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February 28, 2007

QFT of Charged n-Particle: Gauge Theory Kinematics

Posted by Urs Schreiber

Some basic remarks on how gauge theory in nn-dimensions fits into the general framework of the charged nn-particle, followed by a semi-close look at how

Christian Fleischhack
Representations of the Weyl Algebra in Quantum Geometry
math-ph/0407006

realizes, in a continuous (instead of smooth) version of gauge theory, the algebra of observables.

1) nn-dimensional gauge theory: the nn-particle on the nn-group

One advantage of conceiving nn-vector bundles with connection on base spaces X X as functors # tra:P n(X)T n \mathrm{tra} : P_n(X) \to T_n to the structure nn-groupoid T n, T_n \,, and of conceiving all spaces in terms of categories, is that it allows to regard gauge theory on nn-dimensional spaces XX as the dynamics of nn-particles of “shape” par=P n(X) \mathrm{par} = P_n(X) (a suitable nn-category of nn-paths in XX) propagating on the target space tar=Gr n. \mathrm{tar} = Gr_n \,.

Notice the simple but important and potentially delusive fact that an nn-bundle with connection here is regarded as a configuration of the nn-particle in the structure nn-group (or rather in a suitable transport groupoid).

This is quite familiar from Chern-Simons theory (“the membrane propagating on BGB G”) and its toy version, Dijkgraaf-Witten theory (those puzzled by this might find alleviation by looking at the review in section 4.2 of Bruce Bartlett’s thesis) as well as maybe the 2-dimensional version of these (the “string on BGB G”).

2) Gauge Theory with Generalized Connections

One of the most fundamental gauge theories, both theoretically as well as for practical purposes, is Yang-Mills theory. This is a theory whose “fields” are smooth principal GG-bundles with connection on Riemannian base spaces, for GG some Lie group.

Another example is gravity. Since the Riemannian metric appearing in General Relativity may be encoded in a connection on the tangent bundle, it is possible to also conceive gravity as a theory whose fields are smooth principal SO(n)\mathrm{SO}(n)-bundles with connection.

The observation of

Abhay Ashtekar
New Variables for Classical and Quantum Gravity
Phys. Rev. Lett. 57, 2244 - 2247 (1986)

that encoding the metric in terms of its Levi-Civita connection seems to simplify the notoriously elusive quantization of gravity, gave rise to an entire field: “Loop quantum gravity”.

For much more on this see John’s TWF 18 and many other TWFs.

Curiously, the gauge-theoretic reformulation of gravity did not lead to a convergence with the familiar quantization methods usually used in the highly-developed field of quantum Yang-Mills theory.

While the latter is usually handled in terms of the smooth 1-forms encoding the connection on a principal bundle, researchers in the loop quantum gravity program chose to focus on the equivalent description of bundles with connection in terms of smooth parallel transport functors.

Locally, these are functors tra:P 1(X)Σ(G) \mathrm{tra} : P_1(X) \to \Sigma(G) which send paths to elements of the group GG.

For some reasons not really known to me (I am hoping that John will help me out here as he reads this!), there was apparently early on agreement reached in the LQG community that it would be useful to drop the assumption of smoothness on these functors, and pass to a larger realm of “generalized connections”.

(My impression is that the motivation behind this is a) the fact that it makes the space of all connections more tractable and b) that this was regarded as harmonizing well with the expectations of what quantum gravity should be like. But I do not really know.)

As Christian Fleischhack briefly reviews on p. 12 , these generalized connections are functors tra:P 1(X)Σ(G) \mathrm{tra} : P_1(X) \to \Sigma(G) obtained as inductive limits of continuous maps from the collection of edges of graphs embedded in XX to GG, under the obvious inclusion relation of graphs. This induces a topology on the space of (field) configurations conf=[P 1(X),G] \mathrm{conf} = [P_1(X),G] induced from that on GG.

(I am not, at the moment, sure what that generalization really amounts to, with respect to the situations the theory based on it applies to. Another generalization of smooth transport functors to a merely continuous setup appears in the theory of directed homotopy. It would be useful to understand if and how this is related to the generalized connections considered here.)

As usual in quantum theory, a measure on this configuration space has to be chosen by hand (which is a little sad – but we are working on it I, II).

