Thomas Thiemann: Reduced Phase Space Quantization & Dirac Observables

[marcus]

Thomas Thiemann
Reduced Phase Space Quantization and Dirac Observables
18 pages
http://arxiv.org/abs/gr-qc/0411031

Abstract:"In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose a new way for how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme."

 

[selfAdjoint]

I am truly impressed. Bianca Dittrich's paper only appeared this month, and here's Thiemann with a 31 page commentary and extension of it already. Scanning several sections of Thiemann's paper I see how upon reading the earlier paper he must have struck his forehead and cried, "That's JUST what I needed!" Wings for the Phoenix!

 

[marcus]

I am truly impressed. Bianca Dittrich's paper only appeared this month, and here's Thiemann with a 31 page commentary and extension of it already. Scanning several sections of Thiemann's paper I see how upon reading the earlier paper he must have struck his forehead and cried, "That's JUST what I needed!" Wings for the Phoenix!

You are some ways ahead of me, just now, on this one. I don't yet understand what Bianca Dittrich was doing with Rovelli's relational quantum mechanics ---- the "complete" observables correlating the partial or component ones. And I don't yet see how this helps Thiemann with the master constrain program.

Had a long rehearsal last night, concert this week, looked at both papers when I got home but too sleepy to follow them.

It does seem a bit of luck that we had that visit from Edgar1813 last week and had occasion to discuss "partial observables". that was before this Thiemann and Dittrich thing came up and helped prepare us.

I will look at those two papers and hazzard some guesses, and if you would care to, please see if they square with your take on them.

 

[marcus]

In T.T.'s paper there is reference [4]

[4] B. Dittrich, T. Thiemann, “Testing the Master Constraint Programme for Loop Quantum Gravity”, parts I. – V., to appear

despite this not yet being posted on arxiv one can see a sample in a talk by Bianca at Penn State, in spring 2004 semester seminars. The presentation is better than the usual slides, it reads like a brief paper, and there is audio, though the audio is often interrupted by preaching from someone in the audience.

I will get the link.

One goes here
http://www.phys.psu.edu/events/index.html
and changes what it says in the timeframe box from "this week" to
"spring 2004"------there is a slide menu that makes it easy to select
events from a particular time interval

then one clicks "show" and gets a list of seminars from spring 2004

then one sees Bianca Dittrich down at 13 February
and one clicks on "audio and presentation"

then one gets
http://www.phys.psu.edu/events/index.html?event_id=850&event_type_ids=0&span=2003-12-26.2004-05-31

that gives a choice of downloading the audio (which takes 20 minutes or so) or downloading the lecturenotes-type "presentation" which is very quick because it is a simple PDF file.

I like the Penn collection of seminars. Just in that one semester they have talks by Thiemann, Dittrich, Lewandowski, Gambini, Okolow,...
The audio download can be done in the background while one is working on something else, and sometimes hearing the emphasis and inflection adds something

When you click on Bianca's presentation then you get lecturenotes titled


"Testing the Master Constraint Programme for LQG"
and the table of contents where it goes down a bunch of worked examples

these are not empirical tests! they are just working thru cases the way a mathematician would do to get experience and see how it goes. The table of contents lists the various cases tried

1. Review of master constraint program
2. Finite number of Abelian constraintss
3. A second class system
4. SO(3)----compact Lie group
5. SO(2,1)---Noncompact Lie group
6. Maxwell field
7. Gauss constraint for YM coupled to gravity
8. Conclusions

 

[marcus]

Reference [5] in the T.T. paper is to the Bianca preprint you mentioned a few posts back, namely

http://arxiv.org/gr-qc/0411013

and he draws on this throughout, so I need to try to figure out what is accomplished

here is a short paragraph in T.T. conclusions section, right at end:
"The proposal of [5] shows that the issue of the construction of Dirac observables for General Relativity is not as hopless as it seems. While there are many open issues even in the classical theory such as convergence and differentiability of the formal power series constructed, we now have analytical expressions available and these can be used in order to make the framework rigorous in principle. Physical insight will be necessary in order to identify the mathematically most convenient and physically most relevant clocks especially for field theories such as General Relativity..."

so I want to know why Bianca's [5] makes it more hopeful to construct Dirac observables

and I am curious about identifying physically most relevant clocks.

