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Jorge Pullin: Okay. So our speaker today is Joe Kelbruga, who will speak about complex saddle points and gravitational path integrals.

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Job Feldbrugge: Right?

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Job Feldbrugge: So thank you very much for inviting me as just told us. I will be talking about a complex set of points. And even though I was invited to talk about complex setup points. I will talk a little bit more general in in setup points in general, and not only the gravitational, but I also will talk a little bit about part control in general.

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Job Feldbrugge: because I believe that in order to understand the gravitational part in the bullets, it's good to take a little bit more of a broad view and see how all these phenomena appear in different

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Job Feldbrugge: fields in different areas.

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Job Feldbrugge: So I've decided to keep my talk as elementary as possible, so I hope that anyone who has an interest will be able to follow the the rough lines of my talk.

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Job Feldbrugge: Oh, if you have questions afterwards. I guess that's easiest with these online talks.

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Job Feldbrugge: Hit me up. Send me an email. I'm happy to talk.

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Job Feldbrugge: So let's start.

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Job Feldbrugge: So what are these complex cell points all about, and why are we caring about them? So the first thing to to notice is that there are roughly 2 ways to do quantum gravity.

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Job Feldbrugge: So the the original way or the the first way can be summarized in the canonical quantization. So what you have is the the Schrodinger equation for quantum gravity. So the wheelers of its equation.

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Job Feldbrugge: And so you have a set of differential equations, and you can try to solve these differential equations as you would with any set of differential equations.

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Job Feldbrugge: And if you now look into something which is known as resurgence theory, then you can do perturbative analysis of these solutions, and what you find is that they will generally follow kind of a trunc series. And this Trum series is nothing more than saying, well, it's not only perturbation theory, like perturbation theory, as in this form, so let's say, powers in H. Bar.

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Job Feldbrugge: But you should really consider a more general series

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Job Feldbrugge: which has a sum over the so called settle points which will be either real or complex. And for every settle point you have to perturbative series where this S is basically the Einstein Hill production or the action in general. For that settle point which I will talk about. So resurgence theory basically tells you that if you want to do a perturbative analysis of quantum gravity in economical framework, you will

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Job Feldbrugge: encounter these complex cell points and you can study counter gravity this way, or in any kind of system, for that matter.

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Job Feldbrugge: So well, this is a well defined question. You have to be careful, because, like any differential equation, in order to make this work, you will have to set boundary conditions.

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Job Feldbrugge: and for that reason it is also common to look at a partens growth formulation instead.

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Job Feldbrugge: So you can also look at the patterns of all gravity just like the just like you have the patterns flow propagator for quantum mechanics.

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Job Feldbrugge: and which in this case is do is you say? Well, let me consider 2, 3 geometries, and let's call me consider all space times interpolating between these 3 geometries, and for every space time in this sum.

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Job Feldbrugge: which I will talk more about you calculate the classical action.

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Job Feldbrugge: and you consider that as a pure face, and then you integrate over all these phases, and then you get an a number which is known as the propagator which tells you a lot about the transition to go from one on 3 geometry, one snapshot of space-time to another snapshot of space-time notice. You have to integrate over the lapse which is basically saying that you have to integrate of all possible times between these 3 to 3 geometries.

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Job Feldbrugge: and also in this formulation of quantum gravity. Do we see cell points coming? So it goes a little bit differently, but is strongly related. So how Picaleif's theory tells you that this highly oscillatory integral

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Job Feldbrugge: can be understood by constructive interference at the settle points.

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Job Feldbrugge: So basically, what picalosis theory tells you is that if you have this oscillatory integral, then most space times won't have a meaningful contribution to your particle.

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Job Feldbrugge: Only this contributions which satisfy the several point equation, which is near the earlier grounds equation, which is basically the Einstein field equations. Only these space times will and the the space time around it will lead to a major contribution to your patent group, and these are the same kind of settle points as we saw here in the economical quantization.

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Job Feldbrugge: And in both cases you have to ask the questions like which sell points? Should I include? Should I include all saddle points select few cell points which I should include

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Job Feldbrugge: are all relevant, real several points relevant

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Job Feldbrugge: which complex sell points should I include? So I will talk about these matters in this talk.

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Job Feldbrugge: So the part interval for gravity is the main subject of this talk.

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Job Feldbrugge: and it was first envisioned by Wheeler, which he says, well, maybe classical trajectories

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Job Feldbrugge: emerge in quantum. If classical trajectories emerging interference in quantum mechanics, then maybe classical space times may emerge in the interference in super space. So super space is the space of all possible tree geometries. And what's basically a wheel face. It is to find there where you go from one tree geometry to another tree geometry. And by studying the interference of this oscillatory integral.

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Job Feldbrugge: he said. Maybe we can try to describe quantum gravity that way

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Job Feldbrugge: and see how the merges of space-time would look like. So that's basically the Avenue I will be taking.

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Job Feldbrugge: So let me give a little bit of an overview of what I will be talking about. I will first talk about oscillatory integrals in general, and this will be mainly in the final dimensional realm. So this should tell you something about which cell points to include and which settle points to ignore.

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Job Feldbrugge: Then I will talk a little bit about some of our histories, and how to actually define these particles, and it's my belief that the settle points are not only a way to approximate the pattern school, but could be actually a key cornerstone to define them in a meaningful mathematical way.

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Job Feldbrugge: and then, finally, I will talk a little bit about how to apply this framework to practical problems, and how you can see that you have to study caustics and Stokes transitions. In the study of, for example, quantum tunneling, but in in quantum processes in general, this is all very much in line with the original data of these subjects which are Feynman and Wheeler, who already talked about this and so many years later. We are still learning new aspects of these questions.

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Job Feldbrugge: So interference. Interference is one of the most universal phenomena in physics. And it's basically everywhere. However, it's often delicate to defined and expenses evaluate. So what do I mean with this

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Job Feldbrugge: an interference? Phenomena in in most cases, in many cases. can be written as an oscillatory integral of this form.

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Job Feldbrugge: So what you do is, you say, integrate over X, let's say, from minus infinity to infinity, or maybe a multi multi dimensional version of this. and for every X you calculate and and face.

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Job Feldbrugge: and you have to add all these faces consistently in order to get your number. But the biggest problem of this equation is that it's only conditionally convergent.

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Job Feldbrugge: And what this means. And I will talk about this in more detail in the next few slides. But just as a heads up. It means that if you take the absolute value of the integrand, then the integral diverges, and that's clearly the case in this integral, because if you take the modulus of the integrand.

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Job Feldbrugge: and you find that it's generally one for a real valued function. F, and that the measure of the real line is infinite

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Job Feldbrugge: the other problem is that if you would have a meaningful definition along the real line, then it will numerically say, sensitive to all the oscillations. You see that you have to add something, you have to subtract something, and only in the the residue do you find?

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Job Feldbrugge: The the answer, and that's that's the second reason why these integrals are highly difficult to evaluate and the the beautiful thing is that picalicious theory solves all these problems in one go under the assumption that F is actually a miramorphic function. So it's locally analytic.

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Job Feldbrugge: So let's talk a little bit about what this means.

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Job Feldbrugge: So conditional convergence can already be seen in sums so many, many of you have studied elementary mathematics, and then you come across sums of this form, and what you often assume is that not only does this converge.

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Job Feldbrugge: but you will require that the absolute value of the integrand or of the summons also converges.

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Job Feldbrugge: However, if this is not the case. We are talking about a conditional convergent sum. and these sums are very peculiar.

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Job Feldbrugge: and you can see that directly from this example. So if you take the logarithm of 2 and you do a Taylor series, and you find that this is equal to this alternating sum.

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Job Feldbrugge: notice if he would be removed, the minus sign and replaced by plus sign in the sum, then it will definitely diverge. However, now you can play around with this sum.

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Job Feldbrugge: and what you can do is you can say, well, let me not do one minus a half, plus the third minus fourth, where I call it all the alt numbers in reds, and all the even numbers in blue. You can say, let me start with the first odd one, and then subtract the next to even ones, and then take the next odd one in the next 2 even ones. And you can basically reorganize your sum. And after a little bit of algebra. You can see that that's equal to a half times the original sum.

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Job Feldbrugge: So you see that when you have conditional cover, convergence sums the ordering of summation is crucial, because if you submit to the the direct way you get a locative way to, if you order them slightly differently, you get half the results.

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Job Feldbrugge: So, and this is a key property of conditional convergence, sums which is absent in absolute convergence for absolutely convergence sums. Where this is finite, you you can reorder them in any way you like, and you will always get the same result.

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Job Feldbrugge: And this is not only a property of

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Job Feldbrugge: conditionally convergent sums, but there it also merges in conditionally convergent integrals. So the the Fresnel integral is a very good example of an only conditional convergent integral, because if you take the absolute value of the integrand, then you see that that's one

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Job Feldbrugge: and the the measure of the real lines infinity. So the the if this is to lead to a well defined results. It has to do with all the oscillations in order for that to come about. So the traditional way of defining this for now to pull is to take a partial interval

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Job Feldbrugge: and then take a limit of R to infinity. And we are very lucky because this Gaussian integral can be solved exactly in terms of an error function. and then you will get this results, which is great. So this is often done. Use canoe spirals.

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Job Feldbrugge: However, now you can ask yourself the question, does this generalize to higher dimensions, and how does the property of reordering appear in these intervals? So what you can do is, you can say, Well, let me take an n dimensional version of this for now, so let me do just the product of N of them.

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Job Feldbrugge: And then you can define the integral in multiple ways.

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Job Feldbrugge: So one way is to say, II regulated on and basically a cube and let the queue go to infinity. And this is a very transparent way of doing it, because you see that this integral factorizes into different ways. And you can just take the product. So then you get this to the power end.

