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Hal Haggard: All right. Welcome, everyone. It's my pleasure to introduce Wolfgang Wieland, who will be telling us about a Planck luminosity bound in quantum gravity.

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Hal Haggard: Wolfgan.

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Wieland, Wolfgang Martin: Okay, welcome everybody and Happy New Year. I'm very grateful for the organizers of the Lqg. Idqgs series for the opportunity that they gave me to present this research. Yeah. And as the 1st speaker of the year, I'm again very happy to tell you. Wish you all a Happy New year, and

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Wieland, Wolfgang Martin: all the best for 2025.

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Wieland, Wolfgang Martin: I will speak about research from last year. Some of you have already heard some parts of it, maybe. And yeah, I'm looking forward to the discussion that we have that we will have at the end of this talk for sure.

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Wieland, Wolfgang Martin: let's just go into. Let me just make sure that.

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Wieland, Wolfgang Martin: So you see again you see my screen. No.

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Abhay Vasant Ashtekar: Yeah. Yes.

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Wieland, Wolfgang Martin: Alright so topics to be discussed. I want to speak about the following themes in this presentation. 1st of all, I will give you

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Wieland, Wolfgang Martin: an in general motivation about the role of Planck luminosity in classical general relativity, and the possibility of a luminosity bound in

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Wieland, Wolfgang Martin: emerging from the quantum theory, and then I will give some evidence for this scenario, and I will give evidence to this scenario by looking at the radiative phase space and the particular non-perturbative approach to the quantization of radiative, non-initial data.

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Wieland, Wolfgang Martin: and then we will see that upon using the hoist action that underlies loop quantum gravity and the loop quantum gravity, discreteness of area.

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Wieland, Wolfgang Martin: Something will happen at a particular

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Wieland, Wolfgang Martin: critical luminosity that is, of the order of the Planck luminosity, and we will close with an outlook and conclusion.

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Wieland, Wolfgang Martin: Now, 1st 1st of all, let us consider the

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Wieland, Wolfgang Martin: general motivation. What is the Planck luminosity now?

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Wieland, Wolfgang Martin: From Newton's constant G. Newton Planck's constant and the speed of light. We can build planck units. So Planck, energy, Planck, time Planck length and so on. And from there we also can consider derived quantities, such as the Planck luminosity, which is nothing than the planck energy per planck time, divided by planck time.

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Wieland, Wolfgang Martin: And now something very peculiar happens in D equal 4 spacetime dimensions, namely, that the resulting expression is independent of H. Bar. So H. Bar cancels out of the fraction of the planck energy divided by planck time. But this only happens in D equal 4 spacetime dimensions. And now you can say, Well, this is just a numerical coincidence, but it is more than that

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Wieland, Wolfgang Martin: because this has immediate immediate physical implication.

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Wieland, Wolfgang Martin: imagine a merger of 2 black holes.

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Wieland, Wolfgang Martin: so they spiral. They are in common orbit around each other, but the system emits gravitational radiation, and therefore ultimately the 2 black holes merge to form one single black hole.

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Wieland, Wolfgang Martin: and during this merger gravitational radiation is emitted. Initially. The emitted power is very low, it will increase as the black holes move closer and spin faster and faster.

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Wieland, Wolfgang Martin: and eventually it rings down, and no gravitational radiation will be emitted anymore, because finally, a black hole has settled and merged.

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Wieland, Wolfgang Martin: Now, during this process, the profile of the luminosity will reach some peak luminosity, and because in Dq. 4, spacetime dimensions, classical Gr has a constant that has units of power, namely, the Planck

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Wieland, Wolfgang Martin: power. We can always write this peak luminosity emitted

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Wieland, Wolfgang Martin: that appears in this process as the Planck luminosity times. Some function, and this function can only depend on dimensionless quantities, some scale independent observables, such as the

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Wieland, Wolfgang Martin: mass ratio or certain spin ratios, and this immediately means that the peak luminosity is intensive, that it does not

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Wieland, Wolfgang Martin: double when you double the size of the system, and this can only happen in equal 4 spacetime dimensions, and it will be different in higher dimensions. And

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Wieland, Wolfgang Martin: since the theory has built in this fundamental power. It is not surprising that during gravitational processes we can reach this power. In fact, during we have observed, during gravitational during Black Hole mergers, luminosities, that come 3 orders of magnitude close to the Planck luminosity.

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Wieland, Wolfgang Martin: And this is very exciting, of course, and it has

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Wieland, Wolfgang Martin: been argued in the past that perhaps there is a bound on the luminosity. Perhaps the Planck luminosity is, in fact, an upper bound for the emission of gravitational radiation, or perhaps for the emission of any radiation in any process

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Wieland, Wolfgang Martin: in the universe. And what is evidence for that? Well, a simple evidence from that comes from the following estimate. So take the quadruple formula for the emitted power.

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Wieland, Wolfgang Martin: So the emission, the the emitted power, is proportional to the 3rd time derivative of the reduced quadrupole. Moment. Squared times. G. Divided by C to the 5.

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Wieland, Wolfgang Martin: And if we imagine now and this argument can be, if you look into Misner Thorn Wheeler, they have a nice little paragraph on that. And now let's estimate this. So let's consider a system of point masses or test bodies in gravitational equilibrium.

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Wieland, Wolfgang Martin: then we can estimate the mass. The characteristic frequency at which this system oscillates and its radial extension.

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Wieland, Wolfgang Martin: But now we bring in the self interaction of gravity that this system needs to be bound by gravitational forces. And this tells us that there is a relationship between the kinetic average kinetic energy of the system and the potential energy. And this allows you to write the luminosity

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Wieland, Wolfgang Martin: only in terms of the mass of the system and its spatial extension.

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Wieland, Wolfgang Martin: But if you now also take into account that the emission of gravitational radiation can only happen

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Wieland, Wolfgang Martin: until you form a black hole, because at that point no radiation is anymore

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Wieland, Wolfgang Martin: really emitted. Then you get a bound, namely, that the emitted power should be bounded by the Planck luminosity.

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Wieland, Wolfgang Martin: which is G divided by C to the 5, obviously a very large quantity.

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Wieland, Wolfgang Martin: and all this is a classical argument.

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Wieland, Wolfgang Martin: and so one could argue, why why should quantum theory have anything to do with this?

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Wieland, Wolfgang Martin: But I would like to argue that this is, in fact, this can be in fact, the case, and

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Wieland, Wolfgang Martin: but if but let us be a bit cautious, namely, that we.

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Wieland, Wolfgang Martin: we cannot expect such a bound to arise from a perturbative approach, because in the if we write down Planck luminosity, G. Newton is downstairs in the denominator, not upstairs, but in the perturbative approach we look at the formal perturbative expansion in terms of the coupling constant of gravity, which in the perturbative regime, is square root of 8 pi. G.

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Wieland, Wolfgang Martin: So we look at small G, but for small G, the plank power would diverge

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Wieland, Wolfgang Martin: for the same reason. In fact, there is no such.

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Wieland, Wolfgang Martin: no such bound seems to exist on the classical radiative phase, space on the classical radiative phase. The radiative modes at null infinity are characterized by the asymptotic shear.

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Wieland, Wolfgang Martin: and the time derivative of the asymptotic shear gives bondi news

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Wieland, Wolfgang Martin: from which we can obtain a classical equation for the emitted luminosity

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Wieland, Wolfgang Martin: in terms of data at future null infinity, and at that point on that phase, space on the radiative phase space. There is no reason whatsoever a priori why

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Wieland, Wolfgang Martin: there should be any bound on the Planck luminosity. And similarly, and for the same reason, nothing peculiar happens at the quantum level. If you look at the perturbative approach, because perturbatively. The shear just assumes some canonical commutation relations for harmonic oscillators, and there is no boundedness in the frequency.

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Wieland, Wolfgang Martin: no boundedness in the emitted luminosity.

