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.


1 Answer 1


The lack of separability in the kinematical Hilbert space in LQG is a mathematical artifact; it doesn't affect the physics, and it disappears when tweaking the mathematical setting.

In more detail, according to the abstract of this paper by Fairbairn & Rovelli in 2004:

In the standard construction, the kinematical Hilbert space of the diffeomorphism invariant states is non-separable. This is a consequence of the fact that the knot-space of the equivalence class of graphs under diffeomorphisms is non-countable.

They add:

However, the continuous moduli labelling these classes do not appear to affect the physics.

And then:

We investigate the possibility that these moduli could be only a consequence of a poor choice in in the fine-tuning of the mathematical setting.

To which they affirm:

We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappears, the knot classes become countable, and the kinematical Hilbert space becomes countable.

  • $\begingroup$ Thank you! And what was originally this set parametrizing the basis of Hilbert space? $\endgroup$
    – truebaran
    Dec 11, 2017 at 20:00
  • $\begingroup$ @truebaran: according to the linked paper, these are abstract spin networks - these are the graphs referred to in the abstract. $\endgroup$ Dec 11, 2017 at 20:05

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