Well, you all know that LQG has different kinds of Hilbert spaces (4 in the sake of truth). You start with the kinematical Hilbert space that is the vector space of all possible quantum states of spacetime. However, all these spacetimes are not physically real, not all of them make sense. Then you have three constraints that "select" all the spacetimes that are physically possible. First you apply the Gauss constraint to the Kinematical Hilbert Space to obtain the gauge invariant Hilbert space, to it you apply the diffeomeorphism constraint to obtain the diffeomorphism invariant Hilbert space, and to it you apply the Hamiltonian constraint to obtain the physical Hilbert space, this is the "correct" space of quantum states of spacetime. Then you know that the kinematical Hilbert space is an infinite dimensional vector space, but my doubt is if the physical Hilbert space is also infinite dimensional. It should seem that not, since the number of quantum states is less than in the kinematical Hilbert space, but given my problems to understand the concept of infinity (you know the set of natural numbers has the same cardinality that the set of rational numbers even if it seems paradoxical) I'm not sure. So the question is, is the physical Hilbert space infinite dimensional, and if not, what's its dimension?(adsbygoogle = window.adsbygoogle || []).push({});

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# Hilbert spaces in LQG

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