Spacetime Covariance in Canonical Relativity

0:00 It's a truly great pleasure to see so many people who have come here to celebrate Chris's 65th day. It's really been quite overwhelmed actually. There have been over 200 people. It's also been a great pleasure to organise because so many of the speakers just agreed immediately to come. It's all a sign of Chris's influence as a degree to which he's been respected in the field. I'm sure most of you know it's normally so difficult to get people to come to a conference and speak, but this one has been really very easy. Just a few practical announcements. There's a small book here that if anyone would like to write down some greetings, please do so. It will be either on the reception desk outside or on the tables just opposite, and we'll be able to dinner this evening. So do please write all your wishes in this book. anyone who wants to check email there's a piece of paper that I gave out with details of an account the computer tunnel was just opposite you do need a swipe card to get in but we've arranged for a series of students to be standing close by so just look around and look hopeless someone will look at the swipe card so I think that's all preliminary announcements detailed questions please call me during lunchtime or coffee breaks and I'll be happy to try to hold. So let me pass you over to Jeanette Halston who's the chair of this session. Well we have one speaker this morning but I'm sure you all know Carol Kukash who's going to speak about space-time covariance in canonical relativity. Hi, Chris. We didn't see each other for a long time. In fact, the last time we saw each other is about four years ago in Warsaw. And these were not the happiest circumstances at that moment. So each time someone is passing by who met you more recently, I am asking you, how is Chris? And when I was a Jew in Berlin, Yanis passed through, and I put that question to him. And his answer was, Chris is the same as he always was.

2:30 He only grew the beard to look older. so I took this from him and I realized that there was a Czech poet who in the 19th century also wanted to look older and he wore the beard as all the people at that time did and there was a celebration of his birthday and his students came and just wished him good luck addressing him venerable elder. Now what is in fact nice in this story is that it was his 40th birthday. So somehow, age was considered to be good in the real life. But Janis brought me to an idea. Namely, when you are looking at that picture, you have a mythological analogy. And I learned that Janus was in fact originally not the deity associated with time. He was a gatekeeper. He somehow watched the inside and the outside of the house So that nothing be stolen from there And now intruders will come from the other side But then of course It became the temporal association What I wanted to say is That I would like to follow this picture And I would like to look today speakers will probably do, by looking into the future and talking about their most recent work. I would like to look into the past and I would like to discuss the things that we were both interested in and the things that we worked on together and that we thought about together. So my dog is a retrospective and only the last third of it if I ever get to that last third will be looking into the future

5:00 Now I try to remember when we met and of course one never remembers these things without going to the records but I think that we met the first time on 24th of May of 1973 there was a seminar at that time that I gave at King's College and you were at that seminar and that's the first time when I remember that we talked together and of course shortly after that Dries David came with that idea that we may write a book the story of course, but let me start with it and let me start 10 years earlier before I met you with the admonishment which I got from Bryce at the Lezuch summer school in 1963. So Bryce started with a stern warning which is of course well known. A chief goal of these lectures is to develop a framework within which the quantization of fields possessing infinite dimensional invariance groups may be carried out in a manifestly covariant fashion. The requirement of manifest covariance means that we shall adopt an overall space-time view from the outset and ignore canonical formulations. So what I am doing here, of course, is what is forbidden because, as we know from those days, there are two ways of quantizing reality. The covariant one and the canonical one. And they are totally different from each other. And, of course, what I want to do is to argue that it is not so. that we can in fact look at canonical gravity and general covariance as going hand in hand. So, let me give you that story and before going to its details, let me just go to a single picture which emphasizes that idea.

7:30 It is an ancient picture and it comes from the fathers of general relativity. In fact, it comes from David Hilbert and from Herman Wilde. It's the concept of the space-time to be split into space and time, and to be split into them by structures put in from the outside, by the foliation of the space-time, by the leaves of instantaneity. by the congruence of lines which brought in the spatial reference frame and these things brought in my hand thought in about as a tenuous type of matter which in fact doesn't influence the gravitational field but provides the The orientation within the space-time is the basis of most of the difficulties that we encounter in the canonical formalism. So what I want to show you over here, and just look at the top of this particular picture, is what is called the Time Out. Certainly not that way. It's the thing called the Time Matter. What is the time-manifold? We have the space-time manifold, which I call light, and there is another one-dimension, oriented manifold, the time-manifold. Now we have a mapping which goes from the event in space-time to a moment in time. And that mapping, taking the event in space-time and sending it into a moment in the time manifold is what provides us in space-time with what we call the instants. So these instants in general are supposed to form a foliation. And an instant consists of all the events which are met in the same moment. Another instant corresponds to another point. And of course the time procession is how we go within this family.

