Okay then, shall we start
Welcome everyone to today's meeting and our speaker will be ugly candle Koichi about chronic against theories with sonorous
Okay. Hi.
Thank you for being here first comment is that I'm actually having a different title for the doc. So it was Albert ISIS me talking about the biological
Theories, but I'm actually going to speak a little bit more broadly and give you some maybe or some comments on the genetic genetic analytical capabilities in the principles of boundaries and we'll
Talk a little bit on the topological theory is that
We also consider at some point. But I will tell you that later on. This is work I'm
Currently finishing with my collaborator Tatiana Sheena Kiran in Malaysia.
So let me go to the to the plant is this is like number two. So first I would like to
Ask some some questions. As a matter of motivation for why we are doing this.
And then I will
Present from scratch, so to say. I mean, this may be very obvious for most of you, but this is, I think, an important step to to start from from zero on how to construct the Hamiltonian formal listen and especially since we are considering boundaries and this is the main subject of the
Of the talk I will then introduce or comment on exactly what happens to this Hamiltonian formerly someone when we are some boundaries.
This is a I'm going to look at the canonical Hamiltonian formalism as opposed to the Guardian Hamiltonian formalism.
Now this is important to distinguish because sometimes in in the recent literature, we have seen that many systems. One is switching
Somehow from The Guardian to the canonical one one uses some call volume techniques to a ride for instance of a hammock.
The syntactic structure and then one goes with a company, having a story canonical description but
One of the us some information from the audience. So here I want to be very systematic and just focus on the canonical picture and not other Tyson, that this is the true way of doing things. So I'm just, just want to do everything consistently with economical formulas
Now once I tell you what the genetic structure of this of what happens when we have this this boundaries. I will then focus on a particular example which is taking the Maxwell theory and building upon the mentor.
I'm here to just put a sane Maxwell and quantitative without actually there are two versions of the same thing. One is to consider Maxwell aponte again, which will
Give him by
By
Actions. I mean, some terms in the action which are defined over the space time
But we know that the dragon theories at the political theory. And this is where there is a contract with a previous work.
And we know that this is equivalent to a chair and Simon's theory on the bomb that he went there is one which is our case precisely
So we can have to description. So, or we should have. We should be able to construct to descriptions, which are equivalent and which gives us the same physical information.
For the system, one in which we only have bound bulk terms and another case in which we do have about them and the boundary
And that's something that we, we should be able to address. And that's precisely why I want to build on the canonical formalism when there's what they're about, then I will finish with some comments regarding with whatever we have found here.
Okay, so that's the plan of the other talk
Then we just go to Slide number three and and ask some some questions. So the first one is, why should we bother about boundaries. Right, so
So why can't we just leap happily in theory in which, as we always do it. We just neglect every boundary term that comes around and when we are questions emotion and everything works very fine, right. So, but then this is a lobbyist questions that we cannot do that.
If we want to answer some physical motivated questions particular, we know that, for instance, if we have a synthetic conditions.
For instance, syntactically flat boundary conditions for theory from space time or less in total emptiness either any as in 30 conditions, and we have two to three. Those are some of the conditions as if there was a boundary and
And therefore we have to deal with the fact that our, our manifold now has a boundary and and we have to
Know how to do the description there. Of course, there are also other fiscally motivated examples. For instance, black holes. We know that with our isolated horizons boundary conditions, we can somehow mimic
Many of the features of a black hole. This is, of course,
A mathematical boundary, because we don't expect for it to be a physical boundaries, but it's a mathematical boundary we impose some boundary conditions.
And then we hope that we will have a consistent description of the six in which we will have something on the Balkan something on the band.
So, so that's what this or two examples that I'm sure there are many more which are
Also physically motivated for simple theories. So, so it's this is hopefully have convey the idea that this is some, something that we need to face at some point when we deal with gateway theory.
Now, one question that arises when when we deal with we do these kind of things is what happens with the vulnerabilities of freedom. And this is actually something that has been
In fashion. So, to save for the past couple of years, and the many colleagues, I don't know how many of you are in the audience right now. But some of you have have looked at several systems in which
There are boundaries and there are some boundaries degrees of freedom that that appeared and that depends on on some choices over makes along the way. I will comment on that.
At the end, but this is this an important question to be to be addressed. Right. So what happens when we have a boundary of their new degrees of freedom that appear there their
Degrees of freedom that were already in in the bulk theory and just somehow worked in character into the boundary or do we have to introduce new ones to make something consistent. Exactly. There are many possibilities, or
Another important feature of what happens when we love and daddy is what happens to this complex structure, we have seen also
Or in the later.
Which are some examples in which sometimes is simplistic structure on the horizon other claims that their
Contributions to the simplistic structure coming from the horizon, and that also introduces some some questions. Exactly. How do we do deal without. How does that interact precisely where the boundaries of freedom.
And then the next question, which is very much related is whether we have to modify the standard deregulated title prescription
Knowing that direct prescription, which is what we know is the standard one for dealing with constraint field furious.
Well, directed and consider boundaries but one could supplement the data prescription with this great your title one principle, which is basically the idea that every time that we have to deal with differentiable functions, which means that every time we compute
The gradient or or the variation of some function, we have to
Kill all the contributions from the boundary
From the band that it is this was has been very successful in in many theories. For example, in looking at gravity in ingesting syntactically flat.
Boundary conditions, right, so here the question is we are going to ask whether we this is enough, whether we have to this prescription is works well in all the cases.
That we may encounter or whether we have to extend or find new prescriptions for how to consistently deal with with HDR is
Abundant
And then there is a last question has to do with. It's a bit more technical and it has to do with possible relations between boundary conditions and constraints.
It has been also some, some, a lot of literature in that ask these questions. And there are many viewpoints on that. So I will come to that later on and will comment more on
His boy.
Okay, so let me go to my slide number four.
And I will start by by recalling and constructed from scratch, how to how to construct a canonical Hamiltonian formulas
Now, of course, the
The starting point is is a configuration space because we always have some physical system in which there is a configuration space. This may be this is genetics. It could be a mechanical system or field theory.
Here the structure is basically the same.
