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Jorge Pullin: Our speaker today is called Hagar was going to speak about effective spin phones on the phone.
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Hal Haggard: Good morning, everyone. Thank you so much for being here. It's, it's a pleasure to get to spend some time with you.
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Hal Haggard: I'm grateful for the invitation to speak about recent work with Bianca Dietrich and Seth Assad on effective spin phones and the fatness problem.
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Hal Haggard: I am continuously impressed by how well this seminar series is organized. So thank you so much. Jorge and everyone who's involved in the organization.
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Hal Haggard: As I said, this is joint work with two collaborators at the Perimeter Institute. It appeared on the archive earlier this spring and the link all the references in the talk are linked as you can click on them and go to the archive for the reference
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Hal Haggard: I want to speak today about discrete areas and geometrical path integrals, better known to us as spin phones.
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Hal Haggard: And in particular, I'm going to be coming largely from Reggie calculus perspective. So in Reggie calculus we approximate space time by discrete triangulation each of the pieces of the triangulation are flat and but it's in gluing them together that we get curvature
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Hal Haggard: The advantage of doing this is that it cuts the number of degrees of freedom of the gravitational field down immensely and allows us to study them more directly. It's also essential for doing numerical work where which I'd like to present to you today.
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Hal Haggard: So this audience knows Reggie calculus very well, but it's helpful to kind of illustrate some salient points from Reggie calculus by climbing a dimensional ladder.
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Hal Haggard: So, first in 2D. We have flat triangles be these together. If one triangle around a point is missing when we glue. You can see that there's curvature at that point. In this case it's a conical singularity.
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Hal Haggard: This shows the the curvature is concentrated on the D minus two dimensional bones of the of the triangulation. So this technical word bones is useful because we can refer to it without saying which dimension where
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Hal Haggard: In 3D will be gluing tetrahedron around a edge which is our bone here and you see a really nice kind of confluence of the metrical geometry of the triangulation, and the syntactic geometry.
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Hal Haggard: The curvature is all concentrated along an edge length, which also parameters is the metric and, on the other hand, we see this deficit angle around that bone.
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Hal Haggard: And quantization of that contact angle is going to lead to quantization of these metrical length variables in the in a sort of standard picture of how quantization comes about. So 3D is really interesting for this match between the syntactic and the metrical structures.
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Hal Haggard: For D. On the other hand, is is even richer. So in 4D. We work with for simplicity's. These are usefully sort of visualized in the plane under a projection as just five
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Hal Haggard: five vertices that we connect every single vertex to every other and that gives us a for simplex. In this case, the bones are two dimensional triangles here I've kind of highlighted one bone and pink so that we can look at it.
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Hal Haggard: And this time, we don't have this nice match between the metrical in the syntactic so if you work with the apparent metrical length variables. The conjugate variable in the syntactic structure is quite complicated.
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Hal Haggard: On the other hand, if you work with the curvature angle which is now a deficit angle around the two dimensional triangle that the simplest. These are glued around
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Hal Haggard: The areas are the things that become contest and and we need to work kind of with a metric described via areas. So this second choice is very harmonious with Luke quantum gravity, which also has discrete geometrical spectrum and is the the route that will follow today.
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Hal Haggard: So with that sort of context. I can tell you an overview of what I want to share with you today.
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Hal Haggard: Our motivation was to create as simple as spin phone model as we could think of, in particular the ingredients we want to build in where the discrete areas spectrum of loop quantum gravity.
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Hal Haggard: I've written it out here gamma times the planck length squared times the square root of J AMP j plus one were large big going to be thinking about this in an awesome topic limit where the jays are large and so we can approximate the square root
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Hal Haggard: Gamma here is the barbero Emirati parameter and from the perspective of this talk is really interesting because it controls the area of separation. The, the spacing between the eigenvalues of the area.
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Hal Haggard: To work with spin foam that takes this discrete spectrum really seriously we cast the action in terms of these area variables.
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Hal Haggard: And something really interesting comes out of that which is that an unconstrained action has only completely flat classical solution.
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Hal Haggard: So if we worked with the unconstrained theory, we would, there's no way we would recover general relativity. So this leads us to consider constraints that we're going to impose on this action framed in terms of area variables.
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Hal Haggard: The tricky part about this is that if we're going to take this discrete areas spectrum as input when you try to impose both the discrete areas spectrum and these constraints.
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Hal Haggard: You're going to run into trouble. The issue is that only certain discrete area values are going to be allowed. So we only get certain
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Hal Haggard: Triangulation configurations that are possible and those may not be ones that satisfy the constraints. So we're going to investigate a strongest possible weekend position of the constraints that's consistent with this area spectrum.
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Hal Haggard: And the model is very interesting to us because it's very accessible. It's very easy to understand what we've done and it's very computer ball. So I want to present some new metrics, as I mentioned before,
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Hal Haggard: So here's the sort of guide of how I'm going to tell you about the model. First I want to show you explicitly how the area variables lead to flat classical solutions, just for context.
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Hal Haggard: Then I want to study the constraints and will study them at first at the as imposed on the classical theory and and we'll see clearly why it's so difficult to impose them strongly in the quantum theory.
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Hal Haggard: And throughout the talk, I'll be working in Euclidean signature and I'll define the Euclidean signature model that comes out of these inputs.
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Hal Haggard: It has the same structure as a standard spin phone with weights on triangles and force embassies and the main new ingredient is these this way of imposing the constraints that are described
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Hal Haggard: In particular, what we're going to do is when we glue to for simplicity's through a tetrahedron. We're gonna make sure that the 3D die. Hey triangles match as well as they can and be consistent with the LPG face space.
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Hal Haggard: Or healthcare space.
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Hal Haggard: With the modeling hand, I want to tell you about these new metrics and both going to present the new metrics that appear in the archive paper that I referenced before
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Hal Haggard: But I will also give you some hot off the press and numeric for more complicated triangulation in the archive paper we studied three
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Hal Haggard: For simplicity's that are glued around one bulk triangle in the new work we've been studying six for simplicity's that are glued or long an inner bulk edge. So this makes the comparison between the area formulation and the length formulation, much more interesting.
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Abhay Ashtekar: Can I just ask a quick question here is about. So the you're working Euclidean signature.
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Abhay Ashtekar: But you still want to have with the Barbary mercy parameter or that, or just a motivation, because in the previous signature, you can just use this Abdul variables. And then there's no Barbie measure parameter. So I just want
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Abhay Ashtekar: To give you pointers.
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Hal Haggard: Yeah, so, um,
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Hal Haggard: You know, maybe we don't necessarily have to call it the barber emergency parameter here because what I'm really using it as is, as a parameter to control the area spectral gap.
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Hal Haggard: But that it's so natural to view it that way because of the what you mentioned that we that we think about it, in theory,
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Hal Haggard: But you're, you're absolutely right that the rule that it's playing here is is truly as a control of the area spectrograph
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Abhay Ashtekar: No, but the body image upon him. It also then plays a role in dynamics and you're not going to do that because
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Abhay Ashtekar: That's calling it as an area gap so to say. So what do you need isn't a gap that's that's a statement is that right
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Hal Haggard: That's exactly right.
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lsmolin: Thank you do
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Hal Haggard: We have no why map here and there's no gamma and because we have no why map so
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lsmolin: Question.
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lsmolin: Is it yes just building things without matter is it's just feeling these constant
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Hal Haggard: I didn't hear the last part Lee, but
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lsmolin: If you don't couple to matter. Is this the same as scaling is constant.
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Ah,
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Hal Haggard: Well, yeah. So we saw in the area spectrum that the L plonk is coming along with gamma. So I suppose that's correctly.
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Hal Haggard: Alright, so that's the outline for how I'm going to present the theory to you, and let's begin.
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Hal Haggard: So let me describe standard length Reggie calculus. I'm going to be referring both to length and area, Reggie calculus so much that I'll abbreviate them both.
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Hal Haggard: Length Reggie calculus for the rest of the talk will be LR C and later I'll introduce area, Reggie CALCULUS, WHICH IS A RC.
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Hal Haggard: So this is the standard Reggie action, which is the area of the triangles, times the deficit angle around that triangle summed over all the triangles and a 4D triangulation delta, I'll call it
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Hal Haggard: The deficit angle, as you know, is defined as two pi minus the some of the di he draw angles in each simplex that that meet at that triangle.
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Hal Haggard: In the length Reggie calculus. We view these areas as functions of the length of the triangulation.
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Hal Haggard: So often when I write an undecorated symbol. I'm using it to refer to the whole collection of length variables. So I could calculate an area by using the three
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Hal Haggard: Length variables that are around that area and go through the whole complex that way. Similarly, the die, he drill angles in the simplex are computed directly from the links.
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Hal Haggard: So if you vary the length Reggie calculus action with respect to one of the bulk links you obtain the equations emotion that appear on the bottom left of this slide.
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Hal Haggard: And many numerical studies have have looked at this and with finer and finer triangulation you recover the Einstein equations.
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Hal Haggard: As you expect
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Hal Haggard: In contrast, in length, Reggie, sorry, excuse me area, Reggie calculus will work with the areas as variables.
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Hal Haggard: And here is one place where simplicities play a role. So for simplex, of course, has 10 edges and 10 faces so locally, the functions that give the areas as a function of the links and be inverted to give the links as functions of the area. So I'll denote the functions for the
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Hal Haggard: Length of an edge with this notation here. He has the edge sigma as the simplex in which that edge is being computed and as little as a collection of area variables for that simplex
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Hal Haggard: So when we think of Reggie calculus as a function of these area variables will write the Reggie action in this forum. So the area times the deficit angle viewed as a function of the areas
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Hal Haggard: Question. Yes.
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lsmolin: When you don't have one for simplex which now the host international conflicts. These numbers are not the same. It's not necessarily
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Hal Haggard: A. So the system.
