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Jorge Pullin: Okay so speaker to this mother and mother engine will speak about space time covariance on propagation and canonical Congress
madhavan: Okay hello everybody in this very strange surreal time I, at least, I find it comforting to come back to something
madhavan: Is is is shared between all of us, so welcome. This is the title of my talk space time covariance and propagation canonical new quantum gravity and
madhavan: Basically, this talk is to give a kind of broad overview of the work I've been involved in over the last many years, either in collaboration, or on my own and all this work concerns.
madhavan: The open problem of the quantum dynamics for quantum gravity, namely trying to fix the various ambiguities, which are present. When one tries to define the Hamiltonian constraint operator.
madhavan: So this is the rough plan of my
madhavan: First just review what the basic problem is with the quantum Hamiltonian constraint.
madhavan: Is that you have infinitely many choices in its construction and therefore the strategy, which I have been trying to follow is to constrain the choices by imposing
madhavan: non trivial physical requirements on the quantum dynamics, which results from these choices and the two requirements, I said, is
madhavan: One as one is that of space time covariance and the other is that of propagation softer basically reviewing what problems are and why they occur. I will then
madhavan: Translate to the second part of
madhavan: Space Time covariance and by spacetime covariance. I mean,
madhavan: The existence of an anomaly free representation of the quantum constructive.
madhavan: I will start with some general remark about space and coherence in energy, and then I'll translate to a particular setting.
madhavan: Of a model which I call the yuan Q model. And this is a model which is obtained by replacing the tribe rotation as you to grow.
madhavan: Lydian gravity by three copies of you one and this model can also in by a novel Newton's constant going to see on demand of Ukrainian gravity, as was shown by me a long time ago.
madhavan: For our purposes. This is a very useful to our model in order to study these issues for FaceTime parents and propagation
madhavan: And then pending on time available, I will go to the third part which is an account of propagation. Again, I'll first make some general remarks and then to illustrate things are in the context of the you want to model.
madhavan: So let me start before going to the Hamiltonian construct. Let me with at least what my graph general viewpoint is and
madhavan: I think each of us has a slightly different point towards what Luke quantum gravity is and maybe we share most of the things which I'm going to say except perhaps for the last, last point, the very last sentence of the sponsor. Let me say what my viewpoint is my viewpoint is that
madhavan: He is a non productive generally Kobe and quantization of general relativity, the quantum quantum kinematics has very well understood and an understanding of the quantum dynamics is still in progress. The three properties of flow quantum gravity.
madhavan: Our first that the representation of the tribe operators picture of districts patient geometry with area contest with a not smallest nonzero eigenvalue, which is approximately the blank area.
madhavan: Second do to spatial data from autism invariance, as we'll see in the next slide.
madhavan: The local connection operator simply does not exist, and only exponential functions of connections exist as operators.
madhavan: And so the deep ultraviolet degrees of freedom are therefore not local connection fields, but there are some new strange degrees of freedom, which are discreet and non local graphical excitations really quantum dynamics dynamics of these degrees of freedom.
madhavan: So, to your point is that the fundamental theory, some degree and we approving it through continuum tools and canonical energy
madhavan: And at some stage one will have to jump and confirm the discussion is on its own terms, I feel that there is still a lot to learn from the continuum structures before make this jump. So that is the viewpoint, which will pervade what I'm trying to do. Let me go to the next slide.
madhavan: So let me now come to the problem of the Hamiltonian constraint, the quantum dynamics of look quantum gravity in its canonical form is driven by the Hamiltonian constraint operator and the following problem arises in its construction.
madhavan: The classical Hamiltonian constraint depends on local feels like the curvature of the connection.
madhavan: But of course the basic content operators are non local economies and one would like to write down an operator version of this constraint. And so, one would like to write down an operator version of the curvature
madhavan: So classically, there is no problem which can be extracted from the whole army of coordinate size delta Lou to a shrinking limit. So the limit
madhavan: as delta goes to zero or minus one divided by the coordinate area of which causes delta squared.
madhavan: So this limited exists in the classical theory and there's no problem.
madhavan: But if one wants to replace these classic quantities like the whole on me by operators and quantum mechanically this limit does not exist.
madhavan: And it does not exist for a very good reason it does not exist because our Hilbert space is background independent
madhavan: And the human space norm cannot distinguish between the smaller and the still smaller still smaller loops which occur in this process. So this operator limit does not exist and one proceeds therefore as follows.
madhavan: One first replaces all the local connection dependent fields in the Hamiltonian constrained by alanna means of small loops of coordinate size del
madhavan: Some of expression to the Hamiltonian constrained which agrees with it as we take delta going to see no
madhavan: One looks at non zero delta one replaces the
madhavan: Enemies and trials and this approximate each delta of n by the corresponding operators and one gets an operator valued approximate age hacked delta. And when is the laps.
madhavan: And one then attempts to take the Delta going to zero limit of this operator approximately in some sense. And the hope is that even though individual
madhavan: Bits and pieces which make up this operator do not have delta going to zero limits the conglomeration of approximately
madhavan: Together, which make up this approximate operator to the Hamiltonian in this conglomeration actually has a limit as delta going to zero. So, that is the hope.
madhavan: And toes. See whether such a limit exists or how such a limit exists, it is useful to do the following exercise. So I will go now to the next slide.
madhavan: So I call this exercise as counting overall factors overall explicit factors of delta in the approximate
madhavan: Age delta. And so in order to do this, of course, the Hamiltonian constraint itself is an integral. And when one writes down an approximation then typically this is what happens DQ decks, which is the coordinate measure is supposed as a factor of delta Q, the electric field.
madhavan: Can be replaced by Fox operator, which has a well defined action on the kinematics but space divided by delta square
madhavan: Square root of Q operator by volume of a small region divided by delta q its volume and the curvature as a horror on me minus one divided by the coordinate area of the law, delta square. So, when one puts all these things together.
madhavan: One can find out what the overall factor of delta is one just cancel various factors of delta
madhavan: And what one finds for the density wait one constraint is that there is no overall factor at all. And what you are left with our
madhavan: What I call finite operators, they may be parameters by some loops of size delta or some surfaces of sight of size delta squared, etc.
madhavan: But they all have a well defined action on the kinematics space and one can then try and deal with this. The operator, which has no explicit factors of delta and
madhavan: As Thomas showed a long, long time ago, one can show that in a suitable sense the limit of this operator does exist in a certain topology.
madhavan: And as you rip and model showed the limit of your of the operator can also be shown to exist, almost suitable vector space, which they call the habitat. In any case, there is a certain
madhavan: There are ways in which one can take the letter going to zero limit of these are my office and get some action.
madhavan: Unfortunately, the operator action which one gets them continuum limit depends really on the infinite choice autonomy approximates do I take
madhavan: Triangular loops or circular groups or do I take some groups which entangle with the spin networks take their acting upon in some particular way each choice one, one makes them gives you a different choice of continuum limit action.
madhavan: So the problem is that the action of the Hamiltonian constraint is infinitely ambiguous.
madhavan: And therefore, the idea of strategy, which one would like to follow is to try to constrain these choices by extremely non
madhavan: Physical requirements. The two requirements. I will be talking about that of space time covariance and of propagation. Some now transiting to the second part of my talk, which is that on quantum space time covariance. And I will start with some general remarks.
madhavan: So classical space time covariance is encoded in the characteristic form of the constraint algebra. This was shown a long time ago in classic PayPal, by which man who cash and try to avoid
madhavan: And therefore, given this this particular form of the constraint algebra, which encodes spacetime covariance in the classical theory.
madhavan: One x as a definition of quantum space time covariance its implementation in quantum theory without anomalies. So one would in our words.
madhavan: One has various pause on brackets and one would like to implement them in some technical sense as committed as in the quantum theory.
madhavan: Comes non triple points on bracket, which is the one I will pick for the purposes of this talk is that between the Hamiltonian constraints. And that, of course, gives a different monetarism constrain
madhavan: Which is smeared with the shift which depends on the lapses but also on the metric, which is a dynamic variable.
madhavan: So their structure functions in the algebra, and that is what makes this very complicated in the stock. I will call
madhavan: The Hamiltonian constraint commentators, the left hand side and this different more ism, which is dependent on the metric as the right hand side.
madhavan: So in order to implement this FaceTime forbearance and quantum theory, I would like to
madhavan: Look for a choice or a class of choices of the Hamiltonian constraint operator which yields and normally free commentators where the left hand side operator committed to is equal to Irish bar times the right hand side operator.
madhavan: Okay, so let me then go to them.