The choice made here is called the Ashtekar-Lewandowski measure, μ 0. \mu_0 \,. It is specified by the property that its push-forward along any generalized connection yields the standard Haar-measure on the group GG.

Finally, this allows to define the space of states of the quantum theory to be that of square integrable functions L 2(conf,μ 0) L^2(\mathrm{conf},\mu_0) on configuration space conf\mathrm{conf}.

Notice that this corresponds to saying that our nn-particle propagating on Σ(G)\Sigma(G) couples to the trivial line bundle on Σ(G)\Sigma(G).


3) The Weyl Algebra on the Configuration Space of Generalized Connections

The aim of math-ph/0407006 is to construct and study the Weyl algebraof abservables” in the present setup. Technical details are abundant. I will try to extract the main idea and point out the connection to the discussion in QFT of Charged n-Particle: Algebra of Observables and Isham on Arrow Fields.

The main point is the identification of morphisms in configuration space.

The morphisms in the functor category conf[P 1(X),Σ(G)] \mathrm{conf} \subset [P_1(X),\Sigma(G)] are natural transformations of parallel transport functors, hence gauge transformation of connections.

In other words, a morphism between functors tragtra \mathrm{tra} \stackrel{g}{\to} \mathrm{tra}' is a GG-valued functions g:XGg : X \to G such that g(x) tra(γ) tra(γ) g(y) \array{ \bullet &\stackrel{g(x)}{\to}& \bullet \\ \mathrm{tra}(\gamma)\downarrow\;\;\; && \;\;\;\downarrow \mathrm{tra}'(\gamma) \\ \bullet &\stackrel{g(y)}{\to}& \bullet } for all paths xγyx \stackrel{\gamma}{\to} y

For comparison, in the smooth setup, an object in conf[P 1(X),Σ(G)] \mathrm{conf} \subset [P_1(X),\Sigma(G)] would be a smooth 1-form AA on XX with valued in the Lie algebra of GG, and a morphism AgA A \stackrel{g}{\to} A' would be a smooth GG-valued function gg in XX such that A=gAg 1+gdg 1. A' = g A g^{-1} + g d g^{-1} \,.

Every “arrow field” on configuration space, i.e. every suitable assignment of a gauge transformation at each point in configuration space, induces a corresponding translation operator on L 2(conf,μ 0)L^2(\mathrm{conf},\mu_0).

But, as familiar from the Weyl algebra of the ordinary point particle, among all those translation operators we want to single out a special subgroup.

The subgroup that Christian Fleischhack chooses is that of those gauge transformations w g S w^S_g (compare prop 3.19 and def. 3.21 on pp. 22,23) which, on every point of configuration space (i.e. on every parallel transport functor tra:P 1(X)Σ(G)\mathrm{tra} : P_1(X) \to \Sigma(G)) is given by one and the same function gg, which is required to have support on a codimension-one surface SXS \subset X.

This isn’t exactly the way Christian Fleischhack puts it, but up to plenty of technical details that I won’t bother with here, it is true that he is considering translations on configuration space that are given by gauge transformations supported on codimension 1 surfaces.

(The deeper reason for exactly this choice, which, from the point of view of general gauge theory, comes out of thin air, is really to be found in the choice of dynamics that Christian Fleischhack secretly has in mind: namely the Einstein-Hilbert action regarded as an action functional on G=SU(2)G = \mathrm{SU}(2)-connections. It is this implicit choice of dynamics which governs the design choice in the Weyl algebra in this paper – and justifies its title.)

In the language of flows on categories, we hence find, for each hypersurface SS in XX and each 1-parameter familiy tg(,t)t \mapsto g(-,t) of functions g(t):SGg(t) : S \to G a flow w g S:Σ()(t) Id conf g(t) conf Ad g(t)Flow(conf). w^S_g : \Sigma(\mathbb{R}) \ni (\bullet \stackrel{t}{\to} \bullet) \mapsto \array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ \mathrm{conf} & \;\,\Downarrow^{g(t)} & \mathrm{conf} \\ & \searrow \nearrow_{\mathrm{Ad}_{g(t)}} } \in \mathrm{Flow}(\mathrm{conf}) \,.