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Hal Haggard: Oh, I see what you're saying. Yes. Um,
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Hal Haggard: Let me continually and if you still have that question. Can you ask it again a little later on I think it'll be clear what I'm doing. Yes.
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Hal Haggard: I'm
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Hal Haggard: So that I he drill and deficit angles are obtained using the formula for the day he drill angles expressed as a function of the links which we've now expressed as functions of the areas. So this gives us the die, he drill angles as a function of the areas
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Hal Haggard: And the thing that's really interesting about area, Reggie calculus. If you've never studied it before is that variation of this action with respect to a bulk area.
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Hal Haggard: Gives us the equation emotion that the curvature around that triangle is zero. So it imposes flatness on the whole triangulation.
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Hal Haggard: And the key reason for this is that the Schlafly get identity guarantees that the variation of the deficit angles is exactly zero in this case the I should say that more carefully the some of the areas times creation of the deficit angle.
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Hal Haggard: So this is a sort of classical Reggie calculus version of the flatness problem.
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Hal Haggard: That is we when we work with area variables are classical equations emotion give zero curvature on our triangulation.
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Hal Haggard: So this is why we have to start to consider constraints.
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Hal Haggard: So we can understand this difference in exactly the way that he was just raising as we start to glue for simplicity's one to another.
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Hal Haggard: There's different number of degrees of freedom between them. You can see this, because we're gluing two tetrahedron one into the other.
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Hal Haggard: And so here I've chosen the the tetrahedron with orange vertices to glue along just to give you a picture in your mind.
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Hal Haggard: And and in the length Reggie calculus, we would be gluing the six edges of that tetrahedron. So we'd be gluing six variables as we put these two simplest things together.
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Hal Haggard: But in the area, Reggie calculus. We're only matching four of them. So the theory is area matched, but the language we often use in the community is that it's not shape matched, there's no reason that the those areas, line up with each other perfectly
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Hal Haggard: So we can resolve this mismatch by introducing 3D die, he drill angles in our tetrahedral.
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Hal Haggard: So we compute the 3D that he drilling goes the same way we did for the 40 day he drawing goals I call them capital fi here.
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Hal Haggard: They're in a particular tetrahedron in a particular simplex and we're computing around a particular edge and the fire is just given by the length formula expressed as a function of the areas
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Hal Haggard: So when we have to neighboring simplicity say sigma sigma prime and we glue them along this tetrahedron Tao.
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Hal Haggard: Will have the same links in the town if the constraints that these day he drone angles match are imposed.
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Hal Haggard: So this is the way of thinking about the constraints that we're going to explore today. It's not the way that's been done in the past in the past shape matching has been implemented by by looking at the two dimensional angles in the triangles themselves and matching those up.
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Hal Haggard: But, but we're going to work with the 3D ones because it's the most natural set of variables to compare to the intertwine or space of of loop quantum gravity.
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Hal Haggard: Okay, so this is the, the set of constraints and and i think Lee this answers your question. Is that correct,
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lsmolin: Yes, it's let's see what have good
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Okay.
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Hal Haggard: Unfortunately, the those constraints that I just mentioned, have the disadvantage that they're not localized on a simplex. So we're going to supplement
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Hal Haggard: Area Reggie calculus with additional variables. In particular, we're going to introduce to die, he drill angles per tetrahedron in our triangulation. So these are now the little five or variables in our theory.
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Hal Haggard: And we, there's an important
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Hal Haggard: Technical point here, which is that we choose these two variables along edges that are not opposite in the tetrahedron. They have to be along edges that are needed a vertex
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Hal Haggard: That you can see that that comes out of the calculation of the tetrahedron, that if you really want to fix all six of its links, you have to use to the heater angles that meet
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Hal Haggard: Today he drills around edges than me.
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Hal Haggard: So here are constraints re expressed in terms of the new variables, the little thighs and the cool thing about expressing them this way is that the constraints of become completely localized on
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Hal Haggard: On a single for simplex now. So we've got the areas expressed, sorry. Got the constraints expressed in terms of the areas and these variables are fixed to those values.
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Hal Haggard: So this localization of our constraints on to a force in Plex is going to allow us to to retain the additive factorization of the Reggie action. So this is a really nice thing.
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Hal Haggard: So these constraints that we're studying here are completely equivalent to the the shape matching constraints that were studied in the past, you can actually show this.
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Hal Haggard: There's sort of higher dimensional version of it. And you can also show that when you impose these constraints on the classical theory, you really do recover length Reggie calculus. So this is a way to constrain area, Reggie calculus and and recover length Reggie calculus, so
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Abhay Ashtekar: I'm just gonna stay the same order that analog this because you said that they're the same, but
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Hal Haggard: You also sell right I just meant that they give the same result of I am sorry. Thanks for the language.
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Hal Haggard: The I just mean that that we really do recover length Reggie calculus out of imposing these constraints.
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Thank you.
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Hal Haggard: In the area angle Reggie calculus ammonium Bianca initially introduced, they were working with constraints on the 2D die, he July angles the angles in the actual triangular face.
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Hal Haggard: So now it's a wonderful calculation that you can do in the intertwining or phase space of loop quantum gravity. The so called kappa that Milton space.
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Hal Haggard: That you can calculate the the pulse on bracket between to die, he draw angles at non opposite edges. And the result of that calculation is a sign of the
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Hal Haggard: Face angle the 2D die, he drawing goal at the vertex, where the two edges meet divided by the area of that face which will express in terms of our area spectrum as we move forward.
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Hal Haggard: All right.
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Hal Haggard: So this set of variables has a particularly nice match with the loop quantum gravity Hilbert space.
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Hal Haggard: However, we see because of this plus on bracket calculation that these are second class constraints. So, so we're going to be dealing in this same sort of setting that up RL and FK models do. I'm thinking about second class constraints on our spin from
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Hal Haggard: Yes.
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wolfgang wieland: He has worked on.
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wolfgang wieland: hyder, could you explain
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wolfgang wieland: The
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wolfgang wieland: The nature of these constraints, from the perspective of the
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wolfgang wieland: Canonical theory.
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wolfgang wieland: So if we start out with SU to Africa, a burial burial Bruce and impose the scale and have to impose Kayla vector and gals constraints. So why do we not see these constraints in there.
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Hal Haggard: So I haven't thought about this from a canonical perspective yet Bianca, do you have a comment on that.
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Bianca Dittrich: Um, yeah, these constraints which we have which the tree. Have I covenant was a sheep matching constraints which were considered earlier and the sheet matching constraints come from the second Eric simplicity constraints. So it's part of the secondary simplicity conflicts.
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Hal Haggard: Good. That's what I, that's perfect. I should have said shape matching right off the bat, because that's really what these are imposing right Wolfgang
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Bianca Dittrich: And so what should I shoot matching optional committee. Yeah.
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wolfgang wieland: Well, I was more pointing out that on the
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wolfgang wieland: Face of su to actually have a barrel variables there just
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wolfgang wieland: Is no shape matching to begin with.
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wolfgang wieland: And so that seems to be this fundamental tension in between the canonical approach and Reggie calculus picture and and. But maybe that's more of a discussion that we should leave put and I just wanted
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Hal Haggard: That's fine. Let's do discuss it more at the end, um,
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wolfgang wieland: Yeah. So yeah.
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Hal Haggard: Alright, so the area, Reggie action factors, admittedly, even with these constraints as I was just saying a moment ago.
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Hal Haggard: So, and we can also easily include boundary triangulation. So this is the standard hurdle and Sorkin term in Reggie calculus.
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Hal Haggard: So, the way I've included the boundary is I've included an index here and t that we take the the deficit angle we expand it in the Reggie action.
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Hal Haggard: We include this index, where the index is one for triangle on the boundary and two for a triangle in the bulk
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Hal Haggard: And you see that we can think of this as a triangle action which I've written here this way and as simplex action which I've written here this way with this simplex action taking this form.
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Hal Haggard: So this is going to be important for our
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Hal Haggard: Our spin phone model because we'll be able to write the amplitude in a factor is a product factor is for
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Hal Haggard: All right, so let me then define the model.
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Hal Haggard: We write the partition function for the model z as a some over the spins of the triangulation, a measure of factor Mew.
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Hal Haggard: Product over triangle amplitude times the product of our simplex amplitude times product over what we call g functions or G factors which are the place where we're going to impose these constraints.
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Hal Haggard: The triangle amplitude is written here and the simplex amplitude here.
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Hal Haggard: And in practice in the numeric. So we're going to take the measure factor to be one.
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Hal Haggard: We just haven't studied the measure of factor yet so we didn't know of a better way to fix what value it should be. But this is certainly something that will be interesting for a future study
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Abhay Ashtekar: Can you say what the
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Abhay Ashtekar: Parts that you're integrating or art. Is it just are you fixing one one for simplex. Are you allowing all possible for simple exist, what are the boundary conditions which are fixed on the spots and so on.
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So,
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Hal Haggard: We're going to, I'm going to describe two examples in great detail in a minute. But the idea is that we have some boundary of our triangulation.
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Hal Haggard: And we fix the data on the boundary and we some over all paths on the interior that are consistent with those boundary data.
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Abhay Ashtekar: Okay, thank you.
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Hal Haggard: And
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Hal Haggard: I just wanted to note briefly here that we could just as well have taken co signs instead of exponential is for these amplitude factors.
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Hal Haggard: It would have resulted in a quite mild complication of our new metrics. So this is not something that would be a problem for us. We would be happy to study it in the future.
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Hal Haggard: And we focused only on the exponential factors because we were interested in the flatness problem interested in seeing if we could figure out what was going on with that.
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Muxin Han: Hi, how
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Muxin Han: About this is missing. So what exactly the boundary data. Is that still spin, spin in the minors, or spin and something else.
