Separable Hilbert Space in Loop Quantum Gravity

[marcus]

Yesterday Meteor found this new paper of Rovelli's and added it to the "surrogate sticky" collection of links.
In case there is need for discussion, it should probably have its own thread as well.
The paper expands a key assertion made on page 173 of the on-line draft of Rovelli's book "Quantum Gravity".


-------from Meteor's post------

"Separable Hilbert space in loop quantum gravity"
http://arxiv.org/abs/gr-qc/0403047
By Carlo Rovelli and Winston Fairbairn

Abstract:"We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot-space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable"

--------end quote--------

 

[marcus]

a new mathematical animal

there is some intriguing mathematics in the Rovelli/Fairbairn paper
(some apparently derives from talks with Alain Connes, some from a book by V.I.Arnold, some seems to be new with Rovelli.)
Among the mathematical ideas I like the "almost smooth physical fields" introduced on page 8 at the beginning of section 3, the section on the "Extended diffeomorphism group".

Here's an exerpt:

"3.1 Almost smooth physical fields

Consider a four-dimensional differentiable manifold M with topology Σ x R, as before. However, we now allow the gravitational field g to be almost smooth, as defined in the previous sections, that is: g is a continuous field which is smooth everywhere except possibly at a finite number of points, which we call the singular points of g.

Any such g can be seen as a (pointwise) limit of a sequence of smooth fields. We say that g is a solution of the Einstein equations if it is the limit of a sequence of smooth solutions of the Einstein equations. Call E* the space of such fields.

Let now φ be an invertible map from M to M such that φ and φ^-1 are continuous and are infinitely differentiable everywhere except possibly at a finite number of points. The space of these maps form a group under composition, because the composition of two homeomorphisms that are smooth except at a finite number of singular points is clearly an homeomorphisms which is smooth except at a finite number of singular points. We call this group the extended diffeomorphism group and we denote it as Diff*_M.

It is clear that if g ε E* then (φg) ε E* for any φ ε Diff*_M. Hence Diff*_M is a gauge group for the theory.

In the Hamiltonian theory, we can now take almost smooth connections A on Σ..."

In the section of Rovelli's book where it would naturally have come---around page 173 of the draft---this discussion was either omitted or implicit. I for one wanted to see it spelled out, and I've been wondering about the almost-smooth category, that now seems emerging. Perhaps someone knows of its being explored in some other context. If it isn't already explored it might be a good small research area in differential geometry, with the potential for becoming a healthy cottage industry (just a thought). Would be interesting to know if the mathematics has already been worked out.

 

[marcus]

seredipitous connection with something Bahns said

In the first post on the "Dorothea Bahns thesis" thread,
I highlighted this quote, which has broader relevance and bears, for instance, on the Fairbairn/Rovelli paper:

"This scheme is based on the idea that the correspondence principle should be applied to physically meaningful quantities only, which in a theory with gauge freedom means that it is applicable only to gauge-invariant observables."

This can be interpreted as indicating that already the classical observables may have gauge equivalences modded out.

In that light the general principle (which Bahns invokes in her
recent paper hep-th/0403108) suggests the appropriateness of implementing spatial diffeo invariance on a priority basis, in the initial construction of the kinematic Hilbert space of LQG.

 

[marcus]

In GR the points of the spacetime manifold have no physical meaning and accordingly the metric g, defined on M = Σ x R, is not physically meaningful either.

the classical object to be quantized is therefore not g, or (in the Hamiltonian picture) the connection A defined on Σ, which do not correspond to anything existing in nature.

Instead, what has physical existence is the class of connection under gauge equivalence. (Logical meaning of Einstein's dictum that space IS the gravitational field: there is no independently existing space.)

Rovelli's Chapter 2 is largely devoted to explicating this, which has become something like the "Pons Asinorum" of QG.

Fairbairn/Rovelli now say: let the classical metric g, and connection A, be almost smooth.
In the Hamiltonian development the gauge group is now the almost smooth homeomorphisms.
(they use the notation Diff* for extended diffeomorphism, but I think their terminology of "almost smooth homeomorphism" is somewhat more self-explanatory)

now since the spatial manifold Σ is merely a mathematical convenience used for purposes of definition, the classical A contains gauge "chaff" to be eliminated before quantizing

before defining any quantum observables, or self-adjoint operators, or constraints, on the hilbert space, one must mod out the physically meaningless gauge stuff.

(as per the principle invoked by the nice Bahns in her paper, that one shall not quantize what is physically meaningless :-) of course nobody is perfect and let him who is without gauge throw the first self-righteous fit)

So one MUST mod out the almost smooth homeomorphisms on Σ. One must factor them out first, from the initial hilbert space. There is no choice about this.

It has to do, I think, with a kind of committment to what is physically real. On s'engage et puis on voit, and let the anomalies fall where they may.

