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

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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.