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Hal Haggard: It's going to be spins for all the the areas right but also these die, he drill angles that I was mentioning, so today he triangles per tetrahedron in the boundary
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Muxin Han: But is that embed code to acuity.
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Hal Haggard: Yeah, the day he triangles are like the entertainers.
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Muxin Han: OK.
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OK.
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Muxin Han: Go ahead. Lee. So those two angles are tetrahedron tetrahedron angles.
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Hal Haggard: That's correct.
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Hal Haggard: Okay, the five, the little Pfizer tetrahedron die. He joins us
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Muxin Han: And what is this and t.
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Hal Haggard: And T is this factor. I just mentioned on the previous slide. It's just a way to allow me to study both boundary triangles and both triangles. So that's an index that's equal to one if you're on the boundary and two if you're in the book.
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Muxin Han: I see. Thank you.
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lsmolin: Yes, just quick question. Is there a linear analysis, which shows you get the right to the field and so
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Hal Haggard: We haven't performed linear analysis. Yet although it's closely related to two things that Bianca, and Seth and I did in area, Reggie calculus. So this is certainly something that we could do Li
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Hal Haggard: Alright, so as, as we were saying the the factors G implement the constraints.
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Hal Haggard: But you see now what the trouble of implementing these constraints sharply would do.
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Hal Haggard: If if G is equal to one if the constraints are satisfied and zero otherwise. We're going to get do fantasy equations for these constraints that I wrote up here, right.
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Hal Haggard: Because we have a discrete set of area labels and these will become the accounting equations, a system of defending equations.
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Hal Haggard: So, um, so that's going to be very hard to meet with our spin labels are discreet spin labels and less unless the configuration is very, very symmetrical.
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Hal Haggard: So if we impose the constraints strongly. It's like we would be eliminating the density of states we would be taking the density of states too low to really have a quantum theory. So this is the key facts that leads us to think of imposing these these constraints weekly
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Hal Haggard: So here we have a fun way of thinking about this. So we're forced. I have a Greek colleague here at Bard in physics. So I asked him how to pronounce pronounce them.
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Hal Haggard: Properly. So if the navigate between skill a reducing too much. The density of states and her of this imposing the dynamics that will not match general relativity. That is the the flat area, Reggie dynamics.
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Hal Haggard: So the way that we're going to implement these constraints is we're going to think of putting coherent states on the tetrahedron.
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Hal Haggard: So the factors in the spin phone model that I'll present today are an inner product of coherent states on the tetrahedral and these coherent states are going to be peaked on the files which are constraints. So this is as strong an imposition of the constraints as we can think of.
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Hal Haggard: Okay, so before I present the the numeric to you it's useful to know that, like other spin phones model spin phone models. This model can be derived from
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Hal Haggard: A constrained topological theory for general relativity. In this case, the, the theory is a higher gauge theory. So in this sense.
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Hal Haggard: The model is completely fundamental spin phone model. There's nothing approximate about it from that perspective. So why did we call it effective spin foams. Well, we wanted to indicate the fact that
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Hal Haggard: That we could implement all sorts of different choices of G in this sprint phone model, if you had. If you thought the as strong as possible constraint idea that we've used is not the right way to go. You could choose a different G.
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Hal Haggard: And you could study your spin phone model that has your way of implementing the constraints. So that was the sense of effective that we had in mind.
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Hal Haggard: In particular, I don't see any reason at the moment, why you couldn't implement a PRL and FK and similar models with a particular choice of G function that is you would find what's the way that you're imposing the constraints and you would implement outweigh
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Hal Haggard: The other reason we thought it was kind of a fun play on words is that it's very numerically efficient. So they're effective in giving us numerical results and they're effective in the sense that you can choose g to be a variety of things.
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Hal Haggard: The derivation from this higher gauge theory, it's a BF CG gauge theory.
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Hal Haggard: Shows that the amplitude factors that I'm using in the spin from these exponential Reggie action can be thought of as sort of
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Hal Haggard: Higher gauge recoupling coefficients. So even if you thought I was maybe making an approximation. When I expressed it as exponential approximation to some three and Jay recoupling coefficient, that's not the case here. You really can view these as fundamental recoupling coefficients.
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Hal Haggard: This also exposes a nice set of connections with earlier work by corrupt and have better a teen and Friday. Well, and what we sometimes call the KBS model.
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Hal Haggard: So these sorts of connections were studied in in a very nice paper by Assad Dietrich directly reality and submit. Plus, it's the first. Excuse me. It's the first of these links on the bottom right of the slide.
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Hal Haggard: Alright so despite the title. These can be viewed as fundamentals and films.
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Hal Haggard: So here's a summary of the spin foam. Again, just for you to be able to look at it. I'm going to take the measure factor to be one in all the numeric that I'm about to present and that's
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Hal Haggard: We're going to study a couple of special cases. So to keep the numeric tractable, we look at symmetry reduced triangulation.
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Hal Haggard: So these won't be a completely general triangulation with every edge for you to have every length at once. Instead we'll pick some of the edges to have the same length, that's going to lead to some areas being equal, and we're going to study that reduce spin from that comes out of that.
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Hal Haggard: That simplification.
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Hal Haggard: I'm going to approximate this inner product of coherence states the G factors that I was mentioning before, I'm going to approximate it
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Hal Haggard: By a real Gaussian. So the, the idea of this real Goshen, is that we already know that the day he's real angles that we're trying to impose with that constraint don't pass on commute.
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Hal Haggard: And we're going to pick the width of the Goshen to be consistent with that non commutation so the sigma squared is going to go like this non communicator.
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Hal Haggard: And as was mentioned at the very beginning of the talk, we're going to consider scaling, not only with Jay, which is what in spin foams. We usually consider the semi classical limit, but also with the gamma parameter the area spectrograph in this context.
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Daniele Oriti: Sorry, I'll ask a question is
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Daniele Oriti: Pleasing Allah. Hi.
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Daniele Oriti: Can you
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Daniele Oriti: clarify a little bit why you use the discrete areas in the classical action because I will respect that and especially involving the the spins, or the, you know, this critize the spectrum of the areas to appear once you respond, the
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Daniele Oriti: Classical looking path integral in the analog a spherical or morning so in a particular basis. That's what happens in in speed for models here instead you directly right
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Daniele Oriti: And especially for the young pitchers that is supposed to be like a integral with this potential of the actual in some measure factor is because you can capitalize it
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Daniele Oriti: But direct in terms of the spectrum.
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Daniele Oriti: Of the area as if there was already some quantization there.
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Daniele Oriti: And I'm a little puzzle that will respect that this type of a special would appear
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Daniele Oriti: Either in terms of continuous areas in a part integral writing of the amplitude or from some semi classical approximation of the actual spin form is function. So once you rewrote the Pentagon again. Dr. All those particular mornings.
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Hal Haggard: Clear and, as I was saying, back here in the key BF slide. Let's see.
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Hal Haggard: Can I pass it.
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Hal Haggard: On and so so Daniel at, I would say, one way to think about this is that we do. We have a spin phone where we're, we've got a delta function on the deficit angle right every Reggie calculus is going to be a flat theory. And so we're thinking of a deficit angle on the on the deficit, sorry.
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Hal Haggard: A delta function on the deficit angle. And now I can, I can expand that in these these recoupling coefficients of this higher gauge theory and and these are exactly the expressions in particular with discrete labels.
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Daniele Oriti: Yeah, and the sun, but then I will not respect if you're special consultant dysfunction. I will not respect and expression of the form exponential have an option.
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Daniele Oriti: In general, that's not what the expression for the outages look like
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Bianca Dittrich: No, it does come out in the States, of course, sign, but it does come out exactly industry. Of course I know he actually
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Hal Haggard: I agree with you that it's unfamiliar Daniela. That's exactly why I was trying to explain is that that's exactly what happens in this theory.
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Hal Haggard: It's maybe the we're not so used to thinking of these higher gauge theories and how they do this, but that's exactly what happens.
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Daniele Oriti: Okay, so when you spend, you get a factor is a product of the call science of the option. Yeah. Is that correct. Okay.
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Thank you.
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Hal Haggard: Alright, so now two more concrete specification along the lines there that are by was asking for. So the first
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Hal Haggard: I'm going to study to triangulation is, as I mentioned in the first triangulation. The
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Hal Haggard: We have three for simplicity's glued around a book triangle. So this is actually the same picture I showed before. Here I've decorated the picture with all the links in it.
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Hal Haggard: So as I was mentioning, an easy way to think about it for simplex is you just specify it's five vertices. So the three for simplicity's are given by these collections or vertices. They share a single both triangle you see 135 appears in all three of them. So that's the red bulk triangle.
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Hal Haggard: The symmetry rejection that we impose is that
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Hal Haggard: Vertices pulled from the set 135
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Hal Haggard: pair of vertices pulled from that set have length x pairs of vertices pulled from the set 246 have the length y and vertices, where one is pulled from 135 and one is pulled from 246 have the length z.
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Hal Haggard: All the boundary
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Hal Haggard: Z edge lengths have the same day he July angle you can check this just by looking at the tetrahedron. All the tetrahedral have the same set of six links and so they all have the same day he drill angles.
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Hal Haggard: So what do we end up with, we end up with a theory that are triangulation that has three independent areas. Two of them are boundary. I'm specifying areas as a functions here so function of x, z, and we get the boundary area B.
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Hal Haggard: Or y and z and z, we get the boundary area. See, and one bulk area which is just AREA OF X x X.
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Hal Haggard: Which we some over this is the, the single variable and our spin phone and it allows us to study expectation values of the deficit angle around that triangle. So that's what we're going to do numerically.