10:00 Now the other thing that comes in is the time map is the space map. And the space map is just shown over here. What we have is, of course, now a space manifold, the space manifold, which I assume to be content. Space is over here. And again, there is a map going into the three-dimensional space manifold and all the points which are mapped into the same point form the reference word line. There are many word lines of course and they form not a foliation but a congruence. And this of course provides us with the reference frame. So this is the basic picture. Now there the mat which goes backward from the time and from the space manifold into the space-time manifold. And this is the embedding mat which is used all the time in canonical gravity when we try to take the space-time events and pull everything down into the split language and it is that split, of course, which causes difficulties. So let me go to that relation between the canonical gravity and general covariance. Here I have the two maps I talked about. What I showed you is the map which goes from the space-time into the time and space. Here is the inverse. What we do is that we take the ordinary Hilbert action, which is invariant under the philosophisms, and we take that action and look at it from the point of view of the pullback on the time and the space manifold. Is this forecast sufficiently or should I try to? Is it better? So we have that pullback and not

12:30 all the quantities obtained by that pullback are dynamical. Some play the role of the Lagrange multiplies, the lapse and the shift, the known object. The induced geometry on the instant is what is dynamical. We perform the Legendre-Duel transformation to the barriers which are sitting at an instant of time. The induced metric gets complemented by the conjugate momentum and there are ordinary equal time Poisson brackets. When we vary the multipliers, we obtain the constraints. But of course those constraints, which are the limitations on the data, should have some property with respect to the Poisson brackets. Now they form what people call the Dirac algebra and the only trouble with this algebra of course is that it is not an algebra and that it contains reference to the objects which define the dynamics itself into the metric and that's the source of our difficulties Now, this is, of course, what was the state of the affairs when I came so luckily to visit you at my sabbatical at Imperial College, one of the nicest years which I had. It was in 83, 84, and you were then very acutely looking into the question of how to build a program of group-theoretic quantization. Namely, you had at your hand at that time what you felt was the appropriate group that describes the kinematics of the theory. The analog of the Weill-Heisenberg group for canonical gravity.

15:00 However, there wasn't its dynamical counterpart The group of dipheomorphisms Which somehow would carry out the objects in a dynamical fashion There wasn't dynamics in the picture Now it was an exasperating but at the same time a farcical situation and it reminded me a situation in the old 19th century comedies there is the bride there are the relatives all of them waiting at the altar but the groom isn't present I somehow thought it would be nice if we try him to fetch in the group that the marriage can be completed. Now, as we know from Victorian England, even if the marriage is completed, it doesn't necessarily mean it will be consummated. So, I did not necessarily say that we will succeed to use this theme to construct quantum gravity. But, we just wanted to look it is possible to introduce the space-time dephalomorphisms into the game. Now we wrote a paper which, in the best of the traditions of writing the papers, was then split into two. And this is the first half of the paper which explains the basic idea on which we then try to implement Now, we try to see what happens in a parametrized field theory. Now, parametrization is another ancient game. Of course, for finite dimensional systems, it goes at least as bad as to Jacobi. And for the field theories, it was first discussed by Dirac. I would like to explain what is BIP.

17:30 Suppose that you have the field theory. And suppose it is propagating on a given space time. So the metric of that space time is fixed. Now observe, the space time action is not invariant under the film of physics. If you put in something fixed and you try to modify the other variables, like the scalar field propagating them, then of course you won't have a different office of invariant action. So that's one important point. Now what one does is that the space-time field, say the scalar field, is pulled back to this time-per-space manifold. And we work with the project, with the pullback versions of the field. And of course, we then do what parametrization reveals. We take the embedding variables, the components of that map, and we are adding them to the remaining variables. This is called parametrization. we canonize the action, we perform the legendary neural transformation, and we arrive at the action which depends on the pullback of the field, the conjugate momentum, the embedding and the conjugate momentum, and a set of multipliers, multipliers which tell you how to go from one point of space to another point of space. We have the ordinary human form and we have here what are the constraints. Now, what are they for the parametrized theory? If you look at them, they have a very simple structure. They consist linearly of the momenta which are canonically conjugate to be in values and then the object which you can recognize as the energy density and energy flow in the space-time.

