Thomas Thiemann: Thank you very much. Um. So I was uh given the task by the organizers of the seminar to somehow summarize
Thomas Thiemann: um. There's a paper that somehow explains a kind of new strategy.
Thomas Thiemann: Um! Two papers that are discussed, models which somehow support the strategy. And then another set of two papers which contain some mathematical tools.
Thomas Thiemann: So um! Here's the table of contents of my talk. First uh the first part concerns the the motivation. Um, uh, So I will discuss the interplay between the hyper surface, the information algebra, quantum non degeneracy, and no notion
Thomas Thiemann: of a non degeneracy of quantum metrics, anomalies, and density weights that have been.
Um,
Thomas Thiemann: then uh, I will um have a look at uh lease molens uh model the u one cubed truncation of uh Euclidean vacuum quantum gravity, and show that here in this model all these notions are nicely fitting together.
Thomas Thiemann: And uh, this then breaks me to um uh the last part of the talk where I discuss renormalization
Thomas Thiemann: uh, as a possibility to arrive at this quantum non degeneracy
Thomas Thiemann: which will see it is probably a key notion that has been much overlooked in the past.
Thomas Thiemann: So let me come to the first part of the talk. Um! So what is the
Thomas Thiemann: uh? I should. I will say at the beginning that this is a lot of
Thomas Thiemann: so um. What is the classical hyper surface? The formation algebra will let me denote by capital. Q. And what follows: The square of the determinant of the spatial metric
Thomas Thiemann: capital. D is the density we wait W.
Thomas Thiemann: Equal to one spatial different of them. Constraint and Cw. Is the density weight W.
Thomas Thiemann: Uh Hamiltonian constraint,
Thomas Thiemann: Then, as is of hopefully well known,
Thomas Thiemann: uh in in January, Covent theory, the quas on bracket between two Hamiltonian constraints is given by a spatial different office and constraint. However, it's not the but the so called open algebra, because the this quantity queue
Thomas Thiemann: appears on the right hand side as A. So in the definition of this vector field which this spatial de humorphism is smeared
Thomas Thiemann: uh
Thomas Thiemann: uh two observations with kind of two observations is that uh, this algebra makes only sense. Uh, if the metric is not the generate,
Thomas Thiemann: unless you have w bigger than equal to two,
Thomas Thiemann: and it's, of course, to real. If the integral of this spatial, different office and constraints has somehow the back manager, Zero support.
Thomas Thiemann: The reason why I mentioned this will become clear, and what follows.
Thomas Thiemann: So then, come, let us come to this choice of density, weights, and which which choices are made in in quantum new quantum gravity in particular, is somehow dictated by the dynamics.
Thomas Thiemann: So, first of all, every single term in the Hamiltonian constraint, be it the vacuum, constant contribution, the Ransen or Euclidean one the cosmological, constant, or any matter contribution. It couples to the electric field
Thomas Thiemann: to look for. Uh Hilbert space representations,
Thomas Thiemann: uh which, of course, should be such that the Hamiltonian constraint be densely defined in the space, and to pick a vacuum which is killed by the electric field.
Thomas Thiemann: Uh. But this, of course, has the uh media problem that then this vacuum is a zero eigenvector of queue.
Thomas Thiemann: So the vacuum itself is, of course, quantum degenerate in this sense
Thomas Thiemann: uh and here are two little limits which you can show uh
Thomas Thiemann: This single one line uh observation already fixes a so-called non-off, or for turing type of representation of which the Ashtaker level Doski representation is a particular example,
Thomas Thiemann: and this is a representation in which one of the variables is a discontinuous cannot be defined, they can only define exponentials, while the other is continuous.
Thomas Thiemann: And the second observation is that um whenever there is at least a quadratic piece in uh for the um uh
Thomas Thiemann: the potential of your gauge field, which is the case for uh young bills, and also for the clean piece of of uh general relativity. And this is such a quadratic piece.
Thomas Thiemann: And if you want the uh hamil and the volume operator to be well-defined, which is kind of
Thomas Thiemann: necessary in uh, in order, for example, to define
Thomas Thiemann: matter terms of the cosmological, constant, and so on.
Thomas Thiemann: Uh and rem on some
Thomas Thiemann: um
Thomas Thiemann: regularization of the Hamiltonian constraint
Thomas Thiemann: over tetrahedral cells of forward and volume absent to the power of D is densely defined if and only if the density rate is equal to one,
Thomas Thiemann: and if a and e are smeared in one and two miles, one the minutes respectively. So in two dimensions. If
Thomas Thiemann: if You' in for space-time dimensions, which is, of course, what we do?
Thomas Thiemann: Um. So let us go back now to the hyper surface deformation. It is well-defined
Thomas Thiemann: if and only if the classical one is well-defined, if and only if uh who is bit? Q. Is bigger than zero for the density weights w equal to one
Thomas Thiemann: um
Thomas Thiemann: so we in the classical view. We want Q. To be bigger than what bigger than zero, so the the matrix should be should be no degenerate.
Thomas Thiemann: On the other hand, in the loop quantum gravity, Herbert Space. We have the spin network functions as a basis
Thomas Thiemann: uh which define a domain capital D,
Thomas Thiemann: labeled by Graphs late finite graph, with final number of vertices, and therefore the quantumatic in this sense is actually
oops.
Thomas Thiemann: The quantum magic is in this sense uh completely degenerate, so it has zero volume, the back almost everywhere.
Thomas Thiemann: And that's that explains why the absent to zero limit in our we month some approximation of the Hamiltonian constraint is actually a quite delicate thing to do, and then knife uh the knife limit would be would be infinite if Q is uh allowed to be. Zero
Thomas Thiemann: can be done in particular by uh the to shawn off our regularization, or using some for some bracket identities which are being used in uh blue quantum cosmology all over the place.
Thomas Thiemann: So um
Thomas Thiemann: how to take this limit. And there have been basically two proposals in the literature. Um. One is using an operator topology that exploits the fact that, uh, we have at our disposal the notion of, If you're more in advance,
Thomas Thiemann: so uh early ideas among those lines would be uh expelled by, and then later also by myself.
Thomas Thiemann: And uh, the second option is to go on to the distributional dual of these uh dense domain of uh, the span of spin network functions, and this has been called habitat kind of spaces first introduced by with all. For
Thomas Thiemann: uh all go
Thomas Thiemann: um you work, and uh,
Thomas Thiemann: but in both cases one gets uh um one why one gets closure. So there's no mathematical consistency in consistency. But there are still anomalies. So we get in the a option we get non-trivial, but wrong quantum structure functions,
Thomas Thiemann: and in the B road we get to real and therefore also wrong quantum structure functions.
Thomas Thiemann: So what is the origin of this T.
Thomas Thiemann: This, or the origin of this tension becomes obvious when you look at the um right hand side of the Poisson bracket of two Hamiltonian constraints which I brought here. Top line again. One more time for W. For density, general density rate. W.
Thomas Thiemann: Uh, and here we see that? Uh,
Thomas Thiemann: you see that, uh,
Thomas Thiemann: the classical Riemann sum contents an order of absent to the miles. Deep terms
Thomas Thiemann: of size, absolute power to our D.
Thomas Thiemann: The quantum sum, all the contents and gamma terms where and gamma is a number of vertices of a spin over function
Thomas Thiemann: and of size, Epsilon, because this derivative here, which in the class in the casting here is actually a derivative becomes a difference one hundred and fifty,
Thomas Thiemann: and the differences of all the epsilon.