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Hal Haggard: It's a very interesting that very similar time to when we put this workout Pietro da na GA and good Simeon Sarno put out another paper on
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Hal Haggard: On the archive where they were studying the same cemetery reduced triangulation from a very different perspective from SU to BF theory. So we can discuss that more at the end if if people would like
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Hal Haggard: Okay, so
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Abhay Ashtekar: Um, so what I was asking before. So the
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Abhay Ashtekar: DNC are being fixed
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Abhay Ashtekar: Right, because they are the boundaries and as being some little as being some low
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Abhay Ashtekar: Exactly like
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Hal Haggard: Okay, so it turns out to be a little bit easier to study triangulation without boundary. So we just take two copies of the complex that I just described. Those are each individually like balls four balls. So we glue those around. And we get an S for glue those around their boundaries.
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Hal Haggard: So we can think of the book area and the first copy and the Bulgarian the second copy. We're going to fix the Bulgarian the second copy a prime and think about what happens for the the bulk area that we're something over
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Hal Haggard: So we perform exactly the spin from calculation that I outlined. So there's some over these were, as I said, we approximate the G factor as a as an exponential
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Hal Haggard: With a particular spread gaps, an exponential with a particular spread and the spread itself is given by the commutation relations. I mentioned earlier.
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Hal Haggard: So this is a completely explicit characterization of the model. I don't know that it's possible to follow this on the fly. I just want you to see that there are explicit details under everything I'm about to show you.
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Simone Speziale: Can I asked you before you show the results question about the setup.
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Please.
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Simone Speziale: With these configurations in which there are no clubs Gordon and complicated expressions.
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Simone Speziale: It shouldn't you also be able to study the an arbitrary configuration and not just a symmetry reduced one. In other words,
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Simone Speziale: If you, if you allow for these quantities to be function of all the spins independently or not to choose a music response model, you can still perform you know medics right
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Hal Haggard: Absolutely. It's just a question of how quickly the computational time is going to scale. We did everything all the numeric. I'm going to present today. We're done on home laptops. We haven't done anything with high performance computing. So there's
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Simone Speziale: These specific one. We data through this any way only one internet
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Simone Speziale: Spin
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Hal Haggard: That's right.
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Simone Speziale: You're not really reducing anything. Maybe in some more complicated triangulation, but so far it seems like the restriction to mini super space is not strictly necessary for your
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Hal Haggard: I completely agree.
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Simone Speziale: I think
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Hal Haggard: So here are results. This is a bit dense, so I'll go through it in stages and
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Hal Haggard: The first thing to note is that the amplitude factors oscillate. We saw that there as was pointed out their actions they carry a factor of I in front. And those are going to be awesome Ettore factors.
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Hal Haggard: On the other hand, were approximating the G constraints. These G functions by real gal oceans. So if you think of the product of all of these factors, you're going to have a complex action that comes out of that.
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Hal Haggard: And here, what I'm doing is kind of plotting the separate parts for you in order to look at them separately and understand how they interact with each other. So along the horizontal axis here the variable is that single book area that we were just discussing
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Hal Haggard: The vertical scale is various depending on which part I'm looking at for the red, green and purple lines, it's the real part of the total Reggie area action and for the dashed blue part. It's the, the value of the G factor.
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Hal Haggard: So you can see that that the Reggie action depends on the value that we choose for the area spectral gap.
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Hal Haggard: And depending on that value, of course, the action is going to oscillate, more or less. So if we take large values of gamma, then we get lots of oscillations as the red curve.
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Hal Haggard: If we take a factor of gamma reduced by a factor of five. We get this green curve which is slower and it's oscillation, if we reduce again by a factor of 10 we get the purple curve which is very slow and it's oscillation
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Hal Haggard: And so the significance of this is that we're going to take the product of these two factors and and we're going to do some over everything right and so it's clear that if your action contribution is oscillating too much.
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Hal Haggard: The G factor is non negligible. You're just going to average out the value of the G factor so you won't really be effectively implementing your constraints, you're going to be
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Hal Haggard: averaging them out.
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Hal Haggard: And you're going to get very little contribution to the some
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Hal Haggard: On the other hand, if you choose a value of gamma that's small enough, then the oscillations will be much more gradual as you go through the region where the constraint is non negligible and you'll end up with some that has some some meaningful value.
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Hal Haggard: So here I've on the left. I've shown a plot where the boundary areas are all fixed to the value 99.5
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Hal Haggard: But as we often do and spend films. We can also study what happens as we increase the boundary values. So the plot on the right is very similar same thought, except for the boundary areas being modified to be 10 times larger roughly 10 times larger.
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Hal Haggard: Alright, so
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Hal Haggard: Remember that what the point of imposing these constraints, was it was to come back towards a length Reggie calculus picture. So we can ask
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Hal Haggard: For the these boundary data. What's the nearest length Reggie configuration and what is the value of the bulk deficit angle for that length Reggie calculus and in these choices of parameters we have that the bulk of deficit angle is around 0.5
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Hal Haggard: So in the upper right, you see the values that we get out of our spin phone for the expectation value of this bulk deficit angle. So here are the values of the boundary spins in the first copy of the triangulation. The second copy of the triangulation, and for the second copies both deficit.
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Hal Haggard: And then you see in each of the columns. The debt various values of gamma that I was mentioning, and you see the expectation that we get as we go to larger and larger spins.
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Hal Haggard: So what's remarkable and lovely is that we do indeed recover a curved configuration from this model.
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Hal Haggard: That curved configuration has a value for the deficit angle that is very close to the classical length Reggie calculus value.
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Hal Haggard: But you also see that there's something interesting, which is that because we haven't strongly imposed the constraints we also get a small imaginary part to our expectation values.
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Hal Haggard: So one of the things I'd like to go on and and illustrate for you a little is where that imaginary part comes from
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Hal Haggard: I'm going to skip this slide. This is very interesting to me. It helps to understand why we
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Hal Haggard: Need this kind of trade off between large j and small gamma. It's the scaling argument that shows why that happens. But I'm concerned that I'm running low on time. So I'm going to skip the scaling argument.
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Hal Haggard: I guess one thing I will highlight emotion, since you're on the call is that we find exactly the same sort of condition that you were finding in in some of your older work.
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Hal Haggard: OK, so we also studied a second more complicated complex in this complex we glue six for simplicity's around a single bulk edge again it's a symmetry reduced and but this time. At least there's this bulk edge that length Reggie calculus would also have
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Hal Haggard: In particular, once you do the symmetry reduction we have three boundary and three bulk areas in the area description or four boundary and one bulk links.
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Hal Haggard: There are three simplest sees that all have the same geometry. I'll call them type one here and three simply sees that share geometry with one another, which is type two, but type one and type two don't have the same geometry.
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Hal Haggard: The path integral involves one bulk variable in length, Reggie calculus, but three area variables in our constrained area, Reggie calculus. So we have a much larger set of things that were something over in the book.
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Hal Haggard: However, if we make use of the fall off of the G functions. We can significantly reduce the range of the summations that we have to carry out in these bulk areas. And this is one of the things that really helps the numeric proceed quickly.
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Hal Haggard: I don't want to take you through this in detail, I think it would be a little bit overwhelming. I just wanted you to have it in case you want to look at exactly what our simplex configuration is
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Hal Haggard: I've used the same sort of notation and strategy for describing it as I went through in detail on the last triangulation.
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Hal Haggard: But in particular, let me highlight the areas that we were just mentioning. So these two areas are shared in in their value.
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Hal Haggard: And have an X sexy and an X x THE PRIME. SO WE HAVE THREE boundary areas. Those are the ones I was mentioning, just a moment ago, these two bulk areas share value and we have three bulk areas that are independent. So it's three boundary and three bulk areas in this configuration.
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Hal Haggard: Okay. So I mentioned already that the G constraints.
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Hal Haggard: Are going to narrow the summation, but we also have generalized triangle and qualities of course if if you violate the triangle inequalities, that's not going to be in allowed simplex
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Hal Haggard: There's a generalization of that. That looks at the foil for simplex. And you can look at
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Hal Haggard: Its Graham matrix and look at any sub determinant that Graham matrix and you get conditions for the existence of that for simplex it's related to the embed ability that Wolfgang was asking about at the beginning.
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Hal Haggard: So those generalized triangle inequalities allow us to say that some parts of the parameter space are just not allowed. So the plot on the right here shows the parameter space of the bulk areas, the x x t the azz T and the Z prime z prime t
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Hal Haggard: On it's three axes and the colored regions in the plot are the ones that are generalized triangle allowed so we don't have to consider any parts of parameter space outside of these colored regions.
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Hal Haggard: In fact, if we now add in the G constraints that I was mentioning that is we put some threshold and we say below this value the G constraints are just going to kill the summation completely
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Hal Haggard: Then we can restrict to the single colored region in the lower left part of this plot.
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Hal Haggard: So that's what that's what the ways that are numeric proceeds more easily.
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Hal Haggard: Here's another slice of that here. I'm plotting it as just a function of two of the book areas xx t and ZTE again the blue region is the generalized triangle allow the region.
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Hal Haggard: And within that region. We're plotting the region where the G factor is greater than 10 to the minus 10
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Hal Haggard: So outside of the gold strip. There's no reason to do the summation because the G factor is just going to kill your son so are constrained area, Reggie calculus is essentially going to be a spin from model along this gold strip within the blue region.
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Hal Haggard: This is the second major factor in the numerical numerical efficiency.
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Hal Haggard: Okay.
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Hal Haggard: So I just want to tell you a little bit about our methodology, just so that no confused. Well, so fewer confusions might arise.
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Hal Haggard: The as we described. There's three boundary and three bulk areas. There's four boundary and one bulk link.
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Hal Haggard: In effect, the G limits the swath of parameter space around the bulk length variation by keeping us near to shape matched configurations. So we're near to length allowed length Reggie calculus allowed configurations
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Hal Haggard: So far, only through expectation values. That's the numerical output that we've studied we investigate for large spin larger boundary values.