Translation of functions in L 2(conf,μ 0)L^2(\mathrm{conf},\mu_0) along these flows is the analog of the action of the translational Weyl operators exp(iλp) \exp(i \lambda \, p) for the 1-particle on the real line tar=P 1()\mathrm{tar} = P_1(\mathbb{R}).

Christian Fleischhack shows that the Weyl algebra on the configuration space of generalized connections found this way has, with respect to the chosen Ashtekar-Lewandowski measure μ 0\mu_0 pretty much the same nice properties as the ordinary Weyl algebra for the particle on the line.

In particular, he shows that H=L 2(conf,μ 0)H = L^2(\mathrm{conf},\mu_0) is, up to isomorphism, the unique irreducible representation of this algebra.

As discussed in great detail in Christian Fleischhack’s paper, that this is supposed to be equivalent to, or at least to play an equivalent role as, the so-called “LOST theorem”, due to

Lewandowski, Okolow, Sahlmann, Thiemann,
Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
gr-qc/0504147.

This is regarded to be central, as it uniquely fixes the space of states of the gauge theory one wishes to quantize.

The crucial assumptions going into this is, as far as I understand at least,

a) that we pass from a space of connections to a space of generalized conections

b) that the translation group in the Weyl algebra is taken to come from gauge transformations supported on hypersurfaces.


4) comparison with other work

A while ago Freed, Moore and Segal had a series of two papers on Hilbert spaces of L 2L^2-functions on configuration spaces of connections. So something very closely related to the issue in loop quantum gravity.

They consider different flavors of connections, namely just abelian connections, but also “higher” versions of them: we would say that they consider smooth nn-transport with values in the nn-groupoid Σ n(U(1)). \Sigma^n(U(1)) \,.

Of course Freed-Moore-Segal don’t use these words, they are instead thinking of these functors, equivalently, as Cheeger-Simons differential characters (whose definition comes, in fact, pretty close to that of a smooth nn-functor on closed nn-paths).

Apart from these technical issues, the problem they study is very much the same as in the kinematical part of lqg: they investigate the L 2L^2-Hilbert spaces of functions on the space of all these connections.

Their main point is a pretty interesting, because rather subtle, cohomological effect one sees here. And of course, instead of “cohomological effect”, we might say “quantum effect”!

This concerns a certain uncertainty relation, if you wish, concerning the “electric” and “magnetic” parts of our connections.

I once went through the trouble of summarizing their work:

Freed, Moore, Segal on p-Form Gauge Theory, I

Freed, Moore, Segal on p-Form Gauge Theory, II

This is very closely related to some work by Gomi, which appeared around the same time:

Gomi on Chern-Simons terms and central extensions of gerbe gauge groups

Gomi on Reps of p-Form Connection Quantum Algebras

It is remarkable

a) how very closely related this is to the starting point of lqg

b) how little interaction there has, apparently, been.

But it is all part of one idea. One could give a discussion about how Freed-Moore-Segal’s description of pp-form connections is an example for a “quantum theory of a charged nn-particle”, too.

Posted at February 28, 2007 10:08 PM UTC

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10 Comments & 3 Trackbacks

generalized connections

I am not, at the moment, sure what the generalization from smooth parallel transport to “generalized connections” really amounts to, with respect to the situations the theory based on it applies to. Another generalization of smooth transport functors to a merely continuous setup appears in the theory of directed homotopy. It would be interesting to understand if and how this is related to the generalized connections considered here.

Can anyone say anything about this?

Posted by: urs on March 1, 2007 2:56 PM | Permalink | Reply to this

LOST theorem

In particular, he shows that H=L 2(confmμ 0)H = L^2(\mathrm{conf}m\mu_0) is, up to isomorphism, the unique irreducible representation of the [Weyl] algebra.

I should point out, as discussed in great detail in Christian Fleischhack’s paper, that this is supposed to be equivalent to, or at least to play an equivalent role as, the so-called “LOST theorem”, due to

Lewandowski, Okolow, Sahlmann, Thiemann, Uniqueness of diffeomorphism invariant states on holonomy-flux algebras gr-qc/0504147.

This is regarded to be central, as it uniquely fixes the space of states of the gauge theory one wishes to quantize.