madhavan: And let's see what happens with density with one constraints, because these are the constraints which was shown to admit nice can continue on limits. So let me look at my hand side and do this counting of powers of delta. So,
madhavan: One has DQ X over here than one as the laps typical lap combination and then one has these face face functions. So one is delta q from the two decks and then from the two
madhavan: Electric fields. One has a flux over delta square hole square from the Q A delta Q by volume hold squared. Then from the ear of flux over delta squared and then from the curvature Ilana me minus one.
madhavan: So if you put all the factors of delta together, what you find is that you get an overall factor of delta times a finite operator. And so, more or less, no matter how you regulate the right hand side as delta goes to zero, you are guaranteed to get a zero answer and this
madhavan: Simplified in in a calculation, a long time ago by Jorge Rodolfo Don and Jurek and indeed the writer is equal to zero. Therefore, if one wants to avoid anomalies. The left hand side must also manage. However, it turns out, as shown in
madhavan: In in dawn and and eurex habitat paper. This can manage for many different actions of the Hamiltonian constraint and
madhavan: The question is how can we discriminate between these various different actions. And as a side note, I would like to also say that
madhavan: The left hand side as is in Thomas's constraint, maybe it is vanishing due to the wrong reasons. So for example, if the second Hamiltonian constraint does not add on the spin network deformations created by the first time tuning in space. So let me say that again.
madhavan: In more detail.
madhavan: That the Hamiltonian constraint, because of the factors of the determinant of metric x only at vertices of spin network states.
madhavan: So let's say we act by the first Hamiltonian constraint default to spin at some particular vertex of the sprint network.
madhavan: And then the second Hamiltonian constraint x on it. So the first Labs is evaluated at the initial vertex we and the second Hamiltonian constraint now has to act.
madhavan: No, supposing it does, it cannot act or does not act on the deformations created by the first time constraint, then the labs, which is the second lab is also evaluated at the first word x where x and the whole expression for the complicated than simply vanishes by anticipated.
madhavan: So I think that this sort of an action then of the Hamiltonian constraint hides an anomaly, because the typical term which is the M one derivative
madhavan: Should come really from the first Hamiltonian constraint labs acting at vertex evaluating at somebody.
madhavan: And when the second level to knit constraint x, I'd like it to act at some different vertex or some deformation of this vortex created by the first time constraint, so that the labs argument is v plus delta and that is kind of a discrete analog of this one. They'll enter
madhavan: In any case, the question is how can we use the constraint algebra discriminate between choices.
madhavan: And so the idea is to somehow not let the right hand side trivialize
madhavan: One way in which to do it is to use high density Hamiltonian constraints. So we can use high density Hamiltonian constraints by scaling them up by powers of square it of cues. So you take the Hamiltonian constraint density or
madhavan: Density one and you multiply by appropriate power scarab square root of three.
madhavan: If you do this you can compute the pause on bracket between two Hamiltonian constraints and you find that the metric in the right hand side is then scaled also by appropriate powers of square few
madhavan: Cents in accounting square root of Q is like the volume divided by delta cute, you see that every power of square root of Q gives you
madhavan: One over delta q power. And so if you adjust the pause appropriately. You can bring the right hand side approximate operator from triviality into non triviality
madhavan: Because you can arrange for this overall factor delta to go away. So if the idea is that if I could do this if one could get a non trivial, right hand side approximate operator.
madhavan: Then one could now try and see whether one could implement space time covariance in a non trivial manner that is the left hand side right hand side would then both be non trivial and one more time, see whether they were equal or not.
madhavan: So that is the general strategy. But recall. Then when one look at the density with one Hamiltonian constraint on the factors are delta cancel.
madhavan: So the moment one is going to reschedule by any power of square root of Q then already the Hamiltonian constraint itself will have overall factors of one over delta
madhavan: And at looks very singular in the Delta going to zero limit. So the question is, when confronted with such a problem that is a first problem which says, What can we do
madhavan: So in order to answer this question. Let me make a brief digression, and talk now about not the Hamiltonian constraint, the spatial different more physical constraint.
madhavan: So I'll go to the next, right.
madhavan: Now,
madhavan: In Luke quantum gravity, only the finite different morph isms are represented as a unit finite unit three operators on on the kinematics about space.
madhavan: And the generators, namely the different more physical constraints smeared with appropriate ship these operators are not defined.
madhavan: Kinematic in that space. And we somehow makes sense of this generator can we construct this generator can we convert the different more physical constraint operator itself by following the methods which
madhavan: Team and develop from the Hamiltonian constraint. So that's what to briefly recall for you. So let me do accounting of powers of delta for the different Marxism constraint.
madhavan: smeared with shift. And so again we have over here, dq X and shift and trial and the curvature. So we have an overall factor of
madhavan: Delta cube coming from the coordinate measure than a flux over delta square from the electric field.
madhavan: And roughly speaking, all over me minus one divided by delta score from the curvature. And so you can see that this is one over delta force and a delta cubed in the numerator. So you get a finite operator divided by delta which is a singular operator.
madhavan: Despite this, one can construct a satisfactory continuum limit. So this is what I talked about in long time ago in the in the Madrid loops conference.
madhavan: And so one secondly follows strategy.
madhavan: Tries to be more flexible with it as what what Thomas and before him, other workers in the field lay down. So what constructs approximates
madhavan: Me dump of Salamis and plexus and I just remind want to remind you that one can do this in such a way that the approximate operator acting on a swing network access follows acts as the unitary operator.
madhavan: For the final different more physical generated by the shift.
madhavan: And I find parameter delta. So, that is the first thing over here, minus the identity divided by delta with some factor of mine side. So this is how the operator acts on a spin network state.
madhavan: And of course it's in complete accordance with accounting of delta. This is a single operator. These two operators are finite or it isn't can mannequin but space.
madhavan: And therefore, of course, there is no continuum limit on the kinematic space. Nevertheless, this of course admits continually limit on the Lewandowski Mirage habitat. So let me just give you a flavor of how that goes.
madhavan: The habitat states are labeled by default prison.
madhavan: Transfers in network, then what was causing the different class of this spin network. Can you hear me still.
Jorge Pullin: Yes, you brought up briefly but
madhavan: OK. So, did you hear me when I talked about the different Marxism operator being represented as you minus one by delta
madhavan: Yes.
madhavan: Okay, so this object doesn't admit a continuum limit or on the cinematical but space, but it does admit one on the Lewandowski morale habitat. So the habitat state is labeled by a different, more efficient processing networks and what they call a vertex mode function.
madhavan: So given a different, more efficient class of networks. One is embedded characteristic is the vertices of the sprint network. So let it have n burgesses so every member of this classes in bonuses.
madhavan: And then this is the state, which is a state in the algebra dual, namely, this is a state which is a
madhavan: Linear function on the finite span of spin network states to the complexes and it can be formally expanded in terms of these bra.
madhavan: Spin networks. So the, the coefficient of the bra S is them. The is obtained from this vertex mode function. So the word smooth function is a function smooth.
madhavan: And which is a function on MPs of the spatial manifold into the complexes and we simply evaluate this function at the end vertices of this network state. And then we some over all the elements of this different Marxism class and we get this algebraic state.
madhavan: And then it what one can show quite straightforwardly is that when you act, find the dual action through
madhavan: This expression of the approximate what one gets is a new habitat state which is labeled by a new function g of delta
madhavan: Which is obtained by taking to evaluations of this of the Vertex mode function one on slightly move vortices of the spin network state simply they are moved by inverse of this defeat modernism.
madhavan: FINAL FEW MORE system generated by the shift that is this toma year
madhavan: Minus the evaluation at this original spin network vortices. So this is quite easy to see because this guy over here moves the spin network state over here and in exactly this manner. This is what the result is. This is the minus one contribution.
madhavan: And if you take the Delta going to zero limit this give some sort of a derivative of the
madhavan: Vertex mode function over here with the shift over here. So, that is also not difficult to see the main point is that the Delta actually gives you a derivative and that's completely well defined.
madhavan: And in the limit delta going to zero, you get a new Vertex mode function, which has given us
So let me make some
madhavan: Yes.
Abhay Ashtekar: Just, just a quick look.
Abhay Ashtekar: At just to finish. Also, just to state which is level by this House, saying that as a result of this operation, you get a new column state which will be labeled by G. Is that correct,
madhavan: That's correct.