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Wieland, Wolfgang Martin: and that and therefore what I

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Wieland, Wolfgang Martin: consider here is a non-perturbative approach, such as the one that we have in loop quantum gravity at hand to explore the role of the Planck power in such an approach. Now

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Wieland, Wolfgang Martin: one may now ask, well, wait a minute. There's no h bar in the Planck luminosity. So

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Wieland, Wolfgang Martin: why why should we ever expect that non-perturbative quantum gravity has anything to say about a possible bound on the emitted gravitational wave luminosity? Well, consider the following analogy.

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Wieland, Wolfgang Martin: Consider an atom carpet to

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Wieland, Wolfgang Martin: the radiation field and then using Fermi's Golden rule, we can compute lifetimes for atomic transitions. But now, to good expression, to good approximation, we can write these lifetimes, which are essentially matrix elements of the interaction. Hamiltonian.

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Wieland, Wolfgang Martin: in terms of classical quantities, only involving the emitted emitted radiation

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Wieland, Wolfgang Martin: only involving properties of the emitted radiation. In fact, we can, to good approximation, write the transition rates just as the power of the emitted wave, divided by the total emitted energy.

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Wieland, Wolfgang Martin: And all of this can be, of course, expressed in terms of the classical energy, momentum tensor. And there is no discreteness at that point in this expression.

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Wieland, Wolfgang Martin: just because there's no discreteness in the, in the radiative phase. Space in the quantization of the

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Wieland, Wolfgang Martin: radiation field.

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Wieland, Wolfgang Martin: But then atomic theory comes in and tells us, tells us that the energy levels in the hydrogen atom are quantized, and then energy conservation links the energy of the emitted radiation field to the energy

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Wieland, Wolfgang Martin: in the hydrogen atom, and at that point H. Bar comes into the game and

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Wieland, Wolfgang Martin: guarantees, in fact, that the ground states have infinite lifetime. So atoms are stable, and what I

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Wieland, Wolfgang Martin: think is happening here is that something similar happens

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Wieland, Wolfgang Martin: in the non-perturbative approach. So, upon adding the wholes term

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Wieland, Wolfgang Martin: to the Hilbert Palatini action. What we obtain is a possibility to compactify at the classical level certain directions on phase space. So instead. And this compactification

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Wieland, Wolfgang Martin: effects, in fact, only certain edge mode variables living at the boundary of the region under consideration, and in that way

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Wieland, Wolfgang Martin: the conjugate variables, which are certain charges turn out to have a discrete spectra. But now the these charges satisfy certain flux balance relations, and this then affect also the quantization of the radiative data, and in that way the

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Wieland, Wolfgang Martin: quantization that we obtain for the Lqg discrete quanta of area affect also the representations at the quantum level of the radiative data. So there is a bit of an it's a bit similar to what happens

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Wieland, Wolfgang Martin: in the example for the atomic theory where you look at the commutation relations for the radiation field. And you would say, Well.

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Wieland, Wolfgang Martin: nothing should happen to this quantity. Its spectrum should not see any discreteness. But then you realize that the radiation has to come from somewhere, or must be compatible with another quantum system, namely, the quantum system of the hydrogen atom, and these together imply

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Wieland, Wolfgang Martin: constraints on the emitted radiation field and something simple, and we will see below how something similar can happen in in gravity.

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Wieland, Wolfgang Martin: What about a few words about the methodology? So what I'm considering is

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Wieland, Wolfgang Martin: quasi-local quantization of the gravitational field.

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Wieland, Wolfgang Martin: If you look at finite domains on a generic cauchy, hypersurface, initial data

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Wieland, Wolfgang Martin: is characterized by the extrinsic curvature and the 3 metric on a cauchy hypersurface, and these have to satisfy certain constraints, namely, the scalar constraint and the vector constraints. And together these 2 constraints generate gauge redundancies on phase space gauge symmetries, in fact, namely, the Dirac hypersurface deformation algebra.

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Wieland, Wolfgang Martin: So if we start, let's say, at the classical level with some initial data that satisfies the constraints on some given initial hypersurface sigma one. But then

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Wieland, Wolfgang Martin: slightly deform this hypersurface and ask

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Wieland, Wolfgang Martin: what? To what hyper, to what initial data, what initial data would then live on this slightly deformed hyper surface? Then the 2 are related by the theomorphism which turns out to be a gauge transformation on phase, space.

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Wieland, Wolfgang Martin: and solving at the quantum level. Solving the constraints amounts to characterize gauge, equivalence classes or gauge orbits under the exponential map of these constraints, but now, if we think for a moment.

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Wieland, Wolfgang Martin: one way to characterize a gauge orbit is to just pick one representative of the entire equivalence class. Now, in this spacetime picture. One way to pick such an equivalence, to pick such a representative from that equivalence class is to push this

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Wieland, Wolfgang Martin: unphysical time evolution to its extreme.

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Wieland, Wolfgang Martin: and by that what I mean by that is to push

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Wieland, Wolfgang Martin: this time evolution to the boundary of the future Cauchy

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Wieland, Wolfgang Martin: completion. Cauchy evolution of this finite region, and in that way we end up with a state that is no longer defined on a space-like hypersurface, but on a null hypersurface on a null boundary, the null boundary of the

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Wieland, Wolfgang Martin: future cauchy development of this region.

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Wieland, Wolfgang Martin: And what then happens is that you have to impose only

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Wieland, Wolfgang Martin: only a reduced number of constraints, because you have, since you have pushed everything to the null boundary, you're left only to impose constraints that generate the residual symmetries that preserve this boundary.

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Wieland, Wolfgang Martin: And

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Wieland, Wolfgang Martin: these are basically just diffomorphisms of this null direction. If, in addition, you fix the light rays that generate this

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Wieland, Wolfgang Martin: this narrow surface, you'll end up only with

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Wieland, Wolfgang Martin: diphomorphisms that that are basically rescalings of reparameterizations of the null parameters. So at the end of the day you have to deal with only one command, and this will turn out to be the Bachardo equation.

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Wieland, Wolfgang Martin: But before going in, so let's now go into the more technical parts of this presentation, and we will proceed in the following steps. So, 1st of all, we

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Wieland, Wolfgang Martin: just give a

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Wieland, Wolfgang Martin: certain type of convenient variables to parameterize the the null geometry at the light cone that we then need to

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Wieland, Wolfgang Martin: to understand the classical phase space. And then we start out with the hoist action that underlies lucron gravity, and we insert this parametrization, and in that way we infer a symplectic structure for gravity on a null con.

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Wieland, Wolfgang Martin: and then the proposal is to quantize this, the resulting phase space.

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Wieland, Wolfgang Martin: But first, st concerning the kinematical structure

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Wieland, Wolfgang Martin: in here, I'm looking at the we're looking at the null hypersurface so, and the vertical lines are the light rays, or indicate the light rays generating this null hypersurface

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Wieland, Wolfgang Martin: and the resulting geometry. If there are no caustics, as the structure of a fiber bundle

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Wieland, Wolfgang Martin: where the fundamental fibers are the light rays generating the null hypersurface, and the base manifold is, is equivalent to any spatial

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Wieland, Wolfgang Martin: section of this null hypersurface.

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Wieland, Wolfgang Martin: If you now take the pullback. If you now take the spacetime metric and take the pullback onto the null hypersurface, we end up with the signature, with the generate metric, of course, with the signature 0 plus plus metric on the null hypersurface and the generate 0 direction is, of course, the direction. The vertical direction in this fiber bundle, namely, the direction parallel to the light rays themselves.

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Wieland, Wolfgang Martin: And just as we have tetrads in 4 dimensions and triads in in 3 dimensions on this null hypersurface, we can introduce a natural dyad, a dyadic

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Wieland, Wolfgang Martin: basis in cotangent space, intrinsic, traditional hypersurface.

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Wieland, Wolfgang Martin: And now we would like to parameterize this any such code, I add on. So the the fixed universal structure, if you wish. In this manifold is this fiber bundle structure, and on top of that we put now fields.

