Gravitational waves are oscillations in the geometry of spacetime caused by the movement of mass or energy. In loop quantum gravity, geometry is quantized so spatial quantities such as area and volume take on discrete values. We worked on quantizing plane gravitational waves, which propagate in one direction and have uniform wavefronts perpendicular to the direction of travel, using the method of canonical quantization on general relativity.
For canonical quantization, a theory is expressed in terms of its canonical variables and constraints on those variables, then the variables and constraints are taken to quantum operators and the states of the quantum theory are those that satisfy the constraint operators. The canonical variables of general relativity are Ashtekar variables, which represent gravitational fields in a way that reveals their similarity to electromagnetic fields and other gauge fields. The symmetries of general relativity are formulated as constraints. We used vectors to visually represent the constraints on a spacetime diagram in a manner that is consistent with their actions and their algebra. We represented the constraints as either a Lorentz boost, a directional shift, or a propagation along a light cone. We verified the algebra of these constraints for gravitational waves. A correct quantum theory will have algebra that is consistent the classical constraint algebra.
We found possible quantum operators for the unidirectional constraint, the constraint that ensures the plane wave moves only in one direction. We found quantum states satisfying the different versions of the operator and checked that these states were normalizable. We determined that two different versions of the operator had nontrivial normalizable solutions. These solutions could be used in conjunction with a consistent quantization of the other constraints as a quantum formulation of plane gravitational waves