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Job Feldbrugge: However, you might as well say, well, I take the product of a few exponentials in the product of exponentials is the sum of the exponentials.

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Job Feldbrugge: So what you see is that this is basically an an an radial integral, and you can use spherical coordinates in order to evaluate. So instead of having a box which goes to infinity, you can take us n dimensional free sphere and let the n dimensional sphere grow to infinity.

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Job Feldbrugge: And now the surprising thing is that again, this integral can be solved exactly.

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Job Feldbrugge: but it will not lead you to the same result. So if you take the one dimensional version, it's it just coincides with this finale. Let's call it the one dimensional one.

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Job Feldbrugge: If you don't, don't do it. In a 2 dimensional case, you will find that you will oscillate around the well desired result. To square of this number.

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Job Feldbrugge: If you go to higher end you will actually diverge.

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Job Feldbrugge: So you see, that's by changing the way. If summation, by changing the regulator, you get a different answer in in principle, you could change the shape of this regulator and obtain any number you like.

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Job Feldbrugge: This is a little bit concerning because we use these conditional convergent integrals all the time, but still we have to get a unique answer. You cannot

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Job Feldbrugge: wake up the next day and get a different answer. You should have a well defined way to calculate these things and to define them.

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So the origin of this problem

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Job Feldbrugge: can be traced back to the property of the Fubini's theorem and the dominated convergence theorem.

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Job Feldbrugge: So these are terms we use all the day every day, and you don't even notice that there are calls this way when you're working with integration theory. But th, the important thing is that if you have an interval which is conditional is absolutely convergent, so both the intervals, convergent and absolute value of the interval is convergent.

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Job Feldbrugge: then it doesn't matter in which order you sow. So you can integrate, or first over X and then y, or you can integrate over y, and then x, but that these 2 are identical.

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Job Feldbrugge: Moreover, if you take the limit of a series of integrals.

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Job Feldbrugge: then you would like that to be equal to the integral of the limit.

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Job Feldbrugge: And again, this property is only true when all these FM's are absolutely convergent.

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Job Feldbrugge: and when we work with a conditional convergent integrals. And we use these tricks, these these ways of manipulating our intervals, we can run into trouble, because if you have a solution. The the order of integration actually matters, if can matter, and in in particular the the limits of the function. That integral is not

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Job Feldbrugge: any longer the inspiration of the limits.

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Job Feldbrugge: So how to solve this, how to make progress? How do we actually define these conditional convergent intervals?

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Job Feldbrugge: Well, there is a method which you have all learned about which is the settle Point method, and that gives some some clues. How to solve it.

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Job Feldbrugge: So what we can do is, you can say, well, if we have this highly oscillatory, integral, and we assume F to be a meromorphic function so locally analytic. we can use Cauchy's interpol formula to deform the original integration domain.

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Job Feldbrugge: and and maybe improve the properties of the integral, and maybe make it even absolutely convergent. So you start with something which is only conditioning convergence. and you go to something which is actually absolutely conversion.

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Job Feldbrugge: And and what you here see is that you started with, let's say, the real line along which you integrate, and you might be able to deform the Integration domain in some funny way, using some magic trick to this red contour, and maybe along this red contour integral, we will be better behaved. In particular it it will might go through a few settle points which are the red, the black points.

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Job Feldbrugge: and you might be able to do settle point approximation of this thing. And if you basically tailor, expand the interval as an Gaussian approximation around. If we sell point, you get the approximation. And this is a well defined trick, and it's it's it's beautiful formula.

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Job Feldbrugge: However, the first question is, this is actually exact. And in general it isn't because you do this Gaussian approximation at every cell point.

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Job Feldbrugge: And the next question you can ask is, what's the optimal contour? Can I choose different contours and get different results? Just like you could take, take different regulators and get different results?

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And if not, which set up points to include which set of points to ignore. So there are the conflict. Setup points come back in so.

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Job Feldbrugge: And there is this beautiful theory which is known as picnic theory, which sources all in one go. So it's basically the no local version of the cell port approximation. So it's exact. And it tells you exactly how to deform your integration domain.

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Job Feldbrugge: and it tells you exactly which set of points are relevant to your integral. So what is the speaker lifted serial about. So let's assume an analytic action, or just analytic F here.

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Job Feldbrugge: So the first step in Piccolic's theory is to say, well, let me analytically. Continue F into the complex. Explain and let's do it here for the one dimensional case. But it'll also generalize to the n dimensional case.

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Job Feldbrugge: So what you can do is you can say, well, let me take the exponents and consider that in the complex, explain and write it as the real part in the imaginary part we were free to do that

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Job Feldbrugge: which then can do. She can calculate all the settle points of F, and if it is a saddle point of F. It will also be a settle point of H in capital to H. By the Cauchy Riemann equations. And what Pika Lipsy, seeing now tells you is that you can start with your original integration domains, let's say the real line.

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Job Feldbrugge: and you can deform it to a set of leftist symbols which are steepest descent manifolds off a set of relevant settle points. and what pica lips in addition, tells you is that

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Job Feldbrugge: a cell point is relevant if and only if the steepest ascents, manifold intersex or regional Intubation domain.

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which is a very powerful tool.

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Job Feldbrugge: So what he basically tells you is that in order to

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Job Feldbrugge: reform or resem or reformulate your highly oscillatory conditional conversions. Integral. You calculate all the setup points of the F in the complex plane, so both real settle points and complex set of points.

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Job Feldbrugge: For every settle point you calculate the steepest descent and steepest asset manifold, which back to the small H function which copy serves the magnitude of your integrals. So here you see in J the steepest descent manifold, and KC. To C. Steepest asset manifold. Now that's along the steepest descent. Everything converges really quickly, because integrals is exponentially suppressed.

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Job Feldbrugge: and along the steepest ascent direction it would be exponentially diverging. Then you intersect the case with the R, with the original integration domain, and that tells you exactly how to deform the interval.

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Job Feldbrugge: Moreover, along a steepest descent, manifold along A. J. You can prove that capital H is constant. and you can so write the integral as a sum of the relevant several points which have determined using this intersection

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Job Feldbrugge: times, e to the power IH at the sell points. which is basically the face of that settle point times an integral which basically goes like a Gaussian. So for every cell point you find an integral which goes like this.

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Job Feldbrugge: and which is absolutely convergent because it doesn't oscillate. And and and you have to find your integral without any trouble using the proxy that the original integral was analytic

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Job Feldbrugge: or metamorphic. And this is basically what picola Serine does.

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Job Feldbrugge: So let's see it in action.

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Job Feldbrugge: So, as I said, if you take the Fresnel interval and you take the partial interval, you take the limit and you get this result, which is the result you learn in many textbooks.

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Job Feldbrugge: if you know, take the 2 dimensional case and use would go from 0 to R, using spherical or polar coordinates, and you find this results, which is oscillating, and it will never converge, because you see that you have each to power. R. Squared where R. Goes to infinity.

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Job Feldbrugge: However, if you would do picolithis theory, you would not do none of this regulator business. We go for a partial integral, and then go to infinity. Rather, you say. Well, let's take these integrals, and certainly that a little bit more so. The exponent is, IX squared. You calculate the saddle points. In this case there are only one cell points, which is X is 0.

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Job Feldbrugge: You say, well, that's several point has to be relevant because it's already on the real integration domain. So the steepest assets has to intersect or regional integration domain calculators, deep, decent manifolds.

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Job Feldbrugge: which is just a straight line tilted by 45 degrees. So that's basically this line.

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Job Feldbrugge: And now you can change coordinates to you, and you find that you are left with a Gaussian interval

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Job Feldbrugge: which is easy to evaluate, and you get the same results. which is, of course, for the one dimensional case. Just a very nice trick, and it looks like a coincidence. Does these 2 have to agree?

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Job Feldbrugge: However, if you now go to the 2 dimensional case, you can do the same trick. And you find basically that you have to rotate both X and Y by 45 degrees to complex plane. And we're still only one several points, which is the origin.

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Job Feldbrugge: and you find the results just using the same Gaussian trick.

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Job Feldbrugge: So you see, that's where, as the the spherical regulator oscillates around Ipi.

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Job Feldbrugge: He, that epicalicious theory doesn't deal with any oscillation. So you don't have to follow the oscillations in the Knush Bible.

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Job Feldbrugge: and he gets the results straight away, which is very neat. So this is the the way piccolic theory proposed to define these oscillatory intervals. And it's basically coincides with the I actual prescriptions. And all these analytical tricks which we use all the time.

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Job Feldbrugge: which is neat.

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Job Feldbrugge: So another example, where you can use speak a list here into great effect, is lensing in wave optics. So I've included one small example. So if you take and thin lens.

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Job Feldbrugge: so you say that there are some light rays coming from a star, and they hit and gravitational lens or a plus one lens in the same plane. And you can describe the interference pattern which emerges from this using this one dimensional, integral. So if you see that it looks a little bit like this for knowledgeable, because there's an I, and there's an X squared.

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Job Feldbrugge: and in addition, you see, there are some face variation which captures the lensing of the thin lens.

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Job Feldbrugge: And what you find is that for this configuration for Alpha is 2.

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Job Feldbrugge: You find that for y is 0. So that would be looking straight on at the start. You see that there are 5 seller points. So you take this thing, you calculate several points, find 5 seller points to red points.

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Job Feldbrugge: 3 of them are real. 2 of them are complex. You find 2 polls at plus or minus I, which are because the denominator it it was 0. So these are these black points.

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Job Feldbrugge: You can calculate the steepest ascent and steepest descent manifolds of all the bread settle points.