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Wieland, Wolfgang Martin: and one of them are autonormal frames to parameterize the intrinsic metric on the null hypersurface, and we parameterize this in the following way. So 1st of all, we split off a conformal factor that characterizes the overall scale of the geometry.

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Wieland, Wolfgang Martin: and then we are left with the shape, degrees of freedom, that character that that describe the all possible angles on spatial hypersurfaces

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Wieland, Wolfgang Martin: between 2 tangent vectors on a cross section of the hybrid of this

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Wieland, Wolfgang Martin: null hypersurface. And these degrees of freedom we can package into a matrix in into a

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Wieland, Wolfgang Martin: a group is group valued data so we can package it into a sl 2 R. Valued field on this null boundary, which I call a Holonomy.

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Wieland, Wolfgang Martin: and these 2 fundamental variables, namely, the conformal factor. And this Sl. 2 R group valued

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Wieland, Wolfgang Martin: data. They define the physical diet relative to some fiducial background structure that I fixed once. And, for for instance, if we assume that the sections have the spatial sec. That the spatial sections have the topology of a 2 sphere. We can just put a round 2 sphere metric on

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Wieland, Wolfgang Martin: on this 2 sphere and use

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Wieland, Wolfgang Martin: an autonormal frame with respect to this fiducial background metric.

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Wieland, Wolfgang Martin: But the physical metric

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Wieland, Wolfgang Martin: is is different from that, not only by conformal factor, but also by some shape degrees of freedom

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Wieland, Wolfgang Martin: that are packaged into this sl. 2 r.

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Wieland, Wolfgang Martin: But

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Wieland, Wolfgang Martin: and this describes the metric geometry of all possible spatial cross sections of this cylinder in spacetime, but on top of that there are also the vertical directions

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Wieland, Wolfgang Martin: that that allow us to go from, to distinguish points on the same light ray.

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Wieland, Wolfgang Martin: and to distinguish points on the same light ray. We need a sort of time coordinate on this null ray, or more generally on the entire null cone.

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Wieland, Wolfgang Martin: and a kind of natural choice would be to use an affine time. So where the covariant, where the covariant derivative

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Wieland, Wolfgang Martin: with respect to the bulk geometry of these tangent vectors with respect to D by du are parallel.

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Wieland, Wolfgang Martin: but this is not so parallelly transported along these light rays. But this is not what I want to use. I want to use a slightly different choice of clock along the null ray, where the affinity, or rather non affinity, of the null generators is actually proportional to the expansion, and this has a technical reason, because it simplifies

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Wieland, Wolfgang Martin: constraints that we will investigate in a moment. Notice that this definition means that now the time variable becomes actually implicitly dependent on the fundamental fields, on of spacetime, because the expansion.

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Wieland, Wolfgang Martin: the expansion of our null generators is different depending on which metric geometry we have in the bulk. So in that way this time coordinate knows actually about the metric geometry, and in this way is itself field dependent so on phase space. Variations of U will no longer

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Wieland, Wolfgang Martin: vanish.

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Wieland, Wolfgang Martin: I call this clock sometimes teleological, because the null hypersurface itself has a boundary. So it is not an infinitely extended null boundary like scri plus. It has actually a beginning and end, which are just 2 cuts. 2 space, like cuts of this null hypersurface, where this clock

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Wieland, Wolfgang Martin: assumes some boundary values, for instance, minus one and plus one. And that's why I call it teleological, because this clock knows at which value

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Wieland, Wolfgang Martin: it has to terminate in the future.

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Wieland, Wolfgang Martin: And now we go on.

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Abhay Vasant Ashtekar: Can I? Can I ask a question here.

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Wieland, Wolfgang Martin: Yes, please.

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Abhay Vasant Ashtekar: Yeah. So I mean, you're spending this careful time, careful explanation about the null surfaces and so on. I appreciate that very much.

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Abhay Vasant Ashtekar: But of course, if I take a null con that you had in the 1st transparency on this subject in real space time, even schwater space-time, as soon as you have got spacetime. The geodess is cross, and you never have such null cones and such thing right now at scry and on the horizon you do. There are exceptional surfaces. So are you building it up only to work at scry? Or are you really thinking in terms of the physical space time? Because the physical space time. I don't think these considerations are

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Abhay Vasant Ashtekar: are valid because you'll just not have such null cons in a you know, any realistic spacetime.

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Wieland, Wolfgang Martin: What I have in mind is is the quantization of patches of null hypersurfaces, and to then glue them together. To build

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Wieland, Wolfgang Martin: initial data bit like

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Wieland, Wolfgang Martin: what is used in classical relativity in the gluing procedure of initial data. For instance, if you, if you there are procedures to glue inhomogeneous initial data to Kerr spacetime, and these procedures use null matching. So

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Wieland, Wolfgang Martin: they take matching null hypersurfaces that that are glued together at coda mentioned 2 surfaces. So this is work by crucial and collaborators to

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Wieland, Wolfgang Martin: to build good

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Wieland, Wolfgang Martin: to build new initial data from patches of given initial data.

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Wieland, Wolfgang Martin: And that is the procedure.

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Wieland, Wolfgang Martin: So.

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Abhay Vasant Ashtekar: But you are concerned.

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Wieland, Wolfgang Martin: So that is that is the reason why I also cut this this null hypersurfaces, assuming no caustics form at that point.

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Wieland, Wolfgang Martin: and then the idea is to to

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Wieland, Wolfgang Martin: that. This is a 1st step to understand the resulting quantum geometry, and then to build richer States by gluing such cylinders in in a crisscross

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Wieland, Wolfgang Martin: way. No, but there's.

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Abhay Vasant Ashtekar: Quantitative difference between the 2 cases in this space. Like case, you do this gluing. At the end of that, you get initial data, and you can evolve it. And then that initial data will lie

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Abhay Vasant Ashtekar: in some in some 40 min of space time. Because, just, you know, because of the Cauchy problem.

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Abhay Vasant Ashtekar: If you take little patches of null surfaces and glue them somehow, then

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Abhay Vasant Ashtekar: the final thing still doesn't have any intersections. Geodesics are still not crossing.

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Abhay Vasant Ashtekar: and so, therefore you could never really embed it in any realistic spacetime.

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Wieland, Wolfgang Martin: And

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Wieland, Wolfgang Martin: in this gluing, what I'm speaking about is, they start with some domain in which the constraints are satisfied, and some outer domain where spacetime is assumed to be. The initial data is supposed to be that of Kerr, and in between there's a transition region. And one way to construct this transition region is to match 2 null hypersurfaces

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Wieland, Wolfgang Martin: in this in this region.

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Wieland, Wolfgang Martin: I think you should go on because it's

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Wieland, Wolfgang Martin: on on top of that. I think I mean.

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Wieland, Wolfgang Martin: how should we know? I mean, we do not know. At this. The the

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Wieland, Wolfgang Martin: in in luke quantum gravity. We have states

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Wieland, Wolfgang Martin: of of 3 geometry, which are very different from classical smooth geometries. So

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Wieland, Wolfgang Martin: just like we have on a spatial hypersurface, just like once we quantize

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Wieland, Wolfgang Martin: Su 2 ashtaker-baro variables on a spatial hypersurface, and we end up with configurations that are not at all smooth, three-dimensional manifolds.

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Wieland, Wolfgang Martin: I think one could expect the same also, for or one should perhaps expect the same for the quantization of the geometry on another hypersurface. That generic quantum States will be

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Wieland, Wolfgang Martin: very far from a smooth Nile cone. And then the question is, how to

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Wieland, Wolfgang Martin: how to reconstruct within the resulting quantum theory such states that represent good semi-classical null geometries.

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Wieland, Wolfgang Martin: But let us now go back to the, to the initial, to the, to this, to the program, so to say.