Thomas Thiemann: And now what happens is, of course, is that in the classical theory these two powers of Epsilon just cancel each other, and you get an order, One contribution. So when you get to classical, integral, which is where you find a non vanishing.
Thomas Thiemann: But in the in the quantum theory we get an order of epsilon to the power. One times the finite and constant number of vertices which goes to zero in the limit absent, goes to zero.
Thomas Thiemann: And this is exactly what those people that introduced the habitat found. So in some sense they were completely correct.
Thomas Thiemann: Uh, because
Thomas Thiemann: what we do in the quantum here is somehow to replace the the argument of this stuff, especially if you m offers, and constraint by something which has support on a Lebeck measures you will set,
Thomas Thiemann: so that is of course not satisfying, and therefore, uh, one can
Thomas Thiemann: one can think what what to improve. So one one option is to go non degenerate to non-degenerate back here. In fact, there has been a proposal to
Thomas Thiemann: long time ago by Um, Tim Koslowski and Hano Zaman. It's for the consider uh a vacuum representation. There's a vacuum that has a a condensate. So the queue expectation value is different from zero,
Thomas Thiemann: or one can try to
Thomas Thiemann: get there out this problem that uh, you have this Epsilon to the power one contribution Times and Gamma going to zero. So we want to try to cancel this one. This Epsilon, by something which is one over absent. This forces you to go to non-standard density weight between two and one. This is
Thomas Thiemann: uh one of the aspects of the recent work by by mother one.
Thomas Thiemann: Um! So when you when you try to do this uh uh, in the first option, you will unfortunately find that the backup is not in the domain of the Hamiltonian constraint.
Thomas Thiemann: Um. And also
Thomas Thiemann: in order to get something close to the right hand side of the possibility of two Hamiltonian constraints you would like still to uh get, not an constant number of terms, but an infinite number of terms, in order to get uh something that uh resembles the classical interval.
Thomas Thiemann: Um,
Thomas Thiemann: With respect to the changing of the um
Thomas Thiemann: and density. Wait. This is,
Thomas Thiemann: or this electric shift strategy by a by and another one,
Thomas Thiemann: which it seems to be to at least to me at the moment very much tailored to Euclidean, back from quantum gravity. Possibly you can extend it to the Lorentz in case using a big transform.
Thomas Thiemann: Uh, I don't see how this could be uh uh transferred
Thomas Thiemann: the for W equal to one.
Thomas Thiemann: The classical hyper surface deformation. Algebra requires that the metrics be non-degenerate,
Thomas Thiemann: how unfortunately, our look quantum gravity. Spindle functions are quantum degenerate almost everywhere
Thomas Thiemann: and that's the reason why, in quantum representations of the hyper surface deformation, algebra, we meet severe difficulties.
Thomas Thiemann: So somehow our quantum representation violates the necessary assumption about the very definition of age,
Thomas Thiemann: and therefore a a a strategy which seems to be promising is to try to find new representations which are quantum non degenerate.
Thomas Thiemann: So, in in other words, it might be a a good idea to make quantum degeneracy a part of the definition of an normally freeness.
Thomas Thiemann: And this um um rents the further plan of my talk. So
Thomas Thiemann: in the second part um I
Thomas Thiemann: Lee, I hold your one cube model
Thomas Thiemann: and show that in a non digitized uh representation, which is slightly different from the I could be at Qg. Representation, but still closely related. You get uh, you can implement all of this. You get an exact, unnormally free
Thomas Thiemann: quantum nondegenerate quantization of this model, which is very close to being quantum integrable,
Thomas Thiemann: and in the second or last part of the talk, then I will go beyond this model, because this model is still much simpler than uh one hundred and fifty
Thomas Thiemann: full quantum gravity. And here I suggest that we we use methods of randomization
Thomas Thiemann: to um derive a systematic constructions of uh quantum non-generated representations.
Thomas Thiemann: So
Thomas Thiemann: let me first go to the you one cube model, then I'll show uh how these features, which I mentioned before nicely fit to each, are here.
Thomas Thiemann: So what is the
Thomas Thiemann: what is the uh you want? Cube model for those who who don't know it.
Thomas Thiemann: So take Euclidean vacuum, so you can. You can define it Both Hamiltonian and Lagrangian way. Hamiltonian definition is the the one that has been uh given by the in the first place. So you take you played in vacuum uh general relativity in
Thomas Thiemann: and drop the a squared term from the density one
Thomas Thiemann: Hamiltonian constraint,
Thomas Thiemann: or, if you prefer the uh lagrangian definition, you do it almost the same thing. But you start with the self to a back from general relativity, of course, in our Euclidean signature,
Thomas Thiemann: and again drop the A squared from from from the lagrangian.
Thomas Thiemann: So this is almost you believe in vacuum gravity, but with a building structure group. So instead of su two, you have v one cubed.
Thomas Thiemann: Um. What is nice about this model? Is that uh the hyper surface deformation algebra is completely unchanged. Um, in particular, we still have non non-polynomial structure functions,
Thomas Thiemann: and therefore it is an ideal test laboratory. For many technical and conceptual issues of quantum gravity.
Thomas Thiemann: Uh, and there's been a lot of work by about um Madavan in the past. Um! And he had. He, he was able to show that uh, you can go much further with this uh opinion more than in uh, and within the actual quantum we have to
Thomas Thiemann: to you, because it is simpler,
Thomas Thiemann: but it still has this uh not toable hyper-surface deformation either. And therefore it it enables us to shed light on the uh set of questions that I was was raising before.
Thomas Thiemann: So now i'm going to quantize this model, but not using charge network states uh that have been used in the past in particular in the world by uh
Thomas Thiemann: another one. But i'm using a slightly more general uh set of functions,
Thomas Thiemann: i'm using uh a more general form factor, as people call it. So i'm smearing the the connection, not with some
Thomas Thiemann: singular uh functions that are supported on one-dimensional lines. But i'm using more general speaking functions.
Thomas Thiemann: Um, I'm still using a vacuum which is killed by the electric field
Thomas Thiemann: and W. Of f here denotes so-called generalize for now always hold on if you want, and this is still a a quite uh discontinuous representation, which you can see from the fact that the expectation value of a generalized hold on me,
Thomas Thiemann: this uh one, if smearing function is equal to zero and otherwise. Thomas I'm. Confused in the last uh integral is the three X. What? What's one dimensional there? I I got something wrong.
Thomas Thiemann: So in in the um, if you want to, if i'm not sure if you are familiar with charge network functions.
Thomas Thiemann: Okay, charge network functions are somehow the abuse in analog of uh, the spin network functions with the support on the one line is that the same? Exactly. Okay. Got it? Yeah, thank you. This is the that's really a one-dimensional integral at the end of the day.
Thomas Thiemann: So if I want to. If I want to have a singular sphereing functions corresponding to a spin network or a charge network functions I would have f support on one-dimensional lines one hundred.
Thomas Thiemann: But what i'm doing here is I'm. I'm. Looking at more general experiences which do not necessarily have that property.
Thomas Thiemann: So, as I said, this in this can be viewed as generalized for long ways. If you want
Thomas Thiemann: erez Agmoni and his s in in in in the quantum gravity, these w of F is how long we are discontinuous. So the the operator, corresponding to the connection cannot be defined by fluxes, are continuous, or the electric field can be defined one hundred and fifty
Thomas Thiemann: um geometrical operators, like the volume or the the error error operator uh have the property that they are diagonal in this representation, so i'm
Thomas Thiemann: trivial, almost trivial, to compute the eigenvalue of the volume operator for the region R. In this representation it's just the classical integral over the square of the modulus of the determinant of F.
