Hal Haggard: Welcome everyone. It's my pleasure to introduce Bianca Dietrich who will speak on the continuum limit of spin foams. Thank you, Bianca.
Bianca Dittrich: Thank you, Han, and thank you for
Bianca Dittrich: cargo organizing this. And as a committee to invite me to talk about the continuum limit of spin forms. It probably should be refinement limit.
Bianca Dittrich: That is the limit of having many degrees of freedom. Many bodies like you have many fish, but I'm
Bianca Dittrich: possibly using continuum limit just because people are used to that so to give you a plan.
Bianca Dittrich: I will motivate why we should be looking into the refinement limit, and my motivation is to film off in symmetry. We will then look at various ways to
Bianca Dittrich: construct discretizations which have different morphine symmetry as perfect discretization, and the consistent boundary formalism
Bianca Dittrich: which motivates certain cost training algorithms. I will then comment on peculiarities on the renormalization flow in different environment theories.
Bianca Dittrich: And hopefully, I have time to actually speak about some results on the continuum limit
Bianca Dittrich: for spin forms. And so
Bianca Dittrich: where this work goes back
Bianca Dittrich: a couple of years, and there have been a number of people involved in that. If you want to have a recent review, there was an article we put out last year with Ses and Sebastian
Bianca Dittrich: for the handbook of quantum gravity.
Bianca Dittrich: So the femoral symmetry we would like to use as a guiding principle for quantization. and indeed, it shows the correct number of propagating degrees of freedom.
Bianca Dittrich: and by that
Bianca Dittrich: the correct dynamics at larger scales it, and also ensures constrained implementation. So the constraints come from having the femorphin symmetry.
Bianca Dittrich: and, as we will see, if we have took him off in symmetry, in the discrete.
Bianca Dittrich: it actually ensures discretization, independence.
Bianca Dittrich: and that not only means it's independent of a choice of triangulation
Bianca Dittrich: or any other discretization. But it does resolve discretization, ambiguities, and artifacts.
Bianca Dittrich: And in particular, we will see that it basically trivializes to take the continuum limit.
Bianca Dittrich: But with all these new kind of wonderful things, it's possibly not surprising that discretizations typically break the film morphine symmetry.
Bianca Dittrich: And so the exceptions we know of are topological. Q of t's and 0 plus one dimensional systems. If we choose to go with appropriate discretization.
Bianca Dittrich: And so to explain like these.
Bianca Dittrich: 0 plus one dimensional systems. So that just means usual mechanical systems which you can make carry a notion of different morphism, invariance, or reparametisation. If you do at time as a variable.
Bianca Dittrich: and in that case, in the continuum, you have a a gauge symmetry which is busily a free choice of time function.
Bianca Dittrich: And if you look at a reparameterized particle in a potential.
discretize it in kind of by choosing piecewise
Bianca Dittrich: linear pieces for the trajectory, you will find basically such an action. such a discrete action. which actually looks the same as for the non-parametrized particle. But the differences is that you would vary this action with respect to both variables. QN, that is a position and tn for all TN.
Bianca Dittrich: Now, typically, for if you have any non trivial potential, you will find a unique solution.
Bianca Dittrich: So this is different from the continuum where, from which you would basically expect you have a free choice of T, and that would determine the queues. Here you find a unique solution. But at the same time
Bianca Dittrich: you find that if you look at the Hessian evaluated on the solution, it has basically a large eigenvalue and a very small eig value. So it's almost 0. And so you have almost a symmetry, but not really a symmetry. But overallmatization! Evidence of
Bianca Dittrich: of the continuum is portion. In this case.
Bianca Dittrich: however, you know, you see already the system kind of might wants to be invariant.
Bianca Dittrich: and so, indeed.
Bianca Dittrich: in this case there is a choice of an action which carries this enviance, and we call it the perfect discretization.
Bianca Dittrich: Oh, perfect action!
Bianca Dittrich: This happens if you take the Hemidakobi action, so the action evaluated on a solution
Bianca Dittrich: for given boundary data and use it for the discretization. In that case you can easily produce solution by taking really the continuum solution and choosing any subdivision of this continuum trajectory.
Bianca Dittrich: By your your point.
Bianca Dittrich: so you can move these points along the trajectory in any way you want. And so this shows that you have a gauge symmetry.
Bianca Dittrich: And indeed, if it now checks the Hessian, it has not direction. So reporter reparitization, invariance is restored.
And
Bianca Dittrich: and so you basically to reproduce the continuum trajectory.
Bianca Dittrich: And so you see see that
Bianca Dittrich: this perfect
Bianca Dittrich: action perfectly miller's
Bianca Dittrich: so continuum dynamics at arbitrary scales. So you can also choose the distance between these points arbitrarily large, and you will still get to continuum the same result. As for the continuum.
Bianca Dittrich: so in some sense the continuum limit becomes trivial.
Bianca Dittrich: Now you might ask, what is the case for Vecchi?
Bianca Dittrich: Gravity which underlies spin forms.
Bianca Dittrich: And so for 3D. Reggae and vanishing cosmology constant. They, you know that all solutions are flat
Bianca Dittrich: and effect any triangulation of lead space gives a solution. Particularly, you can move again the vertices around as much as you want.
Bianca Dittrich: So these vertex translations
Bianca Dittrich: acts are basically a representation of different morphine symmetry in the discrete. and again they allow to probe solution at any scale, so he can kind of move all what he says
Bianca Dittrich: in one corner.
Bianca Dittrich: and have very large fields in the other corner, or very large edge lengths in the other corner, and you can even move vertices on top of each other, and in effectly cause screen your triangulation.
Bianca Dittrich: But in 40, Reggie. you can have curvature.
Bianca Dittrich: And so A while ago we
Bianca Dittrich: computed explicitly a family of solutions with and without curvature.
Bianca Dittrich: And so we evaluated the hashing on the solutions. And here you see a plot on the of the lowest ein value, and so at 0 curvature. You indeed have a 0 ein value.
Bianca Dittrich: meaning you have a symmetry. So that was also kind of a reduced triangulation where you only had expected a dipmorphine symmetry in one direction. But if you turn on curvature by changing the boundary data.
Bianca Dittrich: you find that this lowest eigenvalue goes. And so you see that you get something quadratic in curvature.
Bianca Dittrich: But as an example, you also see that this lowest eigenvalue, the size
Bianca Dittrich: here is basically the difference between the lowest and the next one is kind of 7 orders of magnitude.
Bianca Dittrich: So it is a broken symmetry.
Bianca Dittrich: And here we would expect that we have. Well, there are only solutions for small edge lengths.
Bianca Dittrich: approximate values of continuum.
Bianca Dittrich: If you think about what we do in the past, integrated along quantum theory. We integrate over configurations with arbitrary large at edge lengths. And so that means we take into account also the configurations and also solutions
Bianca Dittrich: which could be quite far away from continuum solutions.
Bianca Dittrich: So how can we construct discretizations with different symmetry?
Bianca Dittrich: And the first clue I showed already. That's a hundred. And Jacobi function
Bianca Dittrich: that it's that is the action evaluated on solutions.
Bianca Dittrich: And I mentioned one notion that you get discretization independence.