And then there's a fundamental object which will be which will be very important for for the discussion here and this is the momentum function that I'm defining in the third probe and this is basically a mapping that takes a
Dungeon that doors on the configuration space and gives us
A real battle on a on a point and it gives us a real number. This is basically a one for on on the configuration space so acting on tiny and vectors and giving us back numbers and then we represent the
Momentum as this one form function p. And then we have the vector field as as as the then we normally represent that functional this contraction between the end
Now, one goes one starts with a configuration function and one defines a face space which is the normally they go tangent bundle.
Over the configuration space. So basically we take we put together the configuration space and the space of one on one forms basically given by this way, the peace.
And then we can define from this momentum function that we got before on configuration space going to find that one for there's no medical data and this is already on Facebook.
And equation one is gives us a prescription of how this one form a spaceship should operate and it's basically the telescopes that they actually have these one for on a tangent
Tangent data, but now on face space.
Has to be equal to whatever what whatever the action of the of the momentum function was some configuration space when we acted
On velocity, which is what I'm calling q.on configuration space so so that's the basic idea. So we elevate something we. That was the final configuration space to the to the face space and from there we define a one form.
And that is called a simplistic potential and normally we choose some standard coordinates and so on. Then it can be written as, as I've done in equation number two.
And then once we have this one form, then we can immediately construct this eclectic to form on face base just by digging exterior derivative of this, of this quantity and that's what I'm doing. At the end of Question for
This is of course the standard way we there are different expressions for instance for for equation to this is just one choice. There are several
Different possibilities that give the same a simplistic structural on it.
And I'm not proving it, but these are the omega, so defined is satisfies all the properties of is invalid extraction industry has to be closed on the way, assuming it's not the yen at a time zone. So we, that's how we obtain
The simplistic before. So now let me go to Slide number five. And now that there's a fundamental equation here in faith based now we are living Facebook's and that's a given by Question three. And this is going to come all over the top.
And this is just basically contains all the information about venture, leave a comment on the questions and this is telling us that we have a function f, and we have something which we call the Hamiltonian vector field which is this x of f.
Then these two quantity satisfy this this equation which is in number three sentences that on the left hand side we have the gradient of the function
And that has to be equal to the contraction of the simplistic to form with this vector field. So, so both sides. We do have one form and that that's exactly how these two objects on a relic. Now we can a
Contract is one form, we will not be 20 vector, which I'm calling here why and this gives us precisely information. How, for instance, how they function F changes along way which is given by the derivative on the extreme left hand side and then on the right hand side, we just have the the
syntactic structure saturated with both the audience and Hamilton and Garfield on this arbitrary vector field why
If you now the vector field. Why is the Hamiltonian vector field associated to the second function, which I'm calling gene.
Then wait, we can define something which is called the bottom bracket which is just a mapping between these two functions. I mean, which maps to functions and gives us
A new function and this is just the definition is given by by equation fight. This is just
A contraction of this eclectic to form with a door corresponding having done in vector fields associated with both of the functions f and g
And we can use that to for instance.
A great Hamilton's equations in one of the of the of the
Corresponding functions is the Hamiltonian which in that same evolution. So equation six gives us how any other function changes with time.
And that's just given by standard procedure. But here, the important thing that I want to stress in this which is very trivial and very well known to all of you is that I'm using GA simplistic structure to define the
bottom bracket. I'm not defining the inverse of the simplistic structure and I'm just using the simplistic structure and the Hamiltonian vector fields and that this religion is going to be important.
Okay, so that's the standard
The scripture on the how we get from how we construct a simplistic structure and how we get done how we've done bit of sales function something time abolition given a particular
Function which is normally done the antigen that it's done.
Okay, so let me go to my slide number six.
Now we are doing it so that the videos discussion was rather genetic I'm now going to
Say what happens with the filters. So this geometric ideas. The fact that we have selected stock children haven't done a bit of CF function that's that's all there.
Is the only thing that we have to be aware, is that the face basis now infinity dimensional. So you want us to be thing has to do things consistently when when has to be careful about so functional analytic issues which I will not do, I will just be
Practical and not to not to deal with those things. So I'm assuming that everything I'm writing is well defined. So
So for instance, here I'm just looking at the simplest case of a space for our field which is would be just a scale or field, which I'm calling fi and corresponding momentum, which I'm still calling
Now this this momentum mapping on configuration space is given by equation seven. So here the V some arbitrary vector field on on space of functions and B is is now the now has a deal that because it has to be a density function.
And Mila density discuss function, but then see the object. And now we can construct the simplistic potential data that will be given by something
In like. Question number eight and him and looked at home just every all these expressions imply only integral server sigma which is the
The hyper certified switching the four dimensional cases just a three dimensional hyper surface.
And I'm not considering right now boundary. Right. So this is all everything so just expressions on the bulk
Now we take the exterior of the liberty of equation eight, which defines the omega. Now I'm using this weird the d which is like
It's meant to be the exterior that anybody on face space just to distinguish it from the exterior of the virtual space time or on the hyper surface.
So we do get we do get an expression for this eclectic structure asinine. And again, this is one case in which there is only bulk contributions to the
Simplistic structure, but I just wanted to hear an interview some some notation and terminology. So let me go to my slide number seven. And now let me consider what happens when we have boundaries in the field theory case right
So as I was saying in the introduction in the standard analysis of casualties one normally disregards the boundaries and all around the terms.
That appear as one integral parts as simply discard here, we cannot do that. And we know that there is a prescription introduced by a radio title one which which is completely correct and gives us a consistent description in in many cases. And what they did was basically to
To ask us, I was saying before, that all the functionality of functions on on face space. The differential
This means that when we are computing the gradient. The F of the function there should be no contributions from the boundaries. So there's only one. I mean, when crisis functional derivatives and so on. And there's only expressions which involves the Volcker terms.
And this is a I mean this in terms of geometrically terms of the gradients and so on is completely equivalent to what we do in practice, which is that whenever we take the variation of the function it all this boundary terms for sure, but
And then the stand up. I don't know if this is assumption or something that one doesn't even question very much is that this approach.
So the regular diet alone approach to fill furious with boundaries you sufficient to lead with all the cases that the metal is completely generic
So the question that I'm asking here is whether this is this a valid assumption and whether we really have to do something else and extend what we know.
Very well with it with the standard cases for which read your title works with. So let me go now to my slide number eight.
And I will done. So what happens. How do we get the boundary contribution to the simplistic structure right
How does this arises well here. The important thing is to recall this momentum upfront momentum up that originated in configuration space as this mapping between vector fields on configuration space to function.