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Hal Haggard: And particular values of gamma
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Hal Haggard: They give us expectation values that are consistent with length Reggie calculus. So we get curved length configurations out of looking at a nice set of parameters and j and gamma
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Hal Haggard: If we were doing a saddle point analysis that is approximating the sums that we're actually doing by integrals.
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Hal Haggard: We would expect that these would be saddles of the area, Reggie calculus action, but where we're taking it only along the constraint directions only along that gold strip that I was just illustrating
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Hal Haggard: So perhaps it's useful to distinguish these from the area, Reggie calculus saddles themselves and we could call them send me saddles their saddles along the constraint.
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Hal Haggard: Swap swath
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Hal Haggard: By construction. The semi saddles agree with the length Reggie calculus adults as well as they possibly can on the week constraints.
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Hal Haggard: Alright so here are the results for this more complicated in our bulk edge configuration.
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Hal Haggard: I put several plots here just so you could look at different parts of our analysis, but I'm only going to go through one because I'm running short on time.
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Hal Haggard: So here's a case where we fix the boundary areas to these three values. The, the nearest length Reggie calculus solution that would complete this complex are given by these values.
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Hal Haggard: Instead we perform the sum over the bulk areas and ask about what our expectation values are. And here I'm plotting the real expectation value of the first area.
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Hal Haggard: Xx T first area in the boat as a function of the gamma the spectral gap parameter
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Hal Haggard: In the second plot. I'm showing the imaginary part of that same area again as a function of gamma
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Hal Haggard: Notice that the two scales are centered at different positions the imaginary part is centered on zero. The real part is centered around for 60 something
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Hal Haggard: And indeed we are an excellent agreement with the length Reggie calculus that we get for 63 for this swath of gamma that is small enough
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Hal Haggard: I'll skip some more area expectation values. Here's a bulk deficit expectation value is the both both deficit angle around that same triangle that we were just looking at.
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Hal Haggard: Again, it's real and imaginary parts as a function of the value of gamma and the length Reggie calculus value for this book deficit is 3.22 radians.
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Hal Haggard: And you see again that we're getting excellent agreement in a region of the gamma values. It's interesting that gamma can't be too small or too large here.
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Abhay Ashtekar: Should I be worried that the imaginary part is can quite large and you want it to be zero, is that
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Hal Haggard: Good. So I want to comment about exactly that next. So let me go ahead and say it and see if it's
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Satisfactory
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Hal Haggard: So I'm skipping over many of the plots that I supplied for you just to look at
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Hal Haggard: Here's a quick characterization of our run times this is very rough. Just to give you a sense
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Hal Haggard: This was done on our laptops using Mathematica each run we computed six expectation values. The three bulk areas and the three deficit angles at those both areas for 40 different values of the Emirati parameter. So we're doing 240 spin foams per run
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Hal Haggard: And here I'm specifying a table with the values on the classical areas.
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Hal Haggard: The maximum bulk spins that we have to some over and the runtime in the right. So of course the boundary spins determine the range of the summation and but it's striking that we can do these calculations and in hours on our laptops.
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Simone Speziale: How many are the internal faces of these
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Hal Haggard: Triangulation the internal triangles. There's three
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Simone Speziale: There's only three
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Simone Speziale: Yep. I see. So you're, again, you're not doing a symmetry reduced mother really your something over as many
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Bianca Dittrich: Nodes, or five into some reduction.
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Hal Haggard: Oh, sorry. I thought you were asking about our cemetery reduction. But yes, there are five in general and three and
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Simone Speziale: Five internal feces.
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Simone Speziale: Yeah. And so with these competing times be very strongly affected if you renounced the to the symmetry reduction in some over all five of them.
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Hal Haggard: Yes, they were.
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Simone Speziale: In you can get very different results from the mini super space has been from kind of thing, including also important things like they're very agencies are such. This is why I'm wondering, I think it seems
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Bianca Dittrich: Similar. Let me comment just honestly
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Simone Speziale: You can still do this calculation. Also in the more general setting, I suppose.
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Bianca Dittrich: We can do it, you know, it is five, but the G constraints. But again, reduce it to something which is almost one dimensional
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Bianca Dittrich: Yeah, so, so it should, should I mean a big piece of I'm trying, but it should not be too much.
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Hal Haggard: Of course, it just depends on what you asked us for two right if we make the bulk spins larger it's it's much larger. Right. So, but yes, there's a there's a range of parameters, where we should be able to do something.
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Seth K Asante: Maybe I can say one thing i i think in general can even reduce this time to because in the competition. We didn't paralyzed. Many things so
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Seth K Asante: Yeah, definitely. If we have five parameters and above, you can still do these competitions in a short time you feel paralyzed and compare many things. Yeah.
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Hal Haggard: I'm trying to emphasize
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Hal Haggard: That we really have not optimized the computing at all by saying we're doing it on our laptops.
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Hal Haggard: Right, it's completely on optimized in every sense
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Simone Speziale: Yeah, I just wanted to emphasize that think your result is even stronger than would you say, because I think it takes enormous effort to to go beyond these mini super space in which case it's
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Simone Speziale: I would say
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Hal Haggard: I'm trying to carefully not report anything we haven't done
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Hal Haggard: But I completely agree there's there's plenty to do here.
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Hal Haggard: So going to what I was just mentioning. And we've not explored complex values for parameters at all.
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Hal Haggard: So everything that I'm about to say is, is purely conjecture and and I find it very interesting conjecture, but it's a conjecture.
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Hal Haggard: So in the in the two examples that we've presented the, the three sympathies simply sees with one both triangles six simplicities with an inner edge we find complex expectation values as exactly as I was pointing out
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Hal Haggard: And we conjecture that these complex expectation values are from saddles that are slightly off into the complex plane.
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Hal Haggard: So we believe we may have identified a regime in which a saddle moves close to the real access, but it's not on the real axis.
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Hal Haggard: And this may be one characterization of what's desirable about our combination of gamma, not too small, not too big, somewhere in the middle and J fairly large and and not to large either
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Hal Haggard: So it's the, the, the fact that we've brought the saddle close to the real access. So the picture that I have in mind is a sort of Picard Lifshitz picture or maybe a more
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Hal Haggard: Down to earth way of saying is just at Steepest Descent picture where I start off with an integration contour along a real access for example.
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Hal Haggard: And then I use the correct complex analysis theory to deform that into the saddle into the country that goes over the saddle that I'm interested in.
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Hal Haggard: So I follow the steepest ascent contour that crosses my initial contour to deform my contour up into the complex plane and then we know that the integral would be dominated by this single saddle and the complex plane. So,
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Hal Haggard: We're, we're not doing that here. We're not doing integrals. We're doing sums, we're not we're not
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Hal Haggard: Investigating complex values, but I would conjecture that if we did do those things that
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Hal Haggard: That that would explain the nature of the complex expectation value essentially our sourdough point is complex. So we're evaluating the deficit angle at a complex parameter. And that's how it's ending up with a complex value.
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Hal Haggard: All right. But the main thing I wanted you to take from the talk is that we have a spin phone model that can avoid the flatness problem in a range of spins area values and
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Hal Haggard: Area spectral gap gamma. The just to pick out one example to report from the more complicated configuration here the bulk expectation values that we get from our new metrics and here the length Reggie calculus values and they're in pretty darn good agreement.
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Hal Haggard: So here is a summary of the model again, just so you can look at it. If you have questions.
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Hal Haggard: The three things that I think are the big takeaways from the talk, or the constrained theories with discrete geometrics spectra can
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Hal Haggard: Suffer from to lower density of states. And the reason for that is that the constraints can impose diet granting conditions that that just kill your density of states.
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Hal Haggard: Nonetheless, by imposing those constraints, weekly, we can find a set of parameters in which the dynamics of general relativity, it emerges as indicated by the numerical results we've we've got
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Hal Haggard: This model is
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Hal Haggard: Numerically, efficient and is a really interesting playground for us to start looking at more questions in quantum gravity. Thank you all very much.
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Jorge Pullin: Questions.
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Abhay Ashtekar: Few, few questions here. I mean, that hopefully so
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Abhay Ashtekar: The first is
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Abhay Ashtekar: You know this this idea about this, this, this, the way that you put this slide. It looked almost like, well, we should really do. The first thing, but then that kills it conceptually, we should really
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Abhay Ashtekar: Yeah, we should really do the constraint theories with discourages spectra. But then that's too small of thing and therefore we are imposing these constraints weekly
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Abhay Ashtekar: I mean, it doesn't seem to be kind of a systematic thing and it just sort of saying that this is what we should do.
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Abhay Ashtekar: But isn't that a kind of a simple way of saying that, well, if your second class constraints. And this is something that I look at it quite a while ago and I think it isn't.
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Abhay Ashtekar: Jonathan either thesis or writings angles that per second class constraints we can always just go to the environment, kind of representation
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Abhay Ashtekar: And if I went to that I can impose a strongly and then what it does for, for example, you know, supposing I got x one x two p one, p two. And my constraint is x, y equals zero, p one equal to zero, then I can just
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Abhay Ashtekar: Impose a condition x x one minus IP one equals zero. And that is just excellent plus ip one equals zero. And that is just in homomorphic representation just says that
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Abhay Ashtekar: The state is the vacuum state with respect to the x 21. I mean, it is a gulshan Victor the zero value, so to say.
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Abhay Ashtekar: And strongly in position in the sense actually making the complex linear combination just those that
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Abhay Ashtekar: It seems to me that that's what you are basically doing. I mean, we're looking at this quarter states, I mean,
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Abhay Ashtekar: That's what you would like to do it for like me, they may be further approximation that you might have done by when you went to some cautions or something like that. So could we not just say that
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Abhay Ashtekar: This is just you know the the yoga second class constraints that we can impose them and you can impose them by while that say it says is that that one needs and have to have a gulshan big that the zero values are poisonous take Victor the zero values.