Abhay Ashtekar: Okay, so
Abhay Ashtekar: Is that for the operation.
madhavan: Yes, yes. So I'm not hearing you completely. Well, but I could make out what you were saying, I guess I'm not able to access the controls. Okay, so in future trend when when you are talking of it.
madhavan: But yes, that is correct. Okay. You
Will get a very defined
madhavan: Okay, so let me make the following comments when
madhavan: I'm sorry, the second time using zoom, so please bear with me. I am not yet really comfortable with it.
madhavan: He saw the following okay I make sense now. So, one can show
madhavan: As is not difficult to guess from the form, which I wrote down that this continuum operator action provides a representation of the constraint algebra is really algebra, the different prism group because what one got on the previous slide was very much like
madhavan: A
madhavan: Lead racking up this early derivative of this function in each of its arguments. So if you compute the commentator two different offices, you will get the right the algebra representation
madhavan: But not triviality of this representation clearly rests on the cinematically singular nature of the constraint operator, which has a limit on a different
madhavan: Space, not the kinematics but space, but the Lewandowski model habitat. So I'll just go back over here. And what I want to save this delta was not there. All then you would simply get zero in the live data went to zero. So you just get a trivial, you get the action of the
madhavan: Constraint of the defeat modernism constraint operator to just be zero. And of course, then the the the algebra would also completely be trivialized. So this delta is extremely important in getting everything to be non zero. And again, when one looks at
madhavan: The Algebra one one takes two streams. Again, it's important that one gets second derivatives and all this has to do with this factor of delta in each of the most operators, so this this niche is important to get something on preview.
madhavan: Secondly, in order to get the different offices minus one over delta action of this operator.
madhavan: To choose the economy flux approximates in such a way that they are a tune structure of the sprint network state. So not only are they are tuned to structure the graph on the line display network, but also to the label of
madhavan: The momentum labels. So this is actually this is a statement that that is what
madhavan: They should I mean expected in hindsight, because, for example, supposing one wants to move G H along the orbits of the shift we somehow need to cancel the enemy in the general presentation.
madhavan: And then create a new law know me with this displaced also in the JIRA presentation.
madhavan: And it turns out there for that need to know where the sages line so that you want to cancel it and you also need to know its representation so that you produce a new edge which exactly has the same representation able J.
madhavan: So the switch you use our Taylor, who bought the spin and the graph labels.
madhavan: So it's not a constant spinoff representation of anonymous or justice think David J. Fixed representation better tailor it to stay. You're acting upon
madhavan: And the point I want to make is that it would have been impossible for to guess the correct approximates if we did not understand what the classical constraint gender gets
madhavan: You pretty a wall. We do the curvature pronouncements depend
madhavan: On the flux is as well.
madhavan: So we really understand union vector field of the classical function well before trying to convert the operator into quantum and so I'm going to try and use these lessons for the Hamiltonian constraint. So if there any questions. I can take them. Now at this point.
madhavan: Okay, so let me proceed, since there are no no questions. Yeah.
madhavan: So let me then outline based on this a strategy for time to implement the constraint as an operator.
madhavan: And trying to get a non trivial commentator, which has the right properties know quantum gravity.
madhavan: So first I want to get an overall factor of one over delta. This is just mimicking what happens for the different Martin and spring and therefore I will scale.
madhavan: Density wait one and unconstrained density by Square to to the one by three, which gives me to the one but over delta nine gives me the overall pattern one over there.
madhavan: The strategy. I'd like to follow is to look for approximates to the Hamiltonian constraint again since there are factors or volume. If the Hamiltonian constraint Brockman
madhavan: Follow the same choice or three monitors and it will pick out only vertices, the skin network state. So there'll be some overt is this will be the evaluation of the labs.
madhavan: And then what I'd like to ask this expression is into a form where there is some finite operator I you which deforms the vertex we on which is x
madhavan: deforms it in some neighborhood delta have quiet neighbors delta of this vortex minus one divided by delta. And then there are some more efficient which level kinds of informations are involved. So that is the structure. I want to make the Hamiltonian constraint have
madhavan: The structure, I would like to find defined the Hamiltonian constraint approximate on a suitable space of option states.
madhavan: So an option state will be again like Lewandowski models habitat state, so roughly very roughly speaking apps label by a function f.
madhavan: On suitable copy or one or sorry unsuitable copy and sigma and we will refine this much more as we come to the you want to theory. But basically, again, some grass with some efficiency which depend on suitable evaluations of this vertex moves function.
madhavan: And
madhavan: These what I will then do is I will evaluate the dual action of this constraint operator and schematically, what I would like to happen is that I would like
madhavan: fumbling combination and you had minus one over delta to yield a contribution to this complex number which is up the form of the evaluation of the laps at the vertex at which it's
madhavan: The Hamiltonian constraint is acting
madhavan: Up would like the function to be evaluated at roughly speaking, a displaced works which is obtained by the action of you had on s displacing vertex
madhavan: Minus the one we just gives me the evaluation of the function at the vertex itself by by death, and some of them take that are going zero
madhavan: Which would get given no function and then St. Louis by and derivative of f. So this is just schematics will see sort of how
madhavan: I will not go to implement it such a nice, but this is kind of what I want to happen and that spiritual lesson I think will be implemented and what I will do in in detail in the you want to carry
madhavan: I would like to happen when I get the second Hamiltonian constraint acting is similar. I will just get an M derivative of this, and the left when I take the commentator, then I will get this nice and minus and and and turn which what I and it's essential, therefore,
madhavan: That, again, the second Hamiltonian constraint actually act on deformations created by the first one in this, although I don't want to right over here, but that is what must happen but this this this thing to actually have this form.
madhavan: So that is the rust on how to get the left hand side which is non trivial, let me now go to the right hand side.
madhavan: Yes.
madhavan: You
Abhay Ashtekar: Like that.
Abhay Ashtekar: I'm
madhavan: Not able to get my
Sorry.
madhavan: Not able to access
Zoom controls.
madhavan: Now I'm trying to mute it or not able to mute my
madhavan: Okay, maybe I will not. I'm not able to mute it away, but
Abhay Ashtekar: Please. Please continue. Oh yeah, it is.
It.
madhavan: OK, one moment of I'm just want
Abhay Ashtekar: You don't need anything just
This
Abhay Ashtekar: So,
Abhay Ashtekar: I just wanted you to explain
Abhay Ashtekar: That you have law.
Abhay Ashtekar: Mesa know V. V.
Abhay Ashtekar: I think that is you deformation minus that we have
Abhay Ashtekar: Visions I understand, particularly, there is a index he paid for me. Delta. What does he
Abhay Ashtekar: That
madhavan: Yeah. Okay so that is over here, right.
Abhay Ashtekar: Yeah. What a stand for before we go.
madhavan: So these coefficients.
madhavan: Would be made up of various labels of of this network state which. So this is a complex coefficient and it would just wait. Each of these operator deformations by some numerical coefficients and perhaps since which depend on the jays etc.
madhavan: So I do not
madhavan: Put into the you have because I'm on the you had minus one form over here.
Abhay Ashtekar: The Windows is arbitrary is complete.
Abhay Ashtekar: Freedom in
madhavan: Terms of a
madhavan: Understand. Understand, so the point is this is a general strategy, I would like to bring the hammer constraint approximately into this form. And when I try to do so, of course, these will be completely fixed by the particular approximates I'm choosing
madhavan: In order to implement and make this phone. So I'm to be completely determined. Once I tell you what particular proximate, I'm going to be using for
madhavan: Constraint. But what I want to be guided by is this general form. I want to be able to get a general form when I have this particular structure acting
Abhay Ashtekar: In this
Abhay Ashtekar: Okay, so
madhavan: I will go now to the
madhavan: Next slide, let me now go to the left hand side to the right hand side. This was about the left hand side.
madhavan: The right hand side approximate
madhavan: Of course side, which was just to be prison, which was mirrored on
madhavan: Independent shift.
madhavan: So it looks at that right hand side, it's the corresponding operator who is single operator would have tries delta which I would have to take to zero.
madhavan: However, if it so happens that the classical right so it could be written as opposed to bracket between objects which were some on to the formalism, then the items. Don't approximate would like the left hand side also have to amateurs delta and delta prime, which would have to be a zero.
madhavan: And
madhavan: Non zero delta delta prime, it would be much simpler to compare the left hand side and the right hand side operators and try to find ways in which to make the two again.
madhavan: So it would be nice if this could happen. And indeed, this happens do miraculous classical points on racket identity, which
madhavan: CASEY Casey Tom and I found in 2012, a long time ago and the identity is as follows.
madhavan: Remember the right hand side there to Hamiltonian constraints. So we have two lapses. So what one can do first is take apps and
madhavan: Multiplied by the densities tried he
madhavan: And divided by some factor in this case of due to the one by three, when one puts in the various density weights of the labs.
madhavan: And the, the electric field and whatever comes from here, one finds that this object behaves like an enterprise vector field.
madhavan: So there are three shifts for I call to 123 and I call these shifts electric chips because they involve this electric field dependence. If you take the electric shift and smear aphorism constraint to get an option, which I will corn d of em.
madhavan: Similarly, you could instead of n, you could use the laps em and you could get d of em i and then if I take this poison bracket.