So, now suppose that I have a field theory and somebody gives this to us. And then I go and do the standard the decomposition and whenever I define the
Function and then I'm
I smeared it with a vector field with
A defined the momentum is merely with a vector field which is what one normally does. And then I suppose that it done that and then I write on expression which looks like 10 and this is expression 10 has
Both done which is standard one that I had before. But now suppose that the we have a boundary contribution which is the second term and
So then what happens now. Well then. Well, nothing happens. Right. So we have to just follow what we know how to how to do. I told you the prescription. I told you before, and this is that from this moment moment popping in and configuration space we can
Lift it up and construct the one form simplistic potential data which is given by equation 11
And that will of course have now two contributions, one from the ball, which is the first term. And one more on the boundary which is a second term.
And then we just follow our gnosis and not doing anything. I haven't done it for now the syntactic structure which will be given by my expression well
Which is just the exterior of the diversity of a space for this collective potential is going to have to contribution. So again, the standard one from the work
And the new one from the Monday, and now I'm calling to this just to distinguish from the wall degrees of freedom, which are where am
I now calling a
With a partial and parcel just to note that these are the fields which are abandoning and living on the boundary of segment right on partial bankruptcy.
And these are well the genetic boundary degrees of freedom.
Okay, so this is
Exactly what what will happen. This is how we are going to recognize and how we're going to see from the strictly canonical perspective that we do have a contribution to the
Simplistic structure. And as I said, this is something that we get when we analyze if your device. So this may be cases in which we, we just have about the Vulcan, which is which is what happens in most of the cases are but they may be instances in which we do have this contribution from
So let me now go to my slide number nine.
So, so what are the practical implications of having such a boundary contribution right now. Let me go back to this basic comments on an equation which I'm rewriting here the question 13
So on the left hand side we have the gradient of a function which you know contracted with an arbitrary vector field. Why, and on the right hand side we have the syntactic structure saturated with both having done in vector field practices basic equation were considered
Now in the case that there are no boundary terms in the right hand side named media on the syntactic structure.
Then of course for this equation to evaluate which is something we want to be to have a consistent Hamiltonian description.
Then, there shouldn't be any boundary terms on the left hand side, right, and that's that's precisely the gray.
And this is exactly the standard read your title case. Right. So there's nothing surprising here. We don't have a boundary contribution to split your structure, then we shouldn't have boundary contributions to the gradient. And this is exactly what what we already knew before.
So, so
Everything that we have done before in greater detail is perfectly perfectly consistent, provided that the
syntactic structure doesn't have a boundary contribution. Now, let me see what happens in the case that we do have about data.
Then of course, everything that it says completely the opposite. So now the right hand side of the equation 13 has some contribution from the boundary
And therefore, to have a consistent TO HAVE A QUESTION 13 to be valid, we need to have boundary contributions to the gradient on the left hand side.
So therefore, in that case, in the case in which the theory consoles comes to us and tells us. Well, I do have a good boundary contribution to the structure, then we can no longer
Have a consistent discussion with the radio date alone prescription. So we know to
It has to be revisited or extended in a sense to incorporate the presence of this is bounded
Okay, so this so far the description that I don't care is completely genetic about how
I mean, Tony in
Theory should should work gauge the results also in the brain. So boundaries. And now I want to illustrate all these words, what I've done, what I've told you in a very generic sense by looking up at a particular example, and that always works to to just
Bring down to it, some of the ideas. And that's why, what I will do now mean starting in my slide number 10
Which is by looking at this, are these theories. I was telling right so I was telling you before we actually have two descriptions of the same theory.
A we can either write it as some security given by the Maxwell action loss upon dragon term which is topological
These two terms are given on the Bounder on the bulk or we could have the same theory, which should be equivalent to a given by Maxwell and Simon's which is a max was done on the ball and mature and Simon's theory on the bottom.
So let me first start analyzing the case of a max will want to again.
That case we know just to set the notation here equation 14 is giving us the this election has two terms. The first term is the max will action.
And the second one is the Pontiac here we have this data. This data shouldn't be confused with a data museum for the simplicity potential. This is the standard data parameter that appears in when we do exactly this right
I'm doing the, the simplest case of an A billion theory. So we have max within IBM data input everything I'm saying, can be generalized in a very straightforward way to the non IBM case so so that's that is not an issue of relevance here some doing, for simplicity, just the IBM case.
So now we go ahead and do with the data analysis because that's what we know how well how to do and when we see that we compute the momentum. And we do have a it that's given by equation 15 the first three terms. Basically, this is the same thing as we had in in Maxwell.
Here, it looks more complicated. It goes, we are doing a generic embedding. So we have labs shift and all things I'm
Not assuming that the slices around. I'm not in ghosts space. My slices are not necessarily anything this is completely generic on any any curb space time with with a genetic slicing
So,
So that's, that's how it looks in in question 15, of course, we see that there is no boundary contribution. This means that the Galactic structure as expected has only the Bolton. I'm not going to repeat that again because I've already went through it twice, so
Given that this is the momentum we get this there. So always only going to be a contribution from the, from the bulk so that's that's
That's okay. I mean, also from the discussion that I had going to get before we expect the like theory with rarely give it a title one prescription to work perfectly well in this case. And as we will see, this is exactly what happens.
Without analysis, there is a primary constraint which is given by equation 16. So on the left hand side we have the thesis.
Local condition this be fi
And on the right hand side is just submitting it a longer segment right so we can when we compute the gradients and so I'm
Kind of going back to office and so on, which would always consider this mirror quantities of course because of a certain function some basis.
Now we can proceed with the data analysis. I will not give you the details and we were right canonical Hamiltonian which is given by equations 17 and 18 I shouldn't have the two names for the same equation is just one equation, but due to live.
So we, as we see that we have some terms which are come just from the, from the Iraq. Sorry for the Maxwell theory and we have a contribution from the point in time, which is precisely where this data term appears right
Now in the in the standard the dark gray a title and procedure, we need to ensure that the primary constrain be per serving time. That means we have to see how it changes.
With a motion or bass, bass generated by the canonical Hamiltonian
And then if, in order for for write down that we need to have a precisely
Both functions well the constraint and they can only canonical Hamiltonian have to be differential, which means that the gray and has to has to be there so we can have this relation between gradient Hamiltonian vector field and and then if we we do that there will be some
Conditions that need to be satisfied at the boundary for for that.