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Wieland, Wolfgang Martin: in the previous slide, I introduced the fundamental variables. That will be the starting point for the quantization. Now, the next step, and what you will immediately realize is that

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Wieland, Wolfgang Martin: they they are so. At that point no

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Wieland, Wolfgang Martin: constraints have been imposed, so they are

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Wieland, Wolfgang Martin: just kinematical geometries at that point. Now, to make these variables that I introduced in the previous slide compatible with the Einstein equations in the bulk. Some constraints need to be satisfied.

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Wieland, Wolfgang Martin: and

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Wieland, Wolfgang Martin: they there are only 2 of them. So one of the 1st one is the Piccado equation that tells us that the second time derivative with respect to the clock that I introduced in the previous slide

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Wieland, Wolfgang Martin: of the conformal factor is given.

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Wieland, Wolfgang Martin: satisfies the following constraint, namely, that the second time derivative of the conformal factor is just minus 2 sigma sigma bar omega squared, where sigma is the shear of these null generators, and how do we obtain the shear? Again? We do not have to refer to a

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Wieland, Wolfgang Martin: to the bulk, to spacetime geometry, to to compute the shear. We can understand it intrinsically in terms of variables living on this null geometry. In fact, the first, st

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Wieland, Wolfgang Martin: the 1st time derivative of the Sl. 2 R. Holonomy, with respect to this time coordinate

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Wieland, Wolfgang Martin: gives some sl. 2 R. Lie algebra element, and this Sl. 2 R. Lie algebra element. We can split it into a u. 1 generator J,

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Wieland, Wolfgang Martin: and 2 translational generators in sl, 2 r. So these are 2 JX. And x bar are 2 by 2 s. Matrices, generators of Sl. 2 r. That satisfy the standard commutation relations of the of the Sl. 2 R. Lee algebra

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Wieland, Wolfgang Martin: and the components that appear in front of these generators. They have a physical interpretation. So the component in front of the complex structure of the u 1 generator has the interpretation of a u 1 holonomy and the other 2 remaining components. They characterize the shear degrees of freedom.

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Wieland, Wolfgang Martin: and the shear then enters into the 1st equation, defining a constraint, and now.

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Wieland, Wolfgang Martin: upon having introduced the kinematic structure and the constraints that we need to to impose. The next step is to consider the symplectic structure, the symplectic structure we obtain from the classical action, so the classical action is

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Wieland, Wolfgang Martin: the Ricci scalar, expressed in terms of 1st order, variables

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Wieland, Wolfgang Martin: shifted by by the sort of tool

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Wieland, Wolfgang Martin: scalar curvature which is obtained by contracting the Ricci.

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Wieland, Wolfgang Martin: The field strength with the epsilon turns on. Now, if the torsionless equation is satisfied, this term, of course, vanishes, but in loop quantum gravity. We work with a more general action at the kinematical level that allows for fluctuations in torsion.

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Wieland, Wolfgang Martin: and the Pabro immersive parameter appears in front of this term, which is akin to the Theta Angle in Qcd.

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Wieland, Wolfgang Martin: And the resulting action has therefore 2 coupling constants or 2 scales, namely, the Newton Constant and the Newton Constant Times, the barbarian immersive parameter.

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Wieland, Wolfgang Martin: and from the variation of the action we obtain the symplectic structure.

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Wieland, Wolfgang Martin: And if you take the symplectic structure on the resulting from from the from this action, and compute the pullback to another hypersurface, you obtain the following structure

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Wieland, Wolfgang Martin: when expressed in terms of the variables I introduced earlier.

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Wieland, Wolfgang Martin: and here we have now 3 contributions. We have contributions from the entire null hypersurface. But then there are also corner contributions from the boundary, from the initial and final cut of this null hypersurface and this symplectic structure is important for us, because it determines the Heisenberg relations from which we infer the classical Poisson

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Wieland, Wolfgang Martin: commutation relations from which we infer the quantization map.

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Wieland, Wolfgang Martin: And there are now these 3 different

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Wieland, Wolfgang Martin: contributions. So let's go through them.

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Wieland, Wolfgang Martin: The the second line gives us a bulk, symplectic structure on the integral of the entire along the entire null hypersurface. And it is expressed so what appears in here, in here we have the fiducial volume area element

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Wieland, Wolfgang Martin: on on a Cut. With respect to the fiducial metric that I introduced earlier. So this this has no variations on phase space. Then we have to

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Wieland, Wolfgang Martin: oh, fuck

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Wieland, Wolfgang Martin: and a trace, a trace in Sl. 2 R. Sl. 2 R. Element and contract it with the shear degrees of freedom. So with part of S. Dot s. Minus one. And this, if we push this part of the symplectic structure to null infinity and expand the Sl. 2 R. Holonomy in a 1 upon R. Expansion.

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Wieland, Wolfgang Martin: This just reduces to the standard symplectics potential at future null infinity.

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Wieland, Wolfgang Martin: The 3rd line gives us the

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Wieland, Wolfgang Martin: this is that this, in the in fact.

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Abhay Vasant Ashtekar: And you're breaking up.

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Wieland, Wolfgang Martin: And this constraint is now conjugate.

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Wieland, Wolfgang Martin: Pardon me.

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Abhay Vasant Ashtekar: Oh, you were breaking up.

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Wieland, Wolfgang Martin: Can you.

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Abhay Vasant Ashtekar: Yeah. Now, we can hear you.

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Wieland, Wolfgang Martin: Okay, sorry. So I repeat. So the second line gives the symplectic structure on the radiative symactic structure. The 3rd line gives us the constraint. What is in the bracket vanishes if the Ricciardo equation is satisfied, and this constraint on phase space is conjugate to the clock variable that we introduced.

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Wieland, Wolfgang Martin: But on top of the

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Wieland, Wolfgang Martin: constraint which is conjugate to the clock variable and the radiative data, there's an additional boundary term.

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Wieland, Wolfgang Martin: and this boundary term is, in fact, or the immersive parameter.

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Hal Haggard: Wolfgang again you broke up. Do you mind repeating that sorry.

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Wieland, Wolfgang Martin: So what we, the what it's awesome

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Wieland, Wolfgang Martin: is that the conformal factor is, in fact, an edge mode, a boundary degree of freedom in phase space that is, conjugate to the u 1 part

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Wieland, Wolfgang Martin: of the of our Sl. 2 R phase space.

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Wieland, Wolfgang Martin: And this is very nice, in fact.

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Wieland, Wolfgang Martin: because now it turns out that all degrees of freedom on phase, space.

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Wieland, Wolfgang Martin: or all degrees of freedom, that we need to characterize the null geometry

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Wieland, Wolfgang Martin: have actually a corresponding phase space. So there's a phase space structure for the radiative part of phase space.

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Wieland, Wolfgang Martin: But then there's also a phase-based structure for the conformal factor. The conformal factor is determined by solving the Reichaudo ray equation.

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Wieland, Wolfgang Martin: But to solve the regiado equation it is not enough to know Sigma Sigma bar the shear degrees of freedom. You also have to to have access to initial data for Omega itself.

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Wieland, Wolfgang Martin: And the face space

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Wieland, Wolfgang Martin: for the initial data that you need for Omega is actually provided by the 1st line that tells us

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Wieland, Wolfgang Martin: that Omega squared

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Wieland, Wolfgang Martin: the area metric, that the area on the so the conformal factor on the cross sections is conjugate to a u 1 part

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Wieland, Wolfgang Martin: of of our Sl. 2 r.

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Wieland, Wolfgang Martin: and notice also that the second line looks very similar

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Wieland, Wolfgang Martin: to a standard symplexic potential for a nonlinear sigma model.

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Wieland, Wolfgang Martin: The only difference is that the structure is way more complicated, because now what spoils the linearity of the phase space, or what spoils the standard expression for the Sl. 2 R.

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Wieland, Wolfgang Martin: Canonical symplectic structure is that there is this conformal factor appearing that depends implicitly upon solving the constraints on Sigma itself. So this becomes highly nonlinear upon solving the constraints.