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Abhay Ashtekar: Am I right about this that we can say that without having to say that what is the best we can do.
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Hal Haggard: Yeah, I didn't mean the best we could do as a as a weakness. I mean, if I understand what you're saying correctly. I mean, the this Bartman would also be doing exactly this, it wouldn't be imposing the constraints strongly, it would be imposing them with some cash and width right so
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Hal Haggard: So all I weren't. It wasn't strongly the sense that you know it.
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Abhay Ashtekar: Is really the operator value of x plus IP. It just becomes Dubai dizzy in the button.
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Hal Haggard: Sure. Okay.
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Abhay Ashtekar: And so therefore it is it really is exactly the by the zero pregnancy equal to zero and that it just killed by the emulation operator.
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Abhay Ashtekar: And a solution to the something we just killed by an elation operate is a vacuum state which is sharp repeat vacuum state farm on cost of medicine which is sharply peaked at x equals zero equals zero. And so it's really a strong position, but in the bottom of representation
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Hal Haggard: Yes. And, and, but you would agree that that strong in position. If I look at it from any sort of X representation would not be
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Abhay Ashtekar: Might not be one value of x, right.
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Hal Haggard: That's all I'm trying to make clear
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Abhay Ashtekar: Exactly not but but on the other hand, the desert. Besides, we are imposing second class constraints which is sort of literature for one. Okay. The second thing was really about
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Abhay Ashtekar: Yeah, I think is more general NPR time. That was what is it
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Abhay Ashtekar: Your viewpoint about doing all this in plus, plus, plus, plus signature. I mean, I
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Abhay Ashtekar: Mean
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Abhay Ashtekar: As a mathematical physics problem. That's fantastic. That's great, etc. But
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Abhay Ashtekar: I mean, how I want to make contact with with the real theory. So to say so.
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Abhay Ashtekar: Do you have a vision towards that, or do you have a
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Abhay Ashtekar: Mini meltdown in picture, for example, that is visions, we say that about you could do the Ukrainian theory, but then there'll be a generalized week rotation, you know, which is really canonical translation on the face space.
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Abhay Ashtekar: Which you know odd things like that. And so is there anything like that here.
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Hal Haggard: Yes, so, um, I should say that I haven't started working on it. So that's the first caveat, but I i think that there's lots of really interesting questions in that direction. So,
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Hal Haggard: Some works have have shown that the you expect the discrete areas spectrum for spatial triangles. But you could get a continuous spectrum for time like triangles.
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Hal Haggard: And and so there's a really interesting kind of question of well, what becomes of imposing these constraints, when you have both kinds of specter of present in your triangulation.
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Hal Haggard: So that's one question.
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Abhay Ashtekar: I guess I'm said, I mean the usual kind of answer both from quantum field theory and from this generalized rotation, etc. Is that you don't perform the integral in the lorenzen domain. Right.
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Abhay Ashtekar: We just some are taking the answers and then do something to the answers nutrient domain to get the physical answers in the lorenzen so I think
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Abhay Ashtekar: Sort of fixing about what happens at each step of the procedure is not perhaps the right procedures right strategy.
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Hal Haggard: That's, that's a very interesting much more general point and i think i agree i by the like the impact of growth. That's what we learned again and again. You just have to understand what the right contour is and then you'll, you'll be able to to do that very, very efficient.
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Abhay Ashtekar: Exactly, exactly. So that does nothing but
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Hal Haggard: Yeah.
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Abhay Ashtekar: So since you since you had this slide here, and this is the question that Daniela was asking at one stage right
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Abhay Ashtekar: I mean, it's true that in this past that you're considering the areas are all discreet way.
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Abhay Ashtekar: But could we not just say that's perfectly fine. I mean that that's exactly what quantum gravity is
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Abhay Ashtekar: On the kinematics of quantum gravity is telling us that don't use some more quote unquote classical parts, but who should read it some more. This concept geometry and there is a way of seeing this, for example, for the particle particles.
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Abhay Ashtekar: On
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Abhay Ashtekar: Circle in one or one Castaneda even if you looked at energy representations
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Abhay Ashtekar: That not part in the in the configuration space in the momentum space, but in energy space or to say that you can just some or the discrete
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Abhay Ashtekar: Values and then still get the correct answer. And so, yeah, could we could just take that point of view. Right. I mean, or is there something incorrect to taking that point of view that well yeah I mean there's a patent doodle.
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Abhay Ashtekar: But already can magically. I know that the areas or contest without any recourse to geometry.
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Abhay Ashtekar: So it's not like to have one class at eight days today. They can magically and therefore I just use some or quantum parts which are all in which they do the contacts good would not take that point of view.
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Hal Haggard: I'm very sympathetic to that point of view, I was struck again recently by exactly what you were mentioning right when we
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Hal Haggard: When we think about the electromagnetic field and we think about the oscillators in the field, we use the energy representation to great effect because it allows us to do a discrete some exam. So I I'm completely comfortable and taking that point of view.
446
01:11:21,510 --> 01:11:28,290
Abhay Ashtekar: But, but I think, but in answer Daniels question. You said something which are not aware of. So I just want to ask one question you got
447
01:11:28,680 --> 01:11:38,490
Abhay Ashtekar: This higher gay share is that a lot and other people that looked at from that is discrete things come automatically is something so. Can you just elaborate a little bit on that. That will be great.
448
01:11:39,390 --> 01:11:50,970
Hal Haggard: Yeah, and this is work that maybe I should pass this to Bianca, because this is not my work, but it's beautiful work where where they studied these higher gauge theories and
449
01:11:51,030 --> 01:11:52,320
Hal Haggard: And they and they
450
01:11:52,530 --> 01:11:57,690
Hal Haggard: Looked at imposing constraints and they get exactly these sorts of theories out Bianca, do you want to add something.
451
01:11:59,520 --> 01:12:06,300
Bianca Dittrich: When we do get spins and the usual innovation over the absolutely because we expand the data function.
452
01:12:07,320 --> 01:12:13,920
Bianca Dittrich: Over some compact compact group elements. So the simplest simplest way to think of this model is to expand.
453
01:12:15,210 --> 01:12:19,140
Bianca Dittrich: Data functions over was a deficit English, which also contact
454
01:12:20,280 --> 01:12:32,040
Bianca Dittrich: And the that gives you basically a discrete parameter in front of the epsilon of the deficit anger and that looks like so much action. And so that's
455
01:12:33,420 --> 01:12:41,760
Bianca Dittrich: What you get and then basically our state and long have ability if type form as a
456
01:12:45,330 --> 01:12:57,600
Bianca Dittrich: Kind of a presentation you point of fire have higher category series that you get it submitted switch it you know i Vicki Melissa's action gets it also be covering some
457
01:13:00,360 --> 01:13:04,920
Abhay Ashtekar: Okay, thank you all, because there are many other questions. So let's just go and I think Liam Laura, I want to say something.
458
01:13:04,950 --> 01:13:10,230
lsmolin: Just a very quick question about the higher gauge your contribution is a bunion
459
01:13:12,330 --> 01:13:14,400
Laurent Freidel: So it's not that hard to understand.
460
01:13:16,050 --> 01:13:17,430
Bianca Dittrich: So sorry, can you repeat
461
01:13:17,670 --> 01:13:19,230
lsmolin: The age part
462
01:13:19,710 --> 01:13:28,650
lsmolin: It's a, it's a cemetery product of the usual gauge cemetery with a higher gates image with the higher gates and she's a female, is that correct
463
01:13:31,020 --> 01:13:42,210
Bianca Dittrich: Yes. Again, I mean it's just it's it's a it's a complicated way to speak about the same era pulled out of a pseudo course or as a focus on translations. Right.
464
01:13:42,870 --> 01:13:45,060
Abhay Ashtekar: Nobody listens, then
465
01:13:45,960 --> 01:13:48,660
Abhay Ashtekar: hoping this is not new one because this is our
466
01:13:48,750 --> 01:13:49,980
Abhay Ashtekar: Because then I don't see
467
01:13:51,360 --> 01:13:51,570
Bianca Dittrich: It.
468
01:13:51,600 --> 01:13:55,590
Bianca Dittrich: As well. But, I mean, there is a need to be open. If you want to get to
469
01:14:03,570 --> 01:14:08,790
Laurent Freidel: Yeah, maybe on the question earlier. It's true. It's a similar product but
470
01:14:09,450 --> 01:14:25,860
Laurent Freidel: The complication is that somehow one layer of the second semi direct product is attached to the area, whereas the next layer. Seems like bugs attached to the edge so it makes it maybe there's a simple way to understand it. I don't think it's it's simple, unfortunately.
471
01:14:27,120 --> 01:14:33,210
Laurent Freidel: I don't question. In fact, a continual progression of have a buy in footwork and was saying so.
472
01:14:33,990 --> 01:14:42,300
Laurent Freidel: I agree that in some sense here, you know, we can take for granted that it's a areas are discrete, I think, you know, the way I see what the
473
01:14:43,050 --> 01:14:52,470
Laurent Freidel: Island Bianca doing is the fact that, well, if you do that, you have to deal with this G factor. The second class constraints and that introduces a
474
01:14:53,070 --> 01:15:04,260
Laurent Freidel: Level of ambiguity. So once you accept this greatness. So now, now you have another China Angelo in lyrical dynamic as, as I was saying, when the way we fix the challenges that we have a notion of what it is.
475
01:15:05,100 --> 01:15:14,370
Laurent Freidel: What is the vacuum. Right. It is, as I was saying, a choice of of cannon G. Once you have second class constraints like a choice of complex torture.