Thomas Thiemann: So it's also quite easy to spot those States, so
Thomas Thiemann: it's quite easy to spot those states um which
Thomas Thiemann: no degenerate. So in this case, of course, this determinant of F should be not zero.
Thomas Thiemann: It is also quite easy to solve. The Gauss constraint is to address those smell functions which have vanishing divergence.
Thomas Thiemann: But this was, of course, already known. Now we come to the interesting part which is spatial diffumor, and the Hamiltonian constraint; and since we,
Thomas Thiemann: uh, we cannot directly define these these operators corresponding to them, because the a operator does not exist in this representation, just as in blue quantum gravity.
Thomas Thiemann: Therefore, uh, we cannot directly talk about representation of the hyper-surface, the information algebra of course.
Thomas Thiemann: But And now this is the uh, the the new insight, and something that really shocked me when I saw this is that you can actually exponentiate them.
Thomas Thiemann: So for the U one cube quantum gravity model of you. Uk and um
Thomas Thiemann: vacuum gravity. You can actually define unitary operators corresponding to both,
Thomas Thiemann: especially if you more often consent, and the Hamiltonian constraint,
Thomas Thiemann: using
Thomas Thiemann: the exponential of both,
Thomas Thiemann: and I can, even in one line, just write down how
Thomas Thiemann: this operate the acts
Thomas Thiemann: it acts on.
Thomas Thiemann: You may call it a generalized spin network function, or a generalized Halloween, by changing the label of the generalized, follow me in the following way: So if this is denoted by the this thing here, which is E. To the X. W. U. And M. You, is this smearing function of the spatial different offence? Constraint?
Thomas Thiemann: M. Is the smearing function of the Hamiltonian constraint and W. Is the density weight,
Thomas Thiemann: And what is K.
Thomas Thiemann: K. Is the uh is a function on the face space of this model which selects from uh, from G and F. Where G is the uh configuration variable, and F is the momentum variable the momentum variable.
simone: Uh, that was a question. But I didn't get it.
simone: Three dimensions. How does that work?
Thomas Thiemann: Um!
Thomas Thiemann: So the the definition of the inner product is in the top line, just the first entry.
Thomas Thiemann: So all you need to all, All you need to know is that the expectation value of a generalized follow me, is the chronic or Delta. Everything else follows from that.
Thomas Thiemann: I can do that.
Thomas Thiemann: But I don't need to.
Thomas Thiemann: This is uh, this is the Gns construction,
simone: so maybe it would be nice if you. Could you explain, then, remind us why, in the usual case we need to do that then.
simone: So in the usual case, if you want a background, independent uh scalar product.
simone: It would be nice if you remind us why, in the usual case, we are forced to do that. And what are you?
Thomas Thiemann: You're not I can. I can. Also I can also, in the usual case, define uh, what I mean by hold on to me and uh spin that work functions. And then i'm just saying that the expectation value of the connect or function is zero unless it's the through the zoom, connect, function. Everything else follows from that
Lee: questions.
Lee: So I I think what we're wanting to here is something about when you're in the so real quantum gravity, then these functions are still too singular, and it's because they're not higher in order terms, because we're in. You want to keep gravity that we can do this
Lee: is that is, that the start of a correct argument.
Thomas Thiemann: So the in in in the nominal building case, of course, if you want a nice or let's say a covariant behavior under one hundred
Thomas Thiemann: uh of functions under the uh. As for under Gauss gauge transformations, then you may wonder, uh, what kind of functions do such have such properties, and then you uh you uh um motivated to go into these uh singular functions,
Lee: right? Which you end up trying to do is smear loops. May, or a lot of things over something like the exponential of the
Thomas Thiemann: right.
Lee: And and that's to singular in the
Thomas Thiemann: yeah. But I think that uh um! The the previous question was was just asking whether one can actually define the representation. Also, you're just using this uh um
Thomas Thiemann: expectation values, and that is, that is indeed the case.
simone: No, The previous question was really uh is a singular distribution, a singular meeting necessary in order to have a background independent uh measure. And you. Your answer is no
simone: including in the
Thomas Thiemann: So suppose we do not want to have an explicit solution of the Gauss constraint.
Thomas Thiemann: Okay.
Thomas Thiemann: So I'm: I'm: I'm looking at
Thomas Thiemann: now three constraints, the the goals constraint, the spatial different, often consent, and the Hamiltonian concerned. I'm. Just want to represent them on some Hilbert space.
Thomas Thiemann: Then I can do exactly what I'm doing here
Thomas Thiemann: in the top line.
Thomas Thiemann: This
Thomas Thiemann: generous meeting functions.
Thomas Thiemann: Okay,
simone: Yeah. Yeah. So your answer is that the background dependent measure does not require singers meeting even in the case. It's only finding a simple solution to the Uh Gauss constraint that requires a singulars meeting. That's what you say? Okay, Thank you.
Thomas Thiemann: I mean, maybe it does not even require that. But we we we choose to do that uh, because it gives you a nice way to solve the galaxy, but maybe it's not the only way to do it. I just don't know.
Thomas Thiemann: But here in this model, of course, one uh one can much simpler. So the cost consent by just using It's meeting functions, Which are,
Thomas Thiemann: we have banishing diversions.
Thomas Thiemann: Can you hear me?
Thomas Thiemann: Yes, okay. Yes. So I I was trying to explain how the Hamiltonian constraint acts on these Uh States in the about space and at that. So by using the uh exponential of the Hamiltonian vector field
Thomas Thiemann: of the some of the different was from often and Hamiltonian constraint.
Thomas Thiemann: And um!
Thomas Thiemann: If you think about it, this is exactly the way. One can also discover how the spatial, different morphine constraint acts on the loop quantum gravity about space.
Thomas Thiemann: And here in this model it turns out that the same can be done
Thomas Thiemann: for the Hamiltonian constraint.
Thomas Thiemann: And um
Thomas Thiemann: to me this is the first quantum realization of what one what people call the Bergmann Quotmar roof
wolfgang wieland: and it you you can, of course, just right. Excuse me. Pardon me, um! Can I just can uh both of you. Could we just jump back to the last slide. I have a quick question.
wolfgang wieland: Maybe it could also clarify some things. So um! When you explain how the exponential of the Hamiltonian and vector constraint acts on the smearing function.
wolfgang wieland: I am kind of puzzled because I I would think of this meeting function as something that is
wolfgang wieland: only dependent on littlelex like the coordinates on uh three space. But uh the Hamiltonian and the diff your constraint. They are um. They act on field space, on on the configuration variables classically. So I don't quite understand how they how this action is defined.
Thomas Thiemann: Could you please clarify, Sure. So the uh thing of
Thomas Thiemann: so K. Of of zero. And F is basically F:
Thomas Thiemann: Okay,
Thomas Thiemann: and uh, what the Hamiltonian vector fuel does is it acts? It acts on on the electric field. And in order that you you get something which is again just depending on the electric field
Thomas Thiemann: in the connection.
Thomas Thiemann: So if you think about a constraint which is linear momentum. So i'm thinking of a as a momentum
Thomas Thiemann: time, something which is arbitrary, complicated, and in the configuration variable. The Hamiltonian vector field preserves the
Thomas Thiemann: functions which depends only on
Thomas Thiemann: uh on the.
