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There. I go.
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Jorge Pullin: Okay, So our speakers
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Jorge Pullin: and three-barrassing models, and it's possible to
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Alexander Jercher: thank you for the introduction first. Of All I want to thank the committee for inviting me for this talk.
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Alexander Jercher: So I want today to talk about the complete barrier train model, and it's called a structure. But to live up to the title that this presentation has been given on the website, namely, Coszantine Spincom. I want to give
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Alexander Jercher: an overview first over the notion of causality in gft and spinforms and rufy theory and spin forms, and then specify to the very crane model and its cordon structure.
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Alexander Jercher: And this is work that was introduced during my master Thesis and Research Research Assistant position afterwards, together with Amelia Weedy and Nbsp. Peters, and soon I was at a Phd. Position at the Massachusetts.
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Alexander Jercher: Okay, So let me come to first of all, a bit of a motivation and an overview.
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Alexander Jercher: Um. The necessity to consider cause a structure in quantum. Gravity comes from first of all from importance in continuum physics. So cause a structure is an integral part of continuum space and physics one
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Alexander Jercher: because It is a characteristic feature of the medicine signature, So we think of light codes, local light cones, and all the things that are important from special relativity,
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Alexander Jercher: but also it encodes all geometric information up to a conformal factor factor which is proven by Malamos theorem.
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Alexander Jercher: And there is also a rich phenomenology associated to causality, and I think one person thinks about horizons or cosmological and black hole horizons, which are another hypothesis that are only present in a Romanian city,
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Alexander Jercher: and therefore, we have the expectation for quantum gravity theory to address the role of causality, either by directly encoding it into the quantum theory, or by showing how it arises in a classical and or continuum limit.
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Alexander Jercher: So before um,
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Alexander Jercher: we are speaking a little bit about the status of causality in copper gravity, or in specific and spin pumps and roofy theory. Let me first give a disambiguation of the term causality, which I use interchangeably
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Alexander Jercher: into the notions of bear causality and time orientation, which is very pedagogically explained in the work by Bianchi at Matadisu and Pierre Matadi. So also had already a talk on that
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Alexander Jercher: so in a continuum picture, we can disentangle the notion of call the structure into where causality and time orientation
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Alexander Jercher: where um locally, this means that causality means that tangent vectors are either time like spite like, or space like, and time orientation means that we can um consider timeline and light, like vectors to be either future or past pointing after the time, direction has been arbitrarily chosen,
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Alexander Jercher: These notions can be extended globally, so two points in space-time have then a time like light, like a space like separation, if they can be connected by a continuous curve with the respective signature.
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Alexander Jercher: Similarly to points, then, have a causal order depending on whether the causal curve connecting those is future, or
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Alexander Jercher: one one then tries to transfer these concepts to a discrete and quantum setting.
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Alexander Jercher: There are some difficulties, and also There are then some refined notions. So, for instance, the notion of back causality can be applied to let's say a four Simplex. Where, then, we say that the tetrahedral of all the blacks have a different signature:
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Alexander Jercher: um causality, notions that are usually called vertex causality, edge, causality and face causality. And these are, for instance, considered by in the work.
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Alexander Jercher: So with these two terms in mind, let's look at what has been already done regarding these two aspects in spin films and goofy theory.
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Alexander Jercher: Let's first consider the time orientation aspect. So
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Alexander Jercher: we observed that the standard barrier, crane, and the Epl model are time unoriented,
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Alexander Jercher: and this can be seen by a time reversal invariance of the kernels. So here I depicted, because this case is more apparent. I depicted the kernels of the berry crane model in its standard formulation, and
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Alexander Jercher: the lack of orientation can then be seen by an invariance when we act with a time reversal onto the normal vectors,
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Alexander Jercher: and as it are the being. And Daniel we noticed this time. Orientation can therefore be imposed by explicitly breaking this invariance,
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Alexander Jercher: and this is where they introduced their model the time-oriented version of the very train model, which is an explicit realization of a quantum causal history's model.
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Alexander Jercher: And um.
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Alexander Jercher: We see that this breaking of invariance is given by expanding the sine in this kernel, and then this epsilon, which is plus or minus, is interpreted as the orientation of triangles.
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Alexander Jercher: And Now the amplitudes and in the spin form model are restricted, so that we have a choice of epsilon, so we do not sum over both Epsilon values which would correspond to an averaging over orientations. But we choose one, and by this kind of atop restriction we obtain a time oriented model,
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Alexander Jercher: now from at least from a Gp perspective. This can be
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Alexander Jercher: bit disadvantages because we we
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Alexander Jercher: set up a gft model. From this there comes out some amplitudes, and then we need to restrict them. So this is some kind of atop, and therefore it would be nice to have a key model which would produce from the very outset piece oriented amplitudes, and I will
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Alexander Jercher: come back to this point at that. At a later point
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Alexander Jercher: Yankee and Martin we saw, have constructed a time-oriented epl model the
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simone: Hi. Sorry. Just a quick question about this
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simone: orientation in the body cream model, if you can remind us.
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simone: So they structure
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simone: up top with the camera with the sign that comes straight from the fact that you one is using characters right
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exactly from simplicity in closure. Yes, you
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simone: so in a sense, if you now break this character, and instead of taking the full trace, you just take one element.
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simone: Ah! Wouldn't you expect any issues with the gauge invariance? In a sense, your model is not implementing any more
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simone: the gauge invariance associated with you know the integral over the group elements of
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simone: delta functions, and so on? Or is there a way to understand these?
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simone: This truncation, so to speak, of the character in a way that will still be?
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Alexander Jercher: We derive the the spin form formulation of this model, and then the breaking of invariance is kind of a talk. So this epsilon is recognized as this orientation, and then the
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Alexander Jercher: the
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Alexander Jercher: kernels are then broken explicitly. So it
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Alexander Jercher: Um.
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Alexander Jercher: So these are the bigger matrices evaluated at all magnetic indices being set to zero as well as the discrete parameter. So I think that's what A. P. At all is referring to right,
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simone: I see. Thank you,
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right.
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Alexander Jercher: Okay. But we will encounter these objects in a more general form later on.