20:00 This is the energy momentum tensor of the field. It is of course being expressed in terms of the canonical barriers. It is projected in one of the indices to the normal to the surface, to the instant of time and it is a function of the variable body. And now we are following the evolution in a spatial direction. We do it by having this Hamiltonian smeared by the multiplier and that's what the action is giving us. Now, the action, of course, pulls the field and pulls the embedding which carries it. And at the same time, it evolves that field in a dynamic fashion. This all is well known. It isn't, however, the way that Dirac interviews the parametrized field. Now what I want to say and what is the basis of those two papers are the following two things. Suppose that you take this object, you smear it by one deformation layer. You smear it by another deformation layer and you form the Poisson bracket. These neat things are, in fact, commuting strongly. And this happens for the prescribed set of these functions. Still no view of the different morphism group. So what is the trick? And this is what underlies, I believe, our present paper. You do not take the prescribed functions. You take a vector field which is sitting on that manifold. But you then restrict it to re-embeddings. And that restricted field becomes a function of the space point, but also a function of re-embedding. Now, if you do this and calculate the Poisson bracket, you find out it no longer vanishes,

22:30 but becomes the same object smeared by the re-bracket between the fields. And it is here that the representation of the space-time different office and group appears in the projected formalism. Now what does it do? Well, as I told you, the Poisson brackets of arbitrary variables in that scheme, the embeddings but also the fields. are carried by that vector field into the future. Now, the embedding itself is simply displaced. The fields which are living on that embedding are not taken as they are. They are dynamically evolved to the new embedding. So, what this object does to the basic variables is that it displaces the embedding and produces the dynamically evolved field on that embedding. So I would like to use this language. Those Hamiltonian constraints smeared by the field sitting on space-time generate dynamics and they represent space-time default officers. but they do not generate difheromorphisms. You can see it simply from the fact that the space-time action is not difheromorphism invariant. So these are not generators of difheromorphisms. These are, in fact, the generators of dynamics. And this is what is wanted, to have that counterpart of the kinematical stretch which he possessed. However, it is extremely important for the later evolution of the subject in which people started to think about the super-Hamiltonian and super-momentum, objects to which I look later,

25:00 as the objects which, in fact, are generating different homophisms and I think that all the distinction between what I call perennials and other people who are observables is based on this fundamental distinction. All right, this is not the way that Dirac did the business. In fact Dirac took the constraints as we have them and he projected them. He projected the deformation vector perpendicular and parallel to the surface to the instant of time as I told you and he did the same thing with the constraints. Now the perpendicular projections are of course the Lex function and the super Hamiltonian of the theory. And he then found out the type of algebra which is far away from representing the space-time What's the geometric meaning of that algebra? It is implicit in Dirac's word. It was spelled out in detail by title point and I described it in the language which simply tells you this. You are using a chart of coordinates in the space of embeddings. And that chart that you are using is a holonomic chart. However, you can introduce an unholonomic basis in the very same space. And that then gives us exactly what is the closing relation between the constraints. So it is this matter of projection and of course what we need to do in order to go somewhere is to un-project, to reconstruct or if you want to redo what you have done, the projection and try to come out to the space-time which is in fact not

27:30 on the projections into the perpendicular and the fungential directions so that's the idea of the first paper now here is the second one we try to do the same thing for canonical gravity now what is the problem of trying to follow Well first, the embedding variables are already in the scheme in fact. We know that what happened there is in fact independent of the embeddings. we have what we call the already parametrized theory I am trying just to see if I didn't skip something that I wanted to say before I think that's all forget that you have them somewhere impressively in the scheme and add them once more, explicitly, to the geometric variables. Now, in an unholonomic basis that I talked about, the embedded variables and the metric variables are discovered. They do not know about each other. You have a double set of constraints. Now to return to the holonomic basis, you need to reassemble and unproject the variables. But the question is how? Namely, we do not know what the unit normal is and what are the tangent covectors as functions of the embeddings. Why? Because the space-time metric on which the whole game in the parameterized theory is based isn't given to us. The very aim of geometrical dynamics is to reconstruct it.