Thomas Thiemann: This is how
Thomas Thiemann: this is why this um action is well-defined.
Thomas Thiemann: Okay, thank you. This is also the reason why this simple solution does not work for actual quantum gravity, because there, uh, we have not a linear dependence of the constraints on a but a quadratic dependence.
Lee: Question. Yes, yes.
Lee: Is there something like a dual volume? That is, if you take the linear the
Lee: and treated like the E. Exactly and right to state which is a function of P, which is the integral over the square root of it. Is that a good state for you?
Thomas Thiemann: Um! So you mean kind of a Jo and Simon's did.
Lee: No, no, it's it's. It's just if I pretend that my is
Lee: that's the determinant of my A, Because I can still take this determinant, even if it is only one queue.
Lee: Um that I can write a volume with that.
Lee: And this is the volume of the exponential of the volume. A good stage for you of the dual volume
Thomas Thiemann: in the to determine on of B
Thomas Thiemann: will be so. That would be something which is, uh uh higher than linear in in a, so that would not be well-defined.
Thomas Thiemann: Um! So you can. Actually, you can, of course, just write down this this model and check that um hyper surface. The information algebra is well represented, but can you can also use it? Just
Thomas Thiemann: a in terms of sinus of A, and so on, and you you derive this representation uh, and this uh action of the Hamiltonian constraint, using,
Thomas Thiemann: uh polymerization and standard tools of your continuity
Thomas Thiemann: one. But you can be more general, and it but it also works for the natural, then, that you wait
W. Equal to one.
Thomas Thiemann: The operators corresponding to Hamiltonian and uh sp spatial, different, often consent, act, in fact, unitarily on the Hilbert space.
Thomas Thiemann: And, as I said before, uh, when you, when you set m equal to zero, this returns for you the action of the spatial. If you'm off in that, we know and laugh from two quantity
Thomas Thiemann: Um, you can implement this uh uh algebra directly on the on, the on the Herbert space. So we do not need to go to this uh dual space,
Thomas Thiemann: and the anomalyfulness is manifest. So the quantum algebra is exactly encoded in the Hamiltonian flow of the classical constraints
Thomas Thiemann: and um. The The fact that we need to be known to Janet is also spelled out quite nicely here, because this uh classical algebra, of course, is again still defined. If the Hamiltonian vector fields
Thomas Thiemann: um uh do not exist, which is the case for
Thomas Thiemann: um
Thomas Thiemann: also nice is that it? It makes contact to a recent uh
Thomas Thiemann: uh discoveries. For example, there is this electrification covariance, and then, if you mobile perspective by a by
Thomas Thiemann: um, you can compute
Thomas Thiemann: the Hamiltonian floor of this uh
Thomas Thiemann: Hamiltonian constraint for arbitrary density. Weight
Thomas Thiemann: of this constraint
Thomas Thiemann: uh delivers that's in order for the numbers of n plus one in F, depending on spatial derivatives of n of so of the
Thomas Thiemann: uh of of of order, and
Thomas Thiemann: erez agmoni. Um. Now let me try to get as close as possible to the loop quantum we are. These states to spend a lot of functions, So it's been that we functions would correspond to this kind of singular one.
Thomas Thiemann: Smearing functions which are supported on the graph,
Thomas Thiemann: so to say, second out spin network functions is that the action X. The actions along the whole graph, not only at the vertices, there is no abrupt loop attachment
Thomas Thiemann: and the action is very non-polynomial on the charges. So, in in other words, the in the obedient analog of the spin that spin labels
Thomas Thiemann: you can also use uh these habitats that were introduced uh um by with all, for at our um, anyway, and then have access also the to the hyper-surface uh the information algebra, not to only to it's exponential, and they are also.
Thomas Thiemann: Uh, we get an normally freeness by construction.
Thomas Thiemann: Um! You can go even further. Uh, so we can try to solve this model, and uh um
Thomas Thiemann: compute the Iraq observers, and so on, so forth. Go to the physical Herbert Space, compute the physical Hamiltonian,
Thomas Thiemann: and and this is, in fact, possible in this model. So group averaging versus reduce face-based quantization, we can. You find relational observvers and the physical Hibbert Space
Thomas Thiemann: Um and the physical expert space and Hamiltonian lead to nonlinear self-interesting electrodynamics
Thomas Thiemann: and self interacting means that uh we can compute the analog of endpoint functions and these are not determined by the two point functions. So the model, the physical Hamiltonian is interacting in that sense.
Thomas Thiemann: Uh, listen to in the similar
Thomas Thiemann: and um.
Thomas Thiemann: Finally, one can also try to define a spin form model from this model, uh, from from starting from the Hamiltonian interior. And here, rather than having formal lead back measures,
Thomas Thiemann: um, we can derive the model um, using discrete and board measures from first principles,
Thomas Thiemann: which I think,
uh, what should you get
Thomas Thiemann: erez agmoni? So the you want cute quantum, all by by introduced by B. Lee is is close to being quantum integrable, at least in this nano for tearing type of representation one hundred and fifty.
Thomas Thiemann: Uh, there's a convergence of ideas, canonical cover, and relational observables
Thomas Thiemann: nicely fit together.
Thomas Thiemann: I would consider this model as a paradigm model, for now on has the harmonic oscillator of loop quantum gravity at which you should test all our ideas.
Thomas Thiemann: Um, It highlights the importance to implement quantum degeneracy.
Thomas Thiemann: Um. The lqg techniques that have been introduced in the past decades work
Thomas Thiemann: um
Thomas Thiemann: particular for density, way to unity, and without using habitats.
Thomas Thiemann: The Hamiltonian vector field of both constraints preserved the momentum polarization of the face base. So, in other words,
Thomas Thiemann: and of course the full con to grab is much more complicated. So here the constraints are no longer polarization preserving,
Thomas Thiemann: and therefore on uh and get needs to go.
Thomas Thiemann: All the techniques have to go beyond that uh, one interesting aspect or idea that has been mentioned by Uh
Thomas Thiemann: in one of his recent papers is that, uh, now that we have
Thomas Thiemann: a kind of integrable model, and it might be a good idea to actually
Thomas Thiemann: and the other uh
Thomas Thiemann: yeah, the uh idea is,
Thomas Thiemann: it brings me to the last part of my talk. So um, going beyond perturbation to you around an international model going trying to define the ue uh non-perturbatively
Thomas Thiemann: it may be a good idea to uh have a look at um renormalization
Thomas Thiemann: in order to uh define representations or find representations constructively, which are which have this property of being a quantum non-degenerate.
Laurent Freidel: So there was two questions, please. Uh hello, uh, Thomas! Uh, It's about the previous slide uh something you you mentioned, so can you explain it a bit more? The fact when you're saying that. Uh,
Laurent Freidel: let's say, yeah me, you know they may usually give me to the constraints on E, you know, generates the excessive over here, which is kind of the connection, and that's that's the basics of the of the quantum dynamics, right? So uh the action of the
Laurent Freidel: uh. So here can you explain why that's not happening. Why, why, the the time derivative of the frame field is not the excessive curvature?
Thomas Thiemann: Yeah. So if you so, why do you get uh? You dot equals to to the extent the curvature. That's because, uh in in the in the actual quantum gravity model uh, we have uh su two gauge group.
Thomas Thiemann: So f the the curvature has. It has a linear part in the quadratic part.
Thomas Thiemann: Uh, in this you you one cube truncation. You drop the ace, the the quadratic part.
