Most of Hilbert spaces appearing in real life are separable ones: even such large for the first spaces as Fock space, spaces of functions in infinite number of variables and so on. However I heard that when dealing with quantum loop gravity one encounter non-separable Hilbert space.
Could somebody explain how this space is constructed? What are natural operators, relevant for LQG, acting in such space?
I would like also to understand some conceptual reason for the apperance of nonseparable Hilbert spaces in this context: since the main idea of LQG is that the geometry is ,,quantized'' I would suspect that Hilbert spaces which occure in LQG would be finite dimensional.