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Job Feldbrugge: and you see that these real cell points naturally are relevant because they already lay on the original integration domain, which was a real line. See that this

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Job Feldbrugge: settle point here on the top and below have no contours which intersect the original integration domain. So they are definitely irrelevant.

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Job Feldbrugge: So you see, that's using and very geometric analysis of complex analysis, you can basically determine which several points are relevant. And in addition, you can calculate the steepest descent manifolds, which are these lines.

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Job Feldbrugge: You see that they are naturally deformable to the original intuition, so you could start with your original integration domain, which is the real line, and now can deform it just geometrically to this blue contour.

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Job Feldbrugge: And now you can analyze your integral along this blue cone, where you find that everything is absolutely convergent, and it converges really quickly, because away from the cell point everything converges exponentially.

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Job Feldbrugge: You can imagine that's in a different situation for a different while. Let's say also, these complex setup points can all relevance. And in particular, what you find is that this is 3 real setup points. So that's in wave optics and and optics in general language, and 3 image region. So that would be kind of the 3 image region. Here

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Job Feldbrugge: we see that the light rays are overlapping.

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Job Feldbrugge: If you now move a little bit to the right, and at some point you will find that 2 of these images will emerge and subsequently have only one relevant ray which correspond to one relevant real settle points.

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Job Feldbrugge: But in addition, we'll find that there are also a complex set of points which will be relevant because the steepest ascent manifold will intersect or regional integration domain and picose theory tells you that one is relevant.

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Job Feldbrugge: So let's now connect this back to the regulators. So

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Job Feldbrugge: what we are proposing to do is to say, well, we have this n dimensional, highly oscillatory, integral, which only conditionally convergence. So this is kind of a divorce case scenario.

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Job Feldbrugge: So what we are doing is we saying, well, let's not consider these regulators where you do take the partial summit, because you see that if you take different partial sums and you might get different answers.

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Job Feldbrugge: So what we are proposing to do is to say, Well, let's introduce a regular to G, which is converges to one as R goes to infinity, so in the limit of R to infinity. You are doing nothing. Basically, it's analytic in the complex plane and dies off at infinity fast enough. So it's a fairly large but restricted class of regulators, which is G, so what we do is, you say? Well, let's instead of taking these partial sums, let's insert this regulator G into my integral.

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Job Feldbrugge: What we now can do is we can do bigger Liftsis theory on this integral because this is for fair finance. R is absolutely convergent.

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Job Feldbrugge: So what you obtain is the limits of our goes to infinity

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Job Feldbrugge: of a sum of relevant several points of the original integration domain of the original integral on the grants and and A A along these intervals everything is absolutely convergence. So everything is well defined. So now, using the dominated convergence theorem. We can move the limit inside of the input. And you can say, Well, G of R, for art going to infinity is one.

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Job Feldbrugge: So it disappears. So it basically tells you that if you take a smooth regulator

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Job Feldbrugge: of your original, conditionally convergent interval. it will be identical to the result you would get with Picasso.

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Job Feldbrugge: which is very nice. So by going away from these very harsh regulators, where you take departure some. And you you find that by difference, regulators, you get different results. If you do it more smoothly, you get a unique results, which is also coincide to speak on the fish theory

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Job Feldbrugge: and and which is unique. So it doesn't depend on the properties of your regular, which is something you always desire. So this is basically the the formalism I will be using.

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Job Feldbrugge: So so what nice and well to do this for 5 and dimensional intervals.

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Job Feldbrugge: But of course we are quantum gravity. We would like to do the real part into

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Job Feldbrugge: which is an infinite dimensionally so, what is this button goal? So the button was, and marvelous formula described by Feynman and Wheeler, and then basically refined by many other people and has been studied since inception for many decades now. And it's basically one formula which claims to capture everything.

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Job Feldbrugge: So. of course, except for quantum gravity at the moment.

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Job Feldbrugge: So this is kind of the button pool for the standard model, where you see that you have gravitational components. You have much world theory. You have the Lagrangian for the hex particle. You have dark matter and and you can try to

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Job Feldbrugge: make this integral mathematically sounds, and try to see. How do we actually define this? Because what you already see is that the integral measure is here missing. So this integral is not well defined as

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Job Feldbrugge: given here. Alfred, that's that's what we are working on.

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Job Feldbrugge: So generally. What you do is you? Let's go to quantum mechanics. Let's take things as simple as possible. Let's say, well, you go from. Consider the pattern to go to go from X 0 to x one in time. T.

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Job Feldbrugge: And you are trying to make sense of this again, as I said. Well, the the id, the hope is that you can stationize this integral meaning. That's the dominant contribution comes from paths around to set up points, which is the Lagrange equation, which is just fsml, a so just Newton's law.

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Job Feldbrugge: And the idea is that classical motion, just like Wheeler proposed.

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Job Feldbrugge: emerges due to the constructive interference of this highly oscillatory interval.

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Job Feldbrugge: But which settle points? Should you include which classical paths are relevant.

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Job Feldbrugge: Can we make these ids really rigorous because these part integrals, the infinite dimensional integrals, are very and mobility beasts? Can we describe. Everything's using classic classical paths or things which are quantum and and cannot be described in this way.

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Job Feldbrugge: So that's something I will be talking about next. So first is to notice, why should we care about these details? Well.

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Job Feldbrugge: yeah, there's a question I see.

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Deepak Vaid: Yeah. So I mean, my question is, I guess, that when you're talking about a path integral for for gravity.

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Deepak Vaid: then in general, one would expect. topology change to also be a part of

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Job Feldbrugge: part of the sum. Right?

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Deepak Vaid: So I mean, in what way is is that being accommodated.

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Deepak Vaid: or is it being accommodated in the normal, gravitational path integral? III mean, it's not obvious to me that it's there.

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Job Feldbrugge: No, I think it's indeed not obvious. So

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Job Feldbrugge: Claudio booster who went first with the name Titoborn as this beautiful formulism of the button gravity which degrees of freedom.

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Job Feldbrugge: But also there. It's not yet clear to me whether it the

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Job Feldbrugge: change of topology is basically taken into account. And I also won't really be talking about that here.

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Job Feldbrugge: So I completely agree with you. But apart from that, even for quantum mechanics, our tool and complex several points which need to be addressed.

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Deepak Vaid: Well, I mean, recently, like this, this, all this work that has been done by

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Deepak Vaid: Net Angle, heart, and others. You know, they they talk about doing the Euclidean path integral, and they say that we we can include wormholes in this description. So

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Deepak Vaid: of.

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Job Feldbrugge: yeah. But then the question is indeed, are these workholes solutions actually relevant to integral or not?

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Job Feldbrugge: And

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Deepak Vaid: okay.

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Job Feldbrugge: thanks for that. You're right thinking in the right direction. But th what I will be talking about is not advanced enough to actually address that at the moment, I think. but hopefully, it will, the future

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Deepak Vaid: sure. No problem.

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Job Feldbrugge: So why should we care about these parts? Well, Feynman already knew that his definition of the part of the pool was a little bit iffy. So he basically describes that even though he is aware of the the potential issues of this integral, the concept might be sounds.

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Job Feldbrugge: So that's what he writes in his book with hips.

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Job Feldbrugge: which is a great book for button tools. So it's good to know that the founder already noticed that these issues exist.

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Job Feldbrugge: So the the, the reasoning which I will be taking this talk is basically nicely ser summarized by Terra. So he says, well, the point of vigor is not to destroy intuition

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Job Feldbrugge: is that it should be used to destroy all that intuition and clarify evaluating good intuition. So basically, what I will do is I will formalize the part of the goal, using the set of points in a minute and talk a little bit of the mathematical problems and the the ultimate goal is to develop a a better formulation of the patent rule which can actually be used to evaluate problems, and hopefully also talk about these face transitions in the future.

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Job Feldbrugge: So before we can talk about the settle points for the part of scroll, we have to talk a little bit about infinite dimensional integration theory. So the first thing to notice is that if you want to define a metaphor well defined

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Job Feldbrugge: integral. You first meet in Sigma algebra.

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Job Feldbrugge: So what is a signal autograph? Some of you might already know this. But let's just go through. It's step by step.

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Job Feldbrugge: So Sigma algebra offers space. Omega

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Job Feldbrugge: is a space of subsets. So just like in topology, you can have a space of all open sets or space of all closed sets. So in measure theory, you have a space of subsets over which you like to integrate.

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Job Feldbrugge: And the first thing you want is that the whole set is in your set of subsets.

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Job Feldbrugge: The 0 set has to be in the set of subset, because you would like the complement of any subset to be already at your sigma.

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Job Feldbrugge: so point one and 2 are kind of trivial point 3 is a little bit more nuanced, so you would like to say, I would like to be able to take accountable set of subsets, and I would like to pick take the Union, and I would like this union to be again in my.

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Job Feldbrugge: So if your algebra satisfies these 3 conditions. then it's a signal algebra ur all good.

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Job Feldbrugge: So why do we need a Sigma algebra. Well, we need a sigma algebra, because we want to define a measure, and a measure is a map from the subsets from the sigma algebra to the real line, or the positive real line, basically including infinity. In some cases.

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Job Feldbrugge: So a Sigma measure is in map from the Sigma algebra to positive numbers

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Job Feldbrugge: which has the property, that the 0 set is of measure 0, which is great, because if you integrate over and 0 domain, you would like that to be 0,

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Job Feldbrugge: and in addition, you would like to be able to say, Well, if I take accountable subsets of the space in my signal right. And I, if I take the accountable union of them, which is well defined, because that is also an element of the signal algebra

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Job Feldbrugge: we'd like to measure to be equal to the sum of the subsets whenever they are pairwise disjoint. So if you take 2 subsets A and B, and they have no overlapping points, and doesn't matter whether you integrate over and be simultaneously, or you first integrate over A and then integrated for B, and then add them together.