The crucial assumptions going into this is, as far as I understand at least,

a) that we pass from a space of connections to a space of generalized conections

b) that the translation group in the Weyl algebra is taken to come from gauge transformations supported on hypersurfaces.

The second point, b), comes from the choice of dynamics on our connections, which is taken to be Einstein-Gravity in these works. It would be interesting to see an analog of this theorem for the case where the Weyl algebra on the space of connections is that corresponding to the dynamics of Yang-Mills theory. Did anyone look into that?

Posted by: urs on March 1, 2007 3:23 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Gauge Theory Kinematics

The idea behind loop quantum gravity is that holonomies around loops can be used as the basic degrees of freedom in gauge theory. To make this tractable, it’s tempting to assume these holonomies as independently specifiable (modulo a few simple relations). This is only possible if we switch from smooth connections to generalized connections.

It’s this assumption which gives space a certain ‘discrete’ quality in loop quantum gravity. But, whether this assumption is physically justified is one of the huge unanswered questions hanging over loop quantum gravity!!!

I spent about 10 years working on this stuff, and this question was always one of my biggest worries.

Posted by: John Baez on March 2, 2007 12:51 AM | Permalink | Reply to this

Hilbert spaces of L2-functions on configuration spaces of connections

Thanks!

Here is a further question and a further comment:

1) a question

I am lacking intuitive access to the way these generalized connections are set up in detail using an inductive limit on maps from graphs to groups. I understand that for a fixed graph, the generalized connection would just be any map from the set EE of edges of the graph to the group GG (i.e. a mere morphism of sets). The collection of all such maps for a fixed graph is then naturally a topological space using the topology of GG.

As I take the inductive limit of that in order to get something that I can apply to an arbitrary path, what is it, intuitively, that I get?

Is it in any way a continuous functor from the topological category of paths to the topological category Σ(G)\Sigma(G) (and hence possibly related to work in directed homotopy theory)?

I would think “no”, right? It seems that, at least roughly, the value of a generalized connection on a path is something like the infinite product of all group elements assigned independently to each point of the path.

(Sorry for going on about this, I am just trying to come to grips with how general generalized connections really are.)

2) a comment

Just for the record (and actually for reminding myself about it) I note that a while ago Freed, Moore and Segal had a series of two papers on Hilbert spaces of L 2L^2-functions on configuration spaces of connections. So something very closely related to the issue in loop quantum gravity.

Of course they consider different flavors of connections, namely just abelian connections, but also “higher” versions of them: we would say that they consider smooth nn-transport with values in the nn-groupoid Σ n(U(1)). \Sigma^n(U(1)) \,.

Of course Freed-Moore-Segal don’t use these words, they are instead thinking of these functors, equivalently, as Cheeger-Simons differential characters (whose definition comes, in fact (as you know better than I do, I am just saying this for those listening to us talking about this…), pretty close to that of a smooth nn-functor on closed nn-paths).

Apart from these technical issues, the problem they study is very much the same as in the kinematical part of lqg: they investigate the L 2L^2-Hilbert spaces of functions on the space of all these connections.

Their main point is a pretty interesting, because rather subtle, cohomological effect one sees here. And of course, instead of “cohomological effect”, we might say “quantum effect”!

This concerns a certain uncertainty relation, if you wish, concerning the “electric” and “magnetic” parts of our connections.

I once went through the trouble of summarizing their work:

Freed, Moore, Segal on p-Form Gauge Theory, I

Freed, Moore, Segal on p-Form Gauge Theory, II

This is very closely related to some work by Gomi, which appeared around the same time:

Gomi on Chern-Simons terms and central extensions of gerbe gauge groups

Gomi on Reps of p-Form Connection Quantum Algebras

It is remarkable

a) how very closely related this is to the starting point of lqg

b) how little interaction there has, apparently, been.

But okay, here at the nn-Café we enjoy pondering universal language and seeing how all kinds of things are special cases of that: maybe I should next write an entry about how Freed-Moore-Segal’s discussion of pp-form connections is an example for a “quantum theory of a charged nn-particle”, too.