madhavan: And I some from, I do want to three with some numerical coefficient, then this turns out to be exact. Pause on bracket identity, where the minus sign goes for the Euclidean theory and the plus sign from
madhavan: An empty one can actually choose any density wait one once and get a similar identity. The only thing which changes factor over here, except
madhavan: For the is when the Hamiltonian contained is of density way one. So this is somewhat intriguing because due to some other reasons about materiality of the right hand side, I'd argue that it might be good to look at
madhavan: Higher density wait constraints and therefore not a density wait one and so classically here what one finds density wait one this right hand side just identically vanishes. One does not have a useful identity. So that's kind of an integration.
madhavan: Okay, very good. So now let me tell you what the status of this program is what I would like to do is to him Hamiltonian constraints, so that it has a particular form.
madhavan: And then pose this identity as, as in some suitable fashion as a commentator identity, the quantum theory. And of course, I can't do it. Even for Euclidean gravity. Yet that is the direction I'm going to
madhavan: Focus. You want you model where we replace su to try rotations by you want to transformations, one finds the model as a constraint Algebra I saw more free to that of utility gravity's one has identical structure functions for them.
madhavan: And as I said, the more sense. Lee's humble Newton constant were given limited up
madhavan: So for this model, one can implement this strategy.
madhavan: One can construct continuum limits of these operators age and deep
madhavan: Suitable space and many technicalities I am not able to go into
madhavan: But in some suitable way one can define a limit of this commentator, which is non trivial and which agrees with this right hand side in Islam paper which I had in 2018
madhavan: So, before going to a few assorted details of the you want to model. Let me make some general comments and and then a prognosis of where this is all all heading and then I'll come back and talk a little bit more in detail about the you
madhavan: So let me make the following comments, then the strategy works and you want to model because the analysis of the classical Hamiltonian vector fields shows that the evolution of classical evolution of the connection.
madhavan: Can be written in terms of combinations or electric dependent mature morph isms electric field dependent gauge transom
madhavan: Since we have a very good understanding of face face independent Detroit office hours and H transformations and can use this understanding to try and write not operator correspondence of peace. Peace dependent different modes engage transformation.
madhavan: Secondly, the detail calculations which I did are on a very small so options states the states have someone's which
madhavan: Spin network summons have only one vertex with a half to train constraint. So these are single vertex distributions. Now it. It's already a long calculation, but I am quite sure, looking at how thing that it should be straightforward to extend this
madhavan: Result of family free non trivial.
madhavan: Constraint algebra to the multi vertex case as well. There are some
madhavan: Other technical issues. I can't go into them technical issues there more about the nature of particular condition choices one makes which I would have liked to debate.
madhavan: But there is no time for that. The broad picture is that yes, or I've done it for single vertex states should be straightforward to extend it to multiple states as well.
madhavan: But the main problem is my action of the constraint which I used in this long paper does not support production and propagation. I'll come to hopefully in the next part of my talk.
madhavan: And therefore, I have not been in this in more detail what I have done is that with a slight modification of the choice of constraint action nevertheless has this same you had minus one form for changes are these different coefficients which are assigned to
madhavan: And, you know, some technicalities one actually does get rigorous propagation. So, this I showed in a paper in 2019 and I will tell you what I mean by propagation
madhavan: And normally freedom for this choice remains to be shown, but over the last few months, I've been working and I think that it's quite
madhavan: Promising that one will be able to show anomaly freedom as well, in a certain sense. So that is the status. Let me go to a brief prognosis of happen in future.
madhavan: And then I'll come to the you want to see where it will become little bit technical. So I hope at least this first and broad part of top you can take away with you.
madhavan: So the next step, which I would like to do, or I think we should be done by by people is to to go to Euclidean gravity.
madhavan: And again, as I will come at the end of the top recent analysis of classical equations shows that the tenement equations can be written in terms of electric field dependent different more fizzles as an electric field depends gauge transformations
madhavan: And roughly speaking.
madhavan: All the graph related problems already arise in the you want to model and in the Euclidean case one will need to confirm, of course, all the non trivial problem from nature of a city.
madhavan: So, at present, I think the domain open issues are just some for you want you to normally free action consistently propagation. I think this should be possible.
madhavan: Then to generalization of this and I'm optimistic that progress can be made the lawrenson case however is wide open. I don't know how to do this.
madhavan: One possibility which I discussed in a more speculative paper was to try and map directly Euclidean solutions to long term solutions are using the team and complex. So far, this is
madhavan: Which was introduced by my mom beautifully long time ago. And then also advocated by eBay, to go from Florida incentive to Euclidean
madhavan: So that is the rock prognosis, let me know. Come in more detail to the you want to model. So the plan any general questions I could take number two is I will go. Do you want you model and present. Just a few details because back time
madhavan: Okay.
madhavan: So let me then go to the you want to module and give you some asserted detail. So this is to just give you a flavor of various things which go on in the you want to model.
madhavan: Pertaining to space time covariance and anomaly free
Constraint algebra.
madhavan: Okay, so let's look at the model itself. The face face variables are a triplet of you want connection is an electric fields which are economically candidate. So these API's and he is for ICO 2123
madhavan: The consumer just the kind of a billion obvious you want two versions of the astute Euclidean constraints. So you have
madhavan: Three God's law constraints which are just the opinions of constraints. Remember here, the electric field is completely gauging barrier.
madhavan: Then you have the different more physical constraint which is just a similar combination of the vector constraint and the gospel constraint.
madhavan: And then the Hamilton time constraint which is exactly the same. So it's excellent i j k where this doesn't have any is just the alternating simple, it doesn't have any structure constant interpretation, then you E. F.
madhavan: In accordance with a strategy, the same content works. So one puts in a factor of 10 to the minus one by three and the labs then becomes a density weighted object or minus one by three.
madhavan: The curvature is just the ability of curvature over here. And what I mean she was just one would take two G's. So in the initial 14 times to add and Q is just the determinant of
madhavan: You know me.
madhavan: So let me know different the key objects which will be present in whatever I talk about, namely the electric shifts. So there are three of them. And I've already said what they are and times eight times goodness one by three.
madhavan: And then one can smell the different autism constraints. So there is a summation with Jay over here. So you are E dot f. And when you implement the gospel, then
madhavan: This part just goes away. So you have the different Marxism constraint which is this number gospel constraints office.
madhavan: And then you have this identity as usual for the astute also holds for you want you can have the pause on record. So the Hamiltonian constraint is equal to this. And this is what we'd like to implement in the quantum theory.
madhavan: So let me know. Good. The quantum kinematics.
madhavan: So, one has just the you want to analog of the sprint network states, which I will call charge network states because the integer value charges which
madhavan: Label the representations of one cube there three of them to want you to acutely and you have states again label by colored rocks and just products of the edge salamis over these various edges and each edges color with these three into the new charges.
madhavan: Since it's a billion theory. These are exact I can state. So the electric clocks operators electric clocks operators just communicate with each other. There is none of the issue non quantitative at
madhavan: Engaging variance simply means that the sum of the outgoing triple of charges at every vertex vanishes.
madhavan: Let me now come to the quantum shift greater which is the key object. And what I want to do. So the quantum shift is, as I said, the labs, times the electric field operator times the inverse matrix over here.
madhavan: These are all completely electric field dependent objects even once when even if we perform the team trick. So, this turns out to be completely electric field dependent and therefore the
madhavan: Child network states are eigen states of quantum shift in one can extract an item value.
madhavan: So this is what what happens, since subject a mantra is non zero only at vertices of the quantum of the talent network, the quantum shift eigenvalues also only non zero charge network workspaces.
madhavan: So let me go in a little more detail to the next slide about the quantum shift.
madhavan: I'll give you more details. I think maybe on the slide. After this, so let me just have some quantum shift. And now let me go to the schematics of how I will use the quantum shift to define the action of the Hamiltonian constraint.
madhavan: So this is going to be heuristics just schematically how things happen and then I will tell you in detail just give you the final results because there is due to essentially due to lack of time.
madhavan: So the Hamiltonian constraint is he, he, an F, and what are done and with this due to the one by three to get like density rate. And so what I've done in red is to isolate the part which is the classical electric shift.
madhavan: Now, when I look what I want to do is I want to quote the action of the Hamiltonian constrained in terms of
madhavan: Different more isms generated by this electric shift. And for that I'm going to use an identity, which is which we see all the time. For example, in the t plus one, the composition
madhavan: Which is if you take a shift vector field and.it to the curvature of the a billion curvature. Now, then you can write it as the leader derivative of the connection minus
madhavan: This object kill
madhavan: This identity also works in the IBM theory with the electrician. So instead of this chip. You just placing by the Electric Sheep over here.
madhavan: And the Hamiltonian constrain them looks like when looks like this lead derivative
madhavan: Coming from here and then this total derivative, one can do an integration by parts. And this of delve deep comes and hits the Eb over here. So you get something proportion to the gospel on those on the gospel surface where the gospel song I can forget about this town.
