Jorge Pullin: Your speaker did as you can. I want you, so I will speak about avenues to spin for phenomenology.
Johanna Borissova: Okay, thank you very much for the invitation and introduction. Today I would like to to present you with a new perspective on the degrees of freedom that are relevant for spin forms and a semi-classical limit.
Johanna Borissova: This is based on work with Bianca, and we expect us to set the stage for a diverse phenomenology, and also a variety of other conceptual and fundamental questions.
Johanna Borissova: Okay, let me see.
Johanna Borissova: Okay, You should be able to see the second slide. Now I will start with a motivation. So spin forms are covariant path Integrals for quantum gravity parts in the case of gravity represents space-time geometries
Johanna Borissova: that satisfies certain boundary conditions, and they are weighted by an amplitude which is given by exponential of 5 times the action.
Johanna Borissova: Now, classical geometry is just an equivalence class of metrics, modular dipomorphisms, but quantum geometries are fuzzy and the reason for this is common to a large class of spinful models.
Johanna Borissova: So at the classical level constraints up here, these are the primary simplicity constraints in the form of first class algebra. but at the quantum level these constraints become partially second class. I would don't normally parameterize, but about very music parameter.
Johanna Borissova: and if these second class constraints would be imposed strongly to what restrict the solution space to much so, therefore, they can only be in post weekly.
Johanna Borissova: But as a consequence, the pattern to define over an extended configuration space. So it is, then, a natural question to ask, Can we parameterize this extended configuration space?
So this space is parametrized by a metrics.
Johanna Borissova: Now my plan will be as follows: I will first introduce to you what an area metric is. I would then provide you with some evidence. Why, Aaron metrics appear in a semi-classical limit of spin forms
Johanna Borissova: I would then ask, Can we derive an on action which describes the dynamics of an error metrics that mechanism that imposes constraints weekly, but as a classical level.
Johanna Borissova: then I will ask the question: okay, Out of the degrees of freedom of the error metric, can you find a subset which corresponds to a length metric? And if yes, how does the effective action for this metric look like? Does it reproduce gr, or do we have to expect corrections?
Johanna Borissova: And so my final? My final slide will then be on geometric interpretation of this part of the area metric which is not the like metric part.
Johanna Borissova: And so, finally, as I said, my message will be that this sets the stage from from phenomenology and also other interesting questions.
Johanna Borissova: Okay. So in this slide I would like to introduce to you what an aerometric is, and I would like to do this by comparing it to the concept of a link metric.
Johanna Borissova: So a length metric at a point on a manifold is a symmetric rank 0. 2 tensor I'm. Assuming non degenerate metrics in the sense that this matrix here is not a determinant, not 0.
Johanna Borissova: A length metric can be used to define an inner product between 2 vectors where these vectors come from a tangent space at this point.
Johanna Borissova: So if these vectors are the same, vector and then Romanian signature. The inner product of a vector with itself, quantifies the length, or in this case the length squared of this. Vector
Johanna Borissova: If these vector are different, then the inner product of these 2 vectors quantifies the to the angle between the vectors that intersect at this point
Johanna Borissova: so very similarly on aerometric which was introduced by scholar and collaborators. An area metric can be seen as a metric, for by vectors where the vectors that are vet together, they live at the tangent space at this point.
Johanna Borissova: So, therefore, more formally on a metric, is around 0 for tensor with the following properties here.
Johanna Borissova: So the first property says that it's a symmetric under the exchange of the first and next pair, and the static second index pair. This must be the case, because if you want to define an in a product, and we want this to be symmetric.
Johanna Borissova: The second property says that it is antisymmetric in each of these index pairs, which also has to be the case, because a bi-vector contains a vet product which is antisymmetric.
Johanna Borissova: and so finally, one can also demand a similar condition. As for the length metric case of non degeneracy that does not degeneracy implies that the matrix here is non-degenerate it allows for a consistent raising and lower ring of by vector indices.
Johanna Borissova: So now the inner product of a simple, by vector which is so just the vet product of 2 vectors, and no linear combination of it.
Johanna Borissova: The the inner product of a simple by vector quantifies the area of the parallelogram spent by these 2 vectors.
Johanna Borissova: On the other hand, if the planes that are to go to these 5 vectors. If they intersect at the line, then the inner product of these 2 by vectors, they parametrizes the 3D that you triangle between these planes.
Johanna Borissova: so one can, besides of these symmetry properties, also demands that the I am a metric is cyclic. This corresponds to this a requirement here, if all other algebraic symmetries are satisfied, this is equivalent to demanding that the cyclic mirror has no totally antisymmetric part.
Johanna Borissova: and therefore the contraction of the error metric with the Epsilon new to which are 10. So is 0.
Johanna Borissova: So okay, it's important also to note that every length metric and uses a cyclic error metric by which of this definition here?
But the opposite is not true in general, and, in fact, only in 3 spacetime dimensions, where length and aerometric have the same number of degrees of freedom, namely, 6
Johanna Borissova: only. In this case every area metric is induced by on length, metric in dimensions higher than, say, a higher than 3
Johanna Borissova: Aaron metrics feature, a a a a true generalization of length, metric geometries, that feature many more degrees of freedom than the length metric. So, for example, in Depos 4, a cyclic error metric has the same as your bride symmetries as the Riemann Tensor, and the effort has 20 degrees of freedom
Johanna Borissova: as compared to the length metric which has only 10 degrees of freedom. So an interesting question is, then, once you have this area
Johanna Borissova: a an area metric or spacetime, or an area metric? Can you identify a subset of degrees of freedom which define an effective length metric? And if yes, what could the other part represent.
Johanna Borissova: So I will ask, answer this question, and in the context of Pfi it's in the Plavansky formulation. It was also an effect. Affection introduced by scholar. But I will not go into more detail on this.
Johanna Borissova: First of all. Yeah, with this, now that you know that an what an area metric is, I would like to give you 2 pieces of evidence. Why, we think that a metrics are relevant in a semi classical regime of spin phones.
Johanna Borissova: So the first one is the results on a lattice continuum limit of the error regions.
Johanna Borissova: So, just to fix notation in the following, I will be the length of an edge, a D area of a triangle and a simplex.