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Alexander Jercher: Okay, So let's move on to the back causality case. We observed that, at least in the standard formulation, the barrier train and the Epl model. They treat space like Tetrahedra. Only
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Alexander Jercher: now the question might arise, if that is a problem. Well, this depends on the perspectives that one has,
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Alexander Jercher: because we we not assured it. If we only treat space like Tetrahedra, then the bare, positive structure, as we know it from continuum space and two weeks should somehow imagine a continuum limit, and we also have to acknowledge that treating only space like tetrahedra means that we extrude from the very outset time-like and lights, like boundaries,
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Alexander Jercher: also inspired from matrix models having more than just space like building blocks,
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Alexander Jercher: the whole theory could have a different phase structure, and therefore a different continuum limit. And this one can also see matrix models where single matrix models have, for instance, a different phase structure than multi-matrix. Models.
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Alexander Jercher: There are, then extensions there have been extensions being developed, taking this issue of where causality.
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Alexander Jercher: So by Alexander and spitial in an fugee setting, they have been through the time, like and light-like boundaries,
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Alexander Jercher: and in a spin form on and gft setting paddas, and we're already considered a better train model with space like all time, like Tetris. So this means that in the interaction term, where one has simplicial interactions,
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Alexander Jercher: one has either only space like Tetrahedron, or only a time like tetrahedron,
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Alexander Jercher: then for the Epl model Coronavir, the Abbey dog, came up with an extension that includes space like n time, like Tetrahedra, So it's a simplicial model, but with all kinds of combinations of signatures,
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Alexander Jercher: and there has been developed some asymptotic analysis from Leu and Tan, and also from Simo and Steinhouse. And there's also more recent work on that. And, for instance, the su one one spin network set a key. Here. They have been treated by a Tlb. In this work that is shown here.
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Alexander Jercher: Um, but what we um
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Alexander Jercher: and observers that the combining minimum, and still it excludes light like tetrahedron, and there is no explicit gfp formulation. So depending on your perspective, this might be a problem or not, because you might say that. Okay, G Oft. Formulation is just another representation of the theory, but for some applications it might be very useful to have a Gt formulation attack,
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Alexander Jercher: and the reason for that is that for the Ehl model one still has an explicit of the simplicity constraint in representation, and therefore one does not have an explicit Gfd formulation.
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Alexander Jercher: So um after this short overview over what has been done. Let me um. Let me then set the C. For the rest of this talk, and also send specified to the completion of the barrier train model.
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Alexander Jercher: The objective is to in the most democratic fashion, construct a gft and spin for model. That includes space light light, like in time, like Tetrahedra, with all possible interactions,
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Alexander Jercher: and I should probably also add what I will not talk about. So beyond the scope is to consider the time orientation aspect I I talked about earlier, but I will come back to that and indicate how this could be turned into a future research research project,
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Alexander Jercher: and also more recently in the work that I already mentioned by decline and collaborators, where causality violations have been considered in the setting of Lorentz. And
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Alexander Jercher: so we have now defined our objective, and before going on to the completion of the barrier Crane model, let me first, maybe
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Alexander Jercher: talk about the Barrier Crane model as it has been formulated before that. So using only space like Tetrahedron, the
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Alexander Jercher: I will introduce an extended formulation. So
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Alexander Jercher: this extended formulation that consists then in having also a space like normal vector.
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Alexander Jercher: So maybe just basic. The The very crane model is a bfronization of personolar Palatini gravity. So without the Ulster and therefore without the emergency parameter, and it has space like hyper services, or the constraints are considered with strictly space, like hyper-services
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Alexander Jercher: and the extension consists of not only considering for s to C elements, but to consider in addition, a normal vector which lies in the upper sheet of the two sheet, A type of
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Alexander Jercher: the simplicity. Constraint is then not the quadratic simplicity, constraint, but the linear simplicity, constraint, and I will show on the next slide how this looks in using by vectors
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Alexander Jercher: the action of the model is given as a sum of a kinetic and a vertex term, so the
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Alexander Jercher: in the kinetic term we observed that the group fields are simply identified by a data function. So we glue tetrahedral together and identify also the normal vector
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Alexander Jercher: in the interaction term. We have simplicial interactions, and the normal vectors are integrated over separately, which indicates that the normal vector is, in fact, not a dynamical variable, but is an auxiliary variable that helps to impose the constraints in a better way.
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Alexander Jercher: This group field satisfies two symmetries. So first of all, we have right covariance, which is an extension of closure, so we have an invariance. When we act on.
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Alexander Jercher: When integrating this condition over the normal vector X, we re obtain the usual closure condition. The
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Alexander Jercher: The simplicity condition is then given as an invariance of every entry with respect to right multiplication. By this you
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Alexander Jercher: and this you, when x is time like this, U lies in the stabilizer subgroup of X, which is then isomorphic to su two.
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Alexander Jercher: Here we have also a pictorial representation, where we see that the normal vector is identified between two Tetrahedra, and here we have a sufficient action where every normal vector is integrated over separately.
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Alexander Jercher: Now what are the properties of this modern
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Alexander Jercher: so closure and simplicity commute in this formulation, and therefore these constraints are imposed by a projector
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Alexander Jercher: and the linear simplistic constraint for gamma to infinity. So, in the which is the case for the Berry Drink case where Gamma is, the the constraints are first class, and therefore should be imposed strongly
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Alexander Jercher: because there has been a criticism towards the Berry crane model, that it imposes the constraints too strongly. But this only applies to the projectic simplicity, constraint, which is a second class,
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simone: and also because we have sorry. Why do you say that the linear Cpu constraints are not second class? Because when gamma is taken to infinity, then they actually commute so well.
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Alexander Jercher: I talk about the discretized constraints
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simone: I see. So when we
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simone: so you the State, my first class or second class, is when you look at the primary simplicity constraints by themselves.
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E.
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simone: What do you mean by primary simplicity? You might
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simone: um fine. So in because in the discrete theory, all you have is the um. The sympathy constraints between the fluxes. And when you the statement that their first class of second class is referring only to these guys, Right?
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Alexander Jercher: Yes,
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Alexander Jercher: and in group representation. This can be seen also by considering, when these constraints are imposed by a projector, so in group representation,
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simone: and then
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simone: analysis with the continuum theory which it doesn't change. If you have quadratic or sex, or linear constraints. That's why I was asking. But your answer is,
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simone: it's clear.
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simone: Okay, also. I I think, that the discussion that is given in this reference here is rather clear concerning the simplistic constraint in the for the
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Alexander Jercher: the discussion. There is rather clear. I think so.
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Alexander Jercher: Yeah. Um.