Thomas Thiemann: So F is just uh the exterior derivative of a
Laurent Freidel: that. That's what I call this a polarization preserving okay and and geometrically. Do you understand what it means that it is not the excess of curvature. What kind of geometry does that
Laurent Freidel: correspond to?
Thomas Thiemann: Hmm.
Thomas Thiemann: It's difficult to uh to um derive um. A geometric intuition about this is just uh the the dynamics. Dynamics is such that, uh, in fact, you can. You can first solve uh a first order equation just for you.
Laurent Freidel: Okay,
Laurent Freidel: Yeah, it's a it's a it's a it's not it's, not. It's not a
Thomas Thiemann: but the nice feature is that it has compared to other models which are also So it has this problem: property that we have a non-trivial and open hyper-surface deformation, algebra.
Thomas Thiemann: And this is what I wanted that what I was interested in in testing.
Laurent Freidel: Okay, Thanks. Sure.
Lee: Uh. Comment.
Lee: Yeah, I i'm sorry if you should tell with this at the beginning. But are you aware of? This? Is a whole community that studies this model under a different name?
Lee: They call it self to gravity, and they attribute it to Palestine. Um to i'm sorry, Townsend
Lee: Johnson, The The town that was that is known for his work on super membranes right,
Lee: and they all point back to Thompson
Thomas Thiemann: while the model was introduced by you
Lee: way way Prime? No, I was way prime.
Lee: So, anyway, I think actually it's an opportunity.
Thomas Thiemann: Thanks.
simone: Sorry. And can you say something about kitchen? But then, because, if understood the previous point,
simone: now we are going to look at quantum states which are not charge networks.
simone: So then, we have this nice representation of the algebra of these algebra. There's a quantum operators. But then, what happens when we post gauge in audience?
simone: You mean for this model. Right?
Thomas Thiemann: I solved it.
Thomas Thiemann: So it's the middle of this slide.
Thomas Thiemann: You just use functions which have vanishing divergence.
simone: So in a
simone: e, even if it's a even if it's in a video model with some lines, it could be still good observable. But here you're not using those as observable to What are you using apart from formulas, Just in words, just to give us intuition of what is going on because it's A, B. And you could say, use the electric field and the magnetic field that they gauge invariant. Here,
Thomas Thiemann: you mean just uh of those with respect to the goals constraint
simone: right? Because if you're If the way to solve the Gauss constraint is by using the electric field, then I think I understand the the what's going on. But I just needed to
simone: confirm if that's what's going on or not.
Thomas Thiemann: Yeah. So in in particular, a Wilson kind of a generalized Listen: Function Will's loop function, which is, be it with a function that has vanishing divergence. It's as if,
Thomas Thiemann: and the electric field, of course, is also gauge environment with respect to the Gauss gauge transformations.
wolfgang wieland: But in the uh, in the non-abelian case, and even more so in the gravitational case, the gauge symmetries are highly nonlinear. So the smearing function, or what? What would it mean for this mirroring function, Now that it is it's a field dependent. Or
wolfgang wieland: could you comment on that,
Thomas Thiemann: please? So, as I was mentioned before. One
Thomas Thiemann: One can define the same representation also in the non-obeling case.
Thomas Thiemann: But then it is much
Thomas Thiemann: less natural. Let's say, to try to solve the Gauss constraint in terms of those variables.
wolfgang wieland: Mhm
Thomas Thiemann: but the representation itself works the same
Thomas Thiemann: also in the normal building case,
Western: unless since there are some some general question before you go to to about the setup. Uh there, there'll be questions about the the the, the the gauge constraints. But what about the different Also for you. It seemed to me that they going to to um.
Western: It's it's main tools that allows us to work with the most invariant things. So here I mean for me that they exist. I don't see why it shouldn't, but
Thomas Thiemann: we we use actually not. I mean the the nice thing is that um uh. So the these uh Hamiltonian vector. Fields. Let let's. Let's take the case that M. Is equal to zoom. Just. We just look at the spatial different offers. And now,
Thomas Thiemann: then,
Thomas Thiemann: the action of this Hamiltonian flow corresponding to a spatial Dishimorphism is exactly the action of a spatial decomorphism. Of course it's not down to one that's how to solve them, how we find that the the basis of states which are the environment
I and they they pretty well defined. I see it.
Thomas Thiemann: Oh, I go in much work, much more beyond that, because i'm going to the reduced phase Space
Thomas Thiemann: It it I've missed it, as it already happens, or you. It has happened. Yes, in my paper.
Thomas Thiemann: Oh, it's not a not on this slides. Oh, not that That's why I was saying in the beginning. I'm sorry for, because the the home this model alone deserves to talk by it by itself. But I have. I was supposed to summarize three or four papers. So it's not here. Okay, that's all right. But then uh,
wolfgang wieland: why, why uh going back to the last slide, perhaps. Why, to the previous slide on me. Um!
wolfgang wieland: Why, then, choose such a singular state for defining Omega. Why not? Uh do the Gns construction with the Gaussian state in F?
Thomas Thiemann: Because you're stopped in line zero uh the operator that you've tried you will find is badly, badly. Uh no, ill-defined and it's not certainly not tensely defined
Thomas Thiemann: that forces you. Because that that's why I was saying in the in the beginning, Um,
Thomas Thiemann: These kind of
Thomas Thiemann: the non-ho Fartherian type of representations are natural when you, Hamiltonian, is such that the electric field is part of every single term,
Thomas Thiemann: so that, uh,
Thomas Thiemann: and a vacuum which is killed by the electric field as a natural choice.
wolfgang wieland: Okay, I see. Yes,
Thomas Thiemann: okay. Should I actually go on?
Jerzy Lewandowski: Yes, please go on. Okay.
Thomas Thiemann: Hmm. So
Thomas Thiemann: yes, in the last couple of minutes. Then we go to um
Jerzy Lewandowski: uh as I try to motivate uh Hamilton as a normalization. So actually an organizational e an announcement. So, eh? Eh? The The chairman is gone. I'm replacing the chairman, and I suspect that even Abai is not with us today, so probably there are no adults just us.
Jerzy Lewandowski: We can do whatever we we want.
Jerzy Lewandowski: Okay, Go, go.
Lee: I I'm definitely not an entail. So
Thomas Thiemann: okay, so um.
Thomas Thiemann: The idea for for this line of line of work comes from constructive quantum theory
Thomas Thiemann: uh as initiated by this gentleman that i'm listing here in the first line.
Thomas Thiemann: Um,
Thomas Thiemann: and uh, i'm trying to cut a very long story short. So uh the script. And here is a set of resolution. Skills!
Thomas Thiemann: Um
Thomas Thiemann: uh! This is supposed to be a partially ordered and direct set,
Thomas Thiemann: and uh, we have at our we're supposed to have at our to dispose of this um
Thomas Thiemann: uh triples of objects. The first entry is a Herbert space; the second is a vacuum, and the third is a Hamiltonian,
Thomas Thiemann: and these are labeled by elements of this label space of this resolution skills.
Thomas Thiemann: And uh, suppose that you have what people call consistent systems of isometric injections. So these are isometric injections of Herbert spaces into each other from course, resolution to fine resolution,
Thomas Thiemann: which are such that if m one is smaller than m two, it's more than three that uh, some of these things work together in the expected way.