Johanna Borissova: an epsilon, the deficit angle at a triangle. So all of the following is an in 40, and so, therefore the curvature is concentrated at triangles
Johanna Borissova: now in length to edgy calculus, which was introduced by Edgy in the 19 sixties. It provides a discretized version of Back from Gr on a piece ofise flat, discretized for the manifold.
Johanna Borissova: the fundamental variables and the length where g action are the edge, length and the action is obtained by just summing all triangle areas times the deficit angle.
Johanna Borissova: So in a case of length, Reggie calculus, the deficit angles parametrize the curvature
Johanna Borissova: concentrated as a at this triangle, so, in other words, they parameterize the angular gap that would arise if you flatten the 4 D simply sees which meet at a triangle if you flatten these to 4 D flat space time.
Johanna Borissova: Now, by contrast and error, energy calculus, the fundamental variables are taken to be the areas of the triangles. So note that within a given simplex the map between lengths and areas can be inverted
Johanna Borissova: just because the single simplex has the same number of edges and triangle areas.
Johanna Borissova: But as soon as you start gluing simply sees together, you have many more error variables than length, variables, and therefore the that the error, energy action describes the dynamics of many more degrees of freedom.
Johanna Borissova: It's also important to note that why your gluing the errors can match, but the shape does not have to match. And therefore, if you vary this action with respect to the configuration variables.
Johanna Borissova: and you get the equations of motion with which say that the deficit angles are 0, then this does not mean that solutions are flat, but it rather means that the combination of curvature and shape M. Matches 0.
Johanna Borissova: So there was as a summary the error as you action. First of all, it describes many more degrees of freedom, and second, solutions are not necessarily flat. so the error energy, action appears in a semi classical limit of spin forms, and the following is a result: by Bianca and Biancana, another Phd. Student.
Johanna Borissova: So they saw that putting the error to action on the lattice on a hybrid, Kubernetes lattice and a flat background expansion.
Johanna Borissova: the error energy, action can be seen as a discretized version for an error, energy, action for an an aerometric action in the limit in which this lattice constant case to 0.
Johanna Borissova: And so importantly, this aerometric is associated to the hypercube, and the degrees of freedom of this area Metric can be split into a part that is massless, and is precisely 10 degrees of freedom for the length metric.
Johanna Borissova: plus a set of massive degrees of freedom, and these are suppressed by the inverse power of the lattice constant by the inverse lattice constant. So if you integrate out these massive degrees of freedom.
Johanna Borissova: the result is an effective action for the length metric, and it's effective action reproduces to lowest order, but it also contains a V. By squared corrections at next order.
Johanna Borissova: So this is an example that area metrics appear microscopically in the semi-classical regime of spin phones.
The second piece of evidence that I would like to give you, for I metrics and and spin forms is a microscopic identification.
Johanna Borissova: It is based on recent work of, and not just a Phd student and bianca they showed that semi classic, the semi-classical spin from the degrees of freedom can be matched to aiometric degrees of freedom.
Johanna Borissova: So, first of all, a classical Simplex, as I just mentioned, has 10 degrees of freedom.
Johanna Borissova: This this can be to taken to be either the 10 lengths or the 10 areas. I can see something in a jet. There's no question
Lee Smolin: I assigned right in the extraction of the
Lee Smolin: the the length area metric
Johanna Borissova: to me. Do you mean the sign of the Why squared term?
Lee Smolin: Yes, and the
Johanna Borissova: so regarding the Einstein hover part. Yes, the other one. I think so. But this is meant to be more formally. It is just meant to say that there's a Y squared corrections, and in fact, a sign will be important later in our effective action there is a sign which is crucial to state that this action is ghostly.
Johanna Borissova: But regarding the error reduction, I I'm: not clearly here this time. Okay, Thank you. Okay, Thank you. Okay.
Johanna Borissova: No. A classical simplex has 10 degrees of freedom. A quantum simplex consists of the gluing of 5 quantum tetrahedra in such a way that it respects the quantum uncertainty relation.
Johanna Borissova: So a quantum tetrahedron has 5. No 4 normals can be parameterized by 5 quantum numbers. These can be taken to be 4 areas. So 4 norms of normal lectures.
Johanna Borissova: plus one in a product between 2 normal. So 1, 3D he triangle and the quantum uncertainty relations. They say that only the areas of the 10 triangles shed by pass of the triangle can be identified.
Johanna Borissova: There was a consequence. A quantum simplex has 15 degrees of freedom
Johanna Borissova: on the other hand, there's also the notion of a semi-classical or coherent or twisted simplex, in which case the 5 5 classical to where he draws together.
Yeah.
Johanna Borissova: So a classical tetrahedron can be time advised by 6 quantities. These are: these can be taken either 6 lengths. either the sex lengths or for areas, and to the he draw angles at not opposite edges.
Johanna Borissova: Then the the quantum gluing in this case means that only the areas of the 10 triangles are identified. So the 10 drivers that are shared by pairs of tetrahedron.
Johanna Borissova: So, therefore, semi-classical Simplex has 20 degrees of freedom and these can be split into 10 triangle areas plus 10 dihedra angles
Johanna Borissova: 2 PET tetrahedron, and a non opposite touches.
Johanna Borissova: So Hosse and Bianca, short at the 20 degrees of freedom of a semi classical Simplex can be arranged into 20 degrees of freedom for an era metric associated to to Simplex.
Johanna Borissova: So with these 2 examples, I Okay, that's right.
Johanna Borissova: Okay. So with these 2 examples, I've hopefully shown you that air metrics appear in a semi-classical regime of spin forms. They are pre and microscopically as well as microscopically.
Johanna Borissova: So by the way, actually aaron metrics were introduced in the context of string theory, but they also pia and holography, and also in reconstructing geometry from entanglement. Now here we see that they also appear in a semi-classical regime of spin forms.
Johanna Borissova: so my goal will be there, and the following to parametrize the classical extended configuration space at a continuum level in terms of error metrics, and for this I will start from the configuration variables in the Plebanski formulation of Gr.
Johanna Borissova: which, in fact, sets the origin of spin from quantization, and which I will introduce in the next slide. And so the question will be, then can we derive the dynamics for an error metric Can we reproduce the continuum effective result of the area reg action on the lattice.