30:00 And therefore, we cannot say the embedding variables are the variables which somehow give to that metric a desired form, which is 6. So what can we do? We do not know how to understand it. Now the answer that we gave was fix the metric as far as you can without actually killing its dynamics. You cannot fix it altogether. What you do is that you fix as much as you can by imposing the coordinate conditions. We chose the Gaussian conditions. However, when you impose the conditions, you temporarily suspend the constraints because you are not allowed to vary the variables that produce the constraints. So what you are doing, you parametrize the theory back by the embedding variables. and by that process you restore the invariance of the action and you get, instead of two sets of constraints, a single set of constraints on unprojected, in the unprojected language and then you play the parametrization game Now, the basic thing is that the chants still vary. The Smyrton jet, the space-time field, represents space-time thermophism and they correctly evolve both the embeddings and the gravitational layer. And on top of it, HU, which is not at this stage, that's going to match. only this combination is preserves the suspended weight QM gravity constraints it means if you happen to start with the data which satisfy them and you evolve they still remain the weight QM data if the constraints vanish

32:30 initially they are propagated by the constraints written here. And hence, in order to recover weight you have gravity, you impose these constraints by hand. Not by varying anything, you just go to the speed bank. Alright. This was the stage at which we ended. And it was done in a poor man's way by quite a lot of algebra. Just to make it lighter, I would like to read the final two sentences from the second paper so that you may get some amusement. Now, these are the sentences. We speculated on the possible implications of the scheme gravity. In elementary particle physics, we are already familiar with the ideas of random loops in string models and two-dimensional membranes in non-Aberian gauge theories. Now we must turn to the theory of random three-dimensional hyper-surfaces in a four-dimensional space-time. One is irresistibly reminded axiom of Maria the prophetess One becomes two, two becomes three, and out of the third comes the one as the fourth Now it has an interesting history. Of course we had those things discussed as jokes in between the two of us but when it came to the idea to impose them as the conclusion of the paper with time in the cheap of course Chris objected he said after all I want to keep my scientific reputation and I do not want those words to come to the paper now I played the devil's advocate and I told him No one reads papers to the last page.

35:00 If you are lucky, then of course they will read the abstract. So you are safe. And moreover, if anyone sees that sentence, he will blame it on me. And therefore your reputation will be conserved. Now, let me reveal to this audience that the sentence and the interest in Maria the Prophetessa was actually crisis. But, how did you ever get to that last page? Now, as I told you, we had that scheme but I was totally at loss what that scheme means, dynamical What is the physical interpretation of those suspended constraints? And what is, in fact, the type of physics which lies behind that stone? I realized that only later when we work with Charles' story on the problem of those modified weak constraints. Now here is the story. This is the space-time version of it. You have the equations which prohibit the 0, 0, the temporal-temporal component of the metric to become anything else but minus 1 that fixes the time foliation. And where is the mixed components which fixes the reference frame? In the data ADN version, you say that the left function is equal to unity and the shift vector is equal to zero. The basic question is, when are those conditions to be imposed? Usually people impose them after the variation, after the full set of Einstein's equations was already derived. And it's the familiar game of de-parameterizing the theory in the Dirac's way by imposing the second class constraints

37:30 and then dealing with the scheme that includes them. other way, to do it before the variation. In that case, the variations of the metric are related and it means that the Einstein equations get modified. Now there is the analog in the usual classical mechanics. And you know that if you have a constraint, say a particle moving on a fixed surface. That you can realize the game in two different ways. You use the generalized coordinates and you get what people call the Lagrange equations of the second kind. Or you impose the conditions by Lagrange multipliers. Now in this second way you get through those multipliers, what are the forces by the distance. Now let me show you what happens when you do it the second way. You add to the Hilbert action the action which imposes the conditions on the variables. These are already the parameterized variables of the parameterized version of the field. And you are imposing this condition in the covariant form. You project into the time function, which is given as part of the variables which you rate. You do the same thing with the second best thing. And you have here the space piece of the memory. Now these objects here, the multipliers and the invariants, are space-time scalars. This action is invariant under space-time thermophysics. And so you can vary these variables that are here, are invariant. What do you get? Now this is what you get. You introduce the space-time quantities which are extracted from your multipliers and the other objects.