Differential related to
To happen and that
takes me to my slide number 12
And then
We look at the defense stability of the canonical Hamiltonian, we get some boundary contribution to that which is a question 19
And then the standard retitled on prescription Salesforce, that we should have this equation is two terms of both the blue diamond director have to banish for the canonical Hamiltonian to be differential
And now I come here to sauce. This is something that has to happen, right. Like, I'm here to some point which is very important with, but I haven't really talked in detail about and this is the issue of boundary condition, Frank.
And it is of course a very important issue when we define the theory part of the definition of the theory comes with a prescription. The what happens at the boundaries.
And here I'm there are basically two viewpoints that we can take or how appear in the literature and the first one is that we do begin with it from the very beginning of the theory.
That's what I'm calling one here, when both physically motivated boundary conditions and I'm just said that these are going to be valid throughout the evolution
For instance, we know that the case of an internal boundary where we want to make, make a black holing in isolation.
We have, we have this isolated for voluntary conditions which are conditions that have to be satisfied. A throughout the complete history of the universe on this. No.
Surface that we are using as an internal boundaries.
And there are there are other examples in which one can one can boast precise in this case of Maxwell. There are some physical demotivated boundary conditions in which one you're says that this, this is going to be satisfied wild and then we have to see what happens with the formula.
So that's, that's one viewpoint. The second would be point would be to say, Okay, I don't know, a priority. What boundary conditions they want to impose
I just want to see how what happens and then along the way. And then I have to go and make impulsive boundary conditions so that
Humans don't formalism is well defined for that would be, for instance, if I take that viewpoint, then I have to say that. Okay, now the question 19 is going to both some boundary conditions because
That is those conditions, both a blue equation and the right equation to be one ish. I have to. Well, the fields have to satisfy some boundary conditions on here we see explicitly that there is
That the on three again contribution to the theory becomes non trivial before. And as we will see, has no contribution to whatever happened. So the bulk but already
Equation 19 is telling us that there is going to be a contribution there because
Because of these equations that need to be satisfied that the longer
Okay, so this is all everything. I'm going to say about the deal so boundary conditions.
If I have enough time. I could give you exactly the what happens with many choices on and so on. But here I just want to focus on the general structure of the theory and not going to
focus very much on on what particular boundary conditions where we are choosing
The only thing I'm doing is, is I am assuming in to follow with with my description of the theory that I had chosen boundary conditions that make equation that Dean bunch and that's good for me now.
Okay, so once we have to we have done that. So we have ensured that the bow, the primary constraint and Hamiltonian. The differential
Now we want to impose their condition that the constraint we present time illusion know to right equation 20 I'm writing it to the standard
Way supposed bracket or remember that that idea. If I put some brackets in terms of this collective structure. I haven't done a vector fields and so on. So again, 20 is telling me that basically that throughout the evolution given by the, the total Hamiltonian now.
Which includes the constraints. Also, we have a
This condition has to has to be a percentage, right, the time evolution has to has to be equal to zero and that leaves that condition 20 leads us to a new constraint which is a equation 21 the left hand side is nothing but the Gauss law.
This is written in this is the canonical moment for for the connection which actually is not the same as the electric field.
Or the Maxwell theory. It has a contribution from the Pontiac game, but we can we can actually define a
chemical transformation or go back to the electric field for the moment of the max Redfield Gauss low looks exactly the same and everything work the symbol here I'm just simplifying things and writing in terms in terms of the canonical momentum for the future.
And then this the right hand side of 21 as usual is just this mere version of the of the constraint. Now of course we have to check that the equation 21 right answer the question 21
This miracle strain has to be also the potential and that doesn't bring any new new conditions on on on the boundary new one. There are no no boundary conditions imposed by condition.
And we have to check question 21 which means that this this mere Gauss law has to also put some commute with the total Hamiltonian. That has to be preserved in time. And that gives us to the to this equation 22 that only has a contribution from the, from the boundary
And that is immediately satisfied on the same conditions that were needed for the differential reality of the
Offer God's law right so if we we may the gases law differential equation 21 which is already warranted by some previous one better conditions.
Then we don't need to add in any new conditions and immediately equation 21 will be satisfied with means, which means that we do not have a new new constraint or not that is a constraint.
The everything that we have in the theory is just as we had in the max will kill us, namely that there is a primary constraint, even by the vanishing of the
PC, it'll say that one component of the momentum and gases law and these two are priceless among each other. And so we have the same constraint structure as we have in the maximal fury, there is no modification to the canonical structure of the theory by adding this country again.
I know Alex is. All right, so
I've
Got several, several things that happened in this case. So are the conditions that in the previous task is already telling me that your little omega should be zero on the boundary or what time
Yeah, so the little omega, not, not, not necessarily
So why is that
I mean, so what what what what why is this this bottom zero
Yeah, well I didn't go into it. You want to go into that depends on what particular boundary conditions and choose
Right. So, but there's one boundary condition, for instance, for which the, we know that the form of they have
Of the one form has to be a gradient to. Therefore, we do have a
Watch what forms.
Me. Look at my
Answer your question.
Yes. I mean, one of the boundary conditions. He says, you say yes, this little blue you see right there at the boundary so that i can i can
Take that one. That's a very simple one. And then we do that, then everything just immediately like that all the boundary conditions which doesn't happen as much more involved but simple as once one in which this is the case right it's done.
Thank you.
Okay, so I'm in my slide number 14 and we know I'm now.
I can, since I have everything I have all the structure. Everything is differential and son. I can just go on.
And write down the equation so motion you symposium brackets and these are questions 23 to 26 and just notice is, it is precisely the Maxwell equations. There's no contribution from the dragon. There's no data contribution here.
And there are no boundary contribution. So the questions from watching since there was no contribution to the founder contribution to the syntactic structure, not to the gradient. So everything in this
Hamiltonian equation which is the main equation we're looking at has only contributions from the book.
And then I say already mentioned, not the only effect of the contract term is still the boundary conditions needed to make this one is inconsistent and that of course
Will depend very much on the details of what boundary conditions we are considering. And that's something I'm not looking in right now in detail right so I'm not going to say more than that other than that's where the confusion comes
From the phone to enter. And then as we expected from the general discussion that we had before a break your title is perfect perfectly well defined and everything works just fine. Okay, so let me go to the, the other case in in slide number 15 which is a maximize your time.