Thomas Thiemann: Uh,
Thomas Thiemann: then uh you get, you get a consistency condition also for the Hamiltonian, which says that if your call screening uh fine resolution, Hamiltonian and you get
Thomas Thiemann: yeah
Thomas Thiemann: uh low resolution, Hamiltonian
Thomas Thiemann: and Um. If such a system of of Uh theories labeled by resolutions is given, then you can, uh you get an equivalent definition of a continuum theory, in which there are two
Thomas Thiemann: uh
Thomas Thiemann: the continuum Herbert spaces what people call the so-called inductive limit of a good spaces,
Thomas Thiemann: and the Hamiltonian is kind of a projection
Thomas Thiemann: uh, or the the The continuum Hamiltonian has projections to uh find out the resolution which cons co coincide with the Hamiltonian defined at
Thomas Thiemann: uh find that resolution.
Thomas Thiemann: And how does it help to find
Thomas Thiemann: quantum non-degenerate representations?
Thomas Thiemann: So here I was had prepared a a technical slide about what people call multi-resolution analysis and wavelet ue,
Thomas Thiemann: and uh
Thomas Thiemann: erez agmoni did Three most important ingredients of this are what actually is a multi-resolution analysis uh associate scaling functions and associated scale uh wavelets one hundred and fifty.
Thomas Thiemann: But I will skip that slide uh, and maybe come back to it. Uh, if there's interest later, let me just say for the moment that the multi-resolution announce is a nested system of uh vector spaces, actually her good spaces,
Thomas Thiemann: uh part of which we are part of the to space over the Sigma and Sigma are the spacious license of our system,
Thomas Thiemann: and you can think of them of these silver spaces as one particle in that spaces and coding the resolution that we are looking at.
Thomas Thiemann: So you can think of them as many functions, form factors, and so on depending on the theory that you are interested in.
Thomas Thiemann: And uh, the second ingredient is a is a scaling function. Wh. What people in wavelengths you you call a scaling function, and such as mother scaling function. There's only one of them, or several few of them, but certainly a finite number of them are such that res scaleings and translates of them
Thomas Thiemann: provide an orthonormal basis of this final resolution uh smelling function. Hilbert space
Thomas Thiemann: uh And organization. With respect to such a multi-resolution analysis uh was um considered by people working in constructive quantum fe three before um,
Thomas Thiemann: and we this use this um These sky functions, these uh um scaling functions as an off-normal basis.
Thomas Thiemann: And it turns out that because of uh, the way these things work together, this uh combination I am I am dagger is a projection in that space of smearing functions which will become a little bit important in in what follows.
Thomas Thiemann: And using such a Mrr structure, mri structure, you can you find a a call course grading system.
Thomas Thiemann: So if if you don't know to understand the details of this construction, that's not important, but all of that is important is to just keep in mind that such an Mrr. A structure is an organizational principle. Did you find what is a course screening map?
Thomas Thiemann: And uh, of course, there's lots of choices of such course grading structures. And
Thomas Thiemann: Erez Agmoni people. So this is a whole discipline in mathematics. Uh Wavelet theories. So people are interested in in such functions which are, or or scaling functions which are somehow local, both in position and momentum space one hundred and fifty,
Thomas Thiemann: and have some scooter's properties, and there's lots of lots of works by by these people here
Thomas Thiemann: uh and for zoom, notice, or even a field metal for
Thomas Thiemann: that has been given to Ingred to Pages
Thomas Thiemann: um
Thomas Thiemann: do this in practice. So we pick an Mr. A structure, so which I said, As I said, it's just nothing else, and a nice way to organize your uh course training um procedure.
Thomas Thiemann: And then i'm supposed to give
Thomas Thiemann: be equipped with uh a system.
Thomas Thiemann: Um
Thomas Thiemann: uh which consists of a Hamiltonian together as a phase space. And I can use this uh, this, Mmr: A structure or course training tools in order to discretize my my face space.
Thomas Thiemann: I can use them to discretize my Hamiltonian
Thomas Thiemann: in this way,
Thomas Thiemann: and this is, of course, uh
Thomas Thiemann: kind of an arbitrary choice,
Thomas Thiemann: and the whole out of the game consists in showing that, uh, no matter what kind of choice you do at the beginning uh you're defining a organization flow which drives you into a kind of unique theory,
Thomas Thiemann: and
Thomas Thiemann: you you find a H about space, a vacuum annihilated by this Hamiltonian.
Thomas Thiemann: You define, for example, while elements corresponding to this discretized harm. Um!
Thomas Thiemann: And then you start
Thomas Thiemann: computing the random.
Thomas Thiemann: So you are supposed to compute a vacuum from from vacuum. So you have some old vacuum at high resolution
Thomas Thiemann: uh at
Thomas Thiemann: uh little, and and
Thomas Thiemann: of your um
Thomas Thiemann: consisting of a vacuum, a Herbert space, and a Hamiltonian at each resolution. And Here's the formula that allows you to compute the new vacuum from the old vacuum
Thomas Thiemann: where M. Is August, smaller than the resolution labels by M. Is more, it's always cooler than the roof, resolution, and prime,
Thomas Thiemann: and the last step then consists in looking for after you have computed this flow looking for fixed points.
Thomas Thiemann: And why does this help to define or to to construct quantum non-degenerate representations? Because, uh at each resolution this wild states are are non degenerate because they are excited everywhere at that resolution,
Thomas Thiemann: and then you then go to the continuum. You expect that this bone degeneracy is inherited also at the
Thomas Thiemann: Uh. It had infinite resolution, so to say.
Thomas Thiemann: And now we want to test this,
Thomas Thiemann: and uh, as you may have seen or realized, um, all of this in constructive quantity is made not for constraint systems, not even for gravity, is actually made for for a nice one.
Thomas Thiemann: Systems with a well-defined Hamiltonian. So not other. So so for for consensus, and has never been defined so my
Thomas Thiemann: well. Our proposal was here to
Thomas Thiemann: never the less apply this kind of kind of ideas, also a to a constraint system. To each of these constraints let's say, for the to to each of the spatial different from some constraints, as if it was a Hamiltonian. So just get copy paste the formulas, but do it for all the smelling functions.
Thomas Thiemann: That's the proposal. And then, of course, the questions that I immediately arise. What about the vacuum? Is there a common vacuum for all the constraints?
Thomas Thiemann: Is it necessary to have a vacuum for all the constraints?
Thomas Thiemann: Should we also discretize the labs and shift functions.
Thomas Thiemann: If, yes, how
Thomas Thiemann: and how does the constraint? Algebra and the anomaly structure react to the renormalization flow?
Thomas Thiemann: And the this this kind of uh techniques have been applied so far only to, and the most uh simple and more complicated the theory that we could think of
Thomas Thiemann: um. It's uh Carl Cookers parameters to be into them two dimensions, which is, uh, classically. So Google. And therefore we um. You look at this model from a quantum perspective, and from the perspective of renormalization.
Thomas Thiemann: So this is now parametized. V. Three on the cylinders or the Time access cross uh uh circle. It's boundary boundary conditions. Oops
Thomas Thiemann: um,
Thomas Thiemann: and i'm just listing
Thomas Thiemann: erez agmoni what we found. So the first lessons that some's degree of smoothing smoothness of this creating function is mandatory in order to have a well-defined one hundred
Thomas Thiemann: algebra of constraints.
Thomas Thiemann: The racket decrease of the Fourier transform of uh these very functions is also necessary,
Thomas Thiemann: i'm mentioning this because people like to work It's a so called ha
Thomas Thiemann: multi-resolution analysis. This is the classic blocks and course grading uh transportation that people use, for example, in in the uh renormalize the ising model. So this would not work in this case.