40:00 You interpret this as an irrotational velocity. And I will tell you what this is to be interpreted. What you obtain from this action by varying the space-time metric is in fact the energy momentum tensor of a specific material system which you know. And that system is an incoherent dust in the motion which is irrotational. But in addition to it, there is this strange term, where fortunately in my very early days in relativity, I studied what is relativistic theory of a heat carrying system. And what you get here is simply what is called the Eckert's energy momentum tensor for a heat-conducting relativistic gas. Now that is the system, which of course, which is not too attractive from the physical point of view But you see what happens under these circumstances namely you come to the modification of the Einstein's theory by introducing the spurious source And when then in the end you say that source isn't there In fact, the dust is only letting a grin of the space. Then you come back with the constraints of the ordinary vacuum and gravity. So that's the way in which I just understood finally what is the physical thing that one is doing. Now let me mention two things. that when we formulate the same thing in the canonical way, doing the ordinary ADM days, then these multipliers in the theory are just recombined into those momenta

42:30 which appear in the version of the theory which we needed to represent the different homophisms. And I would like to say that they are the same things that you can associate with other types of the conditions like the harmonic condition or staying on the hyper-sertices of a constant mean extrinsic energy Now of course then, the question comes This source is very unphysical. Can you do better? Can you really take something that is simple as a phenomenology consisting and just bring it into the game. And that's the story that I and David Brown developed with dust as a standard of space and time in canonical quantum gravity. What you do is that instead of working with the irrotational piece, you simply add in the spatial variables and corresponding multipliers into the definition of what is the four velocity of the system. In fact, this is the well-known decomposition of the four velocity into the crash coefficients. These things that enter there are the crash coefficients, crash potentials. And then you start essentially what we didn't get in the previous meaner as the gauge fixing condition, namely that that four velocity be unity. but we vary only the variables, not the U of loss, but the potentials which enter in the equation. But I want to stress that the coordinates in which we introduce the time function and the spatial counterpart of it, and a clear physical interpretation.

45:00 Capital T is simply the proper time measured on a clock moving along the dust trajectory and the axis which I have here are in fact the objects which provide me with the moving coordinates of the dust lines and also the momenta have a very they have a mass density and momentum density along the dustward lines now there is a slight appendix to that game when you follow how is the motion taking place along the dustward lines parallel to that You get the definition of the energy density of those lines. And curiously, for this system alone, for no other system, you see that it doesn't depend on the dust variables at all. It's only a function of geometry. It's a square root of a quadratic expression. that quadratic expression being the square of the Hamiltonian and then the square of the supermoment. You must define this in the quantum theory by spectral algorithms, of course. And the algebra which implies that the Poisson brackets in this quadratic combination even when the dust is suppressed, it means irrational. Now that was the fact that was quite surprising. Namely, that you have the way of combining the standard constraints of relativity into a quadratic combination which generates an abelian algebra. Now, I never knew what it meant, and that was the other point in which Chris and I entered into interaction. Now, there were students of Chris, Fotini Markopoulou, and Yanis Kuretsis,

47:30 who studied if there are other combinations of the constraints of that time so I wanted to say there was this puzzle, we never got an intuitive feeling about it in the end if there is something that you can do with it in relativity because if you abolish the dust then of course this is a quadratic expression which does improve anything in space-time if dust is there you have the dynamics if you remove the dust you necessarily stay on a single slice but it was a curious aside and this is just another example in which we try to understand the things through working with each other and with the students now I am running of that time that was assigned to me so I must make the decisions I wanted to tell you that there was another time when we were together and that was the time when we tried to understand what is called the problem of time So, I spent three months at Imperial College and Chris spent, I believe, about a month with us in Utah. And unfortunately, it was the only time when Chris visited us. We always wanted to get him back, but of course, as you know, it's not so easy to travel. So, it was a wonderful time and I want only to make a single comment about it It was the second time when we contemplated of writing a book of writing a book on the problem of time in quantum gravity Now we have very intense discussions during that era