Hm…

Posted by: urs on March 2, 2007 11:12 AM | Permalink | Reply to this

Re: Hilbert spaces of L2-functions on configuration spaces of connections

Could you expound on how differential characters,
rather than differential forms, are related
to your approach to connections? or have you explained that elsewhere?

jim

Posted by: jim stasheff on March 2, 2007 2:46 PM | Permalink | Reply to this

Re: Hilbert spaces of L2-functions on configuration spaces of connections

Could you expound on how differential characters, rather than differential forms, are related to your approach to connections?

What I was referring to was that the definition of differential characters is rather close in spirit to the more general idea of an nn-connection being something that sends nn-paths to an nn-group.

A Cheeger-Simons differential character (as briefly reviewed at the beginning of this entry) is defined to be something that sends nn-cycles to the group U(1)U(1).

This is to be interpreted as nothing but the assignment of the holonomy of an nn-connection to that nn-cycle.

The smoothness condition on this assignment is, on the other hand, still encoded in terms of pp-forms: one demands that this assignment of phases to cycles is such that it comes from the integral of an n+1n+1 form whenever the cycle is in fact a boundary.

In this sense, the definition of Cheeger-Simons differential characters is somewhere in between that of Deligne cohomology, which expresses everything in terms of local pp-forms, and the description in terms of smooth nn-anafunctors, which describes everything in terms of parallel transport maps.

All of these, though, are equivalent (while only the last formulation directly generalizes to more general, for instance nonabelian, codomains).

Posted by: urs on March 2, 2007 3:14 PM | Permalink | Reply to this

Re: Hilbert spaces of L2-functions on configuration spaces of connections

Just for the record and for my own benefit, I notice that some aspects of my question above are addressed in

C. Fleischhack, Regular Connections among Generalized Connections.

This shows that ordinary smooth connections form a subset of (Ashtekar-Lewandowski-)measure 0 in the space of all generalized connections, while being dense in that space iff the structure group is connected.

There is a curious construction involved in demonstrating this, namely a non-smooth (possibly even non-continuous) global trivialization of a smooth bundle (i.e. a trivializations at the level of sets: this always exists, unless, as Toby Bartels might maybe remark, you don’t buy into the Axiom of Choice in Set\mathrm{Set} ;-).

Unless I am misunderstanding something, the space of generalized connections never contains any information about any possibly nontrivial principal bundles.

The introduction emphasizes the original motivation for the – apparently – drastic step from ordinary to generalized connections: when quantizing a gauge theory we need a measure on the space of connections, and that is hard to come by for smooth connections.

At least for the space of connections modulo gauge transformations. But should that come as a surprise?

What is, rigorously, known about measures on spaces of smooth connections not divided out by gauge transformations?

Probably this question is simpler in the case of Yang-Mills theory, where the base space carries a Riemannian structure, and hence a measure. This should induce a measure on the space of paths and turn a connection into a function between spaces with non-pathological measure.

I’ll see if I can find any literature concerned with this case…

Posted by: urs on March 22, 2007 7:38 PM | Permalink | Reply to this

Re: Hilbert spaces of L2-functions on configuration spaces of connections

I just consulted my local expert on these matters. Before I forget what we talked about, here some notes.

1)

First of all, my above description of the Weyl algebra is, unfortunately, not quite right. The translation operations are not gauge transformations supported on codimension one surfaces, but something similar with a funny twist in the signs.

So the value of a generalized parallel transport over a path piercing right through that hypersurface is not invariant, but changes by tra(γ) 2tra(γ 1)tra(γ) 2d 2tra(γ 1), \mathrm{tra}(\gamma)_2 \circ \mathrm{tra}(\gamma_1) \mapsto \mathrm{tra}(\gamma)_2 \circ d^2 \circ \mathrm{tra}(\gamma_1) \,, where d 2d^2 is some group element (and not dd 1d d^{-1}, as I thought).

Too bad. This means that these Weyl transformations are not the ones naturally encoded in the configuration space category.

(But possibly this just means that the config space category should contain more morphisms than ordinarily present in [P 1(X),ΣG][P_1(X),\Sigma G]…)

2)

There is apparently \sim 10 year old work by, probably, Lewandowsky and somebody else (we are still trying to track down the reference) which shows that for a fixed smooth GG-bundle PXP \to X, there is a topology on the space of thin-homotopy classes of loops (apparently called “hoops”) such that smooth connections on PP are equivalent to continuous functor from these loops to the group.