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Job Feldbrugge: So that's the property which is required for Sigma measure. and once you have a Sigma algebra and sigma measure, you can define intervals. So what you can do is, you can say, well, let me consider simple functions which are just block functions. So it's basically all for I if X is in IA AI and all by 0.

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Job Feldbrugge: So these simple integrals, these simple functions, can be easily integrated because you can just say, well, the sum of the number of blocks you take, and you multiply by the height of the block, which is Alpha I. And then you multiply by the measure of that se subset. So that's all in 2 different.

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Job Feldbrugge: But now what we can do is we can say, Well, if you have an arbitrary function, which is, let's say, absolutely convergence. and which is also posted, let's say, and we can start to approach that function from below using these block functions using these simple functions. And we can say, well, the the interval is defined in the limit.

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Job Feldbrugge: We consider all functions which are Domini nominated by the function G, and then you say, well, I take the maximum office check. Basically.

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Job Feldbrugge: if you take an options which you can be both positive and negative, you just have to do it for the post apart in the negative part and then subtract the difference.

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Job Feldbrugge: So that that's how you define integrals. It's important to think you have to sigma measure, because otherwise there's no sense to what an integral is.

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Job Feldbrugge: So if you go to an arbitrary textbook of patent school theory like Feynman Hips.

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Job Feldbrugge: then what they do is, they say, well, if I take the time, discretized version of my pattern tool, then I get a finite dimensional interval.

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Job Feldbrugge: and what I do is I describe this measure in terms of le back measures. So it is le back measure of the set. A, B is just B minus a. So if you have n of these measures together, then you have the products, and the idea of the platform goes to take the limits

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Job Feldbrugge: and to infinity where we find your time discretization at infinite.

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Job Feldbrugge: and you get some objects which looks a little bit like this. However, unfortunately, this thing is no Sigma measure. so it doesn't satisfy the criteria which we just described here.

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Job Feldbrugge: and you can trace this back to the translational variance. So if you take this, set, the one dimensional, the back measure and just move the interface a little bit to the right, and the measure is still the same.

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Job Feldbrugge: It doesn't matter where the interval is located, so to say. So it comes translated, and the measure is invariant.

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and you can. You know what you can do is you can say, well, let me take this

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Job Feldbrugge: and studied this infinite dimensional object.

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Job Feldbrugge: And you can say, well, the the measure of the unit n dimensional queue is definitely one, because it's not every dimension. It's a product of once.

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Job Feldbrugge: I can now subdivide this animational queues in 2 to the power and sub cubes of half dimensions basically

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Job Feldbrugge: and due to translational variance. I can move them all to the let's say the left corner cube.

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Job Feldbrugge: and you can say, well, so you find that one is equal to 2 to the power n, which is the number of cubes. Times the measure of this smaller cube.

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Job Feldbrugge: Now, if you take the limit of into infinity.

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Job Feldbrugge: then you find in order for this equation to hold, the measure has to be 0.

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Job Feldbrugge: And this is very problematic, because if you take this, an infinite dimensional object like this, the stage of paths.

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Job Feldbrugge: And you try to define the infinite dimensional back measure on it. You find that it's impossible to do this because of translational inference.

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Job Feldbrugge: So this is the big problem with this Dx. As is traditionally described.

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Job Feldbrugge: so many measures don't work in infinite dimensions. However, there are measures which are actually well defined. Internet, infinite dimensional space. So these are, for example, known as Brownian motion.

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Job Feldbrugge: So if you consider a random walk which goes from 0 to 0 or from 0 to another point.

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Job Feldbrugge: And you can

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Job Feldbrugge: describe the random walk like this, and you can describe subsets in the space of paths by saying, Well, all the paths which go through a set of windows, let's say so slits in the double slit experiment.

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Job Feldbrugge: and for every space which is defined this way. You can describe the winner measure in closed form and also at infinum. You can show that the winner measure exists and has all the nice properties you want.

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Job Feldbrugge: So the winner measure might manages to avoid this problem for the back which the back measures are, because there's no notion of translation, you cannot say. Well, let me split it into small queues, and then rearrange them and post them in the same position, and say, Well, it's 2 to the power end of them. So that's basically the reason that the winner measure

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Job Feldbrugge: is an infinite dimensional sigma measure, whereas the infinite product of ly back measures is not

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Job Feldbrugge: so. We can use this, the back measure. 2 defined passenger rules in Euclidean signature.

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Job Feldbrugge: So this is the final cuts formula which I will just briefly discuss in order to give a sense of how it works. So what you do is start with your original integral, which is this highly. I don't know what to do with this, because it it's it's very delicate to define, difficult to evaluate.

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Job Feldbrugge: So what you do, you should say. Well, let me go to imaginary time, whatever that means. And you basically replace this, I by a minus sign. And you see that this minus sign becomes a plus sign.

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Job Feldbrugge: And you say, Well, this is better, but it's still ill-defined. And why is this ill defined? Well, this Dx doesn't really make sense.

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Job Feldbrugge: Moreover, if you study this more closely, and you find that for Brownian motion. which is what we will get to.

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Job Feldbrugge: and the paths are almost everywhere not differentiable. So what does it mean to have X dot squared? This will diverge.

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Job Feldbrugge: So what basically cuts realized is that you can combine this MX. Dot squared over 2

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Job Feldbrugge: with this Dx

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Job Feldbrugge: into the Brownian motion.

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So you basically take the kinetic turn.

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Job Feldbrugge: And this DX, and even though by themselves they are ill defined

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Job Feldbrugge: together, they lead to a well defined object. So you basically see that the the wellness of the paths and the wellness of the measure basically cancel and give you a well defined measure.

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Job Feldbrugge: So, using the Brownian bridge measure. which is just new. B, you can say, Well, let me define this thing

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Job Feldbrugge: after normalization as just this interval, which is perfectly well defined, and any mathematician will be happy to play with that.

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Job Feldbrugge: So you basically get an interval of the potential. And this is a well defined integral, and it has no problems except for the fact that it's working with imaginary time, and that might not be a concern.

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Job Feldbrugge: However, if you want to, despite evolution, as we typically do in quantum gravity

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Job Feldbrugge: in in systems, then you might want to work with the real time.

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Job Feldbrugge: And and since this trick really relies on the fact that you replace the I by a minus one.

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Job Feldbrugge: you are in trouble.

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Job Feldbrugge: However, box I claim is that if you use, combine this intuition with this winner measures with Pica lifts theory, you can find a way out. So, as we saw before. If you take an n dimensional integral, and you can add this regulator, and you get these speaker lifts into one, even though there's still an eye here around every cell point along the steepest descent and forks. It's like a Gaussian interval.

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Job Feldbrugge: And and what we found while ago, and this is supported by different

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Job Feldbrugge: investigations. For example, researchers. People also move in this direction. What we find is that you can define and well defined, possible in real time. By which we define in this form, where you basically have a set of complex cell points and real cell points

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Job Feldbrugge: which we'll talk about in a minute. And for every cell point, you basically can stick on a winner measurement.

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Job Feldbrugge: And it's not actually you win a measure. But this is something which is defined in terms of the main measure. So basically, the integrals defined along the left symbol, which is the the steepest descent manifold of this set of points allows you to define a well-defined sigma measure, and then you can integrate.

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Job Feldbrugge: So again, you see that these complex setup points are really important, because if you wouldn't have the notion of complex seller points in the relevance, then you wouldn't be able to make this definition at all. So this is basically the 5 and cuts formula in real time. That's at least what we hope.

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Job Feldbrugge: For. Every. Yes.

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Ding Jia: Verification question, what is Delta X.

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Job Feldbrugge: What this

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Ding Jia: Delta? Tell? X. Oh, it's basically just a face factor. So in this W could be approximation, it should the exponent of C

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Job Feldbrugge: product of C

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Ding Jia: no. The the second term in the yeah, this E to the power. IT. You mean next term.

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Ding Jia: See you today.

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Ding Jia: and and see

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Ding Jia: before the measure term C to the Delta X

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Job Feldbrugge: C to the Delta X. Oh, it's is Mc. Is just enumerating the different settle points

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Ding Jia: right? What is Delta X.

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Job Feldbrugge: Oh, the the fluctuation with respect to the setup point

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is a classical path. And then you can say, Well, I integrate over all paths with respect to this. So you basically say, well, I have a test for activation which goes from 0 to 0, which is Delta X

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Ding Jia: dot X is integrated over the whole stiffness, descent counter. Right? Yes. Okay. Thank you.

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Job Feldbrugge: I see another question.

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Cong Zhang: Yeah, yeah. So so, so, my question is like, why it has to be complex. So can I use? Can you use this? this real critical to to do, to to to to get this, this, this, this, this pronoun bridge mirror.

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Job Feldbrugge: Yes, so you will find a set of real cell points which are relevant and complex cell points which are relevant. And you have to do it for both.

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Cong Zhang: Okay, okay. So let's say, I see

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Job Feldbrugge: so the the of course, then the big question is which set up voice include and which set up points not include. So in the interest of lifetime. Let's go a little bit more quick and say, Well, let's consider first the no boundary proposal, which is a very simple problem to address with these methods. So for the boundary also, you get an which looks a little bit like this.

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and you are very lucky because the the the possible skill factor can be done exactly because it's for traffic.

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Job Feldbrugge: So you basically obtain an one dimensional integral over M, which the labs

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Job Feldbrugge: with an oscillatory face, which is E to the is where the classical action is cubic in N,

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Job Feldbrugge: so what you can do is you can now unlistly continue S. 0 into the complex plane.