Thomas Thiemann: Um!
Thomas Thiemann: What work? For example, is the so-called Jewishly
Thomas Thiemann: multi- resolution analysis again. I'm not going to into D. Thirty. I'm just flashing how the uh Orthonormal basis at Resolution Capital M. Looks like, and M. There's a little m denotes the lattice point on on the circle.
Thomas Thiemann: Um.
Thomas Thiemann: One can show that the flow has the unknown continuum theory as a fixed point
Thomas Thiemann: uh the classical algebra is centrally extended,
Thomas Thiemann: and therefore we get to be with all algebra, and you recover that we was over algebra. And this, uh,
Thomas Thiemann: uh, in this normalization procedure,
Thomas Thiemann: and therefore there is no come back from in this case. But it's not necessary.
Thomas Thiemann: It is possible to discretize also labs and shift functions, but it's not necessary
Thomas Thiemann: and the most interesting.
Thomas Thiemann: Our lesson is maybe the last one here.
Thomas Thiemann: Um,
Thomas Thiemann: and it's physically correct that there are finite resolution artifacts here. In this case we can actually compute them quite in close form. So what we do is we take the known one
Thomas Thiemann: quantizations of the continuum constraints which have a central extension
Thomas Thiemann: um,
Thomas Thiemann: we project them to find out resolution. We compute the algebra of these guys.
Thomas Thiemann: You get the finite resolution analog of the commutator of the x-axis commentator, plus the central extension that should be there, because it's central
Thomas Thiemann: at Resolution, M. And
Thomas Thiemann: out of which
Thomas Thiemann: looks like an anomaly, but which is more or less just an artifact that comes from finite resolution,
Thomas Thiemann: which you can see from the fact that um
Thomas Thiemann: the week
Thomas Thiemann: operator topology limit of this artifact is going to zero as you turn up the resolution to infinity.
Thomas Thiemann: Um!
Thomas Thiemann: The natural density of one is not necessarily an obstacle to our uh to close the algebra in at least not in non degenerate representations.
Thomas Thiemann: Um,
Thomas Thiemann: um! The exponential Hamil exponentiated Hamiltonian constraint,
Thomas Thiemann: at least in the you one two model
Thomas Thiemann: has a very different action from what was guessed so far. Uh because it's so nonlinear acting.
Thomas Thiemann: And um,
Thomas Thiemann: it's very nice to see that this can be worked out in a lease you one cube model, and so explicitly,
Thomas Thiemann: and uh, it's. This is called the Hamiltonian, or, in fact, uh general, is minimization.
Thomas Thiemann: Um. It gives you a systematic tool to give to look for quantum non-degenerate representations,
Thomas Thiemann: and it gives you a tool to disentangle uh
Thomas Thiemann: we are discretization artifacts from two anomalies at least in the model that we have
Thomas Thiemann: been able to look at so far.
Thomas Thiemann: Thank you very much for your attention.
Jerzy Lewandowski: Hello! And can you confirm that you can hear me?
Thomas Thiemann: I can. I can hear you. You're okay. Can anybody hear me?
Jerzy Lewandowski: I can hear you?
Jerzy Lewandowski: Okay?
Jerzy Lewandowski: Hmm. Hello! Hello!
simone: Well, maybe for questions.
simone: What do you suggest is the way for the starting to meet in the quantum gravity.
Thomas Thiemann: Um. So at least on my on my computer there was a strong interference. Can you repeat the questions, please?
simone: What do you see as the way forward?
Thomas Thiemann: Um,
Thomas Thiemann: Yeah. So
Thomas Thiemann: hmm
Thomas Thiemann: to me. Um. The The lesson is is the following: I mean if I look at the first part of my my talk, then the the tangent comes from the fact that, uh, on the one hand, we are strongly interested in
Thomas Thiemann: representations in which we want the volume to act only at the vertices,
Thomas Thiemann: in order that, for example, the Hamiltonian constraint is not blowing up on our States of the Herbert Space.
Thomas Thiemann: On the other hand, then, we get a problem to close the algebra, because a classical algebra wants us to have an infinite number of terms
Thomas Thiemann: need for a number of contributions.
Thomas Thiemann: This is where where this tension comes from,
Thomas Thiemann: and the tension can be resolved
Thomas Thiemann: hopefully. If you look for representations in which the geometry is not so degenerate,
Thomas Thiemann: and that maybe it's it requires to have a fresh look at what our smearing functions should be, Maybe we should uh So the Gauss constraint in a different way,
Thomas Thiemann: and so on, so far, so.
simone: Oh, basically just to I think I understood what you say, but just to make it more explicitly, what you're saying is that the way forward should be to stop using the speed network basis
Thomas Thiemann: um
Thomas Thiemann: erez agmoni. That could be one possibility or one one one could um try to find in the same Herbert space a different uh space of of functions which are closer to being non-degenerate one hundred and one.
simone: So so you're saying the Apr space may be the same, but still we should stop using spin network states, or the inverse space is completely different.
Thomas Thiemann: These these are much closer to being a quantum non-degenerate and spin network. Functions. Maybe we should like all the weed States, for instance,
Thomas Thiemann: while one had would have to look at super positions of those.
simone: Okay,
simone: Thank you.
Sure.
Western: Um, Thomas, this is Carla. Uh, I I I found very convincing your the the the part of the of the top where you say that the exponentiation of the misconceptions it's a
Western: it's presumably the the same one has to look at.
Western: Uh. Your main point, however, is not. This one is about
Western: the non sequential no degeneracy. And um uh, I mean. I hear what you say, and I see I see the argument. It's it's general. I would
Western: we couldn't couldn't the problem be the algebra itself. I mean It's not focusing so much
Western: on having
Western: a uh cosine algebra will defined. Is that right? Because at the light of all this, because uh um,
Western: in a sense uh the the overall physical
Western: message of so many years of refugees. That is perhaps a reasonable theory.
Western: Um, which is discrete. So it's a we're used in in like, you see, in in anything to the fact that because of this discreetness you just cannot take derivatives.
Western: Uh, that has been a strength of the full construction, not the weakness. So if you cannot take the reasons, why should you have a well defined uh
Western: uh constraint algebra, which are which are communities,
Western: so I mean this W. All this is going to see, but it isn't isn't that sort of could couldn't but one look also this in in terms of Well forget the the
Western: forget, the constraints algebra, and they find it here in a different manner of we. We have already a Hilbert space that seems to work very well for many things.
Western: Um, we have a number of different ways of thinking of the dynamics. Why? Why? Insisting so much on the constraint algebra.
Thomas Thiemann: So um uh you. You're saying that uh you want uh
Thomas Thiemann: you use. I mean what this set of works um suggests is to look at the exponentiated
Thomas Thiemann: uh objects, not at the objects themselves. Is that,
Thomas Thiemann: Yeah. So that works quite well for the Abelian theory for the non-abelian theory. It's much more complicated
Thomas Thiemann: um
Thomas Thiemann: like this Hamiltonian remarization program, for example, where um we have to define um
Thomas Thiemann: uh invented so far and in the in the
Thomas Thiemann: none of you in case i'm afraid that's not sufficient,
Thomas Thiemann: even if you want to define. Finally the the exponentiated version
simone: Wait. But if I understand your message, if I had to you. If I used the standard refugee methods in this U. One model,
simone: I will not end up in a a corrective presentation of the open surface information on to the right. That was your point. You still need not to use charge network states, but at least in your combinations or whatever you are arguing for,
simone: so that it is not as distribution as in the usual setup right. So what is important is that at the end of the day i'm using two,
Thomas Thiemann: not uh, not new functions which are just uh excited on on graphs, but they must be excited somehow everywhere,
simone: Right?