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Wieland, Wolfgang Martin: You may just a technical remark. You may have noticed this funny index. I here. This just means that there is a u 1 gauge symmetry, which we have kind of removed by introducing appropriately dressed observables. So instead of working with the original Sl. 2 R. Holonomy, I have split off a u 1 factor.

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Wieland, Wolfgang Martin: And this u 1 factor is just this u 1 connection integrated along the null race. So there is.

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Wieland, Wolfgang Martin: you do one

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Wieland, Wolfgang Martin: part that is removed from the original sl. 2 R. Holonomy. And it's not surprising that this u 1 part, because it is a gauge symmetry on phase space, then only shows up as an edge mode as a boundary observable where the corresponding charge is, in fact, the area density.

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Wieland, Wolfgang Martin: all these double stroke D's are ordinary differentials in phase space. This capital D denotes addressed field space derivative. So it's the ordinary which, on the null hypersurface, becomes very simple. It is just the ordinary variation on phase space subtracted.

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Wieland, Wolfgang Martin: Times a correction or minus a correction, and the correction is a diffeomorphism along the null generators

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Wieland, Wolfgang Martin: with a gauge parameter, that is, that can vary along the null ever surface.

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Wieland, Wolfgang Martin: And what is also very oh.

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Wieland, Wolfgang Martin: is that the parameter only appears in a corner term, and this is, in fact, not at all surprising, because we know that in classical, we know from classical, from the, from the classical theory that the change from

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Wieland, Wolfgang Martin: Adm type of variables to triadic variables, and then to Ashtica Barbero variables is just a canonical transformation that cannot affect the classical commutation relations for the radiative data, and that is that this is one manifestation of this fact is

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Wieland, Wolfgang Martin: that the bulk symplectic structure, the symplectic structure for the radiative modes is, in fact, independent of gamma.

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Wieland, Wolfgang Martin: but this does not imply that at the quantum level

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Wieland, Wolfgang Martin: the immersive parameter cannot also cannot also affect the representations that appear for the quantization of the radiative data, as we will see now.

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Wieland, Wolfgang Martin: So now we want to take this phase space and quantize it.

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Wieland, Wolfgang Martin: One observation that I have not discussed so much in here is that

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Wieland, Wolfgang Martin: different light ways or commute. So there are no under commute under the Poisson bracket which makes it

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Wieland, Wolfgang Martin: possible, or to to quantize each light ray by itself, using a sort of polymer quantization

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Wieland, Wolfgang Martin: in the angular directions in along the null directions. On the other hand, I impose a truncation

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Wieland, Wolfgang Martin: to reduce the number of physical degrees, to reduce the to, to go from infinitely many degrees of freedom along each Norway to finitely, countably many degrees of freedom. So the idea is, instead of

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Wieland, Wolfgang Martin: using some smooth profile for the shear along the null generators, we use a truncation

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Wieland, Wolfgang Martin: instead of a smooth profile I use in the following a piecewise constant profile.

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Wieland, Wolfgang Martin: and then the idea is that for each such pulse, which will have a certain duration, duration, and a certain height. In in the in the shear

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Wieland, Wolfgang Martin: we have a corresponding phase space, and each of these phase spaces can be quantized separately, and then the quantization of an arbitrary profile will be obtained by taking tensor products of many such representations and gluing the adjacent quantum numbers.

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Wieland, Wolfgang Martin: And this conversation is highly non-trivial, because there's a highly nonlinear constraint between shear and conformal factor that needs to be imposed at the quantum level.

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Wieland, Wolfgang Martin: And what one then finds is that due to the flux balance loss, in other words, due to the Riccido equation that

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Wieland, Wolfgang Martin: links the data at the endpoints of these pulses to the data that characterize the shear of each of these pulse. Any alteration in the representation theory of these boundary variables will also affect the quantization of the

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Wieland, Wolfgang Martin: sheer degrees of freedom.

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Wieland, Wolfgang Martin: and in this way the quantization of of area can affect the quantum theory of the Chi degrees of freedom.

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Wieland, Wolfgang Martin: Now, in the following, I consider just a single pulse, and for a single, and along the duration of the pulse the shear should be just constant in this time parameter in here again, Sigma, I refer the index. I refers to this u 1 dressing of the fundamental variables. So to this u 1 interaction picture.

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Wieland, Wolfgang Martin: and this has so

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Wieland, Wolfgang Martin: and so it is with respect to that gauge that the shear degrees of freedom are constant along the null generators.

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Wieland, Wolfgang Martin: and for any such constant shear. We can now go back to the Ricciardoui equation and just solve

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Wieland, Wolfgang Martin: solve the equation and get some and get a a profile along each of the null generators, minus.

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Abhay Vasant Ashtekar: Yeah, that's.

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Wieland, Wolfgang Martin: Plus and e plus and e minus in. Here are the initial and final values for the conformal factor.

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Abhay Vasant Ashtekar: Oh, thank you!

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Wieland, Wolfgang Martin: So sigma is the shear and E plus and E minus is the area density or the conformal factor at the initial and final Cross section at which the pulses start and terminate.

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Hal Haggard: Wolfgang. Can I ask a question here, too?

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Hal Haggard: I'm I'm curious if you can give me any insight into

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Hal Haggard: why it is the Rachel Duri equation becomes oscillatory here, or, in other words, why the expansion becomes oscillatory. I understand you've made this gauge choice for your clock variable, and then you're making this impulsive assumption, and those 2 things when you combine them they give this result. I see that clearly, but I don't know what it is about your gauge choice

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Hal Haggard: that leads to this oscillatory expansion. Can you give me any insight? There.

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Wieland, Wolfgang Martin: Maybe oscillatory is a bit misleading. It doesn't mean that Omega squared oscillates in this interval between plus and minus one or something. This is not the case, because

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Wieland, Wolfgang Martin: There is implicitly here already a bound kinematical bound on the shear, that the shear cannot be larger than pi half, or something. So there is. The period is never large.

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Wieland, Wolfgang Martin: the the period is never larger than the interval that we are considering. So you never go through through a caustic

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Wieland, Wolfgang Martin: in here.

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Wieland, Wolfgang Martin: Where did that come from?

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Abhay Vasant Ashtekar: The condition on on the shear is bounded.

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Wieland, Wolfgang Martin: For there not to be a caustic.

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Wieland, Wolfgang Martin: But this is a different. We will see that this is a different. This is a different

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Wieland, Wolfgang Martin: bound than the than the one I derive from the for the Planck luminosity.

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Wieland, Wolfgang Martin: So this is, if if you look at

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Wieland, Wolfgang Martin: if we were to go to future null infinity. All sigma in here would fall off like one upon R. So

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Wieland, Wolfgang Martin: this would be always well within this bound.

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Wieland, Wolfgang Martin: within this pi half pound bound.

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Wieland, Wolfgang Martin: So at null infinity, this, these, these become very small.

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Wieland, Wolfgang Martin: That's what I'm saying. Way below pi half.

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Wieland, Wolfgang Martin: So this is just a way to perm. If there is no. If there are no caustics in this interval, then then Omega assumes the following form.

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Wieland, Wolfgang Martin: and then then this is always, then Omega squared is also always positive.

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Wieland, Wolfgang Martin: and in the same way that we can solve for the conformal factor we can also solve for the Sl. 2 R. Holonomy. So now that the shear is constant along the null generators. It's very easy to solve for the Sl. 2 R. Holonomy. So to solve the differential equation for S,

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Wieland, Wolfgang Martin: which is basically the exponential of the shear. And since the shear is constant.

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Abhay Vasant Ashtekar: I'm sorry I'm confused about this, because it's the same question that Hal was asking. So if you look at null infinity, you're saying that Sigma is upon R. So at null infinity, just 0.

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Abhay Vasant Ashtekar: So the 1st term is going to be. This cost of U.

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Abhay Vasant Ashtekar: The second term is going to be blowing up right. I'm not understanding.

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lsmolin: So.