Now we know that which will have the same theory because apparently I guess we know that they portray in theory is to pull a particular because it is a total derivative. So it's equivalent to a
Something define on the boundary of the space time and that's something is precisely that the actual virtual assignments theory.
So now the same theory, but now written in a maximum Simon's for is given by question 27 the first time is precisely same lesson as before. The Maxwell and the second one is that your assignments on on the bandwagon.
Now we can play with a second time with the same Sterling that in order to make the composition of now the two plus one, the composition and
A guy legs in equation 27 yes this this barbering also includes a quasi services in the future in the past.
Yeah, but that those we neglect because of standard the arguments right that since the variations will be
Suitable that initial finally
We are not going to look at those. But yes, in principle, we do have the SU also those terms right
Okay, I need to have full look well. And so we have a
Okay, but since I let me just make sure that I also understood the previous thing. So if I consider the boundary condition in which is little w zero
Then, then basically you're saying that what people might consider as large as transformations name it where the W is not zero on the boundary
They are just excluded and when is only looking at transformations which don't do anything to the vector potential and the boundary. Is that correct on that previous slide, until the previous slide that I
Don't want to
Yeah, but I saw but i'm i'm not sure exactly about the details of that some
Okay, because you because if you agree that omega is equal to zero and the boundary, then that's what it means, isn't it.
Whether that's that wasn't the boundary of sigma. That doesn't mean that on the on the boundary of for the action so hit him. This is just don't essentials, the actual I'm not talking about boundary conditions they
Like
The last question was separate question.
Just about the dragon, where you just looked at the Hamiltonian framework.
Yes, and
In that Hamiltonian framework, then what is looking at the gate w is zero on the boundary. So we only look at those transformations which vanish and the boundary
Yeah, which are. Yeah, exactly. So don't leave everything embody and
Therefore side.
Yeah, yeah, that's
Something mentioned is zero on the boundary there. Okay, so therefore I mean more to charge for example, who never appear because because all my guys see it on the bomber
Well, yeah, but you could also have this thing about this other sectors tried to as you said it connected by life gets transformation some, some, but we're here, we're only looking at the difference between your decimal things right. So it's not
A large, I don't mean anything profound. I just mean the ones which are 090 on the boundary
Okay, okay. Yeah, those, those. I mean, if you little omega zero, then yeah, you're only allowed to gauge transformations that
Sorry. So yeah, this year there what I did.
So when you send you the recording title. I'm it is recording title by Amanda the condition that
Source of the gay styles emissions. I just zero on the boundary
Doesn't just means that
The throughout the process. I was asking good functions to be differential. I mean, the physical details of what happens when when the bed under the choice have been better condition. So you're focusing on one particular choice of boundary conditions.
Yeah, because that's the only one that you explain so
Yeah. Well, of course.
So with that. So, but, but that boundary condition, then, for example, in general relativity would correspond to having no definitive isms or on the boundary and therefore all the know so that really what we
normally call is constraints.
So, that is to say, you're, you're not going to get generators of space time to him. I mean, so symbolic translations, for example, and that's in the same project leaflet guess
Not those gauge
Because you're
Only getting direct integrations. Yeah.
Oh, you're only getting engaged answers. Okay, so that's what I wanted on the stack.
Here I'm just looking at what it what is gauging what is not good.
But here it just gets at the moment.
Yeah, and this is this is for that particular case of boundary conditions. This is just gauge. Okay, thank you.
Okay, so let me just go back to my slide number 15. And then the question 28 is just to be comfortable as one two plus one. The composition of their
Chin assignment section that's also very straightforward.
And now a go to to the
Prescription see the first step is to see what happens with this momentum up on
A configuration space and then I see that for the for this this scale degree of freedom you want, which is this fi the contraction of a with a bit of demolition vector field that's given by equation 29 there's nothing new there. But then when we look at the
The connection.
Then we see that the
Corresponding momentum mapping has has precisely a contribution as we expect from the world but also a contribution from the mandate that's given by equation fairy right
And here the, this, this quantity little be is just the pulled back of the VM on the bulk. Right. So there's this is just the restriction of the fields on the on the boat to the boundary. But what we do have now a contribution as we might have expected.
And then of course you're following separately that I explained before, this is going to be able to contribution to the syntactic structure.
From the, from the horizon, sorry, from the bonding.
And therefore we are already begun trade your title alone because we explained before, we are not going to be allowed to say that differentiable function sort of those for which the boundary terms lunch. So we have to keep them that it then when we
Take the gradients of of
Various functions. OK. So now let me just analyze the constraints, there is a
Primary constraint 31 if you're sitting there waiting my slide number 17
So this is just same constraint, as we had before. Basically, and
And then we have now a constraint on the on the bundle ID, which is just given by the turn signal theory, which is question 32. And that relates the boundary degrees of freedom. They should have put also boundary review in second term.
And now we can just follow the, the standard procedure. I mean, live as long as we don't encounter any any difficulties, then we are
Allowed to proceed with the exact procedure and that's what I'm doing in the question 33 I'm just defining the
Standard Hamiltonian canonical Hamiltonian, that is given by one time, which is just the max will close the canonical Hamiltonian from from the journal Science theory, which in this case can be written as equation 34. This is just to
Be called by children and and this potential fi
OK, so now in order to proceed with this, the consistency conditions for for for every i mean we have to
We have to ask the consistency condition of the bottom of these two primary constraints, right, that they need to be set reset time evolution, but now. Now, if we are just with the doc, then we start scratching our heads. Right. And we've been doing that for a long time, how do we
Check the consistency of the constraint 32. For instance, how do I describe how to evolve that if that's defined on the boundary. And then if I take the
If I take the name that you have to have this. This is going to be a boundary term. But then how I'm going to compare the person bracket.
I have this thing which is distributional and then all these questions without government very much entangled and one can just think about it for a while it gets very simple. We just look at the equation 35 which is seen
In the slide 18 which is just the same equation. I've been looking at here. I'm thinking I changed the left hand side. On the right hand side. But anyways, on the left hand side we have
Information about the Hamiltonian vector fields.
