Is Singular Quantization Still Necessary for Loop Quantum Gravity?

[Physics Monkey]

I recently bought "A First Course in Loop Quantum Gravity" by Pullin and Gambini. Partly, I was curious to see what, if anything, had changed in the pedagogy. I also got Bojowald's book a while back. In the final section of "A First Course ..." the authors discuss open problems and broad issues and I was struck once again by the fundamental weirdness, from my perspective, of the basic quantization scheme.

No doubt we all remember how old school LQG used a very singular "kinematic" Hilbert space for quantization. Indeed, it was pointed out that in a certain precise mathematical sense, this same quantization applied to a conventional harmonic oscillator leads to unconventional results. Although, like everything in physics, there is a way to (partially) hide this disagreement.

What I'm wondering is this: is such a singular/unusual quantization still necessary for LQG e.g. from the perspective of spin foam theory? I have to say that getting the harmonic oscillator "wrong" is really disconcerting and the very singular structure of the kinematic hilbert space has always struck me as unphysical. Pullin and Gambini emphasize that it is precisely this unusual quantization, even for systems with finite dof, which really let's LQG get different answers, say in the context of quantum cosmology (we came to the same conclusion in another thread of mine some time ago).

Thoughts?

 

[tom.stoer]

The simple argument is this: standard quantization may result in an inconsistent theory of quantum gravity, therefore one has to change the quantization ;-)

I think the problem of the non-separable Hilbert space can be solved, but it's still true that the construction of the kinematical Hilbert space is singular in some sense.

There are some papers on the classical Poisson structure (w/o any quantization) which show that the derivation in the canonical formalism (which starts with an embedding) means that the emerging graphs (embdedded) carry singular curvature located either to vertices or to edges (these two possibilities are somehow dual to each other and show up in the canonical and spin foam models).

I think this can be interpreted such that it's not the quantizationm which introduces this singular structure but that it appears already at level of the classical Poisson structure when implementing the Gauss law constraint.

The question is whether this construction is relevant. There are people (Rovelli, marcus ;-) saying that the quantization itself is not to be taken too seriously but that it's the final theory which must be analyzed. There are many steps where the construction is not absolutely convincing (embdedding of cylinder functions - later we use non-embedded spin networks; reduction of spatial diffeomorphism constraints - later there are only finite ones - and in the spin network space there are none; reduction and regularization of the Hamiltonian constraint with many ambiguities; on-shell closure of the constraint algebra but no control on off-shell closure - which may be irrelevant in the physical Hilbert space b/c there are no off-shell constraints anymore; ...)

It seems to mne that in LQG we have serious difficulties simply b/c a quantization can never be unique b/c there are for sure infinitly many quantum theories with the same classical limit. So at some crossroads we need an educated guess. That's not very problematic, we do something like that in all quantization procedures.

But what is problematic is that we cannot "label" or "count" the different quantum theories. We have no control on the theory space. We do not know this "LQG landscape". And it seems that w/o such a construction we do not have enough indications for the elementary buildung blocks of the theories (we need something like the above mentioned constructions to find a Hamiltonian, to find the SF vertices ...).

That's why I think that some people (Rovelli, marcus ;-) are not right when (partially) ignoring these issues at this state and focussing only on "physical results". That's why I am more with Thiemann, Alexandrov and other's focussing on these weak points.

 

[Physics Monkey]

Thanks for the reply. Interesting as always.

What would you say to the following: LQG with the singular quantization is unphysical because it cannot be simulated on a computer, classical or quantum? Or is this even true?

I know I've seen computer plots in the context of LQC, but how they can achieve this? Is it because they succesfully solve the constraint and obtain a countable or even finite dimensional Hilbert space? I think I used to know the answer to this ...

 

[atyy]

I know I've seen computer plots in the context of LQC, but how they can achieve this? Is it because they succesfully solve the constraint and obtain a countable or even finite dimensional Hilbert space? I think I used to know the answer to this ...

I think LQC (for which they had the plots) is not really related to LQG. LQC seems to be a completely well controlled theory (eg. http://arxiv.org/abs/1001.5147), but it's not background dependent, because of the homogeneity assumption.

The other place I've seen a plot is in spin foam cosmology. The background theory for spin foam cosmology is EPRL. In his Zakopane lectures, Rovelli says the Hilbert space of EPRL is related to SU(2) lattice gauge theory (Eq 22), and states the full Hilbert space in Eq 24 (Does the limit exist?), and says it is separable (!).

I think the difference is that this is the physical Hilbert space, as Rovelli says just before Eq 21, whereas the inseparable space is the kinematic Hilbert space. This would make sense since spin foams are hoped to solve the Hamiltonian constraint.

 

[tom.stoer]

Formally the spin foams solve the Gaussion as well as the diffeomorphism constraint, but they are constructed via a procedure using a "singular representation of diff.-inv. equivalence classes" classically, a non-sep. Hilbert space quantum mechanically and a strange operator topology.

It's not clear whether they solve the Hamiltonian constraint as well b/c a) H is not known in the quantum theory (there are proposals, but no common agreement oin H) and b) it's not clear how spin foams are exactly related to the Hamiltonian.

Anyway - whereas people like Thiemann, Alexandrov and some others stress the fact that parts of the quantization approach are still purely understood, people like Rovelli stress the fact that via EPRL we have a well-defined quantum theory of gravity and that its construction via "quantization" is of minor importance.

This is partially acceptable b/c quantization can never be a rigorosuly defined procedure b/c it constructs and singles out one quantum theory as a member of a huge family of inequivalent quantum theories with a common classical limit. It's like building a house using nothing else but a set of drawings. In some sense there a many house compatible with the drawings and there is no reason why one house should be 'the correct one' whereas all others are incorrect'. Besides the drawing you need the discussion between the architect and the owner, i.e. between theory and the experiment ...

 

[atyy]

Rovelli comments on separable/inseparable spaces on p6 of his Zakopane lectures:

This is the "combinatorial H". An alternative studied in the literature is to consider embedded graphs ... This choice defi nes the "Diff H". A third alternative is to do the same but using extended diffeomorphisms [36]. ... The space Diff H is non-separable, leading to a number of complications in the construction of the theory. The combinatorial H considered here and the extended-Diff H are separable."