Johanna Borissova: and can we do so by that very similar mechanism as at a quantum level but imposing classical constraints weekly.
Johanna Borissova: So yeah, I will first remind you quickly of the formulation of gr.
Johanna Borissova: So the Plavanski formulation in the contains. Yeah. Well it's constructive for them to have configuration in this case the B field a connection, and the lagrange multiplier.
Johanna Borissova: So the B fields and a non-carrow version is an so 4 valued 2 form.
Johanna Borissova: The connection is regarded as an S of 4 value to one form and 5 is a Lagrange. Multiply a feature taken to be with internal indices, and is also required to have the same algebraic symmetry as the Riemann Tensor. So it has 20 degrees of freedom of 20 components.
So the plastic action is then, first of all the Bf. Action.
Johanna Borissova: The B of action on itself defines a topological future with no local propagating degrees of freedom.
Johanna Borissova: One can then also add to the host term, which, coupling constant prima, tries by the inverse Barbara in music parameters.
Johanna Borissova: And so finally, there is this last term here which includes the Lagrange multiplier, and varying this action with respect to this, to the Lagrange multiplier imposes constraints on the b field.
Johanna Borissova: These are the simplicity constraints, because they their solution restricts the if you to be a simple by vector
Johanna Borissova: and in the case of so 4 there are 2 solution sectors. The first one is the gravitational sector. So plugging in this solution into the cleavans section again leads to the Palatini action for Gr.
Johanna Borissova: Or if you include Gamma, at least to the Palatini host action for gr.
Johanna Borissova: The second solution Sector is the topological sector. If you include, if you re-enerate this solution into the Planans section. Then you get a multiple of the horse term.
Johanna Borissova: So now, as I said, primary simplicity constraints are first class, but quantizing the f, and imposing the primary simplicity constraints at a quantum level, they become partially second class, and therefore the and so this mechanism leads to an extended configuration space in a path integral.
Johanna Borissova: Now I would like to do a various to look if some of we can do a similar mechanism at a classical level and suppress parts of the constraints, so i'm. Post them weekly.
Johanna Borissova: And how how this can be done is in the framework of modified.
Johanna Borissova: So the Pravansk action that I just introduced is the sum of the Bf. Action plus simplicity. Constraints on the B field. and the simplicity constraints are imposed by the Lagrange multiplier file.
Johanna Borissova: on the other hand, and modified plastic theories, the simplicity constraints are replaced by a potential which is now an entire function of this field here
Johanna Borissova: so varying the with respect to this, to this field you'll get a new term, which is the gradient of the potential.
Johanna Borissova: So if the second variation of the potential is not singular. So the determinant of this matrix is not 0. Then you can solve this for 5 as a function of P,
Johanna Borissova: and it re and reinserting this again into the Plavanski action, leads to a modified Plavanski action, where now the configuration variables are the B feed and the connection and the simplicity constraints on the beefield are replaced by a potential.
Johanna Borissova: Now, from this action it is possible to integrate out a connection, so to find a solution to the compatibility equation and write the connection as a function of the B field.
Johanna Borissova: So the result is then an effective, modified prevention. But the only configuration variable is the B field, and it is a second order action.
Johanna Borissova: So it's important to keep in mind that this is an effective action in the sense that the connection has been integrated out.
Johanna Borissova: So in the non degenerate case, in which case the determinant of this matrix of second variations is not 0.
Johanna Borissova: This is seen as as just a mechanism of replacing the full full simplicity constraints, but potential for the B Fields
Johanna Borissova: Canonical analysis showed that in the Kyo case, in which, where the algebra is su 2, there are 2 massless, propagating degrees of freedom that correspond to the graviton.
Johanna Borissova: and so these series became known as the formations of Gr. So Jr. Is then obtained in the limit of constant potential versus taking a Ted Delta function potential just reproduces Bf: Again: so in this sense the series.
Johanna Borissova: Yeah. Paramet. Try smoothly between the F. And Gr
in a non-cario case the legend is so 4.
Johanna Borissova: And in this case there are 6 degrees of freedom. More than so, this theories differs substantially from Gr.
Johanna Borissova: Okay. Now, what happens in the degenerate case, and the case where the second variations of the potential are singular. So the determinant of this matrix is you.
Johanna Borissova: So this case should be interpreted in a way that out of the full simplicity constraints. Only a subset of simplicity is of simplicity, constraints it's imposed sharp.
Johanna Borissova: and these are precisely the ones in which the in which the pretend potential is constant directions.
Johanna Borissova: The other directions, instead give rise to a potential for the rest degrees of freedom of the B field, and this potential can be taken as a suppressing potential. So you should notice this is precisely the mechanism that we are looking for, and therefore the question is, Can we use modified clebansk theories
Johanna Borissova: to derive an area metric action.
Johanna Borissova: So yeah. for this. I would like to first to accounting of degrees of freedom.
Johanna Borissova: So in a non Cairo version the B field is an S. Of 4 valued 2 form. So 4 has I mentioned 6.
Johanna Borissova: I can see something else on the shuttle. Is there a question?
Lee Smolin: It's a common.
Johanna Borissova: Well, yeah, the contract?
Johanna Borissova: Yeah, Thank you for the comment. The context in which I would like to talk about this modified is really in the context of kind of versus non kyle and other additional degrees of freedom or not.
Johanna Borissova: Okay.
Johanna Borissova: Now, okay, as I said, s of 4 value 2 form such as this. Beef here has 36 degrees of freedom, out of which 6 are gauge degrees of freedom.
Johanna Borissova: now imposing the 20 simplicity constraints on the beef it reduces the 36 degrees of freedom to 16 for tetrahed out of it Again, 6 are gauge degrees of freedom.
Johanna Borissova: and therefore what you're left with at 10 degrees of freedom for the length metric.
Johanna Borissova: but we would not like to end up with an a length metric. We would like to end up with an error metric which has 20 degrees of freedom. So the question is, we have to impose only 10 simplicity constraints, and which subset, should we impose.
Johanna Borissova: And so to answer this question, to identify such a suitable subset. we use the parametrization of the B field, which is as follows.
Johanna Borissova: So first of all, the so, for the algebra is isomorphic to the direct sum of 2 su tools, and this isomorphism is provided by projectors.