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Job Feldbrugge: And what you obtain is this kind of figure. So you find for several points. And now you see that only several point one is relevant because only several point one has a steep assent manifold which decay

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which intersects the real integration domain.

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Job Feldbrugge: So what you find is that the optimal deformation and the way to make this integral

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Job Feldbrugge: well defined is instead integrated along this chain. And this is something we did a few years ago, and what we find is that the patents will, for the no boundary proposal is well defined.

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Job Feldbrugge: However, the prediction is a little bit different from what Arthur Hawking originally proposed, and we find that large fluctuations

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Job Feldbrugge: are basically more likely than a small fluctuations. Of course, this is within the several points approximately within the

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Mini super space approximation and only using participation. Really.

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Job Feldbrugge: that's something where you see that you can use speaker lists theory in a direct way that what I rather would like to talk about is more nonlinear problems. So let's see how this whole framework can work in action in the last few minutes.

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so

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Job Feldbrugge: you could do the harmonic oscillator. But I decided that it's more interesting to consider the teller potential. So the telepotential is an potential which goes like one over the Cos squared of X.

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Job Feldbrugge: And this is a very nice model, because it's exactly solvable. In particular. It has been well known for many decades that the energy eigenstates of this

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Job Feldbrugge: Hamiltonian operator can be written terms of legendre functions. So these are the piece for some different configurations depending on fee 0. Basically.

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Job Feldbrugge: And if you have the energy eigenstates 5 k. Then you can write the spectral representation of the propagator.

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Job Feldbrugge: so you can calculate the propagator using a one dimensional interval which is bit tricky to do it. But you can deal with it. So now you can calculate the propagator for this

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Job Feldbrugge: autumn system using very elementary techniques which are independent of patentable techniques about settle points. The point is that, you know, can also do the settle point approximation, using these complex settlements and real setup points and see how they compare.

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Job Feldbrugge: So if you just evaluate this integral

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for x 0 and x one as a function of T,

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Job Feldbrugge: and you get these kind of plots. So this is absolute value squared, and the lower plots is the real part for different times. and you immediately see that there is a structure in these blocks which is like here. You see that there isn't place where you see only monotomic evolution. You see a place where you see interference. Loner

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Job Feldbrugge: and these interfinuma are integral to the real time button tool, apparently because you just evaluate a propagator. And this is what comes out.

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Job Feldbrugge: And the nice thing is that you can understand these interference fringes using classical theory. So using the classical ends, the classical equation of motion. So Fs, MO. 8.

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Job Feldbrugge: So what you can do is you can now look, for instance, you can look for cell points. You can look for ways to solve the boundary failing problem. So what you can do is you can say, well, let me start on the left from the barrier

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Job Feldbrugge: and let's try to shoot for classical paths. View classical paths

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Job Feldbrugge: to the right.

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Job Feldbrugge: It's from the left to the left, I mean. So here you see that if you go from minus 5 to minus 6. There's only one way to do it. So you just move to the left. If you now move the final position in towards the barrier, you find something remarkable. All of a sudden you find that there are 3 real classical paths. So 3 real points which should you should include in your definition of your partner.

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Job Feldbrugge: you know. Keep moving this final position towards the barrier. You see that they merge again. So here you see them. They emerge here, in, in, there, and here they fuse, fuse again, and then afterwards you find that there's only one solution to the boundary value problem

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Job Feldbrugge: where the particle has to roll up to the basically the barrier and has to stick there for a while, and then it rolls back down.

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Job Feldbrugge: So you see, that's even in classical theory. There are different configurations where you can either have one real seller points, or you have have 3 real 7 points to department.

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Job Feldbrugge: and these exactly corresponds to these interference. So you find that within this blue triangle there are 3 real classical server points, and outside you find only one classical points.

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Job Feldbrugge: But you see also fringes here, and this will have to do with these complex several points.

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Job Feldbrugge: So these caustics are very nice to describe the the patterns role to describe the propagator, however, they also already emerged in the evolution of particles. So here you have the particle which is a Gaussian stage which interacts, bounces off the barrier and seats at the final time. It's completely bounced by these 2 green lines, which are which are the full costs where you see that go from a transition from

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Job Feldbrugge: single real solution to 3 real solutions.

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Job Feldbrugge:  if you now consider this oscillatory integral, which is the unfolding of the so called kosto stick. You can basically classify the region in space of Mu one and Mu 2, and

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Job Feldbrugge: track, which set up points are relevant to the passenger or to this oscillatory. In school

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Job Feldbrugge: you find that you have a region where you have 3 real, relevant civil points, which is this one which the days are all real. You see, they are deformable to the original integration domain.

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Job Feldbrugge: Then you see that you have this caustic, which is this black line.

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Job Feldbrugge: adds the caustic 2 of the Settle Point merch, which is exactly not like what we saw here and what we see here

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Job Feldbrugge: after you have crossed the caustic, you find that one of these 2 saddle points is still relevant in the complex. The other saddle point is complex, with complex, conjugates and is irrelevant to the integral.

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Job Feldbrugge: So you find that when you cross caustic you find that 2 setup points have merge and become complex. One of them is relevant.

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Job Feldbrugge: In addition, you see these Stokes lines which are these red lines, and if you now pass that line, then you find that

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Job Feldbrugge: and the relevant, complex set of points have become irrelevant. So he see that only one relevant real set of points, if a relevant wheel set up points and then relevant complex set of points.

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Job Feldbrugge: and these are often by these stock slides.

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Job Feldbrugge: If we now apply this to the Rose Moors barrier, which is one of a cost theory.

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Job Feldbrugge: and what we can do is for different H. Bar. We can calculate exact propagator using the spectral representation, which is the blue line.

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Job Feldbrugge: If we do now, a several point approximation, using only the real classical path. You get the black line. So you see that it works rather neatly in the 3 image region.

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Job Feldbrugge: But here you miss all these oscillations. which is to be expected, because that has to do with this complex set of points.

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Job Feldbrugge: If you now look in the initial velocity plane, the complex initial velocity plane. When you can see these 3 real solutions corresponding to the 3 image region with a point on this line in region 1, 2, and 3, which are the 3 real velocities

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Job Feldbrugge: corresponding to these 3 paths.

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Job Feldbrugge: As you now move the final position past this caustic, you will see that 2 of these solutions will merge at the intersection of this blue line and the red line.

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Job Feldbrugge: and afterwards these complex settle points can be seen as complex initial velocities, and they will move along these blue trajectories. If we now include the settle point approximation for that settle point. This complex set of points you get the red line. So you see that in the 3 image region they coincides.

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Job Feldbrugge: But then you see that in the region here you. You capture the oscillations, which is exactly what we saw in this plot. So you see that these interference fringes outside of the caustic region are direct signature of the relevance of complex cell points.

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Job Feldbrugge: which is rather neat, that you can see it so explicitly that indeed, complex solutions

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Job Feldbrugge: to the boundary value problem should be included.

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Job Feldbrugge: So this is one place where you can see it. Another place where you can see it, which is neat is, you can go to the energy propagator. And why do I go to the energy Propagator? Because I want to describe tunneling, and if you want to describe tunneling, then you would like your particle to be at a fixed energy, because then you can say it's below the barrier or above the barrier.

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Job Feldbrugge: So again, you can describe the passenger pool like this.

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Job Feldbrugge: and you find that there are the cell points. Now take this form and this rather similar to real sophisticated quantum mechanics, and also calls gravity, because you basically have an integral over T, which is like an integral over the laps.

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Job Feldbrugge: What you can do is you can say, well, let me put energy above the barrier that's at first, and calculate the exact propagator, which is this blue curve. So you see there are a few oscillations on the left, and it's monatomic to the right, and it means particle. Go, go from the left, and if it goes to the left, and it has reflected that you find this interference phenomenon. If it actually moves across the barrier, it's just monotonic like this blue curve

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Job Feldbrugge: you now do the real seller point approximation you get this black curve. So you see, the black curve is completely ignorant of these oscillations on the left, which correspond to the fact that classically it's not allowed to bounce off the barrier when the energy is higher than the top of the barrier.

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Job Feldbrugge: however, you can find some complex set of points

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Job Feldbrugge: which lives in complex velocities, initial velocity, space. And if you include this, you get the red line. So you see that you nicely cover all these oscillations

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Job Feldbrugge: can also do it when the energy is below the barrier where you describe quantum terminally. You see, that's

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Job Feldbrugge: Cla, classically, it's forbidden to go through the barrier if you start to the left, and you try to go to the right, and there's no real setup points, so you say. Oh, the the probability is 0.

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Job Feldbrugge: However, if you now include a complex seller point, you can actually capture the evolution. So this is the the red line where you see that indeed, you can capture the the, the tunneling through the barrier, using a complex set up points.

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Job Feldbrugge: one caveat which to note is that

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Job Feldbrugge: as you unleash the continue the setup points into complex plane. You can have the phenomenon that your classical path hits the singularity of your potential. So the potential we have. Here's one of a cost squared that's perfectly well defined along the real line. If you go for imaginary X, and you get instead of a cost, you get a co-sign, and if the cosign is 0, then you divide by 0. So you are in total.

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Job Feldbrugge: So these are the polls which are here. So these green points. What you find is that if you start in the left from the left and you push through the caustic, you get a relevant, complex set of points, and if you now keep moving to the right

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Job Feldbrugge: and try to go over the barrier. then you find that your complex clash or path hits this singularity.

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Job Feldbrugge: and after you hit the singularity, you have to do some analytic continuations in order to make sense of it. So if you cannot simply ignore the settle point after it hits the similarity. If you are literally continue, then you recover. The right interferes the the right results. So you you basically can reproduce the turning rate.