Bianca Dittrich: which is that you move 1 point on top of
Bianca Dittrich: another point. And so you effectively coarse grain. Your triangulation gets basically inviance. So the same thing. But also, if you explicitly integrate out or solve for the corresponding variable, you do find that you get again the Hemidobi action.
Bianca Dittrich: So in this sense the handed Nakubi action is
Bianca Dittrich: invariant under changes of the discretization.
Bianca Dittrich: And the same holds actually for the propagator. So for the past, integral amplitude itself, if it's a continuum pass integral amplitude.
Bianca Dittrich: so that means that the handwriting Jacobi function is a fixed point of a core screening flow
Bianca Dittrich: where you start with some guests for your Hematnacobi function or for your action. you construct a new one by integrating out points. And so in this case you can just do that with any pair
Bianca Dittrich: or any triple of points, integrate out the middle point, and you get a new action. And if you repeat that you expect to flow to this fixed point, and so this can be applied in practice
Bianca Dittrich: on a perturbative level, and also in the quantum theory for the pass integral to solve the path integral.
Bianca Dittrich: and it does amount to solving the theory and steps
Bianca Dittrich: but it actually seems to also give you a tool, because, in fact, we used it for the unharmonic oscillator to, in fact, solve the path integral.
Bianca Dittrich: Why are fixed point equations? So you can actually use the fixed point equations to solve the continuum pass integr
Bianca Dittrich: and put it provided with a perfect discretization. Another example where it's headful
Bianca Dittrich: is, for instance, for the Ratchipars integral in 3D.
Bianca Dittrich: You can use these fixed point equation, and these cases they involve. Partner moves to construct a unique triangulation in my end, one loop measure.
Bianca Dittrich: and interestingly, it even includes a pforce phase shift which you have, which you get in the asymptotics of the Ponzano Ritchie model.
Bianca Dittrich: where it's usually explained by the sum of orientations. But here you don't need any sum of orientation. So you just get that by a triangulation. Environment's requirement set
Bianca Dittrich: is quite interesting. So you can it? It shows you that you can fix
Bianca Dittrich: parameters and your discretization uniquely. And so, Zissa.
Bianca Dittrich: examples, where these principles are useful to construct
Bianca Dittrich: discretizations, or even solutions to the continuum and the discrete.
Bianca Dittrich: If you go to higher dimensions or more complicated systems. Well, then, if you go to 4 d.
Bianca Dittrich: And try to construct a measure which is invariant and a 5 one moves. So these are particular. Partner moves
Bianca Dittrich: and the action, the rich action and 4 d. Is actually invariant under these partner moves because the 5 one moves, takes the simplex and subdivides it into 5 simplices. But the new solution is also just flat.
Bianca Dittrich: but it turns out that to measure.
Bianca Dittrich: So one, if you want to have it to be invariant to one loop order only has to be non-local. So you can actually prove that. Furthermore, if you construct these perfect discretization via kind of coarse graining
Bianca Dittrich: for 3 letters field series, with or without gauge symmetries, they turn out to be non local.
Bianca Dittrich: So with kind of in principle, infinite couplings at infinite far away sides which decay exponentially.
Bianca Dittrich: And we have also done this construction for 4 d linearized gravity. Again, the principle is non local because his construction involves free transform.
Bianca Dittrich: So one way to deal with this non-locality is to risk to go to restricted space of configurations, and this has been done by Benjamin Ban Sebastian Steinhaus, for instance. And there is context, and there you've
Bianca Dittrich: kind of again, can fix face weights and also find, for instance, a phase transition, and this restricted, restricted context.
Bianca Dittrich: But in general
Bianca Dittrich: I will comment on later.
Bianca Dittrich: These non-localities are very inconvenient, and they basically go against our philosophy philosophy to have local amplitudes.
Bianca Dittrich: so why do? Non topologically, epimorphism, symmetry, discretizations have to be non local
Bianca Dittrich: so far I should claim that you can prove it.
Bianca Dittrich: For the case of Yf. One moves and the results show that it's not local. But there's also kind of a simple argument
Bianca Dittrich: as a choice view that they have to be non-local.
Bianca Dittrich: And so, if you consider a discretization
Bianca Dittrich: here, for instance, of a cylinder, it's a one plus one dimensional model.
And you have a number of discretization points.
Bianca Dittrich: So if your amplitudes are local, that means you can go from T. One to T. 2.
Bianca Dittrich: Why, I completely look at amplitudes which only uses data at T, one and T. 2,
Bianca Dittrich: and it's the same for T. 0 2 t, one.
Bianca Dittrich: and so if you have different symmetry.
Bianca Dittrich: then you can basically move the points around, for instance, at T one. and again effectively co-screen them.
Bianca Dittrich: So you would have in principle less data. But if it's the femor office menu, and you would expect the same answer. If you go from T. 0 to T. 2.
Bianca Dittrich: But if you insist on having local data. it would mean that you lose information going to T. One.
Bianca Dittrich: and you cannot kind of reconstruct everything you had at T. 2, if all the degrees of freedom are propagating. Oh, if you don't have a TQ of t and so, in fact.
Bianca Dittrich: that tells you that you need a non-local action for this to be 2.
Bianca Dittrich: So you, because the non local action allows you to refer not only to the data of kind of 2 slices, but more than 2 slices to construct your dynamics.
Bianca Dittrich: So these non-local amplitudes, as I mentioned, are very cumbersome.
Bianca Dittrich: Some people even kind of put us one of our axioms that we have to be local. So that is maybe another question. But if you think of non local amplitudes would have to revamp the entire formalism and kind of redoing your canonical analysis.
Bianca Dittrich: for instance, or thinking in a completely new framework about boundaries.
Bianca Dittrich: And so, in fact. But I will introduce boundaries now
Bianca Dittrich: as one method to avoid these non local
Bianca Dittrich: actions. and that's a consistent boundary formalism, which kind of was introduced around 2,012.
Bianca Dittrich: So, and you can understand that as a shift of perspective.
Bianca Dittrich: so usually we take, like the simplest gluey building blocks carrying Min and my boundary data
Bianca Dittrich: like simply says, and glue them together to construct our amplitudes.
Bianca Dittrich: But and we want to shift the perspective, and instead, think of
Bianca Dittrich: more generally boundaries, or, you know, more general boundary data.
Bianca Dittrich: an effect we need a partially ordered set
Bianca Dittrich: of building blocks, or you could say of of boundaries, and the partial order should be with respect to the amount of boundary data.
Bianca Dittrich: and what we will need is consistency relation between amplitudes for these building blocks. And that will give us basically the normalization flow.
AAipad2022: Is, was that supposed to be a strategy to avoid non localities or strategies to to to live in in, in presence of non localities.
Bianca Dittrich: It's a strategy to avoid non localities. In the end it will lead to a renormalization flow which looks local. Still it might evolve with more boundary data.
Bianca Dittrich: We will get some locality. And
Bianca Dittrich: anyway, the the crucial point in these, in in this, in this, having this family of boundaries and partially ordered set
where basically the idea is borrowed from from a Qg. And the cylinder consistent constructions, there
Bianca Dittrich: is that we need an embedding map which tells us how to regain the course data from the fine data.
Bianca Dittrich: So in this example, you can just imagine
Bianca Dittrich: well, you can prescribe how the date, how you have to choose the lengths for this more complicated sphere triangulation
Bianca Dittrich: to regain basically something which looks like a tetrahedron but in quantum theory. There's much more choices for choosing these embedding maps.