madhavan: So let me now go to the heuristics for the quantum theory. So the quantum theory with reasonable choice of operator ordering
madhavan: What happens is that we get the he over here to Atlanta directional derivative with some type items each each bar Richard one then one gets this Absalom i j k l m one gets a lead derivative of the connection with respect to the shift.
madhavan: So let me write it. Now, if I'm going to write it down approximate at finite data.
madhavan: Then what I would try to do is to write this lead derivative in terms of finite different more fun. So what we what the first step is to basically take this
madhavan: Wave function side of the connection. And what this tells you to do is to evaluate it at a displaced argument. So you have a j over here.
madhavan: Plus this displaced argument minus i at AJ. So this is where you get the u minus one structure. This is the scene of the yo minus one structure.
madhavan: And the displacement looks almost like a different than generated by the quantum shift it would have been from autism, except for this epsilon i j k. So, for example,
madhavan: If it was exactly a different office on then I supposing I looked at the G equal to one component over here, I would add an appropriately derivative also equal to one component of the connection but
madhavan: If I'm going to take the one component here. Then I'm going to add the lead derivative of the second component of the electric ship.
madhavan: Of the third component of the connection, minus the derivative with respect to the third component of the electric shock of the second component of the connection. Take that multiplied by delta and then add it over here.
madhavan: So it is kind of some twisting of internal is also going on over here. So that's what the heuristic structure looks like, but we also have to confront the fact that, like in the classical
madhavan: Where the electric shift is adopted in the quantum theory, the electric shift is no longer a smooth object.
madhavan: And then the you want to carry one can really see what is happening because we know that the elect the electric flux or the charge network states.
madhavan: They can be interpreted really visually as want have electric trucks along the edges. So these are these contests electric lines of force as Asian and therefore you can see at the word is that you don't even have
madhavan: The electric field at all, something very similar and one will have to make sense of this object. So the quantum theory, but the main message of this few logistics is that, oh, the Hamiltonian constraint at five o'clock delta
madhavan: See
madhavan: The charge network see it at each of its vertices and some combination of singular defeat more physical lack of any other words and
madhavan: There is some
madhavan: Some start flipping in the neighborhood of
madhavan: Copy.
madhavan: So let me go a little
madhavan: Quantum shift itself.
madhavan: So the point
madhavan: Is this
madhavan: On
madhavan: And because of operator here. It's not the reporter says, and
madhavan: One can easily see that the
madhavan: Contributing factors because it has a
madhavan: Planet Index it contributes actor or national the tangent at the word XV.
madhavan: So the final result for this item is as follows. Do
madhavan: I value contribution from took you to the one by three operator and then the electric field gives you, because it's all
madhavan: Coming from trucks in the flux eigenvalue Q over
madhavan: That is where the index i comes comes from this is a
madhavan: Be more and then you better Long the
madhavan: Evaluation.
madhavan: Enough to actually do this.
madhavan: Because laps in
madhavan: Dependent
madhavan: Waited all j, the density rate minus one by three objects. I have to choose a regulating coordinate patch around vertex charge network at Valley doing eigenvalue.
madhavan: Isn't that coordinate patch that this lapses evaluate and it is
madhavan: A look at you for
madhavan: Vectors on edge and then as some of what the
madhavan: What the Eigen looks like the vertex me
madhavan: So what it does is it adds the sales and mechanics graph deformations alone he tangent down by
madhavan: Each one gives you permission.
madhavan: So that is
madhavan: Give you a
madhavan: Representation of what
madhavan: The final constraint operator does this is building up to that.
madhavan: One who's had city with this one just opens up a box of new technical problems which one is Paris laboriously one by one. And the first one and the most is that one of us needs this patch to regulate and to define the quantum shift. And so, this turns out to be coordinate dependent
madhavan: And I need to choose one minute.
madhavan: Of every judgment.
madhavan: And the immediate question is that, well, what happens when two different variants, because I'm not going to be worried about the pot on racket between the Hamiltonian and I'm on him, but also between me and the different offers and constraint.
madhavan: Want to be around, or is this going to be possible. I'm going to get an anomaly feed representation also have the few more physical constraint constraint bracket.
madhavan: It turns out that this is actually possible and technically complex complicated
madhavan: But no one has to me.
madhavan: Up ending this
madhavan: Isn't constrained commentators in other choices must be consistent with different more and and very roughly due to lack of time.
madhavan: This can be done by choosing patches for one charge. And it's the morphic image.
madhavan: Different systems.
madhavan: What does this and there is again a lot of freedom in doing this, there are amateurs which tell you that this freedom. Doesn't matter.
madhavan: This then it turns out your office and pool variance and then we achieved by tailoring the action of the constraint, a two properties of the option to the ball. Let's say this again.
madhavan: These various constructions of them constraint we have been seeing one Taylor's the Hamiltonian constrained direct stay active with grass structure one one looks at all the Burton isn't one Taylor's attention.
madhavan: This
madhavan: State approximate to the constraints, so that every time.
madhavan: I need things to be it is
madhavan: What I'm saying is that one also.
madhavan: Which is
madhavan: Interested in the actual state which is infinitely the states.
madhavan: And approximate really x
madhavan: This
madhavan: Has to be tailored now.
madhavan: Labels have this option dual state being acted on.
madhavan: In an appropriate way.
madhavan: Taking advantage of the different over and choice of GORDON SAYS, one can show if you're Marxism variance or
madhavan: Constraint algebra.
madhavan: So I just want to a new
madhavan: Which has been
madhavan: Now go to the next slide.
madhavan: Okay, because
madhavan: Of lack of time I'm James and questions.
madhavan: For point
madhavan: And
madhavan: I think
madhavan: The Discourses to give you what is happening and water.
madhavan: One can learn from this exercise.
madhavan: Questions to take them. Now later or talk individually or schedule as well in more detail.
madhavan: So,
madhavan: The calculations are what the number is this. So let me describe what these linear vertices are
madhavan: Because the structure of the constraint which are like to
madhavan: Of the lab, some information.
madhavan: Structure over here.
madhavan: Off shell states.
madhavan: Yes.
madhavan: Go ahead.
Jorge Pullin: Why don't you go ahead
madhavan: Go ahead.
madhavan: Okay. Can anyone hear me.
Jorge Pullin: Yeah.
madhavan: Okay, so I'm going ahead right
Jorge Pullin: Yes.
madhavan: Let me come to
madhavan: These linear says
madhavan: Sorry, as the transparencies side.
madhavan: I have the structure of the constraint.
madhavan: The Hamiltonian constraint, roughly speaking, this is it's
madhavan: How do I get this, I get this object by comparing
madhavan: All the child.
madhavan: And come bearing the
madhavan: Please for text.
madhavan: And the word the evaluation of the function at the vertex of the parent, the parent vertex and the child word exact compare the two, I take the difference in that is what goes into this derivative. Your and I have a multiplicative
madhavan: Commentator I wanted to know this can constraint. He had em on the new actual state. So again, this the approximate operator.
madhavan: State in Malta parish and of times this partial derivative of
madhavan: X of the tile and at the vertex of the parrot.
madhavan: But em over here is a lapse of data being evacuated and
madhavan: So I need to evaluate it and at the coordinate match of the child vertex
madhavan: Coordinates.
madhavan: And these because I'm
madhavan: Requiring the few
madhavan: distinct patches.
madhavan: Complete the competition have to transfer from the child coordinate attached to the parental court for have to
madhavan: Have this transformation.
madhavan: And these Jake opens appear on the left hand side and similarly frightened side, we have to electric different offers and constraints.
madhavan: And it turns out the DJ the calculation only works in a trance and the peons
madhavan: Through this derivative operator over here.
madhavan: The
madhavan: Constant Jacob eons and cons means arise.
madhavan: From linear coordinate transformations. If the quarter transmission between the child and the parent is laid out there on can get a constant Jacoby
madhavan: Is technicalities, to tell you the due to some very abstruse go reason I need an element of linearity in whatever I do. And this linearity.
madhavan: Some items on have vertices which are the property of being linear and that is what am
madhavan: I know it's a bit opaque. But what I want to tell you from this slide is that the competition involved because coordinate which is on board.
madhavan: And the competition comes for really simple and trans. If one has Jacob Ian is there has to be some element of linearity in these coordinate transformations and this in turn districts, the brass summons.
madhavan: All these off shelves dates to have vortices, which are
madhavan: All linear versus to define on the next slide.
madhavan: So let me go places and the nature of these deformations, which will happen to an end constraint actually makes, so I will say that the vertex is linear. If there is a coordinate patch in which all edges at the vertex appear as straight lines. So if you're familiar with this paper or
madhavan: Oh, this is saying that all higher per module I banish at these workspaces.
madhavan: So this is a different, more physical environment concept, the linearity of this vortex.
madhavan: And the coordinate patch patches in the exclusive or as linear, namely in which the the edges explicitly appear a straight lines I will call as Linda or not. It is not true that when a linear, what unique linear coordinate patch. There are many, many of the
madhavan: There is a way of, I mean,
madhavan: choosing one of them and evaluating the quantum shift in in in that and making a choice so that finally everything is different office and corner and I will not go into these details, I will take a linear coordinate patch for the purpose talk
madhavan: Now with this type of these coordinates.