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Alexander Jercher: Because when so, when coming back to these properties, when we have the constraints imposed by a projector, we also have a unique formulation in the sense that the model, then, does not depend on the order of
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Alexander Jercher: imposing the constraints, and also how often these are imposed. Ah. Modular volume, factors of S. To C. But, for instance, in when this normal vector, is not not considered, there are different versions of the barrier train model, so a better grain, A, B and C, for instance; and these then depend on the order of imposing the constraints
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Alexander Jercher: and the constraints are also imposed in a covariant fashion. So this means that the normal vector is taken into account. But when the group elements are simultaneously rotated, the normal vector is rotated as well.
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Alexander Jercher: The four of the constraints is explicitly known in group representation which can be
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Alexander Jercher: advantages, and the results of gft condensate. Cosmology, for instance, are also recovered. So these these results have been worked on first for the epile model. But this also works. If we consider
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Alexander Jercher: this for the for the barric train case,
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Alexander Jercher: and we have a geometric interpretation in terms of five vector variables. So, integrating out the normal vector one can see that this right Covariance condition simply reduces to closure,
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Alexander Jercher: meaning that the by vectors close the
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Alexander Jercher: and the simplicity. Constraint then tells us that this tetrahedral lie in a hypothesis which is orthogonal to these normal vectors. And this is the linear simplicity, constraint. When gamma is sent to infinity, so in the presence of a whole term,
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Alexander Jercher: we would have here a plus one over gamma
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Alexander Jercher: contracted with B, and not starv. But this term then vanishes. If we send Gamma to infinity.
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Alexander Jercher: This is what I mean with linear simplicity, constraint, because a quadratic sympathy, constraint would mean that
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Alexander Jercher: this is, it is squared in B, and then equal to zero. But here we have the linear one.
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Alexander Jercher: So, after having discussed um,
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Alexander Jercher: the the very train model with space like Tetrahedral, Let me now explain the completion of this model.
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Alexander Jercher: So X was assumed to be time like so far, and we have therefore restricted to space in tetrahedron that's now allowed for all normal vector signal,
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Alexander Jercher: and for that it is very useful to study a bit construction of of S. To C. And in subgroups, and how to interpret them. So the signature alpha is either plus for for the normal vector being, time like for lighteline and minus for space,
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Alexander Jercher: and more or less economical choice. For these vectors or examples are these ones depicted here.
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Alexander Jercher: When we now consider the stabilizer subgroup of S. To C. Which stabilizes these factors for the timeline case, we have su two
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Alexander Jercher: for the light-like case we have E. Two. This is the group of isometries acting on the Euclidean, two planes. So these are two translations and one rotation, and in the space. Like normal vector case,
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Alexander Jercher: we have su one one.
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Alexander Jercher: And now the homogeneous spaces of so two, c. With respect to these stabilizer subgroups, we get the two sheet of hyperboloid, the light cone, and the one sheet of type harboroid, and these are exactly the spaces in which the normal vectors.
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Alexander Jercher: Now the second key idea is to allow for all possible sympatheticial interactions, and where we get twenty one vertices.
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Alexander Jercher: So let me then explain the resulting action for the kinetic term we consider the
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Alexander Jercher: we see that it's basically a sum of all signatures. And for every signature we just have the basic interaction term where Tetrahedra are identified, together with a normal vector And what I've depicted here is just the case for a space like light like, and pie like tetrahedron,
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Alexander Jercher: and plus is the number of space like Tetrahedra and Zero, the number of light, light and and minus the number of time by tetrahedral and every part of these interactions for themselves. They look like a C initial interactions where just we have a different distribution of normal vectors. So
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Alexander Jercher: they are integrated almost separately, but they can have a different signature. So,
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Alexander Jercher: and maybe to give a few examples, and also to relate to what we have seen so far. Here, for instance, is in a dual representation, a simplex with space like Tetrahedra,
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Alexander Jercher: just as we know it
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Alexander Jercher: from the summit formulations from the model of Paris and Robelli, where they consider, for instance, five time. Like Tetrahedra, we have the zero point five here,
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Alexander Jercher: but these two in the middle are also interesting from a Cdt perspective, since um in Cdt. The one has two kinds of simply seeds, one with one space like at four time, Anthra and the other one with five time.
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Alexander Jercher: But of course, there are also all kinds of combinations. So, just to give an example of a let's say more crazy configuration, we have the two to one simplex with two space like one, two night like at one time,
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Alexander Jercher: and these make up twenty, one in total
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Alexander Jercher: um.
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Alexander Jercher: So
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Alexander Jercher: we want to consider now the spin representation, not only to derive the spin-form formulation of the model, but also because a primary presentation is crucial. In most of the calculations.
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Alexander Jercher: Other types of applications, then the spin representation is very important, The.
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Alexander Jercher: And for that we consider the canonical basis of unitary, irreducible representations of S. To see in the principle series.
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Alexander Jercher: So these representation spaces are realized as a space of homogeneous functions on C two with a continuous label R and a discrete label,
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Alexander Jercher: and the economical basis means that we expand this in representation spaces into su true representations
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Alexander Jercher: inoded as Qj. And this sum, so these are infinite, dimensional, as one can see from that which simply reflects that S. To C. So on compact.
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Alexander Jercher: There are two casemy operators of S to C. That's the first one, which is also interpreted as the square of the area of the given triangle and the second casemi operator.
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Alexander Jercher: Now, when one translates the simplicity constraint that we have seen the
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Alexander Jercher: so far in group and in by vector representation to spin representation, then we see that it implies that the second car limit is vanishing. So this is specific to the Mirror Crane model. In the Epl case we have that, for for instance, for a space like normal, vector
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Alexander Jercher: basic tetrahedron, that su two, as your true representations are embedded into S to C. But for the very crane case this simply means that either rho or mu is equal to zero,
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Alexander Jercher: which is given as follows: So we have a vigner matrix, and we integrate this matrix over u Alpha
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Alexander Jercher: where you alpha can be. Now, su two is or two or su one one, and then we have this projector, so the projector can be then written as a
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Alexander Jercher: cat, Brown notation with these invariant vectors. I, where these vectors. I are then invariant under the action of U Alpha,
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Alexander Jercher: anticipating results that that are derived using integral geometry later on.