Thomas Thiemann: Still, they can be deep. They can be quasi-excited on the graph, because you can always take a graph
simone: distribution and and smell it out. Some all and make it almost look like a one dimensional right. I guess that would be for me. That would be the next step here, mainly figuring out whether one can obtain the same results. Starting from a charge network theory,
simone: finding a suitable meeting over the graphs.
simone: That would mean that's been. Networks could still be used also in on a billion case. But if you prove some no go, or whatever, some very unnatural
simone: something that one needs to know.
Thomas Thiemann: Yeah, all I can say is that in the you one cube model uh one can play this game uh. So you take you, take some whole um. This case of the smearing function, which is as as as you want to. Uh uh a a a graph,
Thomas Thiemann: a graph steering function which is just excited on the edges of a graph
Thomas Thiemann: from what we thought it should act.
Thomas Thiemann: But of course one keeps. Keep in mind that this is the finite action of the Hamiltonian. It's not just a single action, so it's somehow the the result of of many, many actions. So it's uh it's not that easy to to compare,
simone: but still you said it can also be represented as painful model.
Thomas Thiemann: Yes, uh. So what I'm saying is that um you can.
Thomas Thiemann: You can
Thomas Thiemann: uh,
Thomas Thiemann: as is contained in many, many textbooks, and then you define first of all the path undergo
Thomas Thiemann: on what people call the reduced phase space with the with the physical Hamiltonian, and then you unfold the uh, the reduced phase, space to the kinematical phase space.
Thomas Thiemann: And finally, after many, many, many steps, you end up with something that looks like the some measure times the exponential of the the action
Thomas Thiemann: that can be can be done In this model.
Mehdi: I can see several race hands, but maybe some of them are already. It is so already satisfied. So what about Western? Have you asked your question?
Western: Okay, What about? Mostly
Western: uh the gradient sector, and maybe the Lorentz one or the three D and the forty in the sense that there there is this quantum reduction that connects with the full theory. Is there anything we can learn and import uh from there here,
Thomas Thiemann: so that and their desk can be there. Some kind of interaction between this and uh, what a Manuela was doing in the sense that um
Thomas Thiemann: he is somehow um
Thomas Thiemann: as far as our recall, he's somehow engaged fixing the
Thomas Thiemann: at least some partial gauge facing is going on, which enables him to somehow reduce the gauge group.
Thomas Thiemann: Uh, also to an a little one.
Thomas Thiemann: Uh, in that sense uh his procedure and the one called cube model are quite similar, but of course the
Thomas Thiemann: is ah such that there is an a squared.
Thomas Thiemann: So the one cube model is still simpler in that sense,
Thomas Thiemann: erez
Thomas Thiemann: uh, and this is this is the deformation of of gate groups which is called, uh consistent in the mathematical, The literature, and what he he points out is that? Suppose that one one of the members of this two Uh theories,
Thomas Thiemann: which are consistent deformations of each other, is such that one of them is integrable.
Thomas Thiemann: Then it looks like a good idea to to to quantize the into global model and perform perturbation to you for the full model for the for the full month of theories around that. Uh, in the global model.
Thomas Thiemann: So I think that's something that
Thomas Thiemann: might might be worth looking into.
Western: That's very interesting, Thank you,
Mehdi: and would you like to
Muxin Han: thank you?
Muxin Han: So So first the thanks for the talk. Um! So I I think there was one slice on, and one line about the relation uh with renewal space based organization. So for the you, one queue theory,
Muxin Han: it's a and it's. Are they company equivalent to operator and the the constraint colonization and reduce basic from that is um organization. Are they company equivalent?
Thomas Thiemann: So um
Thomas Thiemann: uh, let me see whether I have something
more specific to stay here.
Thomas Thiemann: No, unfortunately, there's no details about this. But, as you know very well from uh your Phd: this is work. Uh
Thomas Thiemann: yeah, What What one has to do is the following: Um. So here we are dealing with It's U. We whose constraint algebra is not the algebra,
Thomas Thiemann: and therefore uh, one cannot use the rigging map techniques per se. One has to first
Thomas Thiemann: uh reformulate uh the classical theory in such a way that the constraints form a linear algebra,
Thomas Thiemann: and this can be done for this, for this model for the one cube model. And starting from this Abelianized
Thomas Thiemann: version of the consent algebra, and you can show that reduce face-based quantization and the rigging, map, or group averaging construction are equivalent.
Muxin Han: Okay, so um
Muxin Han: But then um. The question is because you know, the
Muxin Han: um we'll, we'll, we'll
Muxin Han: um it doesn't rely on uh the the um uh quantum normally free uh construction of uh constraint algebra, but of course it's um and it's relates to the uh algebra. So i'm. I think my question is um
Muxin Han: whether um so? The the reduced face for the organization is Um, It's more advantage has more advantage.
Muxin Han: Um, then the um
um than the constraint quantization, because um! It seemed to me that
Muxin Han: the red deals by facebook have reduced Space-based colonization is always correct.
Muxin Han: Um!
Muxin Han: And it doesn't see the the um um fundamentally coming from the constraint algebra! And do you think it's uh it's right way to understand.
Thomas Thiemann: Um,
Thomas Thiemann: yeah, yes, and no um. So looking at the details of reduce face face quantizations, one has to take this as a grain of sort. For example, if you, if you, if you're trying to solve your constraints for for a set of momenta, you may have to solve quadratic equations, and then there are
Thomas Thiemann: sign issues. So there are always several branches in your face space, and you have to pick one
Thomas Thiemann: quantizing at the end. Only half of this phase. Space.
Thomas Thiemann: Um.
Thomas Thiemann: Also there are assumptions that you have to make, and when you, when you solve for this momenta then you have to assume that certain matrices are non-degenerate
Thomas Thiemann: uh having made that assumption. And of course, the reduced phase space quantization is more convenient, because you never need to worry about this anomalies.
Thomas Thiemann: But uh, if if you, if you're interested in in in the kinematical quantization and the constraint, quantization, and how disconnect with other than uh, so it can be also the operator quantization over the to operate the uh constraint Conversation is, it's interesting,
Thomas Thiemann: but I agree that uh from from an economical point of view. The reduced face-based quantization for very complicated systems is is probably the most uh promising route to go.
Muxin Han: Okay, is there. Some new license. Um, you inside, coming from this, you one two phone because it's easy to compute.
Thomas Thiemann: Yeah lesson in this sense.
Muxin Han: Um. I mean the advantage of one or the other
Muxin Han: um reduce space based, or or um constraint out of pop
Muxin Han: constraint. Organization.
Thomas Thiemann: Um. So not not not really, because here it was not so difficult to um to arrive at the reduced phase space. The description also is coming from the constraint description. But if I try to extrapolate to a more complicated systems, then
Thomas Thiemann: let's say, Euclidean vacuum general relativity,
Thomas Thiemann: I mean,
Thomas Thiemann: uh approach to define the physical habit. Space is probably a hopeless task.
Muxin Han: Okay, thank you.
Mehdi: Hmm.
Mehdi: If not
Mehdi: with a single speaker again,
and
Mehdi: thank you.
Thomas Thiemann: Thank you. Bye, bye,
Western: right.
Lee: Thank you.
Western: Now you like we will over your shirt, and we hope to have you soon investor with us.