Bianca Dittrich: And so in principle, you can kind of.
Bianca Dittrich: and that's a kind of tetrahedron and an arbitrarily fine
Bianca Dittrich: apparently.
Bianca Dittrich: and so the consistency
Bianca Dittrich: condition is again the same as you are used in loop on gravity. So if you have amplitudes for this entire family. If you pull back the amplitude from a fine one to a coarse one.
Bianca Dittrich: you want to find again the amplitude for the course one
Bianca Dittrich: from the amplitude for the fine one. So that means that basically, you get always the same answer.
Bianca Dittrich: independent of
Bianca Dittrich: how fine a triangulation you use to compute your answer.
Bianca Dittrich: And so that allows us also an explicit algorithm
Bianca Dittrich: and to construct such consistent amplitudes. So we start with an amplitude for the simplest building block. We define amplitudes for more complicated building blocks.
Bianca Dittrich: for instance, why are gluing, gluing these building blocks to the more complicated one?
Bianca Dittrich: what we then need to do is to construct a truncation.
Bianca Dittrich: And for this truncation, however, again, the choice is crucial.
Bianca Dittrich: and you could say that can be done by this embedding map. I will comment that you can actually find a dynamically preferred embedding map which is basically defined by the amplitudes themselves.
Bianca Dittrich: And in practice it's really important to use these dynamically preferred embedding maps.
Bianca Dittrich: because if you don't use these dynamically preferred embedding maps and just an arbitrarily chosen embedding map and and practice, for instance, for instance, if you just choose something which sets
Bianca Dittrich: autism final degrees of freedom to 0,
Bianca Dittrich: you basically force a system to flow to this vacuum state, where all the degrees of freedom
or set possible
Bianca Dittrich: but if you do that, you do get improved amplitude. It's a fixed point for this iterative flow. for the simplest building block.
Bianca Dittrich: And you can basically repeat these steps and between building blocks which are more complicated. And so you would get improved amplitudes on all States.
Bianca Dittrich: And so you basically iteratively, would construct consistent amplitudes with more and more for more and more complicated boundaries.
Bianca Dittrich: An important point is here that the amplitudes in principle, you can expect that they change across all scales.
Bianca Dittrich: So for the meaning, the scale now is basically the amount of boundary data
Bianca Dittrich: and the renormalization trajectory is actually encoded in this consistent family of amplitudes.
Bianca Dittrich: So
Bianca Dittrich: if you have said, then constructing the continuum limit is a small part of this construction again, because you would know that
Bianca Dittrich: using kind of a coarse triangulation, or using a very fine triangulation, would give you the same answer.
Bianca Dittrich: at least for the, for the questions you can ask at using the coarse triangulation. So for sufficiently scars, observance
Bianca Dittrich: and tools, Us. Ideas actually implemented in in tensor network anomalization methods.
Bianca Dittrich: So here you have building blocks which are these squares with amplitudes
Bianca Dittrich: and say, you have 4 variables and you glue these building blocks to a larger square which has more boundary data, which basically are these blue edges.
Bianca Dittrich: And Louis means that you sum over these shared boundary data.
Bianca Dittrich: and the next step. So now we have a basically building block with more boundary data. But to have something iterative, you need again, the same amount of boundary data you started with.
Bianca Dittrich: And so to do that
Bianca Dittrich: to do the following.
Bianca Dittrich: you find basically the truncation of this boundary data, which as best as possible approximates the the gluing between these bigger
Bianca Dittrich: blocks
Bianca Dittrich: to basically a new building block.
Bianca Dittrich: So and there's a serm out there that
Bianca Dittrich: basically says
Bianca Dittrich: the best approximation you can find in this way
Bianca Dittrich: is by a singular value, decomposition.
Bianca Dittrich: So what you do is you takes these
Bianca Dittrich: these building blocks or these amplitudes, which have a certain amount of boundary data, you
Bianca Dittrich: partition them an in and out boundary data. And this you can
Bianca Dittrich: and quote in a matrix. And for this matrix, you do a singular value decomposition. And what you then do is just to carry
Bianca Dittrich: the Eigenvalues with basically a num, the largest number, the largest Eigenvalues. So you choose a truncation size.
Bianca Dittrich: say 4 and you take just the 4 largest Eigenvalues.
Bianca Dittrich: And this singular value decomposition in particular, the unitary matrices which kind of make the spaces transform. You can understand as fields redefinition.
Bianca Dittrich: And so what you do is to order your fields into more and less relevant for this course training.
Bianca Dittrich: So these basically
Bianca Dittrich: unitary Max. they're basically represented here by these maps, which have 2
Bianca Dittrich: 2 in ongoing and one outgoing edge.
Bianca Dittrich: and these you can use us embedding maps. So you glue them to your larger building block
Bianca Dittrich: block, and you get again something a new building, I mean a building block which now has the same amount of boundary data you started with. But the amplitude changed.
Bianca Dittrich: And so this allows you to iterate this procedure. And that's basically the tensor network algorithm.
Bianca Dittrich: How much freedom is there in this? I mean
AAipad2022: to find this embedding mapping right when it's trying to minimize the error.
AAipad2022: But you might find one way to minimize error. I might find another.
Bianca Dittrich: Yeah. So so in this context, Sscm, that you know, single value. Decomposition is really the best
Bianca Dittrich: choice. Given this problem, you know, which I've thought.
Bianca Dittrich: And
Bianca Dittrich: but different algorithms kind of
Bianca Dittrich: differ in how large
Bianca Dittrich: a triangulation, or in the details of of what you want exactly to approximate. No. So that come goes into more the specifics. So
Bianca Dittrich: a criticism could be that this is kind of too much of a local thing to consider this approximation. So you want to do it with larger building blocks. And so there are different algorithms
Bianca Dittrich: to do that which, however, are more costly.
Bianca Dittrich: and so that's how how these things sometimes differ.
Bianca Dittrich: But there's always a step of of finding
Bianca Dittrich: by, typically of using a singular value decomposition, you can still try something else. But hopefully, the details would not be too much. I mean, you can type from minimizing the error of and comparing that explicitly.
Bianca Dittrich: But also, I believe, the singular value. Decomposition is basically the most cost friendly one.
AAipad2022: Okay. But I mean, there's a difference between what one can do computationally and what is kind of the theoretically proved that something exists. And then, you know, then one might be able to. One might try to find the best way to get there, and there may be a lot of freedom, computational freedom. But here the 2 things seem to be a little bit mixed up in the sense that
AAipad2022: as you're saying, you know, one might have to decide as to what exactly
AAipad2022: what aspects of the of the system should be better approximated.
AAipad2022: or something like that. So I'm not clear about whether there's a clear organism.
AAipad2022: that computation may be difficult to to impose or to carry out.
AAipad2022: or it's kind of one has to do it by trial and error.
Bianca Dittrich: where Zisa. Zisa is a kind of most popular kind of tensor network anomalisations.
Bianca Dittrich: which use a singular value decomposition.
Bianca Dittrich: Their more complicated version.
Bianca Dittrich: which uses some concept called entanglement filtering. But, for instance, they have only been in
Bianca Dittrich: developed in 2 dim for 2 dimensional systems.
Bianca Dittrich: And so indeed.