(I would like to better understand continuous parallel transport. We have talked about that here.)

If anyone reading this here knows where this result by Lewandowsky (or maybe somebody else) can be found, I’d be grateful for the reference.

3)

What would enlarging the configuration space from smooth to generalized connections correspond to, for the simple example of the particle on the line? Answer: there the config space is \mathbb{R}, and doing an analogous generalization turns this config space into the Bohr compactification of \mathbb{R}.

While this is not a step that we usually do when quantizing the particle, apparently there exists some people who have thought about how much of quantum mechanics carries over to the passage from config spaces to their Bohr compactifications. (?)

4)

In a way the underlying reason for this enlargement of config space is that one completes the algebra of functions on config space with respect to a unusual norm, namely the supremum norm (as opposed to an L 2L^2-norm as one usually does in quantum mechanics).

Let’s see if I can reproduce this more precisely:

For aa the space of smooth connections on a fixed smooth GG-bundle PP, and H(a,) H(a,\mathbb{C}) the space of smooth Wilson loop functions on this space (which evaluate a connection over a given loop and trace the result with respect to some representation), the completion of H(a,)H(a,\mathbb{C}) under the supremum norm H¯(a,) \bar H(a,\mathbb{C}) is isomorphic to the space of functions on generalized connections. (A generalized connection is a parallel transport functor with no condition on smoothness or continuity).

Or at least roughly. I am probably missing lots of details here.

5) While the above Weyl transformations don’t respect any subspace of smooth connections inside generalized connections, it’s dual action does respect the space of cylinder functions on smooth connections, i.e. of those functions that evaluate any smooth connection on at most finitely many paths.

This dual action is the obvious one: for example let f γf_\gamma be the cylinder function f γ:Atra A(γ) f_\gamma : A \to \mathrm{tra}_A(\gamma) which evaluates the parallel transport of the smooth connection AA over the path gamma, and let γ\gamma pierce a codim 1 hypersurface such that it splits in two parts γ=γ 2γ 1\gamma = \gamma_2 \circ \gamma_1, then acting with the “translation” operator on f γf_\gamma turns it into the assignment Tf γ:Atra γ 1d 2tra γ 2. T f_\gamma : A \mapsto \mathrm{tra}_{\gamma_1} \circ d^2 \circ \mathrm{tra}_{\gamma_2} \,. This is still a cylinder function on smooth conenctions, certainly.

Now, I don’t remember what the topology or smooth structure on the space of cylinder functions is taken to be, and whether this action is continuous or smooth with respect to this translation operation.

Posted by: urs on March 23, 2007 2:18 PM | Permalink | Reply to this

Re: Hilbert spaces of L2-functions on configuration spaces of connections

(I’m not sure if you are still interested in this question…)

The paper you were looking for is probably Lewandowski Group of loops, holonomy maps, path bundle and path connection .

However continuity of the holonomy as a map from the space of thin-homotopy classes of loops to the structure group is not enough to yield a smooth bundle and a smooth connection. For this some ‘smoothness’ conditions are required which are looking very similar to the convenient setting of Michor and Kriegl. (I’m very interested if this resemblance can be made rigorous.)

Posted by: Tobias Diez on February 8, 2014 2:34 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Gauge Theory Kinematics

As was kindly pointed out to me by email, the above links to the entries on Freed-Moore-Segal’s work were broken.

Should be fixed now.

Posted by: urs on March 2, 2007 3:02 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field
Weblog: The n-Category Café
Excerpt: The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
Tracked: March 5, 2007 4:23 PM
Read the post History of Understanding Bundles with Connection using Parallel Transport around Loops
Weblog: The n-Category Café
Excerpt: A list of some papers involved in the historical development of the idea of expressing bundles with connection in terms of their parallel transport around loops.
Tracked: March 23, 2007 6:41 PM
Read the post The First Edge of the Cube
Weblog: The n-Category Café
Excerpt: The notion of smooth local i-trivialization of transport n-functors for n=1.
Tracked: May 4, 2007 8:59 PM

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