For it gets gets for the total Hamiltonian and an arbitrary one way. And on the right hand side we have the gradient. So the canonical Hamiltonian
And the two constraints. The first one is a constraint on the bulk and the last one is a constraint on the boundary
And now we have to have any problems because we can just to natively say write down right on site and left hand side.
And equate and compare and see what happens. Right. And then from from dollar question. Just read out whether Hamiltonian vector field.
Is in this case and that Hamilton battlefield. A if everything is consistent and he's going to give us time it will know somebody is going to give us the conditions for for the preservation of the of the constraints. So we do that.
And we have now some contributions to the boundary which are here, returning in before equation 36 there is one which which is this here. I'm this x with the sub index a high
A is just a means to the component of the Hamiltonian vector field in the direction of of the field, a
Little a, is, is the space time index.
And then they order the questions. It seems just means that now the contribution at the boundary of the hundred million vector field in the direction, be a that that is this expression, even on the right hand side.
And now we find that we have in order for this to be well defined for for have a equation 35 to be valid. We need to impose an additional constraint on the boundary which is given by a question 37 in my slide 19
And the
And, well, that's something that with to deal with right and now we so we go and ask what they consist of the conditions of or the
Primary constraints are and that is a great given by the question after question 37 which is just basically seeing how this how this evolves and we as we expect we get this is a contribution on the only from the bulk and this is just scousers law we get as usual girls law us
As a consistency condition for the primary constraint. And then the second equation that hustle doesn't have a number which is now the preservation of the
Of the boundary constraint is going to be able to, well, some conditions on the right hand side of those conditions need to be need to vanish for for the for this.
To be preserved. Now say we're selling the we we got the Hamiltonian, the gases law and we have a now on your constraint on the boundary which is the equation 39 right that needs to be needs to banish for the preservation of the primary constraint.
And I will see that the, again, we now we want to check the consistency of the preservation Mughals law that is that gives us equation 40 and that's the story forward to see that precisely we we get again.
Preservation provided that again that the boundary conditions were such that the other cases were certified
Now the thing is that we can just look at all these equations of the consistency and song and then we see others. For instance, we can solve for this new new a which appeared in the equation 36 in
Slide 18 that was one of the components of the Hamiltonian vector field. And then we see that we can get that you equation 41 that does just
Like continuity of the Hamiltonian vector field on the bulk as it approaches the the boundary. So we do get now an expression for that and that immediately tells us that those boundary primary constraints that were given by equation 32 which is in
In
Slide number 17. Those are second class.
Okay.
So I'm ceiling in slide 20 and so after a question for you. I'm I'm
Concluding that those new boundary constraints that appear or second class. And now we can still play and see you find that the components of the boundary Hamilton and Victor filled out the form
Given in 42 so we can actually solve for not just the forms of the haven't done a bigger field on the ball, but also if
They haven't done in vector field on on the boundary. And those are given by equation for it all. And just for consistency check they satisfy question 43, which is what we expect because of the primary constraint given by
By 32
Something I didn't write down, but we can we can look. I mean, also find what the Hamiltonian vector fields on the board car and we find exactly the same as we had in the previous guest, namely in maximum quantity it. So we do get the same
Equations of motion on the bulk
Of it, but now we get something also information on how to deal with, with the boundary degrees of freedom at the end of the day we do, we do get exactly and
Haven't done this in detail because I also running out of time, but what we see is that precisely we do get the same degrees of freedom, regardless of whether we started with a maximum volunteering or the max will
Turn Simon's important thing here is that in the case of my work, Max. What's your assignments. We had to give up the idea that the
Notion of the instability plays that nothing survives. So they are right, the boundary. But we do have to not only present. I mean, still live with.
Terms of the gradient of the boundary, but also don't have too much with those terms that came from the boundary contribution to the syntactic structure. So, everything becomes at the end of the day, consistent and completely equivalent to the Maxwell Pontiac in
Theory. Okay, so that's basically the end of of this example that I wanted to consider. And let me just summarize now. Okay. Sorry, let me go to
Make some comments on slide 21. One of the things that we see here. And this is something that already happened before. I mean, something I didn't mention a lot. But the first thing we did was
To go see that by itself, the Pontiac. Pontiac in theory and the chair and Simon's theory and compare them right
Now the thing there. In that case, it's more tricky, right, because we only have Andre is defined on the bulk
Max a chance, I must be defined on the boundary. So how are we going to compare a furious on the Balkan the boundary
And then furthermore, if we look at the constraints structure of the two theories that very different. And this is also have manifested here in this case.
In the ball theory. For instance, in point today and also here in Knoxville country, in theory, we only have first class constraints. One is
The primary and the other is a secondary constraint, but they are the first class.
Now in the case it in the other case in the boundary case it is already happens for sure and Simon's by itself, but you also have been seen maximal student Simon's
The additional that we have an additional constraint that appears in the laundry. That was not there before in the world theory, but then if this turns out to be second class.
And this, this happened in just pure chance Simons, and this also happens in Maxwell turns
Now there's another issue with a How To Treat boundary conditions. And as I mentioned that before.
And there are some people in the letter literature for reasons that happens to them us this constraints in the traditional direct centric.
And then they're subject to this consistency conditions and in some cases in some examples. One one says, for instance, the dataset. The Tower of conditions that need to be satisfied on the boundary for this.
For these under the conditions to be preserved on time, that may or may not have an impact strong impact on on how the degrees of freedom change in the
In the boundary input that's also depends very much on the details of what was, what are the boundary conditions. We started with and what are these external conditions that we that we find
And that also depends on on this viewpoint. I was mentioning before, whether I want to impose boundary conditions to begin with and ask them to be satisfied. They were all the time where I want to build boundary conditions by asking the furious. I go along to be consistent.
So this and you're saying that this is an important issue that the one needs to to understand better. And then of course we can understand it in some particular cases but
I feel that we, there's some need for a more global understanding of what happens in general. And this is has also to the with the previous point one in black, that also, some authors have have a
Pointed out that the patently this boundary conditions are seen as constraints or when there is their boundaries that they seem to be a bit and always a second class constraints.
And not as, firstly, let's go straight here in the, in the example we have seen this is, this happens, it isn't exactly the case.
But I still don't have a very strong understanding of exactly if this has to happen all the time of this is just some particular feature of the of the examples we have looked at
Okay, so now let me go to the third, the common which is in blue. And in this viewpoint with this extended the dark right you're willing formalism.