Well this is a kind of paraphrase, of how I understand the thrust of this paper. Mathematically it seems to hold a lot of goodies, like a Mexican pinyata. The grad students and postdocs will be blindfolded and given cudgels to swing at it till it let's down its shower of results.

 

[marcus]

Fairbairn/rovelli kicks butt

I've been reading more in the F/R paper and like it more the more I do.
they don't prove that the isotopy classes of embedded graphs (iso-knots) are countable, which I wish there were a reference for (must be well-known in knot theory)

the big thing with this paper is almost smooth homeomorphisms.
that is too long a phrase so since "almost" is the same as "quasi"
I will call such things
"Q-morphisms"
A Q-morphism of the manifold Σ is a homeomorphism which, together with its inverse, is infinitely differentiable except possibly at finitely many points.
that is, a Q-morphism is an almost smoooth homeomorphism in the sense of Fairbairn and Rovelli.

They prove that if two embedded graphs are isotopic they are also
Q-morphic to each other.

The Q-morphisms form a group, which F&R make the gauge group in setting up the kinematic state space for LQG.

The kinematic hilbert space turns out to be separable because
the basis (of Q-knots) is countable.

 

[marcus]

there is a place in Fairbairn/Rovelli where they do not tell you why a certain sum in finite, but the answer is on page 171 of Rovelli's book.
maybe I did not already say this in this thread.
the line in the book is right under equation 6.44 on the second page of section 6.4-----page 171 of the online draft.

in the paper, on page 4, they just say equation (6) which defines the projection map to the invariant states
and right below that they claim "the sum converges and is well-defined"
but do not explain why it converges. however the sum is finite.

I still want a simple explanation of why the iso-knots
in Σ are countable. I don't own the right book for that and wish a source were online.

 

[marcus]

experimenting with Rovelli/Fairbairn notations
the way R/F organize the paper they define things using the usual diffeomorphism group (as on page 4) and then
on page 8 they say go back and do it all over again with the extended diffeomorphisms. I'm finding how to imitate their notation in latex, and going over that.

the 3D manifold is $$\Sigma$$

the space of almost smooth connections on $$\Sigma$$ is denoted $$\mathcal{A}$$

the cylindrical functions on $$\mathcal{A}$$ have an inner product $$\langle , \rangle$$ (defined on page 3) and their completion under the corresponding norm is denoted $$\mathcal{K}$$

The local SU(2) gauge invariant subspace is $$\mathcal{K}_0$$.


The spin networks are SU(2) invariant so they belong to $$\mathcal{K}_0$$ and indeed (by the Peter-Weyl theorem) span. The spin networks are taken as a basis and the subspace consisting of their finite linear combinations is denoted $$\mathcal{S}$$

Any element of $$\mathcal{S}$$
can be viewed as a linear functional on $$\mathcal{S}$$
by means of the inner product $$\langle \Psi,\Psi' \rangle$$

F/R define $$\mathcal{S}'$$ as the algebraic dual given the topology of pointwise convergence----essentially comprised of infinite sequences of elements of $$\mathcal{S}$$
which converge pointwise

Fairbairn/Rovelli call the gauge group Diff* for "extended diffeomorphisms" of $$\Sigma$$,
that is the almost smooth homeomorphisms of $$\Sigma$$

on page 4 they refer to this gelfand triple
$$\mathcal{S} \subset \mathcal{K}_0 \subset \mathcal{S}'$$

they define a projection onto the almost-smooth-invariant states
$$P_{\text{diff}}:\mathcal{S} \rightarrow \mathcal{S}'$$

$$(P_{\text{diff}}\Psi)(\Psi') = \sum_{\Psi'' = \phi\Psi} \langle \Psi'',\Psi' \rangle$$

the sum is over all states Ψ" for which there exists an
almost smooth homeomorphism φ taking Ψ' to Ψ"

I guess the point here is that there is a subspace of
$$\mathcal{S}'$$ consisting of those states which are invariant under almost smooth homeomorphisms (I considered calling them "Q-morphisms", Q for quasi-smooth.) These would have to be linear functionals, real members of $$\mathcal{S}'$$ not just
members of $$\mathcal{S}$$ moonlighting as members of the dual.

So what we are interested in is a projection from $$\mathcal{S}'$$ into that subspace----the Q-invariant states, or the "extended diffeomorphism"-invariant states, or the "almost smooth homeomorphism"-invariant states.

that is what $$P_\text{diff}}$$ is, the projection into the subspace of invariant states.