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Alexander Jercher: One can see already some structure here. So,
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Alexander Jercher: and when the normal vector is timelike, then the discrete label is set to zero, and if, inserting that into the first Casimi operator, we see that the cas in the operator is strictly less than zero, and this we interpret as one,
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Alexander Jercher: the triangles being then spaced like
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Alexander Jercher: also for the light-like case we get a nu equal to zero. So also we have space like triangles inside light like Tetrahedra. And this case is kind of interesting because classically
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Alexander Jercher: um space, like tetrahedral, have space like triangles, but light, like Tetrahedra at least classically can have space like or light like triangles where the light-like ones are degenerate.
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Alexander Jercher: These are included in the in the theory, using
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Alexander Jercher: using um representation, they would.
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Alexander Jercher: So we have here strictly less than zero, and not less or equal to zero. So,
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Alexander Jercher: having having space like normal vectors. So on time, like Tetrahedra, we get a superposition of mu equal to zero and planning that into the first Casimir we see that it is either bigger or smaller than zero, and therefore time by Tetrahedral, contains space like all time like.
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Alexander Jercher: So this is all known classically, which can be easily derived. But
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Alexander Jercher: these are then results quantum geometric results. If you want to put it like that, because we use here representation, the
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Alexander Jercher: and so maybe let me skip the part where I show how the group fields are.
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Alexander Jercher: So um!
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Alexander Jercher: We can define the kernels
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Alexander Jercher: which are bigno matrices contracted from the right side by invariant vector and from the left side by an invariant. Vector So these eyes, I remind, are components in the canonical basis of invariant vectors,
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Alexander Jercher: and these are usual bigger matrices. So maybe to come back to the modest question from before, in the usual case, that we know where we have time like normal vectors. These are:
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Alexander Jercher: It gives us a delta nu zero, delta n zero, and also for this case, and what one obtains by this contraction is simply D with a lot of zero stem.
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Alexander Jercher: So this is the case, we know, but it is now generalized to all kinds of signatures, and also these kernels then
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Alexander Jercher: carry two normal vectors with different signature, and the interpretation is rather simple. So these represent
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Alexander Jercher: These labels are associated to triangles, and these triangles are shared between two Tetrahedra and the Tetrahedra can now be of different signature.
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Alexander Jercher: Um,
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Alexander Jercher: then, the spin representation for the group. It's one can derive the spin from amplitude for it, or it's in too complex. So the
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Alexander Jercher: we have usually, as usual, pace edge and vertex amplitudes. For instance, the phase amplitudes are given by the
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Alexander Jercher: the edge. Applications are set to one and or equal to one, and the vertex amplitude is given by a product over ten of such kernels
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Alexander Jercher: corresponding to the ten triangles appearing in appearing in the for Simplex, and we have for five signatures corresponding to the five signatures of the five Tetrahedra occurring in one.
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Alexander Jercher: Now, although this looks like we have rho F. And what we have is that in these amplitudes either new F or R is set to zero.
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Alexander Jercher: We have space like triangles that are labeled by Rho, F and Nu is equal to zero, which leads to a continuous spectrum, and we have timeite triangles labeled by zero, which have a discrete spectrum.
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Alexander Jercher: Let's consider the properties of these kernels, so by integral geometry one can derive explicit expressions for for these kernels, which then define the whole model. Basically. And these methods have already been used by.
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Alexander Jercher: But for the case where one has to space like to space, like or through time, like normal vectors, and not for the mixed cases, and not for the ah light-like case. So what we have is we have ah for new expressions. So for two light, like Tetrahedra, and for the mixed cases.
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Alexander Jercher: And the method used here is, I said, integral geometry, which is basically a two theory on the homogeneous spaces of Ss. To C. Which has been developed by Gil for archive. And
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Alexander Jercher: now
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Alexander Jercher: let's observe furthermore, the kernel. So we make the observation that also the completion of the Berry cray model is unoriented, which is not very surprising,
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Alexander Jercher: and which is probably not surprising because we consider constraint, Pf. Theory, quantization thereof, where neither Bf. Theory nor the constraints are sensitive to the orientation, and therefore, of course, also, the constraint model is not sensitive.
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Alexander Jercher: And there is also another interpretation that,
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Alexander Jercher: considering
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Alexander Jercher: different from amplitudes, as realizing as being realized by the path integral, we have an integral over the laps function from minus infinity to plus infinity, which might be interpreted as averaging over orientations.
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Alexander Jercher: Um, and similar to to the case I have shown before for the usual Baron trail model, we observed that we have an extended symmetry, so these currents are not only invariant under Sdc. But they are invariant, and the the larger group, namely, the whole Laureate, or one of the Stabilic.
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Alexander Jercher: So pin one, three is one of the double covers of zero, one, three
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Alexander Jercher: um,
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Alexander Jercher: and the whole Lawrence group can be understood as the proper autochronous Lawrence group, semi-direct product with the client group, consisting of the identity, time, reversal spatial parity, and spacetime reversal.
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Alexander Jercher: Now, if we would want to construct a gft model that gives us then the
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Alexander Jercher: extent, the representation theory of Sl. Twoc. To pin one, three, basically by including also how this time reversal and parity act on the representation functions.
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Alexander Jercher: Given such a model, one could then either impose a symmetry or not symmetry, and therefore the model would turn out to be unoriented or oriented.
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Alexander Jercher: It would not be surprising if if this would work exactly like that, but one would need to work that out explicitly, of course,
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Alexander Jercher: but it seems like an interesting research project.
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Alexander Jercher: So um,
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Alexander Jercher: I want now to give some context and applications of that, because, after all, pay, we introduce now a lot of vertices and a lot of new terms. But what can we do with that? And how can we set this into a context, so one hundred and fifty,
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Alexander Jercher: maybe. Let me come directly to the
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Alexander Jercher: A. Maybe. Let me let me give the relation to the Convali me the extension of the Dpm model just a quick comparison. So the converting Nimida extension is a quantization of Palatine evolves, while the cookie very train model is a decision of Palatini.
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Alexander Jercher: The simplicity constraints are imposed in a different fashion, and because of that
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Alexander Jercher: in spin representation also, we have a different simplistic constraint. So the Epi model basically consists of embedding su two representations into
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Alexander Jercher: um. And if the normal vector is based like um as Coronad and you would I showed. This, then, corresponds to an embedding of su one. One representations into Ss. You see, and in principle I think that this could be also extended to the light-like case, and in this case one would then need to embed um by a easel. Two
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Alexander Jercher: representations into ss, you see, and from the canonical side.