Bianca Dittrich: maybe to emphasize that the main step or the main. So step is this truncation step is, and this is kind of really choosing the embedding map
Bianca Dittrich: in some sense is connected to choosing.
Bianca Dittrich: What is
Bianca Dittrich: I mean? It's connected to choosing the to constructing a vacuum state.
Bianca Dittrich: But
Bianca Dittrich: this kind of algorithm has been tested on on at least many examples in 2 2 dimensions.
Bianca Dittrich: And we have done all those 3 dimensional systems.
Bianca Dittrich: Okay. yeah, just go ahead.
Bianca Dittrich: So well, this, this allows you to iterate and find fixed points. So the entire
Bianca Dittrich: algorithm, of course, depend on how many boundary data you have. So that's crucial in in basically estimating the cost and what is doable.
Bianca Dittrich: And so
Bianca Dittrich: indeed, where you could in principle compute it for a given amount of boundary data, and then repeat for more boundary data. But this is technically where basically, the the constraints come from on applying this, this algorithm is how much boundary data you want to handle.
Bianca Dittrich: And so we have worked with these tensor networks quite a lot and developing also new algorithms. So we have done a lot in 2D.
Bianca Dittrich: These models which were supposed to physically act as underlocks of spit forms.
Bianca Dittrich: But we also develop new
Bianca Dittrich: versions of these models to be able to treat with a gauge series.
Bianca Dittrich: There. The main difficulty is again, that you want to save on some out of boundary. But if you have gauge series
Bianca Dittrich: you have redundant data. So you kind of want to only work with cage and my aunt data.
Bianca Dittrich: And these models, these algorithms have all always been tested and also applied with 3D. Spin forms. Again, where there is some notion of simplicity, constraints
Bianca Dittrich: which we constructed. and the extracted phase, diagram, and phase transitions in principle. So
Bianca Dittrich: we also constructed a new tensor network algorithm with a fusion basis which the fusion basis is basically a basis which is dual to the spin network basis
Bianca Dittrich: and is much more suited for for coarse graining. It's basically already in the name. Fusion means that you fuse physically the charges in the spaces.
Bianca Dittrich: and in fact, it's more suited for coarse graining, because it also captures torsion, which you do get by a just coarse graining gauge series. In particular, the spin networks. If you coarse grain them, you do get structures which are more
Bianca Dittrich: generous and spinetworks.
Bianca Dittrich: And this is still the leading algorithm. And in the field.
Bianca Dittrich: and Sebastian also considered coupling matter to interd finorlets
Bianca Dittrich: in particular.
Bianca Dittrich: trying to figure out what is a what is exactly the the function or the kind of
Bianca Dittrich: coupling constant
Bianca Dittrich: for these meta models. And that brings me to the to the next point, which is
Bianca Dittrich: what is different between a series where we have different symmetry and series.
Bianca Dittrich: like letter Sketch series, where we have just a background lattice.
Bianca Dittrich: And so in Letter Skate series, you usually start with a given regular letters which has a lattice constant, a. You define the model at the scale. And so you compute the action at larger scales.
Bianca Dittrich: And particular find basically is a beta function, which is how the coupling changes as a function of the scale.
Bianca Dittrich: however, with a dynamic lattice. So if you have different morphine symmetry
Bianca Dittrich: or just gravity. That means you. These edges can have any lengths.
Bianca Dittrich: and it's not only that they can globally have any lengths, but that 2 different edges can have also 2 different lengths.
Bianca Dittrich: So also I've drawn this, let this very regular.
Bianca Dittrich: I copy a highly irregular letters.
Bianca Dittrich: and so if you want to define like
Bianca Dittrich: any any coupling to matter, for instance, on these letters. or any system on this letters. that we need to know the physics, and these couplings
Bianca Dittrich: for all length, scales, and for all possible inhomogeneous length scales. So we need to already know the Beta functions to define the consistent dynamics.
Bianca Dittrich: So that is a lot to ask for.
Bianca Dittrich: And so instead.
Bianca Dittrich: indeed, what we should see is that we have initial seed amplitudes.
Bianca Dittrich: I wouldn't say they are fundamental, because these amplitudes are defined on all length scales, but we wouldn't trust them on arbitrary large length scales. So what we have we should see as a guess of the dynamics over all length scales.
Bianca Dittrich: Then we do this iterative course, training procedures, and to get improved amplitudes. Then again, the amplitudes on all length, scales could be effective.
Bianca Dittrich: affected.
Bianca Dittrich: And so here, maybe, is another example to illustrate the challenges. to just write down actions or amplitudes which are consistent.
Bianca Dittrich: and that's something. You know, I just called letters form factors.
Bianca Dittrich: So here's just an example. You can take, for instance, the harmonic oscillators. But so this will be a 3 field theory.
Bianca Dittrich: But then you have 2 terms, and you have to come up with basically finding functions of the lengths
Bianca Dittrich: around these vertices you are considering.
Bianca Dittrich: So that
Bianca Dittrich: we get the consistent dynamics which basically is reproduced by choosing arbitrary lettuces.
Bianca Dittrich: And so even for the one dimensional system of the harmonic oscillator.
Bianca Dittrich: where these functions would be ratios of trigonometric functions, so not necessarily easy to guess.
Bianca Dittrich: But you can kind of compute them by using the Hamidniac obi function.
Bianca Dittrich: So these functions already have to take the normalization flow into into count.
Bianca Dittrich: And so that's what you need to compute.
Bianca Dittrich: So my claim is kind of impossible to guess all these consistent amplitudes. but you need to construct them via an iterative course gain in flow.
Bianca Dittrich: and the consistent boundary formalism allows to do so.
Bianca Dittrich: And identifies a dynamically preferred truncation scheme.
Bianca Dittrich: Okay.
Bianca Dittrich: so other questions to this, because I slightly change
Bianca Dittrich: topic.
Bianca Dittrich: Okay.
Bianca Dittrich: so let me comment on some more recent work, and that's actually involving for these spin forms
Bianca Dittrich: cartoon. Yep.
John Barrett: just
John Barrett: in general terms, what you've done is, your iterative procedure ends up with some sort of
John Barrett: presumably limiting dynamics in which all the equations that you wanted satisfied? Exactly.
John Barrett: It was consistent. So can you just sort of think of that as being the like the Hamilton Jacobite
John Barrett: flow that you talked about? In the first place, I mean, does it sort of formerly have the same properties? Or is it something sort of different?
Bianca Dittrich: But
Bianca Dittrich: yeah, it would be kind of this, the same as a Hamid Kobe flow. Just was the same Kobe flow example
Bianca Dittrich: that I presented it in one dimension.
Bianca Dittrich: And then, when when I mentioned, you have this cool, very convenient properties. That's a boundary. The amount of boundary data does not change under course training
John Barrett: right?
Bianca Dittrich: Whereas you know. If you do, if you go to higher dimensions. If you go to 2D. Already
Bianca Dittrich: you have Cisco difficulties that you're
Bianca Dittrich: that amount of boundary data changes, and that's where you need this embedding map and the like, choosing a truncation to allow for this iterative procedure.
John Barrett: but you've computed something like the quantum effective action. Is that is that how you think of it, or I mean, do you also have a picture for this limiting?
Bianca Dittrich: so I
Bianca Dittrich: don't have a picture, because in practice we do the right, basically, for I mean, we did it in practice using these tensor network algorithms.