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Abhay Vasant Ashtekar: Your argument was that well, it falls like one upon a okay. So then.

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Wieland, Wolfgang Martin: Yes.

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Abhay Vasant Ashtekar: And it's 0.

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Wieland, Wolfgang Martin: Yeah, but also e plus and e minus have a certain behavior as.

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Abhay Vasant Ashtekar: And why? Exactly.

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Wieland, Wolfgang Martin: They blow up. In fact, pardon me.

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Abhay Vasant Ashtekar: I didn't. They just fixed? Oh, okay, so it's so, okay, so please explain.

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Abhay Vasant Ashtekar: They blow up.

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Wieland, Wolfgang Martin: Yeah. So if we if we go to null infinity.

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Wieland, Wolfgang Martin: Sigma sigma bar, this expression goes to 0. So this becomes one, and this blows up as r squared.

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Wieland, Wolfgang Martin: So this is just a trivial statement that the area of cross-section of future null infinity is infinite. But then there are subleading terms which don't blow up like r squared, but go like R, or go a constant and further subleading terms.

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Abhay Vasant Ashtekar: Okay, you got it.

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Wieland, Wolfgang Martin: Cool. Yeah.

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Wieland, Wolfgang Martin: there's there's no r expansion at this point. I'm just looking at an abstract null boundary.

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Abhay Vasant Ashtekar: That's what is confusing, that there is no r expansion. But you're still.

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Wieland, Wolfgang Martin: Yes, yes, they they so

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Wieland, Wolfgang Martin: far. Yeah, there is no R. At that point everything is at an abstract null boundary at finite distance. I'm not saying where this is in space time.

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Wieland, Wolfgang Martin: And yeah, just like we can solve the equation for the conformity factor you can also solve for the Sl. 2 R. Holonomy. And again, just because there is no implicit U dependence in here, this exponential is very easy to solve. It's yeah, just a combination of hyperbolic cosine and hyperbolic sine.

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Wieland, Wolfgang Martin: And this is a bit weird, because here Sigma Sigma Bar appears like a frequency, and here, like a

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Wieland, Wolfgang Martin: hyperbolic angle, so to say, and there's this kind of double role, which is.

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Wieland, Wolfgang Martin: And we will see this in the next. Further down in the presentation this double role is a bit reminiscent of the mixing that the immersive parameter does between Euclidean and Lorentzian parts.

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Wieland, Wolfgang Martin: I will clarify this in a moment.

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Hal Haggard: Wolfgang.

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Wieland, Wolfgang Martin: What one? Then?

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Hal Haggard: You've had plenty.

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Wieland, Wolfgang Martin: And then.

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Hal Haggard: So you should feel free to take a few more minutes. But also you may want to start to think about wrapping up.

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Wieland, Wolfgang Martin: Oh, okay, so and now, the

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Wieland, Wolfgang Martin: going to the classical phase space, you find canonical commutation relations. So you find 2 Heisenberg harmonic oscillators at each point on the sphere, and on top of that there's an Sl. 2 R. Algebra, C and C Bar are like creation and annihilation operators. L. Is like l. 3 in. Su. 2.

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Wieland, Wolfgang Martin: And then on this phase space. Now you can relate these canonical variables back to the geometric variables. Delta is just a u. 1 is a u. 1 angle of this phi u 1 holonomy that I introduced along the duration of the pulse. And here you see this shift

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Wieland, Wolfgang Martin: of this u 1 holonomy by terms that are proportional to the immersive parameter here. Here. So this, in my opinion, is the same type of shift we see on a spatial hypersurface, where canonical variables are built from gamma. The analog of the Usu Su. 2. Holonomy is this u 1 holonomy on the light cone shifted by

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Wieland, Wolfgang Martin: Gamma K, and the shift by Gamma K on the spatial hypersurface. The extrinsic curvature becomes on the null hypersurface a shift by terms in

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Wieland, Wolfgang Martin: exponent exponent that are proportional to gamma and go with the shear

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Wieland, Wolfgang Martin: of the emitted pulse, and the area of the initial and final cross section is now related on this phase, space to sums and differences of the number, operators of A and a bar.

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Wieland, Wolfgang Martin: Oh.

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Wieland, Wolfgang Martin: and similarly they can be on this phase base, and this is now. And with this I can conclude then, because the main result will derive from there.

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Wieland, Wolfgang Martin: there's 1 residual constraint.

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Wieland, Wolfgang Martin: and what is kind of amazing about this constraint is that it commutes with the Sl. 2 R. Casimir.

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Wieland, Wolfgang Martin: and that means that the Sl. 2 R. Casimir becomes a physical observable on this phase. Base.

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Abhay Vasant Ashtekar: Where do?

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Wieland, Wolfgang Martin: Couple of spare.

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Abhay Vasant Ashtekar: No problem.

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Abhay Vasant Ashtekar: Where does this constraint come from?

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Wieland, Wolfgang Martin: Where does this come from? Embed the with the phase space, or the symplexic structure I start with into a slightly larger phase space, and this slightly larger phase space is given by the canonical commutation relations on that slide. But then there is one redundancy in there.

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Wieland, Wolfgang Martin: Some of these variables are defined implicitly in terms of other variables, and that turns out to be a constraint on phase space. That is, second class.

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Abhay Vasant Ashtekar: Thank you.

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Wieland, Wolfgang Martin: But it has a simple structure, in fact, because it only involves creation and annihilation operators. So it is basically a recurrence relation at the quantum level something you can solve on one sheet of paper.

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Wieland, Wolfgang Martin: And

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Wieland, Wolfgang Martin: what is interesting now is that the this only one constraint that we have to deal with commutes with the with the Sl. 2

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Wieland, Wolfgang Martin: Casimir. That means that the asset to Casimir is a physical observable.

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Wieland, Wolfgang Martin: but at the quantum level the unitary representations of Sl. 2 R. Are characterized by the value of the Casimir. There are continuous series representations and discrete series representations, and you flip from one representation in the other. Once the Casimir vanishes, when does the Casimir vanish?

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Wieland, Wolfgang Martin: It? It vanishes, or if you fix the area at the initial and final cross section, the Casimir vanishes at a certain value of the shear.

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Wieland, Wolfgang Martin: In fact, you can write the Casimir in the following way, in terms of the area density of the initial and final cross section and the shear that each of these pulses can carry, and the resulting expression looks like that. It looks a bit complicated. But now we can ask, at what critical shear do you transition in the spectrum from one representation to the other?

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Wieland, Wolfgang Martin: And let us ask this question at future null infinity.

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Wieland, Wolfgang Martin: To ask this question at future null infinity. What you have to do is to now relate. And this goes back to the question Abai asked earlier. What we now have to do is to relate the variables that characterize this piles

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Wieland, Wolfgang Martin: in terms of variables at null infinity. So we have to make a.

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Wieland, Wolfgang Martin: We have to find a dictionary how to relate, for instance, the time variable, that we used to the standard bonded time at what you find, for instance, just to give you a hint of what's going on, you find that this vector field, with respect to this time variable, that characterizes the pulses is related to bonded time times, an overall scale factor. Scaling

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Wieland, Wolfgang Martin: caling is just the duration of the pulse in bonded time.

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Wieland, Wolfgang Martin: and similarly, E, plus the difference between the 2. The area of the 2 cross sections blows up, but blows up linearly times. The duration of the times delta u, which is the duration of the pulse in bonded time. So if you then insert the expansion.

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Wieland, Wolfgang Martin: the asymptotic one, upon r expansion into this equation for the critical shear, the shear at which the value of the Casimir flips from the discrete series representations to the continuous series representations you find a critical value.

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Wieland, Wolfgang Martin: and what is very nice is that this becomes now is now very simple. It it is actually constant.

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Wieland, Wolfgang Martin: It's a 1, and only depends on the immersive parameter. So there's a critical news

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Wieland, Wolfgang Martin: at which you would flip from one representation into the into the other.