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Alexander Jercher: The moon has already worked on that in his work on light twisted geometries
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Alexander Jercher: remind that in the very same case we have simple S. To c. Representations basically being rho equal to zero or nu being equal to zero,
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Alexander Jercher: a prediction concerning the continuity of the spectrum. So in the ch extension we get spatial discreteness and temporal continuity for the Berry Crane case. We get spatial continuity and temporal discreet.
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Alexander Jercher: Now, one of the applications that is available
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Alexander Jercher: using the the new configurations, and also that we have mixed configurations, and we have access to the configurations appearing in Cdt. And maybe we can draw some comparisons to that. So I want to show how one can construct the Cdt like T model, being, of course, aware that there are a lot of conceptual differences between.
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Alexander Jercher: So let's recall what are the geometric assumptions before taking the A to zero In t one has fixed edge, length and time, like in space, like X's, are related by an interpolation Parameter alpha.
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Alexander Jercher: Also we have two simplicial building blocks, which are in Ct. Refer to as the for one simplex and the free two. Simplex.
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Alexander Jercher: What is important for us is since we do not have under control edges in G fifteen spin homes, but rather tetrahedra and triangles. It's a number of tetrahedra and triangles inside these different synthesis,
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Alexander Jercher: so as strong before, we have one space like in for time, like a tetrahedron, this for one simplex and five time a tetrahedra in this three, through Simplex,
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Alexander Jercher: and the number of
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space, like the distribution of space, like in time, like triangles, then tells us later, which kind of labels representation, labels we should set to zero.
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Alexander Jercher: And in Cdt, since we have a very restricted or rigid, their configurations are rather rigid. One does not have any topological similarities, and also one has a global
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Alexander Jercher: that is up. We're not present in Gd. And in the diagrams that are generated by G. Of T. Even in the colored version.
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Alexander Jercher: So let's try now to come as close to these conditions as possible in the G. Of key setting,
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Alexander Jercher: so as mentioned, and as also as we work on by Bianca and Thanos, is that in
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Alexander Jercher: I mean, this is let's say this is a bit more older than that. But areas are the more fundamental variables in achieved T. And spin forms, and we also control them in G. Of T. So we have control either over the
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Alexander Jercher: for instance, or the representation labels which are associated to triangles, or, for instance, or by vectors, which are also associated to triangles. So, instead of fixing edge lengths. We therefore fix the area which consists of fixing the representation neighbors.
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Alexander Jercher: We relate the timelike and the space like areas via this interpolation parameter.
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Alexander Jercher: We only allow for two actions, So to get only the four, one and the free two Simplex,
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Alexander Jercher: we introduce a coloring in order to not get any topological singularities, and we introduce a dual weighting which is thought to be,
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Alexander Jercher: which aims to enforce affiliation constraint. So this dual waiting basically tells us that space-time faces. So when we have in the graph we consider a closed loop that we then have only two space like Tetrahedra appearing in that
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Alexander Jercher: this aims at preventing
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Alexander Jercher: preventing topology change. So, for instance, these trousers,
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Alexander Jercher: if one then considers the g oft partition function one can rewrite it as in a rather ct-like fashion, without a limit here. But the effective action then looks at least from the structural wise runner, similar to that of before a big rotation. So one has a term multiplying,
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Alexander Jercher: multiplying the number of space like faces time like faces, and also the two vertices.
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Alexander Jercher: This gives, I know very tentatively. This gives an interpretation of the gft parameters in terms of the bear causal.
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Alexander Jercher: They are cosmological, constant, and they're back gravitational, constant. So,
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Alexander Jercher: and what we at the end of the day we've been up with is a Cdt-like gift t model which has the form of a cause of tensor model. So we end up with a tensor model with three different types of tensors, and
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Alexander Jercher: where the indices being the S. To see magnetic indices.
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Alexander Jercher: So this
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Alexander Jercher: then concludes,
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Alexander Jercher: and I want, as a last point, consider possible applications of that. So the
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Alexander Jercher: one is
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Alexander Jercher: in the setting of condensate cosmology and the relation to physical reference frames. So, as most of us know the background. Dependence requires a relational description, and this means that
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Alexander Jercher: we have to use dynamical clocks and rods instead of coordinates.
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Alexander Jercher: But then, in particular with respect to perturbations, there are some open questions.
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Alexander Jercher: So first of all, if we have drops and rods, how do we distinguish them?
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Alexander Jercher: And this means distinguishing them in some more clear way than just calling them differently. Then also, how do clops and Rods relate to the signature of spacetime? So, for instance, one
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Alexander Jercher: possibility would be that clocks only talk to space like Tetrahedra and rods only to time like Tetrahedron.
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Alexander Jercher: The question is, if one considers them this kind of.
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Alexander Jercher: Then what one can also consider is a landlordgence from analysis of this theory. So a landlord-in-chief analysis inspired by statistical fee theory, provides a rough estimate of the pay structure, and it tells us when the mean field theory is valid
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Alexander Jercher: and recent work by Manchetty, Ovid, Fetus, and Trilligan showed that the Berry Crane model was based at Tetrahedra exhibits a condensate phase which can be consistently described by nifi theory.
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Alexander Jercher: And more broadly, there is a question, What is the pay structure of the complete barrier train model. And can we give a rough estimate of that via a laptop analysis, and probably also is the phase structure different to that of using only space Tetrahedra,
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Alexander Jercher: which would indicate that the having more building blocks that leads us to a different universality class.
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Alexander Jercher: So um,
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Alexander Jercher: I want them to give a summary and present some open problems and perspectives.
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Alexander Jercher: So we have seen a disambiguation of the notion of causality, and also presented briefly the status of causality in gft and uniforms
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Alexander Jercher: we have constructed in G. Of T. And spin for models that faithfully encode spare causality, and we also considered a coloring of this model,
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Alexander Jercher: so not in this presentation, but in the actual work, which doesn't also be needed, which was needed for the Cdt. Like gif key model.
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Alexander Jercher: The quantum geometric results for geometry which have been anticipated by Barrett and Crane already, and also by is that space, like in light, like tetrahedra contains space like faces.
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Alexander Jercher: Tyler, Tetrah, you drop contains space like in time, like faces where the time, like faces, have a discrete spectrum.
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Alexander Jercher: We have determined the behavior and the space and reversal of these kernels, and we have found a one-on-one free symmetry. So the completion of the very framework turns out to be
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Alexander Jercher: unoriented.