Bianca Dittrich: And the highest examples we went to are 3D
Bianca Dittrich: and did it for a certain amount of boundary data on, for instance.
Bianca Dittrich: cubes.
Bianca Dittrich: But then, at least, if you interpret that in the in the lattice gauge series, then, so the tensor network algorithms it can interpret.
Bianca Dittrich: like both in the kind of spin form systems and lattice gauge theory sense. These, the
Bianca Dittrich: fixed points there, indeed, would give you information about the continuum limit.
Bianca Dittrich: And so you can kind of understand it as a. as yeah, as a quantum, effective action. If you want. If you want to have information over all scales, you would have to tune the system to the phase, transition.
Bianca Dittrich: and need to involve more and more boundary data to
Bianca Dittrich: have to not suffer from truncation artifacts.
John Barrett: Okay? Great thanks.
Daniele Oriti: sorry, Bianca. Can I? Just one more question on the Amazon Jacobi case. In the example you gave you had to evaluate on the solution. And you said that corresponds to you know, the evaluation at a given fixed point.
Daniele Oriti: How does the situation change when you, when you have several solutions? And this will so this solution is not unique. You evaluate Theampton Jacobi on. So you sum over all the possible evaluations of different solutions, or or what?
Bianca Dittrich: Yeah, I wanted to avoid this question. I think in one series that is discussed for some morning oscillator, so that you might have to take into account
Bianca Dittrich: and are solutions which do a full period.
Daniele Oriti: Huh!
Bianca Dittrich: Classically, I haven't stopped about it.
Bianca Dittrich: but I think one can work it out with a harmonic oscillator. What one should do. Okay, thanks.
Bianca Dittrich: But but indeed, in that case is kind of okay. What happens if you choose, like your discretization, step
Bianca Dittrich: larger than at the time for one oscillation?
Daniele Oriti: Yes, indeed.
AAipad2022: just following up on John Barrett's questions. I mean this about possibly effective or something. I mean, you got this form factors. And you know here, I mean, it does look very much like, you know, we have some effective action. So is there any. What is the relation between this
AAipad2022: strategy to find
AAipad2022: the refinement or the continuum limit.
AAipad2022: and the strategy of say, for example, the asymptotic safety in which one has this renormalization group flow.
AAipad2022: and then, you know, one is also trying to find some
Bianca Dittrich: in the final theory I can. So what what is the Mo? What is the relation?
Bianca Dittrich: But
Bianca Dittrich: one relation is an asymptotic safety kind of
Bianca Dittrich: where famously runs renormalization flow backwards from from infrared to UV
Bianca Dittrich: whereas for letters models, one would usually say, one runs from UV to infrared
Bianca Dittrich: but actually, my point of saying this year was that it's not very clear which direction you run, and in in fact, and and Gr. And with
Bianca Dittrich: those being become independent where we have difficulties in determining what a scale.
Bianca Dittrich: And you could argue that we rather use infrared dynamics to
Bianca Dittrich: start with our seat amplitudes. and then use the iterative, coarse, graining procedure to
Bianca Dittrich: find to try to find a consistent family of amplitudes over all scales. So you could say that indeed, what we should do is also to
Bianca Dittrich: find a consistent UV dynamics at all.
Bianca Dittrich: But,
Bianca Dittrich: I would say a priori the well, it could happen that we improve over all scales. But, in fact, what we wouldn't want to change too much is at least semi classical infographics, which is pure.
Bianca Dittrich: So in this sense
Bianca Dittrich: we are similar to asymptotic safety
AAipad2022: did this procedure. There's no guarantee that, you know, when you do this going back, this loops that, in fact, your internal dynamics will not be altered. Guarant, you know. I mean, that will be one of the criteria of successful course. Grading procedure
Bianca Dittrich: offer of a successful series.
Bianca Dittrich: I mean, it's yeah. So you know, I can, of course, start with this
Bianca Dittrich: initial amplitudes which are not suited for this exercise, or many things can move on.
Bianca Dittrich: But indeed that it's
yeah. I think, in general, if we start with some
Bianca Dittrich: attitudes, and I think we should be open because it's a criticism that we kind of people say, are we just?
Bianca Dittrich: I don't guess these amplitudes and claims they are fundamental. but in fact. we kind of should expect that the change over oil scales.
Bianca Dittrich: And so that is implementing the the consistent amplitudes and finding a dynamics consistent of our skills, which, of course.
Bianca Dittrich: is super super difficult, and is is a dream which you know we try has never be, has never been achieved in physics. So is is not an easy
Bianca Dittrich: program. But then, at least this iterative course training procedure
Bianca Dittrich: you do. You can improve in steps. So you will. And and you kind of improve in steps. So the liability of your of your calculations. So in some sense you can hope, maybe that's the difference with asymptotic safety. There another criticism is the choice of truncation.
Bianca Dittrich: and which typically is kind of dictated by locality. whereas here you have these procedures, which is kind of dynamically informed truncations. and maybe also more flexibility in how you choose your truncations.
Bianca Dittrich: okay.
Western: there's another question from Western
Western: yeah. Moment. I have a question about different variants. And I just want to understand what?
Western: What exactly do you mean by this? And the the question is actually about by about your first slide when you when you when you say that if a most invariance is broken, I want to understand what what this mean in in the following sense.
Bianca Dittrich: maybe do that after I do the last part of my talk, I'll wait for the end. That's fine. okay, well.
Bianca Dittrich: maybe. Yeah. So so let me say a few words
Bianca Dittrich: on on actual results, on, on the continuum limit of 4 d spin forms. So it's actually not connected so much to the first part.
Bianca Dittrich: realizing all these issues with consistent amplitudes.
Bianca Dittrich: it's a perturbative result. So for this we have to assume that spin forms a metaphase which at larger scales does lead to smooth geometries.
Bianca Dittrich: Baby knows that we have specified kind of at least 2 different phases. This is a Ashika Lewandowski vacuum.
Bianca Dittrich: where basically, you set all the spins to 0. So it's a very degenerate geometry on the other hand, you have the Bf vacuum where you have bf. or be a homogeneously curved geometry
Bianca Dittrich: still, for spin forms where you import some simplicity constraints.
Bianca Dittrich: You might wish something intervene, at least, for the constraints
Bianca Dittrich: have been set to 0 or set to some cautions, as we will see and still, you have kind of geometrically flatness.
Bianca Dittrich: And so and basically need to make this assumption that to construct the perturbative continuum limit, I mean, I can construct a perturbative continuum limit. But it does rely on this assumption that one can actually do that.
Bianca Dittrich: And the main question is, do we obtain linearized
Bianca Dittrich: gravity? Lennox? That's a perturbative order. I will look at
Bianca Dittrich: and to do that. I will use xactive spin forms.
Okay?
Bianca Dittrich: Because that allows to give me an answer and actually even more information.
Bianca Dittrich: And so the effective spin forms there reconstruct them, because they're much more inevitable to
Bianca Dittrich: calculations, and the dynamics is encoded in a quite transparent way.
Bianca Dittrich: So we directly have this oscillating factor
Bianca Dittrich: which is given by the exponential of the ratchet of the area. Rachi action.
Bianca Dittrich: So here we have areas which kind of give you much more general configurations and lengths, variables.