Johanna Borissova: and says that the S. O. 4 value. 2 form can be parametrized by 2 S. U. 2 valued 2 forms.
Johanna Borissova: and which I just as you to be fields. So further, one can then parameterize each of these su 2 be fields, using the band K Theorem in terms of a of 4 D. Metrics.
Johanna Borissova: so 140 metric for each s to be fuel, plus a a matrix of 3 by 3 scalars, which is sunni modular. So, in fact, these are only 8 degrees of freedom here, and therefore 10 plus 8, gives 18 degrees of freedom for an su 2 valued 2 form.
Johanna Borissova: This parametrization was introduced by law, or in the context of modified gravity theories with no more propagating degrees of freedom. So specifically, how does this parametrization look like? First of all, he's Sigma out of plavanski 2 forms. They are self, an anti software, and they in quota 2 space time metrics in each sector.
Johanna Borissova: And then these plabans keep so 2 forms. They are rotated by means of the B fields.
Johanna Borissova: and it'll be fields themselves. They define you new modular, internal metrics in this way.
Johanna Borissova: so that, and summary the 36 degrees of freedom of the B fields can we split into the degrees of freedom of 2 length metrics.
Johanna Borissova: plus a set of 2 times 8 spacetime scalars.
Johanna Borissova: and as as a consequence of the split, the effective, modified Clemenski action ends up, being the sum of 2 effective Pf. Actions, one for each sector, and also remember again, this is the effective P. Of action. Well, the connection has been integrated out
Johanna Borissova: plus the potential term.
Johanna Borissova: Okay. Now, still we Haven't answered the question of which subset should be imposed, such that we can reduce the degrees of freedom after be field to 20 for an area metric.
Johanna Borissova: So how to proceed. We define a cyclic left and right-handed area metrics for each sector. These are just given by the tensor product of 2 B fields, and then we subtract off the totally antisymmetric parts. So that these are indeed cyclic.
Johanna Borissova: It turns out that imposing that these 2 left and right-handed analytics are the same it's equivalent to imposing the 20 simplicity constraints, as was shown by rising barrier.
Johanna Borissova: So this case leads to gr again.
Johanna Borissova: On the other hand, Simona invested the case where the internal metrics are frozen, so they are required to be just. Flat Metrics just corresponds to imposing the Via part of the simplicity constraints.
Johanna Borissova: and at least a biometric theory of gravity.
Johanna Borissova: The just suffers from the same problems as usual by metric theories, so it is unstable in the sense that it propagates, in addition to the masses, Graviton, also a massive spin, 2 particle and a go in a scalar mode, which is a ghost mode.
Johanna Borissova: So, by contrast, what we did is to impose the other part of the simplicity constraints this. This corresponds to the trace part of the lagrange multiplier and importantly, what we did is to set the E left and right hand at length metric geometries to be equal.
Johanna Borissova: so the results for the action is then, and so we we get, then, an action in which the configuration variables are a length metric and 2 internal metrics.
Johanna Borissova: and we view this as an area metric action for the error metrics defined by the sum of the left and right 10 of the metrics. So this a metric is primarily by
Johanna Borissova: by a length metric and 2 internal metrics, and this is also how the split goes. So with this we can fill in the diagram that I showed you before
Johanna Borissova: did 20 degrees of freedom for the length metric split into 10 spacetime scalars plus 10 degrees of freedom for the length metric which actually coincides with the
Johanna Borissova: Okay.
Johanna Borissova: So yeah. But to be able to really claim that this is an area metric action. It would be good if we can invert the relation so that we can write down the
Johanna Borissova: linked an internal metrics in terms of the I metric. So this would mean solving 20 polynomial equations, and it is very likely that an explicit solution cannot be stated.
Johanna Borissova: Therefore, what we did is to invert this relation perturbatively in a flat background. Expansion. So in the following the length metric perturbations are denoted by H.
Johanna Borissova: And the internal metric perturbations are denoted by Kai. These sky fields.
Johanna Borissova: that, traceless because of the unimodularity condition of these on these internal metrics.
Johanna Borissova: So, as a result of this decomposition, the area metric takes the following form: so it's first of all, given by a part that is induced by a flat like length metric.
Johanna Borissova: and then it has first order, corrections, these first order corrections. They can be decomposed into contributions that are sourced by perturbations of the length, metric
Johanna Borissova: and contributions data that are coming from perturbations of the internal metrics. So this is just a linear relationship. It can be inverted explicitly. So we are able to write down H and the Kai Fields as functions of the area metric perturbations.
Johanna Borissova: Therefore, in this approximation we can really write down an action in which the fundamental variable is the area I metric.
Johanna Borissova: Now, again, it takes the form of the sum of 2 effective Pfx actions, but now linearize, so on the fields a PC. Only quadratically
Johanna Borissova: plus a potential, and I will only consider the case of potential for the Kai fields. And, in fact, I will look at the most simplest potential that can be chosen, which is just adding mass to the high fields.
Johanna Borissova: So some effective, modified plan that cleans the action which we view as an area. Metric action is then obtained in the following way: so first of all, it contains for each sector a part which is given by the Einstein Herbert action for the some of these fields.
Johanna Borissova: so the Einstein Herbert action is constructed with a kinetic operator, which just given by the party fits operator. This operator contains the spin 0 and spin 2 projectors
Johanna Borissova: so noted here the Kai fields come with space-time indices, and they obtain from the Ki fields with internal indices by an isometric embedding, so the embedding as such that the symmetric trace less wrong. 2 tensors on su 2
Johanna Borissova: are embedded into the symmetric traces around 2 tensors on space time in such a way that this condition here holds.
Johanna Borissova: and the kind of new fields they are also transfers. So yeah, not also that this redefinition here is actually slightly non-local, because it contains the inverse of the Laplacian operator.
Johanna Borissova: So the action that we have is therefore, first of all the sum of Einstein. How about part
Johanna Borissova: for for the some of these feuds? And then there is this mass, then there are these.
Johanna Borissova: So, as a consequence of this redefinition, it is immediate to see that this action is environment under linearized different offices.
Johanna Borissova: But then, also, if you don't have these mass terms, and look just as a. At a single sector. Then this is just P. F. Theory, and it contains a remnant of the Pf shift symmetry which is linearized.