476
01:15:14,940 --> 01:15:25,800
Laurent Freidel: Right, which is a choice of vacuum. And so here I think once we accept this greatness. I think that's what you know, Alan Bianca are facing, we face a new form of ambiguity, which is
477
01:15:26,520 --> 01:15:43,320
Laurent Freidel: What is the proper choice of this kernel. And so what is, you know, what determines this this clearing states of accurate complex culture. I don't know if you agree on and Bianca that that's kind of some all the focus. One of the focus of here.
478
01:15:44,520 --> 01:15:53,850
Hal Haggard: I think that's, that's a good summary of one of the things we're thinking about, we really are trying to understand how do these constraints get implemented in the spin phone
479
01:15:55,110 --> 01:16:05,010
Laurent Freidel: And so now the question is, In quantum great you what is the, you know, they're the guide to work this backwards torture complex. Yeah.
480
01:16:05,850 --> 01:16:11,700
Bianca Dittrich: Hey yeah I mean we do things as a bit of universality, because in the end is opposite oceans.
481
01:16:12,960 --> 01:16:16,320
Bianca Dittrich: Which are just coming from the faith based options.
482
01:16:18,660 --> 01:16:25,920
Bianca Dittrich: You know, it's a lot cheaper but I don't think it makes a difference. But basically, I think we ought to take that as a lesson for all others before
483
01:16:27,510 --> 01:16:32,970
Bianca Dittrich: The so called fitness program is a very general one is not more dependent comes from to the ski areas.
484
01:16:34,350 --> 01:16:41,460
Bianca Dittrich: And and you, you can. I mean, we show that as a machine, they can avoid this problem.
485
01:16:42,480 --> 01:16:52,590
Bianca Dittrich: And yes, I'm nice things is very interesting question is a complexity is going to complex plane so precise choice of office de facto
486
01:16:53,820 --> 01:17:03,660
Bianca Dittrich: That, yes, indeed, and also in the other model. It's totally precisely implemented several things at the same question. You guys have made some choices.
487
01:17:04,110 --> 01:17:20,520
Laurent Freidel: Your user models. It's essentially hidden Inca tours of queer and states and there's been different choices DISCUSSED SO IT, which in the case of Queens, that is essentially the same citrus of complex structure on on these faces.
488
01:17:25,320 --> 01:17:30,210
Bianca Dittrich: Maybe if I can point out that there's some ice Shelby's ahead, since a long time.
489
01:17:31,020 --> 01:17:34,530
Hal Haggard: Sorry, I can't see the chat at all. So you all have to direct me
490
01:17:36,330 --> 01:17:37,830
Bianca Dittrich: I don't know who's, who's channeling
491
01:17:38,880 --> 01:17:41,520
Bianca Dittrich: And we can understand Ulysse Mona and things here.
492
01:17:44,550 --> 01:17:47,280
Hal Haggard: So, Danielle. I think you were waiting for some time. Why don't you go
493
01:17:50,670 --> 01:18:04,410
Daniele Oriti: Yeah, okay. Yeah, thank you. So just to elaborate on the last point by Bianca any relation with the other models and so do I understand correctly that the conclusion. So get from from from this work is that
494
01:18:05,610 --> 01:18:09,540
Daniele Oriti: The economic model of the other models look flat.
495
01:18:10,560 --> 01:18:25,440
Daniele Oriti: In the connection appearing the ambitious because we are missing constraints that they would allow us to interpret this connection. Indeed, as a metric connection. And so really to interpret the configurations appearing the model as
496
01:18:26,850 --> 01:18:28,500
Daniele Oriti: Metric geometries.
497
01:18:30,180 --> 01:18:40,080
Hal Haggard: That's a difficult question to answer. Precisely. Daniela sometimes, you know, the way different people have characterized the flatness problem in the PRL is
498
01:18:40,830 --> 01:18:57,720
Hal Haggard: Some of them are incomplete. And so the question is, can, can we really pull out exactly what's going on in a PRL. So this is what pier 39 and the team and marcee have been thinking about and they did this numeric that I say that is very nice. You should look at it there.
499
01:18:58,470 --> 01:19:03,210
Daniele Oriti: Sorry, sorry, I didn't mind exactly the work by
500
01:19:05,040 --> 01:19:11,100
Daniele Oriti: Gucci and song. So it seems to me that what you're pointing to is exactly that that you know the
501
01:19:12,390 --> 01:19:19,650
Daniele Oriti: There are the configurations of the current model could be
502
01:19:21,870 --> 01:19:30,660
Daniele Oriti: Matching non flat metal geometries even though the connection that seems to be appearing. The model is flat in the factors are
503
01:19:31,500 --> 01:19:40,050
Daniele Oriti: By the flatness arguments. And the point is, rather, to know be a bit more careful in interpreting terms of metric geometry is
504
01:19:40,890 --> 01:19:52,320
Daniele Oriti: What we get a new corral because it would seem to me that the reason why we should be careful that we may be missing constraints of this this type of the type implementing your model.
505
01:19:53,280 --> 01:20:01,230
Hal Haggard: Yeah, I'm hesitant to say that they're missing or not yet we don't. I would say we don't know yet, but I agree. We're having a very active conversation with them.
506
01:20:01,740 --> 01:20:08,580
Hal Haggard: About whether these configurations are completely shape matched and and what's going on with them. Right, so
507
01:20:08,850 --> 01:20:15,030
Hal Haggard: You know for some time. These have been described as twisted geometries or non shape mash geometries, all the way back to
508
01:20:15,420 --> 01:20:28,650
Hal Haggard: Jimmy and Bianca's work and some money in Ron's work and many, many people have looked at this and I think it's still really to be understood yet what's going on at the saddles that that PF through and others are studying
509
01:20:32,340 --> 01:20:33,120
Daniele Oriti: Okay, thank you.
510
01:20:33,600 --> 01:20:33,990
Yeah.
511
01:20:37,680 --> 01:20:48,630
Ivan Agullo: So hard. I had to leave. So I am I am the chair. Now, there are three more hands app, as far as I can see, I don't know what the order was but let's
512
01:20:48,660 --> 01:20:49,440
Simone Speziale: Let me then.
513
01:20:49,770 --> 01:20:50,730
Simone Speziale: Get them seen
514
01:20:51,420 --> 01:20:52,770
Ivan Agullo: Very good semantic. Go ahead.
515
01:20:54,960 --> 01:20:56,970
Simone Speziale: Essentially the week about gamma
516
01:20:57,990 --> 01:20:59,070
Simone Speziale: You pointed out that
517
01:20:59,580 --> 01:21:16,140
Simone Speziale: If GM if gamma is too small. Your expectation value does not reproduce anymore. The desire the classical value is this valid for the triangulation with the internal edge or it is also valid for the delta three triangulation.
518
01:21:16,470 --> 01:21:32,280
Hal Haggard: We've studied at most in the configuration with the inner edge. The thing that is clear. You know, if we think of that configuration with an inner edge as from the Reggie, calculate the length Reggie calculus perspective, then we know there's one book variable. And in that book.
519
01:21:32,280 --> 01:21:32,910
Abhay Ashtekar: Variable you
520
01:21:33,090 --> 01:21:34,980
Hal Haggard: Have some oscillations. Right.
521
01:21:36,030 --> 01:21:46,650
Hal Haggard: And what happens when we take gamma to small as we saw in the plots, I showed when gamma is very small, there's very little oscillation, even in the the direction of the
522
01:21:46,950 --> 01:21:57,570
Hal Haggard: Three area variations that goes along the constraints, if we take gamma too small. We kill any variation of the action along that direction. And we kill any sort of peeking property.
523
01:21:58,110 --> 01:22:05,370
Hal Haggard: So, so that's where we've studied it in the most detail is that gamma is too small actually kill your, your ability to find the saddle.
524
01:22:06,450 --> 01:22:15,300
Simone Speziale: Right, so the predictions will be that if you do it with delta three. It shouldn't matter too much because in the case of the three there is really no dynamics.
525
01:22:15,780 --> 01:22:21,300
Hal Haggard: I think that's what we're finding. I'm just a little hesitant because we haven't very gamma as much in that configuration.
526
01:22:22,830 --> 01:22:40,770
Simone Speziale: And related to these Bianca said that the details of dysfunction, big G shouldn't really matter. But, but I'm perplexed, because it seems you find the quantitative bounds on gamma in order for the semi classical limit to
527
01:22:42,120 --> 01:22:51,690
Simone Speziale: Be admissible, you shouldn't is bound the dependent on the shape of G. Like, if I look at the plot you showed, if I make G.
528
01:22:52,260 --> 01:23:07,890
Simone Speziale: Broader presumably, I can allow faster escalations and still are not be too suppressed. I don't know. Isn't it isn't there are relation to be expected between the devalue of gamma and the shape of the peak of capital G.
529
01:23:08,820 --> 01:23:13,890
Hal Haggard: I'm not sure which part of what Bianca said exactly that you're quoting, but I think she would agree with
530
01:23:15,030 --> 01:23:25,140
Hal Haggard: Maybe I should just let her say, but I would agree with what you just said that it's it depends on the structure of GE broadly, but as long as we're thinking about a gash in GE
531
01:23:25,470 --> 01:23:30,660
Hal Haggard: And the width is something like what we are quoting here, you should get results like these, right. So the point is
532
01:23:31,020 --> 01:23:31,470
Simone Speziale: I mean,
533
01:23:31,590 --> 01:23:36,270
Simone Speziale: If you change the width of these gals shun couldn't be bound on gamma change.
534
01:23:36,450 --> 01:23:37,740
Simone Speziale: Or you don't expect people
535
01:23:39,210 --> 01:23:42,330
Bianca Dittrich: To this state of mind or is a certain generation.