Is the
madhavan: Shift
madhavan: As them about pulling on the vertex structure along the I knew edge.
madhavan: Shift was a some over age tangents and coordinate edge tangents in the corner. I was using
madhavan: And the an important part of the deformation generated by the Hamiltonian constraint was related to a lead derivative structure with respect to this quantum we have lead derivatives along the edges of
madhavan: Edge engines leader returns with respect to the sentence of the words and
madhavan: On the road because the quantum shift you was zero almost everywhere except at the vertex said
madhavan: One can then visualize quantum theory or the relevant different Marxism at parameter delta to be something like an abrupt all of the text structure along the edge and then some over all these
madhavan: For
madhavan: What one has
madhavan: Political conical deformation, which is well defined because of the linearity of the world as well, then the remaining n minus one edges and vertex which has an violin.
madhavan: Okay.
madhavan: On the page. And so, one gets this sort of a structure where one has these kinks now because was made an abrupt and as delta two zero.
madhavan: Approach the original work so much faster than later. So in the dental going to zero in abroad along the edge and then I will some of these deformations for each of the
madhavan: I equal to one, two images and they will also be charged flips, which I will come to in the next slide here, I'm just giving you a flavor of things, the displaced vertex has the same as the parent will bond because I'm pulling on this along
madhavan: And because I'm using Lynette
madhavan: And because cones are linear structures. It turns out that this Burton new child vertex is also linear as I needed to
madhavan: Do my point. I want to want to want to say is that the the calculation. Because of all this underlying linearity hints. Finally, at some Soto, he smiles linear discrete structure.
madhavan: This
madhavan: truce and technical really making a case of with the simplest spin form also use some
madhavan: Lynette
madhavan: Put to the next slide spawn to wrap up and give you
madhavan: Should have these look like if you're interested in much more technical details I given an earlier top which was much more detailed only to for a and Rodolfo and and that recording is available for people who are interested
madhavan: So this is what the different deformations created by the Hamiltonian constraint look like
madhavan: So be on the phone. I'm just focusing at a particular vertex. So the books, then you have deformations alongside the edge which look like this.
madhavan: And because of various charge flips etc etc finer deformation says, Jay.
madhavan: Looks like this. This is for a particular charge flip and a particular choice of edge tangent along which to pull you get a conical deformation. And it turns out that in this deformation
madhavan: This word is which you create is generically non degenerate and if a second Hamiltonian constraint x, it can act on this vortex.
madhavan: It also turns out due to the details of the flipping that this vertex. The original vertex comes degenerate in the child.
madhavan: So to what happens is that this is a delta neighborhood or the word test outside this delta enable nothing else happens, similar to what Thomas is constrained while Thomas's constraint, but only put one extraordinary edge over here, this difference the entire vertical structure.
madhavan: In a particular way, and what form giant and B form and display vertex with coordinate distance delta where the second number two on a constraint can act and provides the scene for LM and the
madhavan: Structure.
madhavan: One can have what I call this is a downward pointing Co. So in the theory, one can have downward deformed children, one cannot operate the form children as well.
madhavan: Where one extends the graph. And one of the upward pointing code where this is the phone access and it looks like there's only to give you a flavor of what these things look like.
madhavan: This is for what the Hamiltonian constraint does what the different more efficient constraint does is electric different more physical changes is very simple. It just deforms the graph only
madhavan: Have these electric shift type of deformations, with no clips at all. So you have exactly the original charges as before, but you have a deformed graph on which the child six
madhavan: But what you can see is that the electric deformations are very much related to the Hamiltonian deformation and this is what makes it easier to compare the left hand and the right hand sides, and that is why this identity which Casey and iPhone important
madhavan: So this is, again, just to give you a flavor.
madhavan: So let me now.
madhavan: So this was really to give you a flavor of the kind of deformations, which are involved and the kind of manipulations with going to finding what
madhavan: The hammer to an extent action should be the material is there's a lot of technicality. And so I will, I will just stop over here.
madhavan: And then now go to the issue of it's a bit of an abrupt stop. But basically, to give you a flavor of what happens in the US and the expectation is that a student radium theory also very similar things will happen.
madhavan: So any questions. I can take them now. Otherwise, I'll go on to the to the propagation part of my talk.
madhavan: Okay, so
madhavan: Let me then go on to the provocations.
madhavan: Over here, please I'm losing track of time, but please continue. Please let me know when when I'm running out of time.
Jorge Pullin: While you're gone for an hour and 2030 minutes so
madhavan: What, what would you, what would you like
Jorge Pullin: Anything you're not
Jorge Pullin: Too long though.
madhavan: Okay. Okay, so let me do that.
madhavan: I think I should I should be able to finish in 10 minutes
madhavan: Okay, so let me now go to the show.
madhavan: So Hamiltonian constructs constructions in LPG, as we've seen, lead to constraint actions at vertices or spin mats.
madhavan: And these actually obtain the zero continuum actions of the Hamiltonian constraint approximates
madhavan: The Hamiltonian constraint approximate is order again so that you have only vertex contributions and he action whether it isn't what I told you about, or whether it isn't Thomas's action or any other action demands the structure in a delta size neighborhood of the vertex
madhavan: Vertex is completely independent. We haven't done constraint action another vertex. So this is what is called ultra locality of the action of the Hamiltonian constraint.
madhavan: In a very influential and very beautiful paper in the night, we provided. I think the first clear articulation of what proper be in Luke quantum gravity and its potential tension with ultra locality of the action of the constraint.
madhavan: He also argued that article LPG
madhavan: One system with propagation of quantum perturbations from one vertex, or the spin it to another.
madhavan: However,
madhavan: If one looks at that paper. Unfortunately, his dictation arguments are based on almost his claims in the very fresh at that time.
madhavan: He was April, which were only available in pre printed form, there is a lot of tangled history to this which which I will go into if there are questions later.
madhavan: But the main point is that once the fork of confusion on everything is cleared. It turns out that the conclusion and the ensuing folklore that ultra locality of constraint action.
madhavan: precludes property incorrect and this showed in models in parameters field theory in 2017 and then in the US to model in 2019 and there is a loophole in these arguments with the promise.
madhavan: Which is in preparation.
madhavan: The folklore seems to be based on the following fact that the action while the action of the Hamiltonian constraint can create a spirit.
madhavan: Of spin mixtape for a small enough delta, it cannot merge tortoises. So if you have the Hamiltonian constraint that gentleness network vertex. He over here can create the mission over here.
madhavan: And then one can use a few more efficient to transfer it to the neighbor of a second vertex, but there is no way in a delta no sense that you can ever make this vertex
madhavan: This because there will always be a smaller delta where you can only move this to a small data and you know in the limit dental going to zero.
madhavan: You can't run into different more efficient class of this to this, and that really seems to be the problem with electronic collection over here, you cannot do this action and then how could you ever take this perturbation and move it beyond is the question.
madhavan: A little bit more location, the engine to notion of propagation. I've been talking about can be made only just a little bit more precise as follows. In terms of physical states.
madhavan: So any physical state lies in the kernel of constraints and we can view it as an element of the algebraic dual and
madhavan: It admits an expansion, like the option states in terms of spin network brass. There's some coefficients to some more. You get a physical state.
madhavan: Let me you use the nomenclature. We call the set of all these brands summons to be on the brass it and they can't analogues to be the cat set. So, I will say physical state has an associated get set
madhavan: And the elements of this respondents. They are someone's over here with non zero coefficients.
madhavan: And I'll say that the physical state in quotes propagation in the elements
madhavan: Gets it are related by propagation, that is there a subset of these elements which form of propagation sequence where you have a get as a perturbation. I'd say one of it, versus giving you a top stay propagation of this perturbation do a neighboring vertex of as
madhavan: Which is time and then absorption of this perturbation by overtakes we do giving you a better state than an emission of this propagation past vertex we to at cetera so in this, get set, which underlies this physical state has many, many sequences or long sequences. I will say the
madhavan: Biggest populations we
madhavan: shimmer propagation. So this notion of propagation in terms of physical states is lovely distinct from that deriving from repeated constraints landscape state.