Johanna Borissova: If you also now include the second sector and set the length metrics equal to be equal, then the linearize the f shift symmetries
Johanna Borissova: which are separate for each sector. They combine into a single one for for this entire combination here.
Johanna Borissova: But now, as soon as you are at the mass terms, they break the Pf. Shift symmetry, and this leads to propagating degrees of freedom.
Johanna Borissova: So in the Cairo version there's only one Se. To a present, and then it's a to see that you can, just to a redefinition of the metric perturbations. So just to shift this feature by by the Kai Fields.
Johanna Borissova: So in this case the Chi fields will just satisfy auxiliary equations of motion. They can be integrated out, and therefore the result is just to understand her. But action linearized for the shifted graviton field.
Johanna Borissova: But not that. As I said, these kind of new fields, so it's based on indices. They are obtained from the high Ap. Ones by a non local transformation. So this future definition is actually not local.
Johanna Borissova: but nevertheless it shows, or it explains why, at the linearized level. So it explains that the linearized level why these series propagate only 2 degrees of freedom, which was later also confirmed by canonical analysis at a nonlinear level.
Johanna Borissova: Now, in the non Cairo case it is not so clear how which type of degrees of freedom are propagated. In particular. There is no canonical analysis available for these series.
Johanna Borissova: But nevertheless, what one can do is to integrate out to Kai Fields and this approximation, and to look at the effective action for the length metric.
Johanna Borissova: and in particular, look at the propagator structure, which can provide some hints whether one should expect additional degrees of freedom or not.
Johanna Borissova: So this is what we did. We integrated out to Kai Fields. The result here is shown in fourier space for simplicity.
Johanna Borissova: And yeah, this is the final effective action for the length metric perturbations. So there are 3 important points to note you.
Johanna Borissova: The first of the first one is that the action recovers the Einstein. How about action linearized to lowest order.
Johanna Borissova: but then it also contains a full tower of corrections which are quoted in the by tensor, the non-local. So the kinetic opera of the operator is going by such a term half for each sector, and also there goes free in the case, and which the masses are taken to be equal.
Johanna Borissova: So this is interesting, because if we we would have only device where
Johanna Borissova: part, then this would just be similar to quadratic gravity. You might know that you may know that quadratic gravity is unstable. It propagates course modes.
Johanna Borissova: But this is not the case in our situation.
Johanna Borissova: Now also, the third important point is to note that to lowest order the correction terms are just the squared by tensor, and therefore we do reproduce the
Johanna Borissova: the result of the area that you action in the hyper cubic lattice expansion and in the limit, and which the that is constant case to 0.
Johanna Borissova: Okay, now, so this is the part on the length, metric degrees of freedom contained in the error metric and the error metric.
Johanna Borissova: I would also finally like to look at the non metric degrees of freedom in a area metric, and explain to you what they could mean geometrically.
Johanna Borissova: And for this I will look at the onshare connection. the Ontario connection. It's just defined to be the solution of the Gauss law, which is obtained by varying the with respect to the connection.
Johanna Borissova: the Gauss law splits into 2 independent Su 2 Gaussian. And yeah. These are just given in terms of the projections of this, of the connection on the left and right hand, etc.
Johanna Borissova: So the goal is therefore, to solve this compatibility equation for omega plus minus as a function of the B field.
Johanna Borissova: but actually as a function of the parametrization of the beefield.
Johanna Borissova: and in doing this there are 2 equivalent perspectives. The first one was introduced by, and his in the context of modified quality theories.
Johanna Borissova: The way this proceeds is by defining a new connection out of the connection of Cleveland ski to for a plan in a cleavansky formulation.
Johanna Borissova: This new connection is just obtained by transforming the connection via the B Fields.
Johanna Borissova: and one can then compute the exterior covariant derivative of this connection, acting on the plans key to forms.
Johanna Borissova: And yeah, what you will find is that this is your own. So the interpretation of this
Johanna Borissova: is that disconnection is torsion-free. I can see another question.
Johanna Borissova: Would it be possible to define a notion of compatible connection for the aerometric?
Johanna Borissova: Yeah. So this is a very interesting question indeed. This is a way to go. Shula and his original paper has provided some attempts for this.
Johanna Borissova: but, as far as I know, there's no settled, every metric, compatible connection; and, in fact. This is, I I think, 1 one open key open problem to do a and a continuum as well as in a discrete.
Johanna Borissova: I hope this answers the question. I can comment more on this later.
Johanna Borissova: Oh, okay.
Johanna Borissova: okay. So as I said, this connection, here is a transformed one. And this condition, that exterior covariant derivative of this connection, acting on a Plavansky to from vanishes, is interpreted as the connection, being tors and free.
Johanna Borissova: At the same time, the exterior covariant, or derivative of this connection, acting on the internal metrics is 0, and the effort cannot preserve the data metrics. So in this sense this connection is not metric.
Johanna Borissova: Now, on the other hand, it's possible to just not do this transformation. And look what happens, using the original configuration variable. So the original connection of.
Johanna Borissova: and what what one will see is that the exterior, covariant derivative of this connection, acting on a plastic 2 forms, is not 0, and it's not 0 because the scale of our present.
Johanna Borissova: So this says that
Johanna Borissova: connection that is actually contained in a while. At the same time it preserves the data met. Again. The effort is metric.
Johanna Borissova: So therefore, the upshot is the non-metric degrees of freedom of the area metrics source torsion.
Johanna Borissova: Okay. So with this, I would like to give you you the summary and some prospects where one could go from here on.
Johanna Borissova: Okay, First of all, I explained to you that the spin from from parts to go is defined on an extended configuration space.
Johanna Borissova: This extended configuration. Space can be parameterized, semi classically by area metrics, and hence for this the continuum limit of the error at the action on the lattice, and also the geometry of a twisted simplex.
Johanna Borissova: It turns out that the continuum effective dynamics of spin forms can be reproduced by modified non kara plabansk theories.
Johanna Borissova: and they are these not modified non-cari plans to theories are taken such that the area metric is the configuration variable. So the way this is obtained is by reducing the degrees of freedom of the bfield to the ones of an error metric.