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Job Feldbrugge: So just to conclude. if you want to describe quantum duddling, using complex cellar points, using the energy propagate, you can do it. and you find that the instant on the complex desperad is given by this function.

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Job Feldbrugge: So what you find is that it's simply moving into the imaginary direction.

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Job Feldbrugge: But in order to

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Job Feldbrugge: really make sense of this, you have to include this square root of E 0 factor, which has to do with the fact that you cross this port. So it means that you are hitting the singularity. So you have to get an additional factor from that.

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Job Feldbrugge: If you include that factor.

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Job Feldbrugge: then you are complete coincidence with the do we give you approximation of the turning rate. So, using Wb, the approximations of the energy for the time, independent scrutiny equation. You can get an approximation of the turning rate

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Job Feldbrugge: if you do the tele point, approximation of energy propagator. where you have to include complex classical paths.

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Job Feldbrugge: and also have to make do with these singularities. You can retrieve the same formulas. So that's nice, because then you can really say that you have reproduced quantum tunneling with complex classical paths, and these complex, flexible paths are pretty important to describe this.

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Job Feldbrugge: so no link.

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Job Feldbrugge: So we saw some costs for the potential barrier. If you do it for the well, you get even more intricate interference you can see here, for these are very familiar to people who study radio astronomy where you have wave optics. It's kind of fun to see that it also enters quantum mechanics and and probably is relevant to partly for gravity, where you also should have costs, and you also should have singularities where you have the formation of black holes.

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Job Feldbrugge: and these need need to be taken into account to determine which complex solution to the boundary failing problem is relevant to the part of the board in which one should be ignored, just like which complex ways in wave optics need to be included and which one needs to be ignored.

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Job Feldbrugge: And and it seems that the caustics can tell you a lot about which ones you should include and which ones you should ignore, and where the stocks alone are happening.

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Job Feldbrugge: So it's kind of a beautiful blend of analysis and geometry

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Job Feldbrugge: using complex variables.

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Job Feldbrugge: So with this, I would like to conclude. So II would have tried to convince you that interference is really sensible to our understanding of the quantum universe and hopefully is also relevant to the partners, both

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Job Feldbrugge: using picalosis theory and basically, the winner measure we can find in real time analog of finding cups formula where you can properly define the passenger as an integral over paths.

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Job Feldbrugge: using winning measures.

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Job Feldbrugge: So instantones go beyond complex classical paths, and similarity costs need to be taken into account in order to do it. So it's very tricky, but it can be done.

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Job Feldbrugge: And and my hope is that we can use these techniques in the future to study quantum mechanics better quanta field theory, and hopefully, also where we propagate for monthly geometry to later 3 geometry

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Job Feldbrugge: and study this in detail. So it's a very exciting set of new tools which we can use to study all kinds of new and all that.

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Job Feldbrugge: I've been very happy to apply this to different fields and

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Job Feldbrugge: solve small problems which have been in the literature sometimes for decades, which is really fun.

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Job Feldbrugge: Thank you very much for listening, and please feel free to ask me now or send me an email later.

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Hal Haggard: Thank you. You.

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Hal Haggard: Ding. Feel free.

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Ding Jia: Thank you. Very interesting talk.

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Ding Jia: Very fascinating topic. Sorry, for some reason my camera can't not turn on. But I have 2 questions.

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Ding Jia: First, is regarding your new proposal for quantum path integral quantum mechanics. You showed a formula

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that holds for infinite dimensional path integrals.

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Ding Jia: I'm wondering about the assumption behind the formula.

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Is it true that the integral is over? X over the whole real line

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Ding Jia: so restricted domain of integrate. Sorry.

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Ding Jia: What happens if I have a restricted domain of integration? For example, if I integrate over the positive, half real access.

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Job Feldbrugge: Yeah. So of course, if you use Kashi's theorem for deformation of integration domains. And of course you have to keep the endpoints fixed. So if you consider different integration spaces, and this might

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Job Feldbrugge: need to be amended. So if you consider a difference e. Even in the one dimensional case, you find that sometimes you have to add an additional factor. So you say I have to integrate along a call tour, and then I will follow the benefit.

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Ding Jia: I'm wondering about a simple example.

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Ding Jia: For example, take the integral

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Job Feldbrugge: yes, which has a set of point X equals 0. Yes.

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Ding Jia: and that is a relevant set of point. If I integrate over the whole real line.

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Job Feldbrugge: Certainly.

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Ding Jia: Now, suppose I consider a different integral instead of minus infinity to plus infinity, I integrate from one to plus infinity.

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Ding Jia: In this case the previous relevant setup does not belong to the domain. Hmm! And then it seems, there are 2 options, either.

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Ding Jia:  it seems to me. There's only one option which is

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Ding Jia: to say that the new boundary of integration X equals one.

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Ding Jia: There is a new saddle point there.

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Job Feldbrugge: What or

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Ding Jia: and then II can have a new formula that includes this new kind of saddle point. Or maybe you have some other ideas, because otherwise

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Ding Jia: I would have no relevance at the point. And and according to your formula, I would get 0, which probably isn't correct.

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Job Feldbrugge: So what I'm assuming in the formalized that the integration domain just goes to infinity or to a place where basically, the the action diverges. So that that's typically what you find in practical situations. But if you are assisting and say I would like to consider differential interval integrating from one to infinity.

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Job Feldbrugge: then you have to do the complex analysis with a little bit more detail. And you basically find this that you can. In this case you probably can say, well, if I would integrate from minus infinity to infinity. Then I would integrate from along this diagonal line.

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Job Feldbrugge: and in this case you probably could define it in different ways, which, for example, you can go from one a little bit up, and then follow the diagonal line subsequently, and then you get some of 2 integrals

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Job Feldbrugge: where the one part of the integral is open and then find a domain, and the other integral is not the Internet domain. But then the integral is exponentially pressed. But I don't see how you have an additional settlement.

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Ding Jia:  If you don't have an additional set of points, then, in terms of your formula, you only owe some of our relevant set of points.

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Ding Jia: then shouldn't I get 0?

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Job Feldbrugge: No? Then that formula doesn't no longer applies and add additional terms.

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Ding Jia: Right? Okay?

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Job Feldbrugge: So in addition, you also can have logarithms in your integral. And you have different Riemann sheets, and it can be a lot more well than I described here.

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Ding Jia: I see. Yes, okay, that's I understand that answer.

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Ding Jia: I just I took the most simple formula I could write down and

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Ding Jia: try to convey the general spirits rather than understand. If you let me give a final comment on that second question the final comment is, is that for quantum gravity.

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Ding Jia: especially Lorenz and quantum gravity, it seems.

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Ding Jia: is the restricted arth integrals that are most relevant because we're trying to integrate over Lorenzi matrix a for example, the square scale factor.

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Ding Jia: It should be positive, because when it's negative, we have a different signature.

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Ding Jia: And in in one sense we're not doing Lorentz and

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Ding Jia: integral anymore. If we integrate over both positive and negative

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Ding Jia: square disco factors.

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Ding Jia: And that's the comment.

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Hal Haggard: Can you hold your second question? I'll give you Eric a chance, and then we can come back.

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01:06:52.240 --> 01:07:10.160
Jerzy Lewandowski: Thank you. Hello! My question is, is quite, quite in a simple and non expert question. So wh. What is exactly new in what is that new proposal in and in what sense it is different than other old proposals?

413
01:07:11.690 --> 01:07:15.980
Job Feldbrugge: So there are many proposals, and and

414
01:07:16.170 --> 01:07:33.839
Job Feldbrugge: but only a limit of them. Amount of proposals are basically doing continuum buttons globally. So many proposals for the button tool goes for different time slices. So you that time we're gonna realize your pattern tool. And then you take limit to infinity. Right?

415
01:07:35.490 --> 01:07:43.640
Job Feldbrugge: What I am proposing or what we are proposing is to not do that because this Internet product of le back measures is problematic.

416
01:07:43.840 --> 01:08:00.449
Job Feldbrugge: But rather continue regulator. And you continue way off defining departure. And people have been trying to do this for a long time, and so, for example, Joe Chloe has been spending a lot of time on that. But there is a whole history of people trying to do that.

417
01:08:00.590 --> 01:08:06.190
Job Feldbrugge: But they always considered the real domains. They never. We really wanted to complex domain.

418
01:08:06.210 --> 01:08:16.479
Job Feldbrugge: So basically, what we are doing is we combine that history, that literature with using the inside. That's

419
01:08:16.590 --> 01:08:24.479
Job Feldbrugge: if you do pick a if you see for 5 dimensional interval, you're basically left with few Gaussian intervals in a very rough way. And and

420
01:08:24.710 --> 01:08:41.260
Job Feldbrugge: that's that's basically the new insight. That's what we are proposing. And II am encouraged to do this. Because if I look at practical problems, and I find that's is very closely related to this formalism. Because you can

421
01:08:41.319 --> 01:08:48.979
Job Feldbrugge: see, you can basically see these complex setup points, and you can see the merge of settle points in the solution to the starting equation.

422
01:08:51.670 --> 01:08:53.109
Jerzy Lewandowski: I see. Thank you.

423
01:08:53.490 --> 01:08:58.439
Hal Haggard: I will return to you, Ding, but let me kong Zhang, please go ahead and ask your first question.

424
01:09:02.160 --> 01:09:09.160
Cong Zhang: Okay, so so thank you. So so my question is about this, this, this, this, this Euclidean path integral.

425
01:09:09.240 --> 01:09:18.350
Cong Zhang: So the point is like a when you define this Euclidean pass integral. You need to put this this kinetic term into your measure. Right?