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Wieland, Wolfgang Martin: and this critical news corresponds to a critical luminosity, and this critical luminosity is nothing than the luminosity divided by

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Wieland, Wolfgang Martin: squared plus one that makes the immersive parameter.

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Wieland, Wolfgang Martin: Let me conclude

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Wieland, Wolfgang Martin: so. What I presented to you is a non-perturbative quantization of impulsive nullinitial data. This. But this approach to the to the quantization of such data contains both

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Wieland, Wolfgang Martin: quantum data for radiative degrees of freedom as well as quantization of corner data. The corner data characterizes, among other things, the

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Wieland, Wolfgang Martin: the area density at the initial and final cross section. In addition to that, there is also an Sf, 2 R corner data that plays no role in the analysis that I introduced to you today. And on top of that the Planck power

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Wieland, Wolfgang Martin: has a role to play in this quantum theory, namely, it separates the discrete from the continuous series representations of Sl. 2 R. Below the Planck power. You're in the discrete series representations, and above the Planck power, you are in the continuous series representation. Furthermore.

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Wieland, Wolfgang Martin: there are hints, and this is why this becomes very interesting, that states

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Wieland, Wolfgang Martin: in the continuous spectrum are, in fact, unphysical.

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Wieland, Wolfgang Martin: One way to to see that is, that physical states, in the continuous series, representations are built from kinematical states, in which the shear is actually unbounded.

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Wieland, Wolfgang Martin: and and, in fact, will inevitably violate the implicit pull-off conditions, such that the resulting states

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Wieland, Wolfgang Martin: it's very hard to see how the resulting States should be compatible with the implicit assumption of a smooth, asymptotic boundary at null infinity. Furthermore, there seem to be other problems with the inner product and the semi-classical limit for the discrete series representations. It is very easy to build semi-classical coherent states

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Wieland, Wolfgang Martin: because there's basically an oscillator representation at hand for the continued series representations. This seems much more involved. And finally, one more comment.

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Wieland, Wolfgang Martin: What I think, what all this is pointing to is that the Planck luminosity will play for quantum gravity in D equal 4 space-time dimensions. The same role that the Planck mass has in D equal 3 spacetime dimensions, where it is, in fact.

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Wieland, Wolfgang Martin: responsible for the formation of the classical symmetry, algebra giving rise to such approaches as relative locality, with perhaps even connections to non-commutative geometry.

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Wieland, Wolfgang Martin: Thanks for the for thanks for the many questions and and for tuning in today.

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Hal Haggard: Thank you, Wolfgang. Thank you for a very nice seminar.

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Wieland, Wolfgang Martin: And apologies for going over time.

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Hal Haggard: Are there? Questions? We? We have time for questions still Deepak.

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Deepak Vaid: Yeah.

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Deepak Vaid: Hi, Wolfgang. Excellent talk as always. So I've been going around touting your paper as an example of a concrete prediction from Lqg, so you know, that's why I think it's I mean, because we need to concrete predictions. And

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Deepak Vaid: so so I think it's very important to understand

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Deepak Vaid: what are the possible observational signatures of this

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Deepak Vaid: at the let's say in the short term in the long term. Have you considered that at all.

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Wieland, Wolfgang Martin: Yes, but I don't have. I don't want to talk about that. At this point also. At some point Eugenio came with an interesting idea that there is perhaps

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Wieland, Wolfgang Martin: I mean the immersive parameter is

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Wieland, Wolfgang Martin: is in front of a parity violating term. So the 2 it. It violates. Discrete as this, it violates parity.

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Wieland, Wolfgang Martin: and it would be very interesting to understand whether this has any implications also at the quantum level. For

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Wieland, Wolfgang Martin: for instance, yeah, in the S. Matrix, for instance, that the 2 polarizations behave differently.

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Wieland, Wolfgang Martin: that that could could be one very interesting avenue for further research.

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Deepak Vaid: I have another question, but I I'll wait for my turn afterwards.

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Hal Haggard: Good Abby.

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Abhay Vasant Ashtekar: We have had these discussions many times, and

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Abhay Vasant Ashtekar: I find that these ideas are extremely stimulating, but other than I think they are very patchy.

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Abhay Vasant Ashtekar: and the patchiness comes from the fact that if this was a genuine effect, I would. I just like to say, Well.

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Abhay Vasant Ashtekar: look at Scry, and I just do this quantization as cry.

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Abhay Vasant Ashtekar: And then this predicts the following thing.

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Abhay Vasant Ashtekar: and there is all this going back and forth between.

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Wieland, Wolfgang Martin: I think this this won't work it, and it cannot work for any reason. In fact.

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Abhay Vasant Ashtekar: I'm just wondering, because, okay, go ahead.

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Wieland, Wolfgang Martin: Because there are 2 limits, there's the limit going to.

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Wieland, Wolfgang Martin: There's the limit going to infinity. So the out going to infinite limit and the H bar going to seem to 0 limit

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Wieland, Wolfgang Martin: the H bar to going to 0 limit. Fundamentally, there's only the quantum theory. There's no such thing as quantization

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Wieland, Wolfgang Martin: and taking the quantum theory, H. Bar to 0 gives us a classical phase. Space

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Wieland, Wolfgang Martin: going to an infinite boundary gives us asymptotic limit.

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Wieland, Wolfgang Martin: There's no reason to believe, and I think

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Wieland, Wolfgang Martin: this result, or although it is patchy.

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Wieland, Wolfgang Martin: is an actual demonstration of that fact.

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Wieland, Wolfgang Martin: namely, the fact that these these limits do not necessarily commute.

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Wieland, Wolfgang Martin: It is not true that going to

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Wieland, Wolfgang Martin: taking the quantum theory, taking the H bar, going to 0 limit, arriving at the classical phase space, and then, within this phase, space, taking the out to infinity limit, is the same as

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Wieland, Wolfgang Martin: 1st defining in your

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Wieland, Wolfgang Martin: quasi local quantization of gravity and asymptotic boundary, and then taking the limit, H. Bar to going to 0.

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Wieland, Wolfgang Martin: But that conversation and.

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Abhay Vasant Ashtekar: You're making it a virtue, but it seems to be a limitation because you could take. I mean, if the real

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Abhay Vasant Ashtekar: space is supposed to be, as you know, quantum mechanical, then you have to tell me what is the arena in which you are describing gravitational waves.

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Abhay Vasant Ashtekar: and if this arena, does it exist in the quantum theory, or does it not exist in the quantum theory? If it does not exist in the quantum theory, then this discussion about the luminosity bound is is empty.

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Abhay Vasant Ashtekar: If it does exist in a quantum theory, what is it? I don't have to go to classical scry.

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Abhay Vasant Ashtekar: You can just go to content whatever you want to call the replacement of this.

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Abhay Vasant Ashtekar: you just have to tell me what is the arena in which you are defining gravitational radiation

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Abhay Vasant Ashtekar: in the quantum theory, and I would be very happy with that.

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Abhay Vasant Ashtekar: And the second thing is that the I mean for a long time you have been doing this impulsive waves, which is the 1st thing once you try, and I like it very much.

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Abhay Vasant Ashtekar: But I feel that lot of results you're obtaining are artifacts of that

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Abhay Vasant Ashtekar: I mean, for example, you say things like well, only the continuous representation does not have is not physical, because if I go to infinity. Everything will blow up. But no, because it went up on R.

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Abhay Vasant Ashtekar: In the infinity, as you're going to infinity. So if you have unboundedness, it doesn't matter, because it's 1 upon R. Will will kill it right? I'm not sure that.

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Wieland, Wolfgang Martin: No, no.

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Abhay Vasant Ashtekar: Things that

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Abhay Vasant Ashtekar: so so I'm not.

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Wieland, Wolfgang Martin: Just go on.

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Abhay Vasant Ashtekar: Fact that there's a there's a there's a consistent

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Abhay Vasant Ashtekar: content picture that I can actually come come with, in which I can talk about these things. I mean, I just say that in this whole thing there's no r.