50:00 And it was a wonderful experience Now what we found out was That somehow we would need more time to put it together as a joint there Than Chris and I actually had So what resulted from that work were fundamentally two review papers, each of them published separately in different proceedings. My version called Time and Interpretations of Quantum Gravity was published at the proceedings of the Canadian meeting which is very difficult to get hold of It's an obscure address Now the other one called Conceptual Geometrical Problems in Canonical Quantum Gravity was published in another set of proceedings under the crisis name. And, in fact, looking back, it was the best possible decision we could have made. Instead of trying to write a single paper, each of us wrote one version of it. But we did it after a long time of mutual discussions. So we, I believe, mutually feel what the other person is saying. And I must say that I find this a much better way of writing the papers than trying to hammer all the differences and spending a lot of time instead of agreeing to do it independently in a friendly way. So, in a sense, I never wrote a joint paper with Chris after that first paper. But I consider this as, in fact, a joint paper published under two different names. Now, there is one thing that remained of that collaboration, and this is, in fact, a directory which is sitting on my system in Utah, in which Chris would pose and de-pose his files.

52:30 And that directory carries the name of C.J. Jung. It is still there if you want to send the files there, please. It's available. Alright, where am I? I am in this situation when I cannot look forward into the future and tell you what I feel is the thing that I would really like to discuss with Chris personally if I cannot discuss it with the rest of the audience namely Chris changed his interests he changed them in fact by looking at the history version of what previously was just the Weill-Heisenberg group. The idea to get into history group version of the consistent histories interpretation of quantum theory. I believe that Jim will tell you something about it. To lift the things to the level of a field theory. And I believe that I have the version of the action which works in the language, which goes to the phase space that is entirely based on the space-time longings. These are the usual four-dimensional geometry and the four-dimensional momentum and they are complemented by only one pair of other variables, namely the time function, the timing, and the associated momentum. And I think that within that scheme, I know what it means to implement the true thermomorphism in the years and how to connect it with the dynamics of the system.

55:00 I cannot believe it here. Therefore, I would like just to wrap up the thing, to come to the conclusions that I found out for that scheme. and I would like them to tell you why I tried to talk about all of this. Now these are the conclusions. The ordinary GERAC APM action and the history action which I have are actually the very same action on the extended history phase space. and they are expressed nearly in two different symplectic charts. The history Poisson brackets induce the ordinary instantaneous Poisson brackets. And what is wonderful, covariance isn't lost, but it is around the gate with the introduction of each auxiliary structure freely buried in the action. And that's the usual story of Kretschmann. When you introduce additional variables, you gain other additional symmetries. So the theory is invariant with respect to space-time dephemomorphisms, but also the dephemomorphisms of the temporal and the spatial The history phase-space formalism is truly invariant under all of these different homophysms and each of them is implemented by a simplex homophysm on the history phase-space. And finally, the Hamiltonian functional is invariant under these homophysms But it does not, as I told you once before, generate it. So, what is all this good for? And why did I talk about it? Now the canonical history action, which I didn't exhibit you,

57:30 doesn't impose any what is called primary history on the law man. They are completely free, those projections perpendicular, perpendicular, and perpendicular parallel to the surface. Now the Dithelmorphisms echo on all histories, the virtual ones as we call them, not only on the actual histories. And there are many Dithelmorphism invariant objects one can construct on the historic histories. Now, why is this important? Primary constraints in the direct theory impose you with the condition on the state function that it shouldn't depend on the lapse and the shift. However, it's nothing to do with the Diffelmophysm invariance of the scheme. Many people, Jim Hattler and Maguire, would like to take the deferomorphism partitions of the hysterist and to do it in the way that explicitly uses, for example, the left function. Now, this is nothing prohibited by the deferomorphism invariants. And the constraints do not generate them, and they have to do something only with what is going on in the dynamic of the world. So the final thing is perennials, as I call them, constants of the motion are not to be confused with observables, namely things that we can in some sense interpret within the quantum mechanical scheme. So it gives you much more freedom in constructing wanton theories both canonical and in the decoherent hysterics version language So, and I am sorry to do this but it's really the last thing that I would like to do I would like to return back

1:00:00 to that picture here we are with the gate as we are watching the gate passing through it from the past into the future there is that sense of the time that goes inexorably in a single direction from the past to the future So I would like just to show Chris, what we have so many times alluded to in our conversations and like this You can roll this and have that eternal return world in which you robotically the snake is biting its own tail. And I would like Chris to look at this in that spirit. Namely that it's not necessarily the past and the future, but that there is some inherent unity of that. And of course you can combine it in the other picture which we discussed many times of the But, that's for a number of patients. Happy birthday, please. Thank you.