426
01:09:18.720 --> 01:09:20.669
Job Feldbrugge: Yeah, let me go to the slides.

427
01:09:20.729 --> 01:09:21.930
Cong Zhang: Okay.

428
01:09:22.859 --> 01:09:24.180
Job Feldbrugge: have a

429
01:09:29.880 --> 01:09:31.210
Job Feldbrugge: this, is it? Right?

430
01:09:33.210 --> 01:09:45.389
Cong Zhang: Yes, yes, yes. And then you you need to put this connecting term into this manner together. Well, Defund Minor. But my question is like, when you try to find that the critical point.

431
01:09:45.720 --> 01:10:04.279
Cong Zhang: and they you need to take this, this, this, this kinetic term out from this, this, this, this minor right? You can see there's something like you define the integral and and and and add fund this, this this discred upon E. Is there any issue here? II just didn't know.

432
01:10:05.130 --> 01:10:24.469
Job Feldbrugge: So the the the short answer is. Yes, there are issues so well, II haven't really got in any of these data. But you find that you cannot really take these deep set manifolds but for the simplest models and take them just to infinity. So what we define is so-called, which are.

433
01:10:24.490 --> 01:10:38.960
Job Feldbrugge: supposed to be in the same equivalence class. So there, there's a little bit of nuance and details which I skipped over in this talk. So the the short answer, yes, there are complications, but they can be dealt with, we believe.

434
01:10:40.270 --> 01:10:41.989
okay, okay, okay.

435
01:10:43.650 --> 01:10:44.760
Hal Haggard: Western.

436
01:10:48.650 --> 01:11:13.609
Western: Thank you. Thank you. Thank you, John, for this very clear and presentation. It was very good to go through all the construction. Thank you so much. I would like to push you a bit to talk about. The physics. So physics. And what this implies for cosmology, for instance? Well, I have several questions.

437
01:11:13.610 --> 01:11:38.550
Western: so that they go from, how much can we trust this techniques? Because, of course. So how far would you trust the results? And the other thing is more specifically, we decided to defend

438
01:11:38.550 --> 01:12:01.469
Western: that you are. But then, when we were talking. You were thinking about computer transitions from a given state of the universe into another state of the universe. So do you think, in terms of something like a bounce in which you have a downloading from one State to another, or how in general, how do you think about the

439
01:12:01.700 --> 01:12:06.319
Western: direction between the boundary and computing transition between different states.

440
01:12:07.690 --> 01:12:29.260
Job Feldbrugge: So I think there's a number of questions. But so the first thing to say is that if you go to the, to the mathematically. I fully trust these techniques so. But if you apply this to physics, then you have to make assumptions like you have to assume that you integrate over space, for example, for the no boundary proposal.

441
01:12:29.510 --> 01:12:34.620
Job Feldbrugge: And there's not to say whether that is actually a valid approximation.

442
01:12:34.980 --> 01:12:41.719
Job Feldbrugge: So, for example, II believe at the moment another law, but also

443
01:12:42.230 --> 01:12:56.260
Job Feldbrugge: working on numerically federating department to go for the no boundary proposal, using discretization techniques also, Mary to our speaker. And and that might give different results that that could well be

444
01:12:56.270 --> 01:12:57.900
Job Feldbrugge: so.

445
01:12:58.060 --> 01:13:03.579
Job Feldbrugge: even though the mathematics techniques are fully reliable. I believe?

446
01:13:03.700 --> 01:13:08.330
Job Feldbrugge: Of course, when you apply to physics, then you have to be careful with your assumptions.

447
01:13:08.780 --> 01:13:20.280
Job Feldbrugge: The other question you asked is, how do should I understand this in more general terms? And indeed, if you have an an action which, for example, has a classical balance.

448
01:13:20.330 --> 01:13:26.750
Job Feldbrugge: and already for the no boundary proposal kind of copulation when you go. No, from a 0

449
01:13:26.810 --> 01:13:53.789
Job Feldbrugge: skill factor, 2. Method for finance skill factor to another final skill factor, you find that you get the interference of a bouncing solution indeed, and then directs. So you you can just expand or you can first contract and then expand. And so basically, what you find is a fairly similar story as what you see for quantum tunneling and evolution in a barrier where you have multiple points which can interfere.

450
01:13:55.620 --> 01:13:59.080
Job Feldbrugge: So you you see the same phenomena happening in quantum gravity.

451
01:13:59.630 --> 01:14:25.319
Western: Okay, very nice. And since you were mentioning the the work by Bianca, let me say that computation that is meant to be just a part of a larger computation of from one state to another. So we'll be happy to

452
01:14:25.320 --> 01:14:31.769
Job Feldbrugge: thank you so much. So let me add one small point to it.

453
01:14:31.780 --> 01:14:39.829
Job Feldbrugge: If you discretize a theory. then, even though the discretized theory might be a good approximation along the real line

454
01:14:39.910 --> 01:14:44.629
Job Feldbrugge: the the unlimited continuation can be wildly different.

455
01:14:44.830 --> 01:14:50.690
Job Feldbrugge: So this is still, I think, topic people are studying. And

456
01:14:50.890 --> 01:15:17.280
Job Feldbrugge: but it it's it's even though these techniques might apply and probably will apply in some way to these discretized loop contour gravity techniques, as you described. People have to do it with care. Because, typically what you find is that if you discretize, for example, the normal path and just the most vanilla time discretization techniques, you find a large number of additional points in the complex plane.

457
01:15:17.540 --> 01:15:18.550
Job Feldbrugge: So

458
01:15:18.560 --> 01:15:33.279
Job Feldbrugge: just as a, if you do this, and we have to be actually careful that you are really finding the set of points which matter which, if you are believing that you there should be continuing theory which corresponds to the continuum theory, and that you're not basically

459
01:15:33.480 --> 01:15:36.820
Job Feldbrugge: falling into the trap that you have this

460
01:15:36.920 --> 01:16:02.440
Job Feldbrugge: well, large, quickly diverging number of settle points which are just art effective to discretization scheme. So it it it's definitely possible. And people are doing it also with both carbon techniques at the moment. And it's very interesting and very exciting that it can now be done. But it's it's tricky business when you take a discretize theory, and then I'll be continuing to explain.

461
01:16:02.820 --> 01:16:16.730
Western: Well, let me just make a one small comment that when you start with this before theory, I wouldn't say that this is a discretization, so that we have something else. So

462
01:16:16.900 --> 01:16:26.449
Western: discretization and everything is real what you find. But if you take a discretization of a continuum theory.

463
01:16:28.140 --> 01:16:28.980
Western: yep.

464
01:16:29.120 --> 01:16:30.700
Western: okay, thank you.

465
01:16:31.570 --> 01:16:37.299
Jose. I'll come to you next. But there's a question in the chat which I'll read out for everyone

466
01:16:37.410 --> 01:16:45.650
Hal Haggard: when deforming the integration contour. There will be arcs at infinity. How are their contributions dealt with in your work?

467
01:16:46.530 --> 01:17:15.559
Job Feldbrugge: So typically, what you find is that in a sense takes care of it. So for of course, you can always think of pathological examples where this won't be the case, like adding logarithms to your integrals. And so it's not completely rigorous. But typically what you find is that the integral becomes more so. Story, you go out because path leads to rapidly changing actions. And then what you find is that basically

468
01:17:15.560 --> 01:17:24.479
Job Feldbrugge: leads to a small arc at infinity, which is a 0, gives you a 0 contribution. That's not to say that it's the case for all theories, but

469
01:17:25.740 --> 01:17:29.920
Job Feldbrugge: of more often than not. You don't have to worry about it.

470
01:17:31.590 --> 01:17:32.520
Hal Haggard: Jose.

471
01:17:34.200 --> 01:17:38.140
José Antonio Zapata: No, hello my! My! My comment and

472
01:17:38.680 --> 01:17:44.820
José Antonio Zapata: question. It was a little bit partially answered already, when you were talking to

473
01:17:45.150 --> 01:17:45.990
thank you

474
01:17:46.600 --> 01:17:53.519
José Antonio Zapata: and to to be bottom the the the thing is you want to

475
01:17:56.860 --> 01:18:02.690
José Antonio Zapata: integral of quantum field theory, then then you are in this infinite dimensional space. But

476
01:18:03.020 --> 01:18:04.690
you need to

477
01:18:04.980 --> 01:18:16.629
José Antonio Zapata: to make sense of that, not only by your mathematical techniques, but II prefer to think about it in the terms of Wilsonian renormalization. So I needed to impose a cut off.

478
01:18:16.900 --> 01:18:25.459
José Antonio Zapata: If I impose a cut off, then it becomes a finite, dimensional thing. And and then I can use your techniques. And and my question was how those.

479
01:18:26.620 --> 01:18:33.530
José Antonio Zapata: and if if there is a comment that you can give them on. How is that this

480
01:18:33.950 --> 01:18:40.470
José Antonio Zapata: finite dimensional integrals? And with this imposed decision of the cutoff

481
01:18:40.650 --> 01:19:00.809
José Antonio Zapata: interact with a Picard in. So so, Cal, this is the question, and the comment is that it seems that that defining sigma algebra for the quantum field theory is kind of in necessary from my point of view, because II have to define it, that they teach scale and then take a continuum limit in

482
01:19:00.860 --> 01:19:02.839
in the sense of renormalization

483
01:19:06.470 --> 01:19:07.470
José Antonio Zapata: voting.

484
01:19:08.160 --> 01:19:10.170
Job Feldbrugge: That. That's it.