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Alexander Jercher: We have seen that one can also construct a Cdt like Gft model trying to get as close to Cdt as possible, which resulted then in a causal tensor model.
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Alexander Jercher: Open problems that we encounter is what is the role of degenerate or light, like faces, So, as we have seen, light like faces characterized by a vanishing area, so a vanishing a Casim operator
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Alexander Jercher: They did not appear, and they can.
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Alexander Jercher: They are described by the trivial representation of vessel to see, because for that the Kasimi operator vanishes, and the question is, should wanting through those. And are these and interpreted as like faces? Or are they degenerate? And These
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Alexander Jercher: configurations are somewhat obscure, because they also appear in a line of principle analysis. And it would be it. It is required to consider these in more detail
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Alexander Jercher: the perspective, and also in the context of what junk and collaborators have done, which in the actions and configurations should be ruled out. And how should we impose local positive conditions. So, having all these kind of sympathies, we
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Alexander Jercher: we glue them together. We want to have a non degenerate light on structure.
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Alexander Jercher: We would at least have some conditions how to allow them or how to rule them out.
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Alexander Jercher: This is on an an an open issue.
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Alexander Jercher: Then, as mentioned before, It would be interesting to develop an Oriental G of t and spin-pole model based on the Pin group or on the whole Lawrence group,
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Alexander Jercher: also to perform an asymptotic analysis, in order to see whether these vertex amplitudes lead to Lorentzian for synthesis,
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Alexander Jercher: and to study the bearing on cosmology as well as examining the phase structure of this model by a Lano Ginsburg, which is much more harder by a functional renormalization of analysis.
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Alexander Jercher: With that I want to close, and I'm happy about discussions and questions,
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Abhay Vasant Ashtekar: So Hooray asked me to
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Abhay Vasant Ashtekar: take all the chair yet to leave.
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Abhay Vasant Ashtekar: So, please, there are questions I either ask or raise your hands
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in there.
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Abhay Vasant Ashtekar: The
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Abhay Vasant Ashtekar: Weston has a question.
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Western: Thank you so much for for the talk. Uh. So I understand that the uh, the synthetic analysis and the classical limit of this model is uh a work in progress. And yet I would like to ask a question about this, because uh,
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Western: so, if what you're constructing in is something that the that is acidity like. But acidity uh, it's a problem that the duster limit response to something that is not here. But it's more something that is like a Java gravity with the breaking or orange variance. And it seems to me that you're putting all the ingredients there to go in the same direction. But so ah, maybe.
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Western: But ah, how do you agree with this? Do you think that? Ah! Are there indications in your opinion that you would have some of the breaks, or in civilians. And if so, are we learning something about what goes wrong in this kind of imposition of causality.
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Alexander Jercher: So I see the point. I think that it's. Still, i'm not sure about this, but I think that it's still disputed If one gets for Java dish it's in the three plus one-dimensional case at least.
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Alexander Jercher: But yes, so if we more and more cut to or cut down to Cdt by imposing all these restrictions we would then get probably a very similar pay structure to Cdt, and therefore we would also get the features with that.
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Alexander Jercher: This would mean that we have something to compare with also, because we then could import the stuff from as a public safety. But
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Alexander Jercher: But of course, if let's say, this holds true that we get all Java dishes with some kind of variance breaking, we would get the same troubles. So I would agree on that.
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Western: Yeah, I mean the Uh. Sorry. Just a small comment regarding this, because then this is something you may see immediately in cosmology, and in the shape of the uh, as magical perturbations.
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Western: So I mean, we know that today the the the one uh the proponent. You can perturbations from for them by observations because of this not it breaking. So I mean it will be interesting to see that's the same, and I think but again to me, and then the interest will be to see what goes wrong. What are the
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Western: Thank you.
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Thank you,
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Abhay Vasant Ashtekar: Daniel. I think as a comment or question,
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Abhay Vasant Ashtekar: Daniel. It
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Daniele Oriti: I just say it verification. I mean the The model that we construct an Alex
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Daniele Oriti: discussed is is not a Cdt model per se.
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Daniele Oriti: It becomes a Cdt model as he discussed in the last part,
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Daniele Oriti: if you fix the geometric data to be the same, so you between a bilateral translation as the first part, and you also impose this global foliation condition
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Daniele Oriti: that in itself the model is simply a generalization of better training which you have all possible
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Daniele Oriti: light constructions for the geometric synthesis,
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Daniele Oriti: light-like timeline space. Life in itself it does
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Daniele Oriti: nothing to do with any ravages smelling
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Daniele Oriti: modern.
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Daniele Oriti: It shows that once you have all this extra information in a spinful, and you the context. Then you can also proceed. If you are interested in
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Daniele Oriti: imposing the restrictions that produce a Cdt-like Bottle
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Daniele Oriti: So I do believe that
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Daniele Oriti: what I understand is that there was no consensus and agreement in the Cd community. That That's a necessary result in the four-dimensional case it is actually proven analytical in the two-dimensional case that it happens, but they they argue that that doesn't mean
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Daniele Oriti: necessarily that it happens in forty
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Daniele Oriti: that the
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Daniele Oriti: that's that's what i'm understanding about when they say,
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Andreas Pithis: Thank you all right. Can I comment on that, too?
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Andreas Pithis: So they allow for more vertices,
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Andreas Pithis: But they don't impose this foliation constraint, and then they see they get another universality class as compared to Euclidean dynamic. Ah! During relations, and to ordinary, let's say standard
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Andreas Pithis: causal dynamical triangulations,
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Andreas Pithis: the point is not clear or settled whether this locally causal Dt or this causal dynamic triangulations of the standard formulations lead to the same result, and in three plus one dimensions.
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Andreas Pithis: This is also not clear. So what you
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Andreas Pithis: do is and
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Andreas Pithis: you take away the global foliation constraint.
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Andreas Pithis: You add more interactions, basically more vertices.
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Andreas Pithis: So that that is, I hope clarifying. So it's not clear whether it leads to the same phase structure.
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Andreas Pithis: Okay, thank you. Thanks. And there's infected. Sorry just to compliment what what I reply to to.
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Daniele Oriti: But apparently they are also interested in modifying their own construction just in case. And in fact, they've been proposing this more local causality conditions that avoid any global variation, as Andreas pointed out,
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Daniele Oriti: Investigate.
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Abhay Vasant Ashtekar: Okay,
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Abhay Vasant Ashtekar: Thank you. It's a morning.