We don't need to introduce new degrees of freedom, depending on what particular choice of boundary conditions, we might have
Remaining data so freedom that are not canceled by page at the boundary. This is for example, what happens in the, in the case that I was mentioning, if we if we are we have the gauge
Your notification automation is given by God says lower such that they leave the boundary embody and then we have more degrees of freedom as we would expect because we will have these
Would be gauged degrees of freedom that are not canceled by gauge, but they said not your degrees of freedom that was not there before. It's just that we are not canceling temperament.
Yeah, but what we are, know what we are not doing is what many people have done recently in this couple of years, which is to introduce and add extra degrees of freedom, just to cancel.
Things are the banter. This is not what we are doing right in my viewpoint is just will change in the theory because you're you're adding boundary go you're adding degrees of freedom that we're not there, and that
Need to be motivated a case by case. So this is not what we are doing. We're just starting with under the degrees of freedom. We had to begin with, or then we see them exactly what what we have left with
This viewpoint, of course, you say is consistent with letting the theory guide awesome tell us exactly what is gauging what is not gauge and that is
Given by a precise detail analysis of what the constraints are I mean consistent constraints are or if we can also look at the particular instance they gave me the directions of this eclectic structure, which in the case of
Boundary conditions. Well, one has to also check what happens with it again that the directions on the
On on the boundary
So I think that's, that's all I wanted to to comment on here on what we have. And let me just summarize here.
In my slide number 22
So we have a consistent procedure for addressing this case, the artist with with boundaries, the structure of the theory as we started with an action song is going to tell us whether we have a boundary contribution to the simplistic structure.
If there is no boundary condition to a similar structure, then that as we say argue, then the standard definitive related conditions.
Are in our foundation they yield, they shouldn't they yield a consistent description. This is just the regular Title Case.
Now, if the boundary conditions to the back to this black to do structural not zero, then we need to extend this directory title procedure and there's also a canonical way of doing which I I show you here, by the way, this is this idea of how to deal with.
These how to find I'm enjoying and vector fields, including contributions at the boundary is something that was suggested recently by Barbadian collaborators, they were looking at other examples, but so we're just taking
What they suggested and implementing here and the arguing that this is this is gives us the commissioning ingredient to make
Everything consistent fact in the case of a contribution from this and click extorted they will not discuss. I mean, the slightly complimentary what they were going to what we're doing here, but just want to mention that they they they they were doing some parts of that. All right.
Okay, so if we do have a non trivial contribution to laundry from simply the structure there will be contributions, the boundary, also from the gradient and they simply click structured different stability. It is not the notion of the financial reality is not
Does not mean that boundary terms have to watch so that that is important message here and in the in the particular system that we looked at, which I'm colleges Maxwell's laws of the political term we have full control over these issues and we see that everything works.
Just fine right
Now just My final comment here. Is that the case of an isolated horizon present. Another interesting example because there we have
instances in which there are contributions to this eclectic structure from the boundary
And then we we have to have a exactly understanding of it is interplay between what is differential what it's not.
What is generating gauge where it's not and take into account also these contributions from development, but of course I have run out of time. So that probably will come in and say thank you and this is a reference thank
Thank you very much for the talk. We open for questions.
Unclear in your office, how the Audubon who is actually
About 200 GMT program, or more precisely how other boundaries. Then it into space time. So if I look at it in creation or the sequence of equations that needs to come in relation to creating
A modeling.
So you'll have a
Partial cushy cushy hyper surface partial machine I persona sigma with the boundary sigma which is not a boundary of the in space.
But
A very key to me how this partial sigma is actually then extended interest to or some other money forward in space, time, and this seems to be important in my opinion for how to
Do the analysis because in order to understand the preservation of the constraints we have to look how the constraints of preserved under the timer conclusion but not a time of rush depends on how the boundary or pop by boundary
Boundary of hydrocephalus is actually embedded into space time so
I'm confused about this because
So you agree that we have a partial coaches are for sigma right and that this may have a boundary which is just the sparkle sigma
Yes. And so, which sense. Are you so what what is embedding the you're kind of embedding of this
Boundary
How our seat license based on how how we continue anymore. It's
Time for a week.
Or no, sir.
Yeah, yeah. I understand this, this may be an important issue. If you are really looking at space time as fully edited by hyper surfaces and so on.
But if you go to the beer canonical viewpoint, not, not, I'm not saying this is the right thing. But if I take the viewpoint that I mean the pure canonical viewpoint.
The only thing I'm I have a sigma and it's Monday, I don't have a space there and what I do have is sort of fields defined on sigma and now I have Hamilton equations that you tell me how this thing, simple and it's up to you if you want to, at the end of the day.
reconstruct the space time on which you can interpret these fields evolving or not they could just leave and say that my only experience I have an everything is just the final sigma and then I will have a problem.
It from my perspective of how to embed this into space time right
So I'm this is if you want. This is a very nice viewpoint Bureau canonical in which I'm just looking at what happened when sigma. What happens to feel some sigma and it's it's
Yes, I have a question.
Yes, in the isolated horizon case related cases that several of us stayed in the 90s where we talked about to dance teacher and Simon
Ignited whether or not you wanted to. We went to the quantum theory you wanted to have functions on the boundary. He didn't want to have fun, just on the boundary. And that was one surrounding conditions, which I think works similar to Maxwell.
Maxwell match it with you want to have punches through the boundary as the isolator I
Need the part of the gals as well become said advice, by definition, they tie that the field to field value to the function into the field into the room coming out of the function
Yeah, I could end in this particular case when there is a Western same as in Bob. It always turns into a second class because precisely that's also the information that
That constant and this eclectic structure, right, that tells you that the connection doesn't come up with itself.
So, so, yeah. In this case it is. I agree with you completely. And it's very clear that the that the things become second class in the boundary. And I don't know. I mean, I agree, also agree that this is structurally similar
To what happens in your self 12 case with
That you were considering in the in the 90s. Right.
Right. Well, thank you, thank you.
So,
Yeah, so I mean what your main point was that
That's in the examples that you consider at least you don't know didn't introduce additional degrees of freedom on the boundary that the tedious very defined and complete answers.
Yeah, that's one of the main points. Yes.
Okay, so I understand that I appreciate very much
But I like understand better the other part of the thing. So now we have this equation.