madhavan: So therefore it is conceivable even if the latter notion, namely repeated action does not generate propagation duty ultra due to a locality propagation can still put it in physical states and I will be state based notion, actually. He also uses in his analysis.
madhavan: Of course, the operators determines there are no that is determines the structure, the physical states. So, the importance of propagation in that sense is tied to the form that constraint operators.
madhavan: Because the physical state depends on the structure that constraint operate it. Isn't this sense that demanding propagation constraints, the available choices of the Hamiltonian constraint or
madhavan: No one possible route to propagation is as follows. Let the action or the Hamiltonian conceit parents Wynette create or set of different children are they doing almost this case on my case let the structure of the cat said be such
madhavan: That if a parent is in a cat set is in the children. In other words, if you have a particular spin network state in the case
madhavan: Then if you act upon it by the Hamiltonian constraints, their children and all these children are also months in the brand some which defines them become a regional parent was in
madhavan: Similarly, if a child is in the cat set, then all possible so also indicates. So if you have a particular a headset.
madhavan: And if you have state which when acted upon by the Hamiltonian constraint gives you this particular state. I will call that state of possible
madhavan: Maybe many possible parents for your child. And all these possible parents are also the cancer. So, suppose the caps at underlying physical state has this property then it turns out that propagation can ensue to the existence of these possible parents as follows.
madhavan: These conclusions basically pertain to the formation generated by the constraints that is to all of a given pair.
madhavan: And these do not include propagation sequences. Because of the possibility of the vertex merging operation as we saw
madhavan: The get set, namely the summons also include all possible parents
madhavan: One can have possible proper one application sequences as follows. So again, you have a Hamiltonian constraint acting on a Penn State is your child over here.
madhavan: The state is Marxism and manner. So all different morphic images of this child or in the cat.
madhavan: This is where takes over here by default Morpheus. And here, and of course not merging action American strange constraint, but you can ask, is there a state as prime
madhavan: Such that when the Hamiltonian constraint act on this next. It gives me this for this particular child. If it does, then this is a portable parent this child over here.
madhavan: In other words, this child has known unique parent. It has a distinct parent year but both of them occur in the cat said, and then give you a proper sequence. So that is the basic idea. And this idea is implemented in Parliament price field theory.
madhavan: You lose some technicality.
madhavan: And in order to bypass these arguments. It turns out that the existence of these possible parents that is the existence of nominate parentage is crucial. Go back to his paper, then there's an implicit assumption based on the Q st prepayment that
madhavan: Any deformation, which is obtained by the Hanford constraint can be uniquely associated particular parental vertex and Morrison I show is that this is not true. You can have non unique parentage. And then once they are non unique parentage many arguments.
madhavan: How very quickly. It turns out that the end to enter formation is not consistent with propagation and play because if you do form.
madhavan: an embodiment vertex, you get an imbalance child or text.
madhavan: And if you have a neighboring vertex of some other violence, it could never produce this child because it will only purchase a child or four different violence. So these sort of deformations really cannot give you a vigorous propagation and it is possible to define a slightly modified
madhavan: Construction wearing a lot is people who, for the purposes. The children and environment vertex
madhavan: Something like this where you have a fall for balance because I'm running out of time. I wouldn't be able to tell you exactly how this happens. But basically, only three of these edges are pulled at one particular chooses to some over the choice of these three edges.
madhavan: Very similar.
madhavan: To one of the possibilities which Thomas consider in the early days. That is what most tells me
madhavan: One can build a constraint action based on these n before vortices and all, with regard to the section one can show that you get propagation and work in progress, shows that I think I normally freedom also might be possible with these constraint action so that
madhavan: finishes the technical talk. I'm just going to show you a few pictures and then wrap up.
madhavan: So in the US tube theory, one needs to show propagation sequences and I'm just going to show you one certain sequence over him.
madhavan: So here we have a parent stay with two verses of defiance the Hamiltonian constraint x at the first vertex and all three of these edges along the fourth one here gives you a for violent edge over here.
madhavan: Then using the different morph isms we drag this to the vicinity of the next word
madhavan: We can modify this vertical picture by the action of different comes
madhavan: In this direction, sorry.
madhavan: And then it turns out that this particular chart.
madhavan: Which has
madhavan: Been coming from here to here. So remember, these kinks or form but pulling by the Center for deformation
madhavan: These now are directly connected to vertex vs overtakes me as a much higher balances balances in my three older and it turns out my taking this vertex
madhavan: Default of the Hamiltonian by the action of the electric defeat modernism constraint and get exactly this. And all these states are contained in the tech side so in a way you can get a propagation sequence. And let me show you putting all these things together in a pictorial day
madhavan: Or if I put the propagation sequence then really doesn't
madhavan: Look as go upward. See, and finally lands up at sea, and then it can go beyond. So this is just to show that this is
madhavan: And then this was just trying to visualize this is what could simply spin form in some on analog centering
madhavan: And then let me summarize
madhavan: In the last slide, and really sorry to Yvonne so overboard. There's a lot of material.
madhavan: I'm sorry. I'm so over the last decade or so I tried to use the requirements of anomaly free constrict algebra of propagation to home in on to the physically correct
madhavan: concentrator progress has been possible in toy models of increasing complexity certain parameters field theory.
madhavan: And those are just two different more physical constraint, I talked about in Madrid Conference. And then finally, in the you want to model.
madhavan: And this progress is due to the fact that classical evolution can be understood in terms of social reform of this particular spatial different systems in the you want tube theory which I electric field and
madhavan: In Euclidean theory we are hopeful progress because the evolution equations can again be written in very and full form. This is new to me. This is my by actually a few
madhavan: In to visit India.
madhavan: Is denied something on the blackboard for the triad field and then with a little help from me looking at the connection fee equations and
madhavan: The equations can be written in really a very nice form where he is some epsilon i j k. These are the electric Shipley derivatives of he similarly for the magnetic field which is just from the kitchen a curvature and
madhavan: Script L objects are living ordinary literary terms, except that in the ordinary Lee.
madhavan: Are you see an audit of operator you replace it by gauge covariance Arabic and then there's similar revolution equation for the connection.
madhavan: And if you use the cell dual were also these equations are true. I've just shown them for the Euclidean theory and they're so beautiful. There must be some
madhavan: Bundle interpretation for this, which will hopefully help in the analysis for quantum theory. So, so thank you so much. Wherever have stayed on to listen to the longer version of this talk. I'm sorry. I went on.
madhavan: And I thank you all for listening, patiently to see. Thank you.
Jorge Pullin: Any questions.
Abhay Ashtekar: So,
Abhay Ashtekar: If you can tell for me.
Abhay Ashtekar: Can you hear me.
madhavan: Yes, I can hear you on mute.
Abhay Ashtekar: Okay, so the
Abhay Ashtekar: Father, just in general for, you know, not
Abhay Ashtekar: Getting every detail right but just broad picture, I would like to understand the following issue you began by that, then I'll have
Abhay Ashtekar: A problem is that
Abhay Ashtekar: A couple of problems. One is too much studying Hamiltonian constraint that there has been too much freedom and the second was about the the
Abhay Ashtekar: Right, so don't or do a custom there with the concert with me on the north end
Abhay Ashtekar: And then you showed us various ideas and various you know enormous technical progress that has happened at the end of the day, then I go back to the main question about ambiguities.
Abhay Ashtekar: But what is the status and and i know similar very general question, but maybe just some first
madhavan: Okay, thank you. So, um, it's, it's a difficult
madhavan: question to answer, or because it certainly on suicide started it. But, but I find my financing this so firstly classifying ambiguities in in a
madhavan: Way itself is a big job, so I can't say. But, you know, technically, there was a whole function space of ambiguity is which are so much and now there is only so much what I can say is qualitatively. What do I believe
Are
madhavan: The things which may have a chance of
madhavan: And I one thing which which seems to be more
madhavan: Strongly from the calculation is the fact that one has this kind of a linearity or vertices and that underlying whatever structure we have or perhaps there is some want to take advantage, not worry too much about the smooth different offices, but also lean into
madhavan: Into some picture where where you can take advantage of these wise linear kind of structures.
madhavan: That general comment, which I think
madhavan: Probably people have been doing all along, gives additional sort of meat or have to was doing this.
madhavan: I think the fact that, given this linearity, one can focus on clinical deformation, which actually move vertices. I think that is a big takeaway for me on so
madhavan: I think that what seems to work. Are these kinds of conical deformations and
madhavan: Not say that. Okay. Other things would certainly not work. And so I cut down a whole lot of ambiguity by telling you that uniquely one has these conical deformations, which are to work. I can't really say that
madhavan: However, after working on and hard on the problem, at least in the US military and looking at the structure of the theory.
madhavan: It seems that on this lead simply because of the action of the classical Hamilton and Butterfield's and the requirement.