Bianca Dittrich: And so what you also have are Gaussian factors.
Bianca Dittrich: which imposes constraints reducing areas to length.
Bianca Dittrich: but not sharply, but as cautions.
Bianca Dittrich: And so the this of these kind of these co options is determined by the commutation relations between these constraints.
Bianca Dittrich: which, for instance, involves the barbarian music parameter.
Bianca Dittrich: and so the main part is a reti action. And then you have basically, if you want to write it and kind of an action, an imaginary part which basically involves the logarithm of these cautions.
Bianca Dittrich: And so you can basically ask, what is the lattice continuum limit for the area redj action. And that's an open fashion. Since the 90 s.
Bianca Dittrich: And so. and my understanding was, it's widely assumed to not to lead to gr.
Bianca Dittrich: Also. John taught me at some point he was not the one assuming that it does not leak to Gr.
Bianca Dittrich: And you can claim that this is the original flatness problem.
Bianca Dittrich: because if you look at the equation of motion for area Reggae. It does say it does basically say that some quantity which looks like the deficit angle. But it's actually not the deficit angle has to vanish.
Bianca Dittrich: And so this that's the first question we will answer and But you have also the constraints. So you can ask, do the constraints term change anything.
Bianca Dittrich: and so I won't tell you
Bianca Dittrich: much about how this calculation is done.
Bianca Dittrich: So you do take the action and define it on an infinite
Bianca Dittrich: regular letters.
Bianca Dittrich: and but I have taken also various lattices, and the result don't change.
Okay.
Bianca Dittrich: What you have sent to do is to classify the variables according to their scaling, and how how they behave!
Bianca Dittrich: What is the scaling in the momenta in the Hessian momenta? Here means Fourier labels. So the regular letters allows you to do a Fourier transform.
Bianca Dittrich: And then basically, you can compute a serious expansion of the effective action in the lattice constant.
Bianca Dittrich: And so what is basically the slightest continuum limit for the linearized? Have you a rich action?
Bianca Dittrich: So here is a crucial point is, despite having an enormous amount of variables. So for various discretizations, it's 50 or 100 or 300
Bianca Dittrich: variables pair let aside
Bianca Dittrich: so only degrees of freedom which are massless, are really corresponding
Bianca Dittrich: to actually the lengths matrix.
Bianca Dittrich: So you have only always 10 degrees of freedom which are massless. On top of that you always have this linearized dipmorphine symmetry. so always 4 will be
Bianca Dittrich: each.
Bianca Dittrich: And so this is more or less the reason that in the continuum limit, if you just look at the
Bianca Dittrich: I guess. Now lowest order, and the lattice constant. The continuum limit of this action is given by the linearized ancient heard that action. but we could also compute the next to leading order term, when the lattice constant
Bianca Dittrich: and that was given by a Y square term.
Bianca Dittrich: an effect we could show that it arises from an effective area matrix.
Bianca Dittrich: This array matrix we could construct for each hypercube but also Joseim
Bianca Dittrich: has recently shown that you can construct an area metric for each simplex.
Bianca Dittrich: an area matrix has 20 components. So it has more components. If you integrate, grade out the 10 additional components of the area matrix set is what leads to this Y squared term.
Bianca Dittrich: And so here, what is happening is that you have a retic calculus. It has more degrees of freedom.
Bianca Dittrich: In fact, macroscopically, you can extend, understand that as an extension of the configuration space from lengths to area matrix.
Bianca Dittrich: but only the length, metric degrees of freedom are massless. and all the additional degrees of freedom have Planck, mass. and that's the reason why you do get gr on the limit.
Bianca Dittrich: Do these constraints, terms change anything? And so the answer is
Bianca Dittrich: actually no, they don't seem to be essential in this continuum limit.
Bianca Dittrich: Say so. The constraints only affect the massive degrees of freedom.
Bianca Dittrich: There are these degrees of freedom which are in the area metric, but not in the length, matrix, and so they add something to the mass, which is imaginary and but they are parameter dependent.
Bianca Dittrich: But that doesn't change the overall picture.
Bianca Dittrich: So you kind of still gets a YI squared term as the next correction. But it's a coupling constant. There changes. This is complex parameter.
Bianca Dittrich: So that's actually good news, because it means a universality.
Bianca Dittrich: a kind of universality inside.
Bianca Dittrich: because the different models kind of change.
Bianca Dittrich: or they differ in how the constraints are implemented in details.
Bianca Dittrich: But that doesn't seem to matter too much for the continuum limit.
Bianca Dittrich: So I've got a question here. Yeah.
John Barrett: so you you start with area variables. And then you, you want to say that in the limit, basically, only the length variables
John Barrett: survive.
Bianca Dittrich: Actually, it's a length matrix variables.
John Barrett: Yeah, yeah, exactly. So, how non local is that is it? Is it local in each simplex, or does it need neighboring simplexes, or or what
Bianca Dittrich: it needs? A, I think it's it's it's local on each hypercuper.
Bianca Dittrich: or it's hypercube, right? So even in the in even in the cases where I refine, yeah, I refine these Hypercubes.
Bianca Dittrich: it was still just 10 length variables per 10 length matrix degrees of freedom per hypercube.
John Barrett: Right? So it's not local in each simplex.
John Barrett: No, on each simplex actually, area, we have 10 areas and 10 links. So it doesn't mean that those links are the ones you get from those areas.
Bianca Dittrich: No, it's really a construction on the level of each wipe equipment.
John Barrett: I see. So it's hypochond
Bianca Dittrich: like there's a whether something is massless or or not. You need to lose a Fourier transform for that. You need kind of many Hypercubes?
Bianca Dittrich: So, yeah, so, so, wh? What is? I would like to understand the relation between this
AAipad2022: various part various parts of the talk? So in the last part you sort of had this this this cylinder to consistency requirement, you know the flows that went on. So is this, what was exactly done here, or this is something this part I didn't use cylindrical consistency right?
Bianca Dittrich: So now this, you know, they started from a different corner.
Bianca Dittrich: and
Bianca Dittrich: from effective spin forms which you can see may be in general as an effort to make spin forms more computable and more amenable to one day apply this framework.
Bianca Dittrich: but the result is that well, at least, if you assume that none pert well, that there is a suitable non perturbative
Bianca Dittrich: limit for spin forms so that you get
Bianca Dittrich: a suitable face, and you can do on top of it
Bianca Dittrich: a perturbative construction. Then at least, we know that we can get GR. Out of the limit.
AAipad2022: No, I agree. This is this is very strong, but on that, on the other hand, once we have the answer, can I look at the answer and reinterpret it as saying that? Well, somehow this is not what I did, but in the in retrospect I can think of this as being obtained from cylindrical consistency.
Bianca Dittrich: yeah, I would need to sit down and take a story up with that right now, because that's what we need to patch that. Otherwise it's sort of very disparate ideas. And we don't know. So so basically, what I need to specify is kind of what kind of truncations I use. So
Bianca Dittrich: right? And so basically here III said hopefully.
Bianca Dittrich: hopefully, the interpretation of my microscopic interpretation of what we have of lengths and areas does survive in the continuum limit, so that I can do these perturbative constructions
AAipad2022: that but even within pertinent theory, right? Even like we can talk about. Enormous group flows in the peripatetic theory so similarly here in this, assuming that but should be still able to say that well.