And finally F/R define $$\mathcal{H}_{\text{diff}}$$
which is essentially the image of that projection
for historical reasons it is written with subscript "diff" because it is the hilbertspace of SU(2) and diffeo invariant states----except that now the diffeos are "extended" so they can have a finite number of singularities. And this is a familiar notation for the kinematic state space of LQG.

$$\mathcal{H}_{\text{diff}}$$ inherits the inner product
and they write it various ways, as in equation (9) on page 4 and
as in equation (12) for the case where the arguments were originally spin network states.


still don't know how to do "bra" and "ket" in latex.
or even those two angle irons that look like < and >

 

[marcus]

OK so that is a little thumbnail sketch of how the kinematic state space of LQG is defined using Q-morphisms
and the notation for it is $$\mathcal{H}_{\text{diff}}$$
or sometimes $$\mathcal{H}_{\text{diff*}}$$

using an asterisk doesn't make things a heck of a lot clearer
anyway it is homeomorphisms of &Sigma; which together with their inverses are smooth except possibly at a finite set of points.

and one gets hassled about this
by people who say this is not the right way to implement
spatial group invariance
they say to first build a hilbertspace
as if &Sigma; was a physically meaningful thing and its diffeomorphism group was god-given
and then use the generators of the diffeo group as constraints
to isolate a subspace of the first hilbertspace

this is a kind of approach more appropriate to theories
which are not background independent where the space &Sigma;
may be presumed meaningful and where you might actually want to use the diffeomorphism group, possibly viewed as "reparametrizations" of a physically real manifold.

But here, as F/R explain in this paper, that is the wrong group
and the right group may not even have a Lie algebra, it may not even have generators! In any case who wants to be told to delay implementing gauge invariance and then, when they finally get around to it, they should use the wrong group?

About whether the almost smooth morphisms can be given a Lie group structure see the footnote on page 9. Apparently it is an open question.

 

[lethe]

Originally posted by marcus

still don't know how to do "bra" and "ket" in latex.
or even those two angle irons that look like < and >

you can do bras and kets this way:
$$\langle\phi|\psi\rangle$$

while we are on the subject, you should observe the following symbols:

$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$$

$$\cos^2\theta+\sin^2\theta=1$$

$$\ln(\frac{x}{y})=\ln x-\ln y$$

so use \sum for summations, not \Sigma, and use \cos, \sin, \log, \ln, etc for transcendental functions, not cos, sin, log, ln.

k?

 

[marcus]

thanks for \rangle
and \langle

as for \sum, I was already using that as in the previous post
where there is a summation.
Wherever you see a Sigma in the above post, it actually means Sigma,
standing for the spatial manifold, rather than summation.

 

[lethe]

i have seen you use \Sigma many times to stand for summation. this is not an attack, just advice. i don't want you to get defensive about your LaTeX. i just want you to fix it. i don't know why you want to deny this, but for an example, see this post

 

[marcus]

Originally posted by lethe
i have seen you use \Sigma many times to stand for summation. this is not an attack, just advice. i don't want you to get defensive about your LaTeX. i just want you to fix it. i don't know why you want to deny this, but for an example, see this post

I should hope not an attack. You have asked me to go back and fix a February 3 post (at a time when I did not know \sum) so I have gone back and fixed it, since it matters to you.

But in my recent post I was obviously using \sum for summation and you nevertheless told me about \sum, giving the impression that you might not have understood what is a sigma and what is a summation in my post. (The sigmas are intentional.) There is no need for either of us to be either picky or defensive.

I had no idea you were irritated by those early February posts with sloppy summations. But if there are some other old posts which you want me to fix please give a link, as you did to the February 3 one and I will be delighted to oblige by editing!

(Any thing within reason )

Lethe if something in my notation offends you don't wait a whole month to tell me about it! Let's have the benefit of your reaction while the matter is still fresh in mind

 

[marcus]

Lets not waste any more time quibbling about some notation
from 6 weeks back in another thread!

Lethe do you have any response to the content here?
Have you looked at the Fairbairn/Rovelli paper?

 

[lethe]

Originally posted by marcus

But in my recent post I was obviously using \sum for summation and you nevertheless told me about \sum, giving the impression that you might not have understood what is a sigma and what is a summation in my post. (The sigmas are intentional.) There is no need for either of us to be either picky or defensive.

if you have been using summation symbols correctly recently, i guess i did not notice.

I had no idea you were irritated by those early February posts with sloppy summations. But if there are some other old posts which you want me to fix please give a link, as you did to the February 3 one and I will be delighted to oblige by editing!

look, i am not pointing out your mistakes because i want you to go back and edit them, i just noticed that you seemed to be asking about Latex in this thread, and so i thought i could help you learn your Latex better.

Lethe if something in my notation offends you don't wait a whole month to tell me about it! Let's have the benefit of your reaction while the matter is still fresh in mind

on the other hand, i am not your latex tutor. i didn't feel it was important enough to bother mentioning on those occasions. since, in this thread, you seemed to be asking for latex instructions, i thought it was worth mentioning.

i had no idea you would get your panties in a bunch over having your latex corrected.