Johanna Borissova: and this is done in a way that left and right-handed geometries length. Metric geometries are required to be equal.
Johanna Borissova: This imposes. Half of the simplicity constraints the other half of the simplicity constraints is replaced by a potential.
Johanna Borissova: and so, as a consequence, the degrees of freedom of the error, metrics split into a part for length, metric plus scalars which can be interpreted as sourcing torsion.
Johanna Borissova: Yeah. So the the other final result was that the effective action for the length metric reproduces, and then it contains a vice-president, non local ghost fee correction.
Johanna Borissova: So I would like to argue that once we have such an effective action. We can do phenomenology with us so interesting questions to ask would be, what are corrections to the Newton potential, or to Black Hole entropy, or actually, are there any corrections?
Johanna Borissova: What can we say about black holes in these series? First of all, we would have Einstein by gravity to lowest order. But what happens beyond?
Johanna Borissova: Can we constrain the scale of masses, theoretically and observationally. Can this be also a constraint on the barbarian music problem?
Johanna Borissova: Yeah. So conceptual questions can we recover full nonlinear gr within this framework. and for this it may be, it may be necessary to use another parameterization of to be fields.
Johanna Borissova: Another conceptual question is to do a canonic accounting at a nonlinear level to see if we should expect additional degrees of freedom or not. Also another question is to to what happens if we choose different potentials
Johanna Borissova: it on the fundamental sites or directions to go, is first of all constructing aerometric geometry. So for this one as to define an aerometric combat compatible connection, and once this is done, one can construct all types of curvature, 2 and 10 to and so on.
Johanna Borissova: This has to be done in a continuum in in a discrete. So I think this was also what Tank was referring to it as an interesting direction to go.
Johanna Borissova: Hmm. Yeah. So out of these I would like to mention 2 more points. The first one is, Can we? Is it possible to derive the Black Hole Entropy law, in the language of a metrics, so do I. Metric variables in the sense, provide a more natural parametrizational, more natural derivation for this
Johanna Borissova: and the other part is that I would like I would be interested in is, Can we use the scale up potential, or can we interpret it? The scale up potential as the dynamical or emergent cosmological constant.
Johanna Borissova: Here it appears that they had a fundamental differences between the Cairo and the non-ciral version and the Kyle version. One can use another parameterization for a. B field where one splits off a Plavanski to form for another metric, and then a set of a set of deviations. From this Plebanski to form.
Johanna Borissova: One can then show that this modified Plavansky theories in the Kyle version does really lead to gr to full gr plus the potential term for the scale of it, and in this sense solving the equations of motion for the skate of you, it's just a state that the scale of you' at a minimum of the potential, and after this minimum of the potential could be a cosmological constant. But this appears to be a mechanism a priori defined only for the Kyle part.
Johanna Borissova: It is not clear whether this persist also to the non Kyle version.
Johanna Borissova: Yeah. So with this I would like to finish my talk. Thank you for the attention. and I'm happy to take questions.
Ivan Agullo: and and I had to believe, and so I he left me in charge of the questions.
Ivan Agullo: So thank you, Johanna, for being perfectly on time. That was very nice. So it's time for questions. The pack, as you know. You can go ahead.
Deepak Vaid: Hi, Joanna, I don't know His horror is still around. Just wanted to wish him a very related happy good day, or very happy.
Ivan Agullo: I I don't think it's around, but they will. I will pass the message.
Deepak Vaid: So, Johanna. I I joined like 5 min late. I think so. Maybe I missed the definition of what an area metric is that's the first name.
Deepak Vaid: And so if you wouldn't mind going over that, and the second is second call, my question is that where does it appear in string should be? And in what way, if you could elaborate on that?
Johanna Borissova: Yeah. So an error metric is.
Johanna Borissova: can be viewed as a metric, for by vectors
Johanna Borissova: so similar. When you have a metric, it defines an inner product between 2 vectors and a metric to find some in a product between 2 by vectors, and these by vectors. They are constructed with the vet product of vectors and these vectors that the tangent space at this point.
Johanna Borissova: so
Johanna Borissova: The intuitive way to see this is that the inner product defined by this area metric provides some measure for areas of parallel parallelograms and dyhedra angles between planes. So this is just going one dimension higher, if you want compared to the length metric which measures length and 2D angles.
Deepak Vaid: Okay, and so is is it
Deepak Vaid: isn't it possible to like Recast the I 10 Hilbert action in terms of the Api metric?
Deepak Vaid: Is it? Is it possible to do that?
Johanna Borissova: It is possible at the linearized level, as I showed, because the the linearized level, the relation between aerometric perturbations and length, metric perturbations is just linear so you can solve the linu. You can express the Linux the perturbations of the length metric in terms of the aerometric perturbations.
Johanna Borissova: but at a nonlinear level this would mean to solve but 20 polynomial equations to invert them explicitly, and we've not been able to do this. It would be nice if if this would be possible, but we haven't done it.
Deepak Vaid: So this is. This is a completely different theory from Gr, which is based on
Deepak Vaid: the usual length metric.
Deepak Vaid: Okay, I mean, because if you cannot. if you don't have that.
Deepak Vaid: if you can't go from one to the other.
Johanna Borissova: it is different in the sense that I explained error, metrics, and for D have many more degrees of freedom than length metrics. But nevertheless, we've been able to identify a subset of these degrees of freedom which gives you a length metric, and the effective action for this link metric ends up recovering. Gr.
Johanna Borissova: So in in this sense it is consistent with Gr. At low energies, but these degrees of freedom that are in addition, they are suppressed by a by a mass which is actually the plank mass, and therefore they expected to only modify higher energy or gi at time energies.
Deepak Vaid: Okay, what is the intuitive? I mean justification for using an area metric as opposed to let me I mean, like, Why does it make more physical, then
Johanna Borissova: the intuition that I've been trying to give you is comes from spin phones in a semi-classical regime. So first of all, the fact that we know that the allergy action appears in the semi-classical regime, and putting this on a laptop, taking the last continuum limit. It gives
Johanna Borissova: a continuum effective action for on aerometric. So this somehow shows that iometrics do appear in a semi classical limit of spin forms.
