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Operators of quantum-reduced loop gravity from the perspective of full loop quantum gravity

I. Mäkinen, Phys. Rev. D 102, 106010 (2020), arXiv:2004.00309 [gr-qc] [link]

Quantum-reduced loop gravity is a simplified model of loop quantum gravity. At the moment it is perhaps the most promising approach towards using the formalism of loop quantum gravity to address concrete physical questions about phenomena where quantum gravity effects are likely to play an important role (e.g. the physics of the very early universe, or the final stages of gravitational collapse).

The most important practical advantage of the model is the remarkably simple form of its basic kinematical operators, in comparison with the corresponding operators of full loop quantum gravity. However, based on the earlier literature on the quantum-reduced model, it is not at all clear to what extent the simple structure of the operators is an assumption introduced by hand, and to what extent it can be derived or justified from the full theory.

In my article I give a decisive answer to this question by showing that, in spite of their simplicity, the operators of quantum-reduced loop gravity are essentially just the operators of full loop quantum gravity acting on states in the Hilbert space of the quantum-reduced model. In order to obtain the simple reduced operators from the operators of the full theory, it is enough to realize that the states of the quantum-reduced model carry large spin quantum numbers, and -- after having acted with an operator of the full theory on such a state -- discard terms that are of lower than leading order in these quantum numbers.

In my opinion this result considerably clarifies the relation between the quantum-reduced model and full loop quantum gravity, and strengthens the technical foundations of the model's kinematical structure. When definite physical predictions will be extracted from the quantum-reduced model in the future, we can now be much more confident that these predictions are actually well justified from the point of view of full loop quantum gravity.

Ilkka Mäkinen