Originally posted by marcus
Lets not waste any more time quibbling about some notation
from 6 weeks back in another thread!

it's your call, boss.

Lethe do you have any response to the content here?

nope.

Have you looked at the Fairbairn/Rovelli paper?

nope.

 

[marcus]

well then to continue



Einstein said something quite forcefully which Rovelli's chapter 2 spends a long time discussing with illuminating quotes and examples. Instead of trying to say it I will just mention a few key words as a reminder.

In classical Relativity the gravitational field is an equivalence class of solutions to the Einstein equation.
The continuum is a mathematical fiction. A convenience for purposes of definition (the usual gauge thing) but not physically meaningful. Sounds radical but has been accepted by Relativists since the early part of the last century.
The basic thrust of Loop Gravity is taking Einstein at his word

Anyway the manifold is not physically real, points in it have no meaning.
the gravitational field takes its place.
what looks like Minkowski space is just one possible solution of the Einstein equation---without much matter or curvature----one possible field.
solutions to the classical Einstein equation are equivalence classes under the operation of the spatial group. the manifold used as a mathematical
convenience for their initial definition is factored out. classically.
other fields can be defined in reference to or on top of the gravitational field

So we need a mathematical representation of the gravitational field
and it should be combinatorial or at least not require some prior space.
In this paper by Fairbairn/Rovelli the quantum state of all space is denoted by a knot and a quantum number. The quantum number takes care of the coloring of the links and vertices of the knot.
The knot is abstract, not embedded in some prior space. It is space.

$$|s\rangle = |K,c \rangle$$

(see the top of page 5 and also page 9)

The knots of knot theory are abstract--they can be treated as diffeomorphism equivalence classes of networks--so this is nothing new. Why shouldn't space as far as we can see---the mostly flat gravitational field---be a knot. Or at least its quantum states be knots and mixtures thereof.

The F/R new thing is that K is not a diff-knot but a diff*-knot.
It is an equivalence class under the operation of the Q-morphisms, or almost smooth homeomorphisms, or "extended diffeomorphisms" of the fictional space used for purposes of definition.

 

[marcus]

So to pick up where we left off when Lethe dropped in these
diff*-knot states are the basis of the Hilbert space I was talking about:

The basis of
$$\mathcal{H}_{\text{diff*}}$$
is things like
$$|K,c\rangle$$

and sometime the asterisk is left off even when they mean the extended diffeomorphisms---notation and terminology haven't stabilized.

For continuity here's the post prior to Lethe's visit

OK so that is a little thumbnail sketch of how the kinematic state space of LQG is defined using Q-morphisms
and the notation for it is $$\mathcal{H}_{\text{diff}}$$
or sometimes $$\mathcal{H}_{\text{diff*}}$$

using an asterisk doesn't make things a heck of a lot clearer
anyway it is homeomorphisms of &Sigma; which together with their inverses are smooth except possibly at a finite set of points.

and one gets hassled about this
by people who say this is not the right way to implement
spatial group invariance
they say to first build a hilbertspace
as if &Sigma; was a physically meaningful thing and its diffeomorphism group was god-given
and then use the generators of the diffeo group as constraints
to isolate a subspace of the first hilbertspace

this is a kind of approach more appropriate to theories
which are not background independent where the space &Sigma;
may be presumed meaningful and where you might actually want to use the diffeomorphism group, possibly viewed as "reparametrizations" of a physically real manifold.

But here, as F/R explain in this paper, that is the wrong group
and the right group may not even have a Lie algebra, it may not even have generators! In any case who wants to be told to delay implementing gauge invariance and then, when they finally get around to it, they should use the wrong group?

About whether the almost smooth morphisms can be given a Lie group structure see the footnote on page 9. Apparently it is an open question.

 

[marcus]

smoothness arises from averaging

Here's what I think is a key quote from the conclusions of the Fairbairn/Rovelli paper. I've bolded some parts.
-----quote from F/R conclusions----
...are smooth (with their inverse) except possibly at a finite number of points. The space of the knot classes become countable and the kinematical Hilbert space $$\mathcal{H}_{\text{diff}}$$ is separable. The area, volume, and hamiltonian operator are naturally covariant under this extended gauge invariance, provided that the appropriate regularization and the appropriate version of the volume operator are chosen. The spectra of area and volume, in particular, are unaffected. We expect that analogous results could be obtained also using other mathematical settings, in particular the piecewise smooth category.

We take these results as indications that the continuous moduli that made $$\mathcal{H}_{\text{diff}}$$ nonseparable might be physically spurious. Using the setting described in this paper, the theory appears to be cleaner and to realize more completely its purely combinatorial character as well as background independence. If we adopt this point of view, background independent quantum microphysics is entirely discrete and smoothness can be seen, a posteriori, just as a property arising from averaging over regions much larger than the Planck scale.
----end quote----