Johanna Borissova: Now, this is a microscopic microscopic example. So this air metric was associated with the hyper cube, but also there are hints that and metrics appear in spin forms already at a very microscopic level. And this was what I was trying to show in this slide. So you can associate an area metric to the degrees of freedom, of a semi, classical, simplex.
Johanna Borissova: and and a much more, much more straightforward way to say, I would say that every variables are are the fundamental variables. And so it might be interesting to look at in a metrics already from this perspective. This perspective.
Ivan Agullo: Thanks. Thanks so much
Ivan Agullo: that that that that was where? CPU for question. That that's fine and more questions.
Ivan Agullo: No problem. Oh, yeah, go ahead. So
Ivan Agullo: So I see that the let's see the first correction in the curvature that you have is that a while square that you can find to be because free? If there is this equal equality on the mass that you were showing.
Ivan Agullo: Yeah, but normally, I guess you would expect high order corrections if you, if you go beyond in both equations.
Ivan Agullo: so is that the case?
Ivan Agullo: Would you expect them?
Johanna Borissova: Yeah. So first of all, we have not gone to a higher corrections. I think the what is interesting about is that
Johanna Borissova: formally, you would expect also additional operators coming with a similar kinetic term. But that's not the case. We only get the Y squared part, and the reason for this comes really from the decomposition of the error metric in terms of irreducible components. So the length metric is contained in the traces of this area metric.
Johanna Borissova: There's so via parts that i'm a twice but a scalar degrees of freedom, and therefore we only get divide to my part.
Ivan Agullo: Okay. So I mean that my question is, if you expect, or in the case, you expect higher the corrections. Would you expect that this both screen is going to to be able to be realized somehow, because I would. I would expect that at some point
Ivan Agullo: higher the correction will be reinforced them. Right?
Johanna Borissova: Yeah. So actually, here in this case, first of all, this coefficient of one fault is important, and also setting the masses equal is important. This is what gives us ghost freedom here. It's very non-trivial. I'm not able to say if going beyond this approximation scheme we were to recover or we would stay in goes free.
Johanna Borissova: But yeah.
Ivan Agullo: you want to add something.
Ivan Agullo: Yeah, I can raise the hand for a second.
Bianca Dittrich: So so sorry. Yeah, I think you have provided a a good explanation from this, from the structure of the complications at this level one would not necessarily expect higher order, curvature corrections.
Bianca Dittrich: but it's it's basically only catching the effect of having an area. Matrix. So then, it's important to do it to nonlinear order. That would be interesting to see.
Ivan Agullo: Thank you. What's your
Western: nice presentation? Very clear. I have 2 questions which are maybe the same question related, and the on the basics on the fundamental here. The second is a key one. But first, maybe the first I have to clarify what I mean with a second. So
Western: the first question is
Western: you You'll be talking as as it was the same thing. low low energy and the classical limit.
Western: If If I separate this, so
Western: I take each part to 0.
Western: I I I can still look at the result in theory with high energy or no energy. I mean it might be wrong at high energy, physically wrong, but it's a so I just wanted to clear. Be clear here. When you talk about
Western: going away from from the low energy you mean going to quantum theory, right? The quantum effects is that do I understand correctly.
Western: and and the second question which is related to that
Western: it it has to do with the with, the overall logic of all this story do do we have another example
Western: in which the sort of the the effective.
Western: an effective approximation to the quantum theory has more degrees of freedom than the classical theory.
Western: So do we have another? Do you have in mind that at a model typical something simple, in which, going to the quantum theory, you have more degrees of freedom than the classical theory
Western: that would that would help me understand that the overall logic of this story?
Johanna Borissova: Yeah. So to answer your first question.
Johanna Borissova: the split between low and and and high energy degrees of freedom, as really just saying that okay, semi, classically, we can parameterize this configuration space in terms of area metrics.
Johanna Borissova: But this area metrics have additional degrees of freedom which are an online metric. These degrees of freedom are suppressed by a mass given by the plank mass. So in this sense we have to explain why why Don't, we see these scalars at low energies, and the the reason for this is because these are quantum, they are really suppressed by the plank mass.
Johanna Borissova: So definitely, yeah, the the correction or the the influence they have is contain, and these terms here and we don't see them at low energies.
Western: So this is the of freedom which are
Western: not existing the classical limit, so it's good. I got it so now, now, to the second question, do we do? We have? Can Can you give me
Western: an example in which, whatever physics you want You've You've We have this phenomenon that the decrease of we do so to semi-classical. We want to use the freedom that disappear in the classical limit.
Johanna Borissova: I would say this is very similar to the thinking of effective, where at low energies you integrate all all massive fields, and you end up with only the degrees of freedom that are relevant, and this region
Western: it's. It's similar to what
Johanna Borissova: similar to the effect of you. I think this is also the reason why I find this remarkable is because it shows that I could use also compatible it. Effect of you to you, Don't. You don't have more.
Western: The classical
Western: it's
Western: has all the degrees of freedom. About arbitrary energy is that
Western: understand that the dynamics is modified by integrating
Western: the high-frequency modes changes the dynamic. But doesn't change the number of the the kind of degrees of freedom
Lee Smolin: Who not your Hannah, might refer to higher derivatives, which you might count as more degrees of people.
Lee Smolin: But you also could Just look at Qcd.
Lee Smolin: And Qcd. Is meant to replace theories with have no local degrees of freedom by to local degrees of freedom.
Lee Smolin: That's exactly the point of.
Bianca Dittrich: But yeah, Carl, I think we can easily come up with this. Such examples, because, as your Hannah explained.
Bianca Dittrich: it comes from example, it's there's a second class constraints are anomalous. so we can construct such easy. For example, it's for it.
Western: Okay.
Bianca Dittrich: that that will be a health care aside what's going on here? Because it's that that comes from the phones a normally, which which is
Western: but but that's specifically for
Western: so that are there. So you say there are. There are simple examples in this which happen in which the
Bianca Dittrich: yeah, we we can just make up examples where they are. The you have classically constraints, but the quantum on quantum mechanically. Do not compute.
Bianca Dittrich: I mean, you can just take the same structure, but do it quantum mechanically or
Western: okay. I just would like to see them that would help me understand the logic of the creation of degrees of freedom quantum mechanically.