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simone: Yeah, Hi. Thanks for the talk and stimulating ideas. I was wondering. So one, maybe simple motivation. To
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simone: go beyond the standard space, like
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simone: the simply sees, is that these statistically are very few in the sense that in the span of all possible for simplicities. The ones that are all space like are very squashed. And so random sampling would not necessarily select those. So one may argue that if
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simone: and they might be not enough to get a good continuum limit, and I was wondering if, with your investigations you identified or got ideas about had other motivations or
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simone: things you might be able to do with this extension things that will improve the behavior of the party integral So to increase the motivations to
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simone: to look at this generalization.
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Alexander Jercher: So I think one of the motivations was to have let's say, the most general possible model at hand; and then, if one considers a certain application, for instance, let's say,
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Alexander Jercher: and that one then has a model with which one can restrict appropriately, so only consider the vertices one is interested in, and then try to work with it. So this was maybe one of the motivations, so that one does not have to first say, we want to do that, and that's constructed model for that. But to start with the most general form
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Alexander Jercher: when one considers meta-coupling, so to be more precise um meta-coupling is at least currently is considered as placing scalar fields on synthesis or on the vertices, and then it's propagating along along dual edges
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Alexander Jercher: the question is, if we only have space like Tetrahedra, then all edges are timelike.
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Alexander Jercher: And so the scalar fields are then basically forced to only travel along a time like a long time, like cool edges, and the question then arose: if one would need, then probably also space like Tetrahedra, if one wants to have clocks and lots, and to probably distinguish them.
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Alexander Jercher: So this was also an idea of that.
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Alexander Jercher: Yeah. And
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Alexander Jercher: maybe also from maybe also from tens of models and matrix models where one also considers mighty matrix and mildly tens of models, and this gives them a different behavior.
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simone: You consider at some point going
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simone: more dynamically, making the X's variables also something that you can integrate over
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simone: to let the dynamic select. Ah, or for a time being. This looks like it will just be some kinematical information. You have, as you pointed out, because you might be interested in doing
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simone: transition amplitude between certain classes of geometries.
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simone: Is it something you can envisage you also for the future to. So you mean to have X in dynamic work, right? So the diploma includes an integration over the X's.
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Daniele Oriti: Yes, with some way to. We know authors
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Daniele Oriti: sorry by X. You mean the normal vectors as you
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simone: right,
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simone: and then, according to the way you chose them,
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Daniele Oriti: you have a different type of representation.
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Daniele Oriti: No, they are integrated out. What they are not is coupled. They are not coupled in the interactions of one another,
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Daniele Oriti: but they are integrated out. They are not
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Daniele Oriti: in the path integral You integrate over the X's also,
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but they just disappear because they are not coupled to the thing. You can, in fact, absorb them into the connection.
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Daniele Oriti: But it's not just to be clear. You integrate over them, but within a special uh causality class, or we allow them to span different cause it class. You just go for one term of interaction to another, so it doesn't change anything.
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Daniele Oriti: It's not fixed from the start. If you start coupling them for a circle, you will fail to get the P. Rl. Model in the case in which you also introduce the emergency parallel. So it's really a different class. So we's been for models there.
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Daniele Oriti: It would not match any of the known spin for models. It's interesting to do it, but I wouldn't know exactly what coupling I should.
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Daniele Oriti: Okay.
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Daniele Oriti: But so we're going to do a previous question, and if I if I can add a comment to what Alex. Ah reply, I mean Ah, in fact, Alex. Ah had already ah partially mentioned this possible applications of the model. But when he said, that that is interesting to go to repeat the deal and the good analysis we did for the
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Daniele Oriti: and for this space like model.
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Daniele Oriti: To this ah extended model suggested that in fact that we can find that there is a different universality class, and you find a different type of results in terms of critical behavior, and so on. So it's exactly what, in fact, one will expect.
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Abhay Vasant Ashtekar: Okay, So Dj. Has a question.
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DJ: Thank you. Thank Alex for the nice talk
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DJ: this issue. I'm. I'm. A beginner,
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DJ: and to learn to be informed. I want to do some calculations, such as quantum cosmology for tobacco, to make some physical predictions. The question is, which model should I start with? How in principle I could start with all molecules? I do calculation, inspiration to compare results in in practice
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DJ: I need a starting point. What do you suggest? Which model should I start with my physical computation?
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Alexander Jercher: I would say that this depends on your level of on the applications, or, for instance, for cosmology, you would not bother about
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Alexander Jercher: up to you. You would not bother about time like stuff. So, for instance, you could consider space like Tetrahedra, and then to hyper-surface, and then all you feel that was basic Tetrahedra. So you would be fine with just the standard Vpn or the
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Alexander Jercher: um.
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Alexander Jercher: But also, if you want to compute stuff, there are amplitudes that are easier to compute or harder to compute. So I I think that
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Alexander Jercher: where a train model is easier, in a sense, it's easier to compute than the Vpn model. At least when I think of the explicit form of the amplitudes;
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Alexander Jercher: and it may be that if you then want to go to a continuum setting, then the differences between these models do not affect the the analysis, and therefore you want to choose the one that is simpler one.
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Alexander Jercher: So
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um,
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Alexander Jercher: I would say, it depends on that. If you want to have, If you care about microscopic details, then different models give you different predictions. But it might be that if you want to consider something effective, then I would suggest to work with the easiest model, or the ones the one which gives you the best computability,
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Abhay Vasant Ashtekar: even though they just hinted that they might not right when they made the comment about different university classes.
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Okay,
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Daniele Oriti: only call some versus a space like, only there are problem in different universal in the graph. But this I will not be surprised if that's the case. Uh eprl space like and body crane space like only they may, will be. We want to be effective. Ah, continue
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Daniele Oriti: that. That would be my my opinion. But it again.
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Abhay Vasant Ashtekar: Okay, Eugenio,
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Alexander Jercher: I would say that. Um,
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Alexander Jercher: I would say that what one needs to consider next is probably causality conditions. Ah, for the gluing of ah simply seeds! Because um right now, what we do is we consider all kinds of signatures at level of single four Cpcs. But if we want to speak about extended structures, then we would. We probably should assure that we do not mess up completely the causal structure in between. But this is also, maybe some kind of a
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Alexander Jercher: philosophical input if we want to include causal irregularities or not. So this is probably a matter of taste, and if one decides to do that
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Alexander Jercher: to ah, let's say, rule out those causeality violations, or would then need to consider a proper gluing along, and then, if one has a vertex hinge and ah face causality everywhere, I think one can make them ah better statements. Ah, about that.