It sounds intriguing where he is earlier talking about using Higgs field as clock. Perhaps it is amusing, in certain way, that physicists can now not just blandly assume that they have a perfect-running ideal clock called Time, which is external to the theory and always available. As if some physicists suddenly noticed that Nature had slyly picked their pocket as she likes to tease them. And when they pat their pocket, for reassurance, they notice it is empty and that their Time watch is no longer there. It's gone! So they feel around and get a Higgs particle and inspect to see if that will do for a clock.

 

[marcus]

"reduced phase space" quantization

In Thiemann paper he is discussing the order in which you do things when you do a Dirac quantization using constraints.

there is always going to be a stage in the procedure when you squash down to just the physical states

when you first define the range of possibilities (say the "kinematical Hilbert space") it is unconstrained and there is a lot of "gauge" (artificial extra physically meaningless content) that just comes from the mathematical business of defining the possibilities

at some point you have to reduce that down, or factor it out, or kill it by applying constraints

but this process of reduction can look different when it is done at different points, so Thiemann is telling us about a method where you do some of the reduction early

he calls this "reduced phase space quantization" as constrasted with the conventional Dirac constraint quantization.

In the conventional, you build the kinematical Hilbert space and immediately, before things get any more complicated, you define operators on it. Some of those operators are called "constraint" operators and the way you squash down to the physically meaningful hilbert space is you restrict to those states on which the contraint operators vanish.
It is like defining a curve in the XY plane by choosing a function f(x,y) and stipulating that you only want (x,y) points such that f(x,y) = 0.
then the curve is the set of points where the function f vanishes.
Well, you can take a big hilbertspace containing stuff you don't want, and you can define some constraint operators and they let you pick out just the sub-hilbertspace where they vanish. that is conventionally how to get the "physical Hilbertspace" out of the bigger "kinematical Hilbertspace".

Now Thiemann is proposing to do some of the reduction first, and AFAICS he wants to identify configurations which are morphable one into the other by a smooth mapping----a diffeomorphism.
The original Gen Rel was diffeo invariant, so two things that are equivalent by a morphing are physically the same and are eventually going to be merged even in the conventional approach. So why not merge them early?
That will get rid of some of the "gauge" meaningless stuff.

But what one worries about then (and where Bianca D. helped) is if you do that preliminary reduction on the Hilbert space then do you know for certain that you can calculate with it and define self-adjoint operators on it?

Because there will be more reduction to do and you are going to want to define constraint operators there and procede along lines very similar to the conventional routine (only on a possibly simplified hilbertspace).

So you have to be able to find linear operators on the new reduced hilbertspace which are diffeomorphism invariant which is to say well-defined, so that if two states are morphly the same then the operator takes them to the same place.

The Eureka in this paper is described on page 3.

---quote from page 3---
In section 6 we combine the ideas of [ref. Bianca D] with those of [ref. T.T. masterconstraint program] by showing how the Master Constraint Programme for General Relativity can be used in order to provide spatially diffeomorphism invariant Hamiltonian constraints. The important consequence of this is that in the constraint quantization one can implement the new Hamiltonian constraints on the spatially diffeomorphism invariant Hilbert space which is not possible for the old constraints because for those the spatial diffeomorphism subalgebra is not an ideal. As a consequence, the algebra of the new Hamiltonian constraints on the spatially dfiffeomorphism invariant Hilbert space then closes on itself (albeit with structure functions rather than structure constants in general). This might pose an attractive alternative to the previous Hamiltonian constraint quantization [ref. T.T. previous LQG development].
---end quote---

I put in what the references in brackets are pointing to.

Well this is pretty sketchy, but hopefully it gives a glimpse at what is going on.

Actually I see that I am just repeating something that T.T. said in the abstract more concisely

---quote from abstract---
...We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme.
---end quote---

One may ask what the Hilbertspace of spatially diff-invariant states looks like----and it turns out that Rovelli has been discussing that quite a bit in his book, and at least if you have the right idea of a diffeomorphism, it comes down to knots. and linear combinations of knots. So this paper of T.T. has the appearance of being part of a kind of convergence of lines of development. Maybe Rovelli, in his own way and without some of the machinery, was working along this same "reduced phase space" track.