Bianca Dittrich: But so yeah, I mean ex and and my, there's article. There. There are articles on second class constraint quantizations.
Bianca Dittrich: This the J. On loose. So it's. Basically Take any. Take any example where you you have
Bianca Dittrich: second class constraints. but you'll post weekly, and you will get the sympathy effect.
Western: Okay.
Ivan Agullo: but we can. We can think of if I cannot explain something to the discussion. So, first of all.
Ivan Agullo: I might be wrong, but I wouldn't understand these as we go into the classic of limits and we degrees of freedom. It's really going beyond some energy scale when you just don't see them providing any more, or in the same sense. And then you can see the W. Poses at low energies, and you just see a conference 30 interaction.
Ivan Agullo: So, as I was saying, it's very.
Ivan Agullo: and then an example of theory with would have let's say more like this of 51. There's higher energies. You can think of it as general if you think of into the quantum corrections. This was done by and
Ivan Agullo: and maybe more people in the nineties. And then you've been an effective theory with higher order. Curvature terms. You'll propagate more degrees of freedom above the plan Mass. But of course you don't trust the C. Or you on the up of the plan. Mass. Right? So it's known that the theory probably goes. But if you go below this mass they don't show, and to me this is very similar to what's happening here.
Ivan Agullo: And again I don't see that this is going to a traffic, but just go into some
Ivan Agullo: energy scale, which is way below the masks of the in the use of frequency. They don't properly. But of course, if the theory is not to be complete, then you will never be able to see the effect of the provocation of these years of freedom, because whenever they will start promoting the theory, it breaks down
Ivan Agullo: right. But it is the way I I understood this, and I don't know if if that was clear for everybody, or or maybe I just don't get it as
Western: but yeah, but that's that's why I asked the first question before.
Western: Of course we we used to case in classical physics. Well, you you the the some degrees of freedom which sort of a frozen
Western: at the high energy and and the low low energy, and then and I mean, if you think of Fermi theory, I mean, of course, the the theory doesn't have the the gauge of freedom. But these are 2 classical theories, the one with the in which for me it's interact directly.
Western: and the one with the with the young meals theory. These are not created by the quantization. It's just 2 different theories.
Western: So my my my first question was to understand which one which for a dig My, is it? It's here. So you're giving me the opposite answer to the one Johanna gave me a moment ago.
Ivan Agullo: I will not say that the furniture is not. It's a classical figure. You can. You can't. You can't get some corrections. They will break down at the at the but below that you can Con. Give them to corrections to the, to the classic election, and they will they will.
Bianca Dittrich: I think. I think the answer is indeed, in general.
Bianca Dittrich: what we use with an effect to it, is always much more degrees of freedom at high energies. Send it low energies.
Bianca Dittrich: In this case, specifically, it's also related to to the this effect which we discuss here is specifically related to the normally.
Bianca Dittrich: but in fact, the result of the calculation.
Bianca Dittrich: And that is basically it doesn't? I don't think that happens always
Bianca Dittrich: is that a structure of the anomaly in the dynamics as such.
Bianca Dittrich: set this
Bianca Dittrich: extension of of this configuration space.
Bianca Dittrich: The effects of that is suppressed at low energies.
Western: Yeah.
Bianca Dittrich: Okay. So so this is kind of a non-trivial. I
Bianca Dittrich: I don't think any a Normally, you can buy it. And then we have.
Bianca Dittrich: They have this, this, this exciting effect. So it's actually a nice design for us.
Bianca Dittrich: That's a that's a that's the effect of this extension is suppressed at high energies.
Bianca Dittrich: because otherwise we would be in trouble.
Western: Okay, thank you
Ivan Agullo: any further in comment or in question, or your Hannah
Deepak Vaid: Hi, I.
Deepak Vaid: And the question.
Deepak Vaid: So one question was just wanted to ask Johanna if if you might
Deepak Vaid: have any more details on how these area metrics appeared in string theory.
Deepak Vaid: and the the second one is, I guess, for Bianca, because she she mentioned the anomaly, and I miss the anomaly parts so
Deepak Vaid: that could be shown again.
Johanna Borissova: Yeah. So on the string theory, part of not many more information. It's just the idea that it can encode the fundamental fees of string theory into a single field also in an effective sense.
Johanna Borissova: But actually, I think again, I would like to come back to this. The composition of the error metric in terms of irreducible components. And it turns out then that the error metric is then very similar to electrodynamics. One can then do the jump
Johanna Borissova: to other few theories. So yeah, i'm not sure about the this string theory component.
Johanna Borissova: But yeah, what I would like to. Yeah, maybe if you're interested in the elect electrodynamics part, and the error Metrics is very similar to the constitutive tensor, which also appears in lightcom propagation.
Johanna Borissova: and in this case it is just a subset of this constitutive tensor. It's the principal part. And we actually this might also be a way to go if you want to construct aometric convertible connections with this.
Johanna Borissova: Oh.
Bianca Dittrich: yeah, you and I mentioned also on the holography. There you have a matrix appearing in 4 dimensions, very naturally.
Bianca Dittrich: and then 10 them into the construction.
Bianca Dittrich: So it's it's it's it's a string. So it propagates. Well.
Bianca Dittrich: you you get an area from a string by.
Bianca Dittrich: and in any case you asked about the normally the the what we call in a. Normally, it's just a very, very own
Bianca Dittrich: non computativity of of part of the simplicity constraints
Bianca Dittrich: which which is said.
Bianca Dittrich: the B fields are replaced by
Bianca Dittrich: angular momentum operators.
Deepak Vaid: Oh, okay. And if you compute out of the so called non diagnosis, if you come, consider as a commutator of non diagonal simplicity constraints.
Bianca Dittrich: This holds term. You You get some a a term which is proportion to to gamma.
Bianca Dittrich: and, for instance, depending on which version. You choose to the volume of the
Deepak Vaid: Okay. All right. So it's it's because of the non-committee to theop and Zoom Meeting
Deepak Vaid: eventually, you know. Okay.
Deepak Vaid: alright, thanks.
Ivan Agullo: Okay, less than a Johann again.
Johanna Borissova: Thank you.
Ivan Agullo: Alright, yeah, Thank you.
Johanna Borissova: Okay.