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Thank you.
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Abhay Vasant Ashtekar: So in a early day, right? I mean. There was a problem with the parrot train model with respect to drive it on, Propagator, and that was one of the motivations to go to the Prl model.
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Abhay Vasant Ashtekar: So maybe you can just remind us what the problem was and what its status is with with your uh generalization, with your completion. So I don't think that this has to do with the completion. So this applies to the barrier train model itself, and at least as far as I understand,
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Alexander Jercher: Berry Crane model has been compared to, or
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Alexander Jercher: the the mismatch of this graviton propagator from Akg and the very crane model can be faced back to the mismatch of boundary States, which, in fact, I would consider not as very surprising, because um one case we have. Ah, the quantization of
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Abhay Vasant Ashtekar: mismatch So no, but it's not a matter of just comparative. It was a matter of just not having some proper lorry and variance.
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Abhay Vasant Ashtekar: So even if I forget about Epr. I just look at married Crane, Propagator by itself, and I My memory was that it had some.
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Abhay Vasant Ashtekar: It was not acceptable physically.
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Abhay Vasant Ashtekar: Don't remember that. Maybe somebody else remembers it similar. You want to sell them.
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simone: Yeah. Well, the point is that it's It's like in the use of freedom, it inter to, at least in his basic version with S. You two right, he uses a fixed intertwiner because he's lacking those degrees of freedom. So when video genuine, we were computing the propagator meant as in error correlations.
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simone: The a correlations were fine. But then we wanted to look at the angle correlations.
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simone: I mean. This was only one of the issues. The other issue was that it was restricted to tetrahedral vertices
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Abhay Vasant Ashtekar: again, for the same reason that That's where the assembly in the tutorial was defined, and they didn't depend on the
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Abhay Vasant Ashtekar: to do. I get a acceptable propagator, and my question is really about that
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Daniele Oriti: we described a number of issues in the followed from describing first,
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Daniele Oriti: and then try to add them propagate with the body Crane amplitude. And that is what Alex basically said, I mean the body trade out. You simply does not have the su two States at uh as boundary states. So when you try to to at those degrees of freedom, you simply miss them.
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Daniele Oriti: There is a more serious question is in the type of it has a subset of them.
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simone: Those that have the Reisenberg and Intertwiner.
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simone: Those are good;
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simone: but if I did, but if I take spin networks that have but those restricted intertwiners can I not use the butter cream model for propagating them.
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Daniele Oriti: I don't think so. You simply don't have as you to data
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Daniele Oriti: diagonal listening to data in the Euclidean model.
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Daniele Oriti: The Yes, you two are spin up to States like in loophole and gravity.
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Daniele Oriti: Yeah, I didn't, mind you, okay about it.
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Daniele Oriti: Yeah. But even in that case it's not really the right as you to that you're talking about. But again, this, I think, is just, you know, trying to match a certain as part of the dynamics with boundary States which are just not there,
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Daniele Oriti: because first thing locally, if they had a purely local Ah! Encoding, we should be somewhere in the point, but intertwined in the barrackrain states is basically fixed. Not really dynamic is the but a train. One is unique. There's no real interpreter. So If there is an encoding with tensorial data, it must be more no local somehow.
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Abhay Vasant Ashtekar: But you're saying that it's open also in the generalization that
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Daniele Oriti: when you have space like initial and final data, I would say, And so the
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Daniele Oriti: it doesn't really make too much of a difference with the extension.
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Abhay Vasant Ashtekar: I think that the committee actually had said that our seminars go on too long,
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Abhay Vasant Ashtekar: and, in fact, therefore, it can encourage speakers to finish in about forty five minutes and have a discussion for about fifteen minutes, so I think we should just have one last question, and then close. So Carlos has a question or a comment.
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carlos beltrán: Yes, hi! Everybody here. I just have a question for Alexander, because I was thinking about
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carlos beltrán: the time orientation of the things, because and I would like to to
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carlos beltrán: to me a question if you are thinking about implementing not only a cause of order, this thing also time or intention in this juris, for example, to consider also um not only a timeline or a state like determined, but also,
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carlos beltrán: for example, in time, like each of them, are going into the past or into the future, because well, maybe i'm speaking. I'm going to say something bad. But um,
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carlos beltrán: and are coming to the future of what I want to pass, so I recognize that there are a lot of different things to explore here a lot of open questions here, but maybe it would be very interesting to consider also the time orientation.
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carlos beltrán: And I would like to know you have some ideas about how to implement this thing in. The more that you're shopping, or because it could be something more important, especially if you want to consider calling these models to matter. So. Um, I like to know if you have thinking about that you have er or you have some ideas or other.
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Alexander Jercher: So if I get the question right, you're asking if one can consider a time orientation, or how to implement the time orientation. Is that correct? Well, for that I would refer to already existing work by,
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Alexander Jercher: for instance, by this, what I've cited from Daniel and Itiva, and also all by the work of Eugenio and Pierre B. Yankees, and because there they exactly consider this issue
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Alexander Jercher: to construct a model which is, which is oriented. So where um in the space for with space like Tetrahedra um, one has then two ah for interface at a space like face, two tetrahedral meat, and then one considers um the normal vectors, if they are, point in the same direction or in in the opposite direction. So if both are future pointing, or if one is future pointing,
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Alexander Jercher: then they introduce the notions of thick and thin wedges.
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Alexander Jercher: So they do exactly that in these models, and one is for the Berry Crane case, and the other one is for the
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carlos beltrán: okay. And what are the implications of doing that?
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Alexander Jercher: A.
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Alexander Jercher: So very, very naively. I would say that you get then a directed one, which probably lies in the different universality class. I mean, this is not very speculative, but if you have an in post-direction rather than averaging over it, I would say that at least, and the microscopic properties of probably they could change. But i'm not sure about this much.
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Alexander Jercher: Yeah. And also the question is, does this error of time arise as an emergent property, or should we impose it on the microscopic building blocks. This is also a
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carlos beltrán: Okay, Thank you very much.
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Abhay Vasant Ashtekar: Okay, let us thank Alex again very much,
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Abhay Vasant Ashtekar: I guess, until next time. So I mean, The committee has said that future speakers should aim at forty five minutes plus fifteen minutes for discussion, so we can reasonable end in an hour, so let's try to.