485
01:19:11.110 --> 01:19:26.949
Job Feldbrugge: Yeah, IIII think it's a fair comment. So if you are working with the final dimensional integral that you can use these techniques without any trouble. So you can go from an initially convergent integral to an absolutely convergent interval. One thing I would like to note is that

486
01:19:27.070 --> 01:19:44.260
Job Feldbrugge: for quantum mechanics and the bothline quantization there is a very nice Brownian motion in order to define these intervals. and I say that it's really nice, because Brownian motion has the property, that it's almost everywhere continuous and almost nowhere differentiable.

487
01:19:44.280 --> 01:19:45.690
Job Feldbrugge: So it's but it's

488
01:19:46.170 --> 01:19:59.770
Job Feldbrugge: continuous, which is nice and typically what you find. And of course it is not in general, so there are probably ex exceptions to this. But often, if you generalize this Brownian motion measure to the field theory

489
01:20:00.040 --> 01:20:05.880
Job Feldbrugge: you find that you can generate something like that, but it will only have support over distributions.

490
01:20:06.110 --> 01:20:12.979
Job Feldbrugge: So it's no longer continuous. And personally, I find this very troubling and something I'm looking into at the moment

491
01:20:13.000 --> 01:20:15.759
Job Feldbrugge: to see. Well, how do you apply this to field theory?

492
01:20:16.160 --> 01:20:24.820
Job Feldbrugge: Because well, in apartment school, you say I would like to go from A to B, but then there should be a notion of actually just filling the path from A to B.

493
01:20:24.960 --> 01:20:45.870
Job Feldbrugge: If your measure has only has support over distributions, then you could jump around and and you don't really have a notion of going from A to B anymore, because it's not continuous. So there, there are definitely subtleties for the field theory which don't merge in the quantum mechanics. And what type of session.

494
01:20:46.160 --> 01:20:50.919
Job Feldbrugge: But still very much on the subject of investigation.

495
01:20:52.910 --> 01:20:57.160
José Antonio Zapata: Thank you. Thank you. But but the the comment, Would that

496
01:20:57.200 --> 01:20:59.170
José Antonio Zapata: problem that you saw

497
01:21:00.780 --> 01:21:05.670
José Antonio Zapata:  and nice Sabbatical distribution. And

498
01:21:07.450 --> 01:21:11.809
José Antonio Zapata: but you you need us to compute the transition.

499
01:21:11.940 --> 01:21:18.299
José Antonio Zapata: and since you are going to compute it. mobilization. it doesn't really

500
01:21:19.280 --> 01:21:20.629
what what you

501
01:21:23.140 --> 01:21:25.059
José Antonio Zapata: and you just take.

502
01:21:26.090 --> 01:21:28.890
José Antonio Zapata: Don't bother about the

503
01:21:33.880 --> 01:21:45.830
Hal Haggard: I think the Internet connection on my end or your end. I don't think it's you. Yo, but I had the same trouble, Jose. Jose, if you wanna put it in the chat, I can. I can try and make the comment for you

504
01:21:46.170 --> 01:21:52.889
Hal Haggard: in the meantime. Thank you for your patience. Ding you wanna ask your second question.

505
01:21:53.710 --> 01:21:58.269
Ding Jia: Thank you all. Yes, perhaps this is

506
01:21:59.380 --> 01:22:06.759
Ding Jia: run into a longer discussion. So may maybe. Just as a soft starter, I'll post a question, and

507
01:22:06.990 --> 01:22:17.090
Ding Jia: you wanna chat more about it. You should chat more about it.  this is about the no boundary proposal on one slide. You showed that.

508
01:22:17.410 --> 01:22:22.810
Ding Jia: if you apply Picard, you got different results from previous ones.

509
01:22:23.170 --> 01:22:29.480
Ding Jia: And presumably that's referring to some instabilities. That right?

510
01:22:30.190 --> 01:22:41.870
Job Feldbrugge: Yes. So if you take the most vanilla definition of the no boundary proposal, and you apply picolia theory to it, then you find a different set of points to be relevant. The complex set of point to be relevant than

511
01:22:41.890 --> 01:22:44.069
Job Feldbrugge: art on hawking originally assumes.

512
01:22:44.390 --> 01:22:59.260
Job Feldbrugge: Of course, there are many ways to get around this. This. This results you can impose boundary conditions at the Big Bang or there are, has have now been several proposals to get perturbations out of the development proposal.

513
01:22:59.340 --> 01:23:06.680
Job Feldbrugge: and and they are all filleted. However, the the only thing which you should keep in mind with this

514
01:23:06.710 --> 01:23:34.620
Job Feldbrugge: is that? Well, in my opinion, is that if you supply initial conditions to a material for initial conditions, and you should find another way to test. That's the tweak you added, or the the the boundary conditions, you added can be far verified in some way. So what we basically did, we took just the calculus theory and checked what happens. And then, to our big surprise result, came out.

515
01:23:34.630 --> 01:23:41.870
Job Feldbrugge: and subsequently many people have imposed different boundary conditions and different ways of

516
01:23:41.960 --> 01:23:45.759
Job Feldbrugge: teasing the the desired result out of it, so to say.

517
01:23:45.780 --> 01:24:03.199
Job Feldbrugge: that's difficult to determine whether you are just teasing the right result out of the compilation because it's only assigned, which is the issue, or whether you have really solved the problem. Because then you should basically find another way to verify that tweak which you edit is filled.

518
01:24:04.510 --> 01:24:10.839
Ding Jia: Understood correctly. You're referring to no other choice of contours for the labs function

519
01:24:11.030 --> 01:24:15.409
Ding Jia: or additional boundary conditions.

520
01:24:15.720 --> 01:24:32.299
Job Feldbrugge: But you don't even have to take department, will. So in the last meeting I attended where Hartwell talked, he said, oh, let's forget about the passenger and use central and just.

521
01:24:32.390 --> 01:24:43.279
Ding Jia: Yes, and that's perfectly fair, as in proposal. But then you have to somehow test that that is the thing which should be predicted. I perfectly agree with you, and I'm not

522
01:24:43.350 --> 01:24:50.739
Ding Jia: personally support your own choice over laps. It seems the positive unlaps is a is very natural choice.

523
01:24:50.760 --> 01:24:54.380
Ding Jia: Now, here's the question, you have this

524
01:24:54.870 --> 01:25:03.530
Ding Jia: instability at 5 1. So this this exponent grows unboundedly. If I increase 5 1.

525
01:25:03.780 --> 01:25:07.890
Job Feldbrugge: Which quadratic perturbation theory in mean super space. Yes.

526
01:25:07.950 --> 01:25:10.030
Ding Jia: yes, yes, exactly.

527
01:25:10.290 --> 01:25:14.449
Ding Jia: And then in the previous paper you discussed this point, there is a tension

528
01:25:14.520 --> 01:25:20.200
Ding Jia: between 2 reasonings. One reasoning is that if I photos of polymorphic gradient flow.

529
01:25:20.480 --> 01:25:24.309
Ding Jia: The Morse function can never increase, though always decreases

530
01:25:24.400 --> 01:25:27.360
Ding Jia: wherever it stays the same, but it never increases.

531
01:25:27.500 --> 01:25:31.989
Ding Jia: So if I start with a real contour with a 0 Morse function.

532
01:25:32.390 --> 01:25:39.989
Ding Jia: I cannot get a positive Morris function. and then that seems to be an apparent conflict with the familiar Shannon slide.

533
01:25:40.190 --> 01:25:45.919
Ding Jia: And that's because

534
01:25:46.010 --> 01:25:52.900
Ding Jia: singularities offshore singularities. And it's an infinite dimensional phenomena. Is that a fair and characterization of your thought?

535
01:25:53.320 --> 01:26:05.880
Job Feldbrugge: Well, the thing to keep in mind is that the total action is actually has negative age function. So it it it definitely decreases it. It's just that if you only

536
01:26:05.930 --> 01:26:12.819
Job Feldbrugge: now plot the the term which is potential proportional to the final perturbation that you find this plus sign.

537
01:26:15.060 --> 01:26:19.169
Job Feldbrugge: So the total action has to satisfy the condition you just described

538
01:26:20.740 --> 01:26:34.150
Ding Jia: right. And this action, this or this exponential term you're showing. Doesn't that come from? The onshore action. Professor? And it's positive.

539
01:26:35.150 --> 01:26:35.960
Job Feldbrugge: Yes.

540
01:26:36.590 --> 01:26:43.560
Ding Jia: so that's in apparent tension with the reasoning that a whole followed homomorphic, gradient flow like, I can't get a positive Morse function.

541
01:26:44.560 --> 01:26:55.600
Job Feldbrugge: Well, the point is that if you have an an function which is a sum of different terms. It could be that the total sum is negative, whereas an individual component is positive.

542
01:26:57.590 --> 01:27:01.899
Ding Jia: And maybe we should take this offline. Yes, yes, that's better way.

543
01:27:02.900 --> 01:27:06.109
Let me ask if there's any questions from anyone else.

544
01:27:08.720 --> 01:27:13.940
Hal Haggard: If not, let's thank you again. Thank you so much, Yoba. It was really nice presentation.

545
01:27:14.090 --> 01:27:15.189
Job Feldbrugge: Thank you very much.

546
01:27:20.200 --> 01:27:22.150
Hal Haggard: Jonathan. Did you have something?

547
01:27:23.910 --> 01:27:26.680
Jonathan Engle: No, no, I was just clapping.

548
01:27:26.760 --> 01:27:31.389
Hal Haggard: Oh, good! Good! The Deepaka had to leave.

549
01:27:31.520 --> 01:27:42.980
Hal Haggard: Okay, just a quick reminder that went out on the aisle. Qgs mailing list that Deepak is trying to put together a journal club, and you can see the email for that. Thank you. Everyone.

550
01:27:45.120 --> 01:27:46.100
Seth Asante: Bye.