So if I go to
I think he does probably page 15 just let me see what which basic is one second.
Yeah. Ah, well, sorry.
Okay. Yeah. So I mean, here again just to recall, you're, you're looking at the first part, which is really the punch again theory, in which case there is no boundary determine the simplistic structure or
Are the action. So now I just say that we want to make this boundary term.
We need him to impose boundary conditions right here.
So one, one possibility is to make this don't manage this the surface to vanish by just using the boundary condition to be fixing the connection.
So space time connection if you like. So, that is to say, Be as well as fire on the boundary, in which case this Margarita manages and everything is okay. Yeah. And that would be perhaps like like the the the choice. One that yeah, got up here.
I
Will know
That, well that's
That's us.
There's like a third choice because you're killing everything there. Right. You're just saying that you're going to fix all the components.
To fix fine.
But I don't, I don't see what else we could do in order to
Because unless you also want to
Put conditions on the fields themselves on the boundary, not just on the variations
Yeah, but, for instance, I mean, there's one example in this case of Maxwell, there is a perfect boundary conditions. So, you, you, you ask the the the
perfect conductor boundary conditions you ask the boundary IBA behave as a perfect conductor and that tells you that okay the fire has to be
Constant, this and that and and that kills have all these terms and everything becomes well defined and his differentiable job just by asking this boundary conditions which are
Not as strong as killing both, am I
Getting that apply because fires constant. Yes.
And the other like the other term is is skilled, but because of this conditions. Yeah.
So,
Yeah, because they're not not not allowed. I love is also zero and the
Napa five zero. Therefore, I just left with this.
Yes, it is arbitrary.
Yes, it is arbitrary and if one of the conditions. Is that a piece
I think is proportionality or
Well, it makes it fun contracted with our punch.
And
But, but that's that we can condition them and the magnetic field. Lot of the electric field right
Well,
Yeah, here, your last patient components of f.
Yes, yes, this is yeah this is on the magnetic field. Yes, but this perfect conduct commander again issues involved the connection and therefore the magnetic field. So also sits, it's not
Yeah, but what I'm saying is that this choice we choose us sometimes he is such that this person cancels.
OK, so now my question is really that I would like to understand what the differences between the two things that happened right i mean in the in the punk jag in case and your turn. Simon's case. So because I'm action was as a function on the space of fields was exactly the same.
Yes.
So you just chose to write one term as a boundary term in the second, the second part and the first party kept it as a volume. Yes. Okay.
So, therefore, what live. So what I'm still not clear about here. So if I were to take the brutal choice in which delta a, b is equal to zero and delta phi is equal to zero as a boundary condition.
In the first case in the jag in case, then the surface time is zero. And then I don't even have to add. In addition, that little omega or little W should be zero on the boundary because delta zero so digital W also, of course, has to be zero.
And so in this case, I don't really have any boundary degrees of freedom at all.
Yes. And then you kill the assignments, you know, sense if you want to go that way. Right.
So that's that. But the way that you can turn Simon's degrees of freedom is by saying that you keep them to begin with, but
I will kill them. If
You don't kill them. I just do the Hamiltonian malaguzzi that includes them and we get the same constraints on them or the end of the day, which is, which are the second class constraints on some so we
Will we will see it but it's you impose a second class constraints, you don't have any any degrees of freedom on the boundary left. Is that correct or not.
Well, that depends on the on the details of like the believer, the boundary and song right because you, you may have a steal the political degrees of freedom. If the boundaries non trivial and song so so you'll get steal those
Those stickers. What do you mentor, but I don't know.
I mean you got that punches punches, then
No, no, you look, you got a bunch of people you could have been no
Boundaries, which are have a non trivial amount of the grouper.
Okay. But then, then it also
I'm trying to understand why, in one case you you are yet. So you're saying that there are degrees of freedom left in the in the in the term salmons case on the
When I'm saying that
You use that out of degrees of freedom and also the degrees of freedom which which I just Apollo, you can write some I'm just not seeing that
What I'm saying is that if you are
Right. Oh.
Yeah, yeah. In that sense, but what I'm saying is that you could also have it by looking yourself Maxwell country again you could have degrees of freedom remaining on the boundary
Right, just as you were saying in this example in which little omega is zero because you're dealing kill them with engagement as you were expecting.
So what I'm saying is that you do if you are consistent with some boundary conditions in the Maxwell Pontiac and you know the exactly the same.
On Maxwell turn Simon's you're going to arrive at the same degrees of freedom on the Bounder which could be zero or or more. So depending on
On the details of the boundary conditions, but any choice which is consistent in one case is going to hear something consistent. The other case, and they're going to coincide. That's, that's the
Statement here and I was particularly big about not going into the details of what the boundary conditions was going to happen here and there, because
I would take me like twice at that point. But I'm just just they just have the argument or the main point here is that
That we get exactly the same thing, either by starting with the bulk theory or by Max will close turn sense in terms of whatever the remaining the rates for in a day at the bundle. Okay, so
Let me understand why are you saying then that
If I chose some boundary conditions in order to kill the surface them this equation 19 that I was showing before
There's some 12
Yeah.
But just some boundary conditions.
Yeah.
That's one of the
Things that you have to kill it. This is not the only one that because you have to, you have to make sure that their primary constraint is 60 and is also differentiable, and you have to also check that the
21 is skilled and that has to do also with 20 minutes or so, one of the questions is 19 but
Yeah. So look, look. Let's continue. I'm just clarifying that there are more conditional. But yes, I suppose that we had satisfy all these conditions. So,
I do say that is a one to one correspondence. That's my question.
That I choose certain boundary conditions in the punches in theory, in order to get what you say is consistent richer.
Yeah, like as then
There is a one to one. But then the second is that there is exactly one corresponding conditions, I can put in the chance I'm a sector. Yeah.
Yeah, that's
That's it. And then the two things are equally equally as each other. Yes. And, but in both of both these cases they may be
Terms, there may be a boundary terms that are left. So there may be a boundary degrees of freedom that are left yes and and both cases, you get evolution equations for those boundary degrees of freedom as well. Yes. Okay. Thank you.
Questions.
That is not the case that, let's thank our speaker again.
Thank you everyone.
So how do we disconnect from this thing.
Meeting.
meeting you. Okay, bye bye