536
01:23:43,620 --> 01:23:45,630
Hal Haggard: In a model, but
537
01:23:45,930 --> 01:23:47,550
Simone Speziale: There was a choice, but at the
538
01:23:48,030 --> 01:23:51,990
Bianca Dittrich: Very end, you need to enforce it. If you, if you want to hope that it comes from
539
01:23:54,330 --> 01:24:01,110
Simone Speziale: Cannot make any way, the answer to my question is yes. If you change these weekly you may expect that the bounding gamma changes.
540
01:24:02,760 --> 01:24:06,420
Bianca Dittrich: Yes, because of course also aren't into the addiction elements.
541
01:24:06,960 --> 01:24:23,670
Simone Speziale: Of kosher salt. So next question, could we now NBC a situation in which the width of these big function g depends on gun might sell for in such a way that the recovery of the semi classical limited becomes gamma independent or you can rule the situation out
542
01:24:25,470 --> 01:24:29,250
Hal Haggard: We haven't looked at that situation. I don't think we could say, we could rule it out.
543
01:24:30,270 --> 01:24:31,740
Hal Haggard: Right. But it's an interesting
544
01:24:33,060 --> 01:24:40,530
Bianca Dittrich: Play. You can put it in by hand. But the point is that uncertainty relation. And so this does not depend on gamma
545
01:24:47,730 --> 01:24:48,180
Bianca Dittrich: So,
546
01:24:50,040 --> 01:24:51,270
Ivan Agullo: In doing
547
01:24:55,020 --> 01:24:56,820
Ding Jia: So my question is about
548
01:24:57,840 --> 01:25:05,580
Ding Jia: Point discussed about previous by people about something over to squirt variables directly. And do you think there
549
01:25:06,780 --> 01:25:11,280
Ding Jia: Is something that needs to be checked if we couple Cairo firm aeons
550
01:25:13,170 --> 01:25:17,010
Ding Jia: Or do you think no there there will not be a problem.
551
01:25:20,370 --> 01:25:28,980
Hal Haggard: Well, there's been some really nice investigations by Jorge polian can be me and others, looking at that.
552
01:25:30,420 --> 01:25:44,880
Hal Haggard: For me and doubling problem in the context of these kinds of spin. Well, in Luke quantum gravity and and they they have some nice points about exactly this, I wouldn't say that I've thought about it in this model enough to really comment directly
553
01:25:51,150 --> 01:25:52,110
Ivan Agullo: Machine. Go ahead.
554
01:25:53,580 --> 01:25:56,460
Muxin Han: Okay, so I have a question about the
555
01:25:57,900 --> 01:26:07,620
Muxin Han: Scaling of j and and the deputies angle on compared to epi. Oh, and you said you guys exactly the same scaling relation between j and epsilon
556
01:26:09,060 --> 01:26:18,210
Muxin Han: As in epl but then what is the, what will be the advantage of this model compared to epl from the sense of the flatness
557
01:26:20,070 --> 01:26:23,850
Hal Haggard: The advantage is that it's so comfortable mission right i mean
558
01:26:23,970 --> 01:26:24,990
Muxin Han: I mean, conceptually,
559
01:26:26,130 --> 01:26:40,350
Muxin Han: Come. Yeah, definitely. And do your do your model is much better in computational sense, but conceptually in the sense of flatness problem. The human. The site is is quality be the same, where you have some advantage.
560
01:26:40,830 --> 01:26:44,910
Hal Haggard: Perhaps this is where Simone. A was pointing out what we think is that the the
561
01:26:45,330 --> 01:26:55,710
Hal Haggard: This observations were making are fairly universal and maybe it's too strong a word, but they're very broad general observations right that this that is a theory of
562
01:26:56,610 --> 01:27:07,500
Hal Haggard: Constraints will will need to implement the constraints and that implementation will generally not be able to be strong, as long as we're doing discrete sums.
563
01:27:07,890 --> 01:27:20,580
Hal Haggard: And and that the you should expect that kind of behavior we see here. So yes, we're saying that this is there's potentially not a flatness problem in the PRL that it just needs to be understood better
564
01:27:22,260 --> 01:27:33,030
Muxin Han: Yeah, so actually for epl and the the qualitative behavior is also similar to what your model is imposing. There are also these kind of G function.
565
01:27:33,810 --> 01:27:45,420
Muxin Han: Well, it's some kind of potential or culture and this kind of stuff appears in ppl and and which gives you the that kind of scaling relation between Jay and epsilon. Oh.
566
01:27:45,480 --> 01:28:00,060
Hal Haggard: Yeah, yeah. That's why I was saving your work. It's the one other place where I've seen this right and and I think that we could make it even more precise. That is, we could really clearly identify what the G functions are any PRL
567
01:28:00,750 --> 01:28:07,500
Muxin Han: I see, I see. Okay. So, another question is, is it hard for you to generalize to Lauren c&c nature.
568
01:28:09,720 --> 01:28:10,380
Hal Haggard: So,
569
01:28:14,160 --> 01:28:29,130
Muxin Han: For me, I see not, there's not much difficulty from ice put up from what I can see, right, and it seemed to me you the vertex empty to this is just like Reggie action, but you can write a warranty and signature rejection
570
01:28:30,330 --> 01:28:30,690
Muxin Han: Right.
571
01:28:32,220 --> 01:28:33,090
Hal Haggard: Yeah, I
572
01:28:33,300 --> 01:28:45,090
Hal Haggard: Know you know when I asked the same question. I started to go kind of piece by piece. I think they're there. I don't see any in principle difficulties. Let's say it that way. I don't see any blockades already
573
01:28:45,480 --> 01:28:51,690
Hal Haggard: But in implementing it. I don't know whether they would emerge and, in particular, I'm not totally clear on
574
01:28:52,170 --> 01:29:10,230
Hal Haggard: How we would think about the fact that the timeline triangles. Well, the way he PRL is usually study Lorenz Ian he PRL we take space like triangles anyway. So in that sense, there is not a lot of problem, looking at a model like ours from from era and but
575
01:29:11,550 --> 01:29:17,580
Hal Haggard: But if you really were to genuinely include time like triangles. Their subtleties that I just, I'm hesitant to jump.
576
01:29:17,580 --> 01:29:30,000
Muxin Han: Over. They are extend. They are also what will karate have an extension of speed bumps with time like triangles. So it's, yeah. They are also seems easy constrains
577
01:29:31,110 --> 01:29:36,660
Muxin Han: Up constraining the area to be discreet for time like triangle as well. I think
578
01:29:37,170 --> 01:29:37,440
Hal Haggard: Yeah.
579
01:29:37,950 --> 01:29:42,180
Hal Haggard: I'm just trying to speculate too much. I don't see any in principle blockades at the moment I
580
01:29:42,180 --> 01:29:44,700
Muxin Han: See, I see. Okay. Yeah. Thank you.
581
01:29:47,310 --> 01:29:48,960
Ivan Agullo: Bye is your question related to this.
582
01:29:53,490 --> 01:30:00,090
Ivan Agullo: Okay bye has their hand raised. And then do you still have a question or. That's it.
583
01:30:01,290 --> 01:30:02,670
Ivan Agullo: Okay, any other question.
584
01:30:06,360 --> 01:30:11,190
Ivan Agullo: If not, how thank you for these very clear talk. Thank you.
585
01:30:12,240 --> 01:30:12,780
Hal Haggard: Everybody
586
01:30:16,170 --> 01:30:19,800
Abhay Ashtekar: Since you're here. Still, can I just ask. And similarly, maybe just quickly.
587
01:30:19,980 --> 01:30:28,230
Abhay Ashtekar: I mean, a couple of times. Simone. I said something like maybe super space and of version, but it seems to me that the whole thing is a mini super space, right.
588
01:30:28,950 --> 01:30:37,590
Abhay Ashtekar: Because one just has three simply says that also find that number of simplest at both ends, one doesn't vary the number of simply says they're
589
01:30:39,390 --> 01:30:48,390
Abhay Ashtekar: dead ends. And so to me it is much more like you know you can look at cazenove spaces which have kind of three.
590
01:30:50,010 --> 01:30:59,790
Abhay Ashtekar: Coefficients and you can set to have them equal to one and that would be kind of a mini municipal space. So I think in the video talking about what is happening, you know,
591
01:31:00,840 --> 01:31:03,690
Abhay Ashtekar: If, as if you should assign same values.
592
01:31:05,700 --> 01:31:12,210
Abhay Ashtekar: The cemetery reduced reduction Negril talked about. To me it seemed more more like the cemetery reduction from the full casner to
593
01:31:13,140 --> 01:31:21,450
Abhay Ashtekar: You know photo that restriction, the coefficient, that kind of thing. But the voltage. It seemed to me is still quantum mechanics is not infinite number of degrees of freedom. Is that not correct.
594
01:31:22,110 --> 01:31:33,900
Hal Haggard: That's correct. I think the only sense in which Simone a mentor. It was the sense that you identified that I'm taking the lengths and areas to not all be independent in the complex that I'm studying
595
01:31:34,230 --> 01:31:43,530
Abhay Ashtekar: Right I but I just want to emphasize this because I think people are very often think of municipal space and cosmology as being a toy model, but this is on exactly the same footing as being
596
01:31:43,980 --> 01:31:49,410
Abhay Ashtekar: You know, it is still a finite number of degrees of freedom. The whole thing which which is fine, but I just wanted to emphasize that
597
01:31:49,860 --> 01:31:51,810
Hal Haggard: I agree completely. Thank you.
598
01:31:53,640 --> 01:31:54,990
Ivan Agullo: Okay. Thank you. Bye bye.
599
01:31:56,280 --> 01:31:57,060
Hal Haggard: Thanks, everyone.
600
01:31:57,990 --> 01:31:58,320
Abhay Ashtekar: Bye bye.