madhavan: Or creating displaced workspaces, together with the linearity of these vortices, more or less, roughly speaking into these chronic formations, not whether these clinical deformations are these of these n to four or n to n or some mixture. I do not know and i i think
madhavan: So, so I I really cannot answer your question. Precisely. But I think, given all the implications of getting the constraint algebra to work.
madhavan: And the propagation to work strongly moves me towards a consideration of these end to for deformations.
madhavan: And again, the
madhavan: At every step one would have to show actually that there is an anomaly freedom there is propagation. Maybe there are other requirements which are written
madhavan: I have not yet gone into important technical choices which make things look not as good as I did them perhaps and and I'm working to modify those things. But they're all within the general
madhavan: Domain of these kind of conical deformations of linear vertices. So I would say that the takeaway is that perhaps what one should look at. Are these kind of deformations which involve literate linear structures and which roughly speaking, have kind of
madhavan: Kind of a political nature and which displays vertices, that is that is my feeling
madhavan: From what I've been, I cannot classify ambiguous and they look at so many ambiguity is before and now now we have only so many back that I don't think I would be able to do but but I think kind of what we were doing before or probably Thomas would also agree.
madhavan: That would not be satisfactory, from the point of view of the constraint algebra, it's time to look at new things and pretty much. I think if one
madhavan: In more detail. This is where one has led to and and just one more point to add, I was talking to Thomas when I was
madhavan: In London, and he said, especially this into for kind of thing from was one of the things he had thought, thought about when when he looked at what it would do to a vertex and, you know, but it would
madhavan: You know how we define tetrahedral at each vertex and one could one could make use of these and perhaps try to actually get these sort of deformations maybe naturally also in the suitcase. So basically, that is to say,
Abhay Ashtekar: Okay, so I think that the
Abhay Ashtekar: Hormones that there is no wrong in just looking at peace with Genia category and it and I think because during our time.
Abhay Ashtekar: If you have some brief comments make it. But otherwise, you can go to the next point, namely, I think the question is really, whether in the
Abhay Ashtekar: Category of that from the beginning. I don't think anything wrong with our district along that continuum limit we just our words, look at peacefully crops that is a theory. And it seems to me that the procedure is much more streamlined and one may have more
Abhay Ashtekar: To be retractable we show that, and they do these are very, very few are completely controllable, but you can either agree or disagree, but don't quote me
madhavan: Um I yeah I yeah i i would like to talk to you more about this because I haven't. I mean, to, to actually do it at initial from the piece wise linear theory.
madhavan: I, yeah, I would like to talk. I'm not
madhavan: I haven't used because you know
madhavan: Now, we should we should talk a little bit more about this, it would be great. I think if actually it's true, then that would be marvelous
Abhay Ashtekar: Okay, so the last quick question was about
Abhay Ashtekar: This
Abhay Ashtekar: Or about propagation
Abhay Ashtekar: And I really easy.
madhavan: One for the comment of I'm sorry to interrupt just one small part
madhavan: So I also wanted to say that
madhavan: Again you are familiar with some of you with some of the technicalities but someone in this whole business, there is a there is a
madhavan: There is a St. Lucia coordinate patch and a traditional vertex structure.
madhavan: And that permeates things, but there is a way to finally get Babs get rid of this or maybe do it as undesirable and integrate over all choices, etc. But if one does this then at the end of the day, one is led to a picture from here, where to look at the physical states then then as
madhavan: Carlo and and and Norbert or someone else in one of these papers speculated, you could just not worry about these various higher order modulation and I wanted to, I just wanted to make that point that this this emerges.
madhavan: Probably quite nicely from the sort of calculations that was the only item I want to thank you
madhavan: I will mute if
Abhay Ashtekar: I just correct that. Can I still ask one more question, because I'm asking to model as
Jorge Pullin: Well,
Abhay Ashtekar: So the last question was about propagation and the, I mean you you sort of have very nice somebody in I think it will then slide 25 but maybe not about you know you wrote down these two conditions about when you get a bus propagation. Right.
Abhay Ashtekar: And Mike Yeah, I think it was like 23 or 25
Abhay Ashtekar: If a parent is in the in the cat said then, so all the children and the child is and get set them up. So a proper possible parents. So what we would like to understand is really
Abhay Ashtekar: I mean, how easy is it to get this. I mean, can I just
Abhay Ashtekar: Can I just want to start with the graph and look at all these tests and my ex say that, well, let me look at all these children and look at all these
Abhay Ashtekar: Parents of the children of the resulting thing. And can I say that this procedure actually converges. And therefore, of course, a presumably, I will just have an infinite number of what is this. I mean, I'll get a golf Arbitrary Arbitrary lines to borrow vertices
Abhay Ashtekar: And so the question is, can I just
Abhay Ashtekar: Buy some general mathematical construction arrive at such get sets which are admissible in there is robbers dynamical propagation
Abhay Ashtekar: And secondly, if that is the case in and out of this. Is this set of of
Abhay Ashtekar: distributional in the sense of physical state that come from this sketch sets are they going to be rich enough. Do you have any idea about
madhavan: Okay, so let me
madhavan: Let me let the mountain seven seven stages. So I put these two points.
madhavan: Because this is what the structure is in parameters field area as well as
madhavan: What can happen in you want cute. Well, the reason this, this happened. So let's let me go very briefly into the reason why this happens in in you want cube theory and excuse me, and it's simply because of this you minus one over delta kind of structure.
madhavan: Now, it turns out that
madhavan: Father kind of
madhavan: regularization, which I'm using and you want cubed.
madhavan: Solutions are easy to find. Because what one really needs is that forgetting about the coefficients. It turns out that one can forget about the questions and at each vertex, one can only required that this you minus one.
madhavan: Actually vanishes when when you evaluate it with the option state.
madhavan: So, one can easily see what are the different missions, or what are the children and one can
madhavan: Make the statement that I'm making because of the u minus one. The you gives you information about the child and the one kind of tells you that the parent should be there.
madhavan: However, the you want your regular issues regularization I use maybe a little questionable due to some technical points which which one can talk about later.
madhavan: It might be that
madhavan: This is not so clean or that all the parents and the one and two mean we may not be able to show it so cleanly. Even with this structure.
madhavan: So that is the first statement. The second statement. So I'm saying, even in a favorable case of you want you. If I were to do a slightly better treatment than it could be that one cannot infer this structure, one cannot info one and two immediately. That's the first statement down to me.
madhavan: The second, the second statement is that what is really important for propagation is that
madhavan: It is not necessary that
madhavan: Number to actually hold. Okay, so in order to to say this a little bit better.
madhavan: Let me look at the propagation sequence. I'm still sharing the screen and
madhavan: When I look at this propagation sequence from s to s perturbation here and to S, Brian. The reason I wanted
madhavan: Ends is that that I wanted this is prime itself to be there, but it could be that s crime itself is not there.
madhavan: But if you imagine that this vertex structure went on to the right hand side, the right hand side, the right hand side that
madhavan: This deformation actually appeared directly on the right hand side without something appearing or which which had an absorption interpretation. In other words, one could only have children, all the way
madhavan: But what was very important is that non unique parentage be the, namely that you had
madhavan: That you had some parent, which could, in principle, generate this child and could generate another child which had a deformation, to which right the parent itself need not be there. So what happens in Thomas's constraint.
madhavan: Is that one can actually show model again because the equations become technically very complicated, but one can show that there is reasonable grounds.
madhavan: For believing that solutions are of the type that they may not contain the parents themselves so they don't have the u minus one structure.
madhavan: But the container of children, which do have non unique parentage, so you can view sequences as
madhavan: As something as a perturbation of one vertex moving here another one and then jumping across to the next one. So, this is
madhavan: Also propagation. It may not be the strict propagation. Which one is looking at, but it could also be so propagation. So what what I'm trying to say is that the very notion of propagation needs to be a little bit fine tuned
madhavan: And then looking at the structure of the Hamiltonian constraint itself.
madhavan: Does not really Gannon, it's not a quick answer as to whether what you will get is propagation or not. I think what is a quick answer or what would be very quick to conclude
madhavan: Is if your constraint does not display propagation. I think one would be able to see this quite quickly that it displays no propagation at all.
madhavan: However, to show that it displays vigorous propagation, as opposed to only a little bit of proposition. I think that
madhavan: You know, what are the kind of propagation sequences in the physical state space that I think would be a little harder to to show, but I'm optimistic that once one one gets various
madhavan: You know, if one can show that the constraint Algebra one walk on one should be able to very quickly info propagation, or actually happens or doesn't happen. So I'm so long answer, but but I didn't know how else to you actually answer the question.
Abhay Ashtekar: Thank you.
Jorge Pullin: Any other questions.
madhavan: Okay, thanks a lot.