.