AAipad2022: there is still is really is implementing the idea of cylindrical consistency. And just that that's not how it was arrived at. But that's what it did.
AAipad2022: That's, I think would be really satisfactory. I mean, there's some.
Bianca Dittrich: Yeah, III agree. but I would have to think kind of how to
Bianca Dittrich: how to construct the the kind of embedded map.
AAipad2022: Correct, exact, exact.
Bianca Dittrich: okay.
Bianca Dittrich: David Yamulla.
Bianca Dittrich: as a further consistency result. Just let me mention that you can ask, can I get the same effective action or same action which we got, like Einstein Hilbert, plus, while square directly from the continuum.
Bianca Dittrich: And the short answer is, Yes, you can.
Bianca Dittrich: And for that we kind of use modified Plabanski's theory. So that was proposed by Kirill. And basically the idea is that you take Lebansky, but replace the simplicity constraints.
By Mosomes!
Bianca Dittrich: So we modified it further by saying, we don't replace all constraints by mass terms, but we impose some of these constraints strongly, and these should agree exactly with the same, which which are imposed strongly in the continuum, and some of them we impose weakly or by mass terms. And so we made this choice. And then, indeed, directly from the continuum, we basically found
Bianca Dittrich: the same action that is Einstein Hilbert, plus y squared.
Bianca Dittrich: But here we found a momentum dependent prefactor, which we couldn't see on the lattice, because we only went to the kind of fourth order and derivatives.
Bianca Dittrich: So here we find this prefactor. And surprisingly, it makes this action. Actually.
Bianca Dittrich: it makes this go stream.
Bianca Dittrich: You don't have any higher ports in the propagator. as as a further result. Further remark I will throw out here is that.
Bianca Dittrich: but in principle you have now
Bianca Dittrich: or to gamma in your cyst, in your coupling parameters.
Bianca Dittrich: And so if you introduce a cosmos constant. we have 3 coupling constants.
Bianca Dittrich: and the same in some sense holds another product safety. They have 3 relevant couplings and Cdt. And an edit.
Bianca Dittrich: And it seems you need really at least 3 coupling constants to be able to find a suitable continuum limit. Moreover, in CD. T's additional constant is also an unisotropy parameter. It's a difference between its effector between space like and time, like
Bianca Dittrich: edges. And the same holds also in new common gravity, because Gamma appears in the space like area spectrum, but not in the timelike form. Okay?
Bianca Dittrich: So indeed, where I had basically, these 2 different parts of the talk.
Bianca Dittrich: So the first I was introducing
Bianca Dittrich: kind of a framework for for doing coarse graining and renormalization and background independent theories. And so we see that this notion of tip amorphous symmetry in the discrete and truest discretization. Independence.
Bianca Dittrich: in some sense the continuum limit becomes trivial. You can use perfect actions which mirror
Bianca Dittrich: exactly the continuum dynamics, but are non lucal for non topological theories. The consistent boundary formalisms allows you to avoid these non localities.
Bianca Dittrich: but to construct a renormalization trajectory us, a family of consistent amplitudes. Basically. this dynamically preferred truncation scheme.
Bianca Dittrich: And you can use tensor networks to try to construct that
Bianca Dittrich: we have kind of pushed to 3D. And so they need to be also pushed, too. 4 d.
Bianca Dittrich: And the second part of the talk. We looked at the perturbative continuum limit for effective spin forms.
Bianca Dittrich: But there we have very good news, because.
Bianca Dittrich: we find general relativity at leading order and inspection. It's actually very surprising, because
Bianca Dittrich: an eye for
Bianca Dittrich: I particularly thought that every wretch action would not lead to GR.
Bianca Dittrich: And I think many people did believe the same. And so you can actually see that as a source of the original flatness problem.
Bianca Dittrich: but surprisingly, it does lead to general relativity, and that means that, for instance, the Bell chain model is still might lead to general relativity.
Bianca Dittrich: and, moreover, we could go one order further in the continu in in the expansion and find the Y squared term as a correction. And basically that arises from the extension from length to every matrix
Bianca Dittrich: and this seems to be really a strong signature of spin forms in general that areas are the more fundamental degrees of freedom.
Bianca Dittrich: Thank you.
Hal Haggard: Thank you, Bianca.
Hal Haggard: Western had their hand up with a question.
Bianca Dittrich: though that was the notion of diplomophysymmetry, I guess.
Western: Yeah. Hi, first of all, let me say that. I think the second part that there's this
Western: second talk that you gave on the on the area
Western: action. And it's very good. It's super good, and and makes me very happy, of course. So it's great work, I think.
Western: But my question is about the the first. yeah, thank you for putting that slide. And it's about different. That's something that confuses me. So I want to understand what you mean exactly by different movies. For for let me tell you what confuses me.
Western: The the question you have there is the discretization of a formulation of a particle in a potential.
Western: And of course, the continuous version of this has a different symmetry or symmetry whatever, which is the continuous, which is very clear, because what start with a queue as a function of a parameter, Tau and T as a function of parameter Tau, and as an action which is invariant under implementation of Tau which means that if you, if you plot
Western: queue is a function of Tau in T is a function of Tau, you have a lot of
Western: trajectories which are gauge related. That's my understanding of of of different variants, namely, if you, if you make a different tau, you you have gauge in a gauge equivalent trajectory.
Western: Now, I would say, and maybe
Western: maybe that's what it's it's a terminology different. I would say that if you want to capture the
Western: this, a morphism invariant part of Q. Of Tau, T. Of Tau. You have to look at queue queue as a function of T. You have to solve Tau away.
Western: So if you plot in the Qt. Space a trajectory.
Western: all the different moves invariant
Western: solutions trajectories are all mapped to the same quantity. So given a plot in Qt. Space, that's a difficult environment quantity. Because if I now make a different office, nothing changes.
Western: Now you seem to be saying that that's not the case, that once we go down to the Qt. Space.
Western: still, there is something that you still want to call different, and that's what escapes me.
Bianca Dittrich: They don't know. Yeah, I mean, it's
Bianca Dittrich: I could. You could say it's on this level that indeed, if you choose. If you choose the teas to be differently. In the continuum as a function of Tau
Bianca Dittrich: you you find
Bianca Dittrich: kind of the same solution, but you find the same solution. But it's the same continuum solution. Here you don't find the same solution.
Western: That's what I don't understand. What what. So how form of that blue curve.
Bianca Dittrich: Sorry!
Bianca Dittrich: What is the gauge transformation of that blue curve? I would say it remains itself. But you say, doesn't.
Bianca Dittrich: What is a gauge? Does I mean, first of all, gauge symmetry is broken. So if I do change the blue points here, I do get a slightly different curve
Bianca Dittrich: connected by these piecewise linear, piecewise linear pieces.
Bianca Dittrich: but it would be a different depend dependence of Q and T. So that's I would call it physical. Why would call it a change in no, I mean, we can. Yeah. So so again, this notion of different one.
Bianca Dittrich: which then leads to constraints.
Bianca Dittrich: And so, if you want to have this relation.
Bianca Dittrich: You you! You come to call it the femoral symmetry.
Bianca Dittrich: If you want to have really a notion of where, in the continuum, I really have one degree of freedom which is physical, which is Q.
Bianca Dittrich: Whereas here, if you kind of ignore that, that it is a broken symmetry.
Bianca Dittrich: then you have 2 degrees of freedom, which are q and T.
Bianca Dittrich: So from this point of view of having the same number of degrees of freedom, and the continuum and the discrete.
Bianca Dittrich: or having constraints.
Bianca Dittrich: or having gauge equivalent set of solutions which you get on the right hand side.
Bianca Dittrich: you should call it the phimorphic symmetry. and so in the and in the continuum again. you can choose t arbitrarily as a function of Tau.
Bianca Dittrich: and in the discrete you want to do the same, and on the right hand side you can indeed choose T to be arbitrarily.
Western: No, I understand that what comes after I'm questioning so what would be a gauge
Bianca Dittrich: gauge symmetry. You had a paper where you say, basically, oh, actually choosing the discrete values of Tau is a gauge symmetry. Now, the Taos are actually not even variables.
Bianca Dittrich: And they do drop out of this discretization. That's why you say it's a gauge parameter. But the tows are not even variables. So for me. That is not a symmetry, because the towers are just arbitrary parameters.
Bianca Dittrich: and what are variables? Are Q. And T. And you are asking, then at least me. I am asking
Bianca Dittrich: what is a symmetry which kind of maps solutions to solutions.
Bianca Dittrich: and leave the action in my aunt. And so the answer is, is. while in that case you go to perfect actions.
Western: Okay? I'm also understanding. I mean, I understand, in the case of the perfect action, as you say in the head to Jacobi, but I, and understanding the case of the in the left case, what would be a gauge transform of the blue curve?
Bianca Dittrich: What is? Hello! I'm I'm I'm saying there is no gauge.
Bianca Dittrich: Sorry the minute I am saying there is no gauge symmetry, because the symmetry is broken.
Bianca Dittrich: There would be a gauge symmetry if the Hessian, evaluated on a solution has a 0 eigenvalue. But there's only a small eigenvalue, and that's why I say it's actually a broken symmetry.
Bianca Dittrich: And so.
Bianca Dittrich: but for me.
Bianca Dittrich: age symmetries have to also do something non-trivial on the ladder of of C. Action
Bianca Dittrich: of on the level of the of the variables.
Bianca Dittrich: whereas your your notion doesn't seem to do anything on, on.
Bianca Dittrich: on the action, on the variables.
Bianca Dittrich: I mean, you are basically saying there is a notion of dipmorphine symmetry, because I define dipmorphic symmetry to leave this curve invariant.
Western: Yeah, yeah. So I don't understand what is broken. It's a II don't see the symmetry which is broken.
Bianca Dittrich: Well, first, it suggests the the mathematical effect that this is very small. Eigenvalue. Okay, that's what you mean the existence of a smaller value. That's but then you can even go in a in a that if you do the renormalization, of course, training
Bianca Dittrich: you do that, you? You restore the symmetry
Bianca Dittrich: P,
Bianca Dittrich: and
Bianca Dittrich: kind of that's that's a behavior of many other gauge symmetries.
Western: Okay.
John Barrett: aren't you just saying that the action doesn't depend on the intermediate values of T.
Western: So tell me what you say
Bianca Dittrich: in the in the I mean here on the left hand side, or on the right hand side.
John Barrett: on the left hand side.
Bianca Dittrich: on the left hand side. It does depend on the intermediate values of T,
John Barrett: yeah. So on. On the right hand side. It, doesn't
Bianca Dittrich: it? It still does. But what the inviance is is where it's
Bianca Dittrich: what what else I mean. If you if you
Bianca Dittrich: well, let me see him all
Bianca Dittrich: where there is an inviance. But you have to change t and Q,
Bianca Dittrich: and the one thing it's not just tea.
Bianca Dittrich: I don't.
Bianca Dittrich: You have to change
t and Q
Bianca Dittrich: along the trajectory, for instance, said Sir Wright in my arms.
Bianca Dittrich: Right? Okay, not just he.
John Barrett: Yeah. So in the quantum model you would integrate over queue. So it's just
John Barrett: the remaining action is just a function of the teas.
Bianca Dittrich: hmm, yeah.
John Barrett: And you are just saying it's independent of the intermediate teas.
Bianca Dittrich: Yeah, I mean in the end, that is, you could. That's just translate, like vertex translation symmetry.
John Barrett: Yep.
Bianca Dittrich: whereas you know, on the left hand side. Your result would depend on how you have chosen the intermediate values of your teeth. Exactly.
Bianca Dittrich: And okay, I mean, I would say, it's it's possibly not physical.
Bianca Dittrich: Yeah.
Hal Haggard: shall we proceed to advise? Question.
AAipad2022: yeah. So this is almost at the end last end of their talk? Yeah, it has to do with this, this effective spin models that you talked about.
AAipad2022: So, as you pointed out. Now
AAipad2022: that you know there are various indications that we need 3 relevant coupling constants. And we've got G and lambda and some. And you like to think of this gamma as being the third relevant coupling. Constant. Okay. But we also sort of know that from this modern perspective effective field series, and particularly the things which have to do with this.
AAipad2022: this form factors? This idea of form factors, and so on, that the GN. Lambda.
AAipad2022: physically they don't run.
AAipad2022: And and so here Gamma also does physically does not run. Is that the statement?
Bianca Dittrich: I mean, it's the same visa.
Bianca Dittrich: I think it. It might run, I mean in the con. Yeah. So so in Cdt and Eddp. you need 3 couplings constants
Bianca Dittrich: to be able to find you. And so indeed, G and lambda you form into kind of an
Bianca Dittrich: a combination of
Bianca Dittrich: and and then you you need another com, another parameter to to find zoom. The question is that they always run in the theory space as John don't know who explain to us in the theory space that are. But physically they don't run.
Bianca Dittrich: Yeah, Jean Jean Lamna don't run. I mean, I'm talking. I'm talking that these statements. It's in 2 different contexts one is an asymptotic safety, and and one is in Cdt and Ed and I wanted to see in C. Dt. Edt. They're still working on a notion of running coupling constants. So I
Bianca Dittrich: it's it's hard for me to spontaneously say what happens to this Unizol? But with grammar. Yeah, I well.
Bianca Dittrich: I mean, there's one paper by Simona and Andario, where they make gamma running.
AAipad2022: No, but I think there's a difference between running in the theory space and running physically right. And so that that was the big difference that John has been pointed out, pointing out to us last few months.
Simone SPEZIALE: but by running physical you mean a running that you would see with asthmatics, I suppose right
Simone SPEZIALE: log in regularization instead of cut off.
Simone SPEZIALE: then, if you add the cosmod, your constant lambda, and you have fermions, then Gamma would run in that sense. But if you have only fermions, or only it goes logical, constant, then Gamma will not run in that sense. I'm talking
Simone SPEZIALE: pure gravity. So lambda. Gmg, but no form. Yeah. Okay. Good. Good. No, no running that you would see in this matrix.
AAipad2022: Good, good. Good. It's consistent with my understanding. Thank you.
Hal Haggard: Simone, did you have an independent question?
Simone SPEZIALE: No, it was really just to comment on this. Sorry I can lower my hand. No problem.
thank you.
Hal Haggard: Are there other questions?
Hal Haggard: If not, let's thank Bianca again.
Bianca Dittrich: Thank you for listening.
Hal Haggard: Bye, everyone.