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Abhay Vasant Ashtekar: Then what? Where am what is the arena for? For gravitational waves

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Abhay Vasant Ashtekar: you bring in scry in order to make us intuitively convenient. I like that very much. But then you say that no, no, but this is not a limit. That quantum theory doesn't have this limit. All right. I also completely accept that. But then I would like to know.

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Wieland, Wolfgang Martin: I'm not. I'm not saying that quantum theory does not have this limit. I'm saying that there are

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Wieland, Wolfgang Martin: that the 2 limits do not commute.

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Abhay Vasant Ashtekar: No fine. We can.

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Wieland, Wolfgang Martin: Start, all of us.

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Abhay Vasant Ashtekar: What is the limit of the.

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Wieland, Wolfgang Martin: Start out with a infinite boundary

445
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Wieland, Wolfgang Martin: at the classical level, write down a classical phase space, and then quantize that phase space. What I'm saying is that this is different.

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Wieland, Wolfgang Martin: From doing the the reverse.

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Wieland, Wolfgang Martin: You see that.

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Abhay Vasant Ashtekar: Let's just start with the quantum theory and tell tell me what is what are gravitational waves in quantum theory.

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Abhay Vasant Ashtekar: What is arena that you're talking about?

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Wieland, Wolfgang Martin: Well, there is no.

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Wieland, Wolfgang Martin: I mean, yeah. Well, this is the fundamental problem we we have.

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Wieland, Wolfgang Martin: we have a description of gravitational waves of energy, bondi, energy, bondi, flux in terms of an asymptotic expansion. And it's and

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Wieland, Wolfgang Martin: there is no there is no such thing at at finite distance, but what there is at finite distance are quantum states of of geometry, and these quantum states of geometry can now not be

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Wieland, Wolfgang Martin: split into a background, and radiative modes. But the whole thing together exists. I mean, this is the this is the whole philosophy of non-perturbative quantization of gravity, of loop quantum gravity. In fact.

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Abhay Vasant Ashtekar: I appreciate that very much. I'm really very sympathetic, but I kind of feel uncomfortable at the end of the day, because

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Abhay Vasant Ashtekar: I really feel that at the end of that I mean, you can apply these things to finite place, finite boundaries, etc. Etc.

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Abhay Vasant Ashtekar: But then I feel uncomfortable when you say. The result is, therefore, for luminous luminosity, luminosity, which is only defined at infinity.

458
01:15:11.630 --> 01:15:19.050
Abhay Vasant Ashtekar: and I and the 2 limits don't commute, and still but nonetheless, I am going to say that there is actually a limit on luminosity.

459
01:15:19.260 --> 01:15:28.189
Abhay Vasant Ashtekar: And the second thing is again, I mean, the statement is that well, the final limit involves immersive parameter. It is supposed to be not have anything to do with.

460
01:15:28.500 --> 01:15:31.019
Abhay Vasant Ashtekar: You know there's no hbo in it.

461
01:15:31.450 --> 01:15:41.619
Abhay Vasant Ashtekar: But then, in the classical theory, at the finite distance I could have just chosen any emergency parameter. I wanted, as you have said many times, would have gotten different area gaps, different things.

462
01:15:41.870 --> 01:15:44.520
Abhay Vasant Ashtekar: And then, at the end of the day.

463
01:15:45.230 --> 01:15:49.459
Abhay Vasant Ashtekar: why is it, I mean, if the result is classical, as you are saying.

464
01:15:49.710 --> 01:15:54.980
Abhay Vasant Ashtekar: that you know that that Ligo runs upper bound, and so on and so forth. Then

465
01:15:55.990 --> 01:16:00.849
Abhay Vasant Ashtekar: I mean, how can it depend on Gamma? Because classical generality doesn't care about gamma.

466
01:16:01.020 --> 01:16:11.039
Wieland, Wolfgang Martin: No, it's not a classical result. It's a result obtained from the semiclassical limit of of quantum states.

467
01:16:11.040 --> 01:16:11.520
Abhay Vasant Ashtekar: Alright, excellent.

468
01:16:11.520 --> 01:16:15.839
Wieland, Wolfgang Martin: I find I find I find, that something happens

469
01:16:16.080 --> 01:16:19.550
Wieland, Wolfgang Martin: in the spectrum of the emitted power.

470
01:16:19.880 --> 01:16:22.019
Wieland, Wolfgang Martin: Or let's let's be more.

471
01:16:22.020 --> 01:16:25.620
Abhay Vasant Ashtekar: The spectrum time we get, but it's not defined at finite distance.

472
01:16:25.880 --> 01:16:30.759
Wieland, Wolfgang Martin: Wait a second, so I find something happens for the sheer

473
01:16:31.680 --> 01:16:40.540
Wieland, Wolfgang Martin: when the shear assumes a certain value in terms depending on the value of other variables, namely, the area of the initial and

474
01:16:40.920 --> 01:16:59.910
Wieland, Wolfgang Martin: and a slice of this pulse. And then, if you take this description to infinity, you can relate the shear, which would otherwise be just an abstract geometrical, observable

475
01:17:00.440 --> 01:17:06.950
Wieland, Wolfgang Martin: to the energy of gravitational waves. So only upon taking the limit to infinity.

476
01:17:07.440 --> 01:17:13.200
Wieland, Wolfgang Martin: you can say something about the about power.

477
01:17:14.400 --> 01:17:24.339
Hal Haggard: All right. I want to try and put the conversation on hold for just a moment, just because there are other questions, and we can come back to it. John Schiller, you were next.

478
01:17:25.700 --> 01:17:28.369
John Schiller: Thank you, Hal. Thank you, Wolfgang.

479
01:17:28.520 --> 01:17:32.860
John Schiller: You briefly mentioned the Casimir in your last few slides, Wolfgang.

480
01:17:33.060 --> 01:17:34.587
John Schiller: I was hoping that

481
01:17:35.340 --> 01:17:40.410
John Schiller: Maybe you could tell a little bit more about the physical or geometric interpretation of the Casimir.

482
01:17:44.520 --> 01:17:55.479
Wieland, Wolfgang Martin: So the Casimir is. This is just the Sl. 2 R. Casimir on that phase base. So the u. 1 generator minus CC bar. And then I ask.

483
01:17:56.090 --> 01:17:58.680
Wieland, Wolfgang Martin: how how

484
01:17:59.020 --> 01:18:14.129
Wieland, Wolfgang Martin: can this be expressed back again in terms of trivial variables? And what do you find is that it is a combination, or it depends not only on shear.

485
01:18:14.370 --> 01:18:30.059
Wieland, Wolfgang Martin: It's, in fact, it's sort of non-locally observable, because it depends on the initial and final area density of your cross section, but also of the shear sandwiched in between. And what I then say is, when we go to infinity.

486
01:18:30.280 --> 01:18:31.270
John Schiller: And.

487
01:18:31.270 --> 01:18:35.539
Wieland, Wolfgang Martin: All these have a specific fall off.

488
01:18:35.770 --> 01:18:48.210
Wieland, Wolfgang Martin: And something peculiar happens when the asymptotic shear assumes that special value. Because then this becomes 0.

489
01:18:49.320 --> 01:18:50.790
Wieland, Wolfgang Martin: Yeah, thanks.

490
01:18:51.430 --> 01:18:52.299
John Schiller: Thank you.

491
01:18:53.510 --> 01:18:56.529
Hal Haggard: Lee. You also had a hand up for a moment.

492
01:18:56.850 --> 01:19:02.070
lsmolin: Yes, I think it's closely related to Abi's questioning.

493
01:19:02.722 --> 01:19:06.790
lsmolin: You you make, and it's probably just my ignorance.

494
01:19:07.000 --> 01:19:10.489
lsmolin: But at some point you introduce the

495
01:19:10.830 --> 01:19:16.890
lsmolin: discreteness in the solutions by saying, you can choose gauges where the constraints

496
01:19:17.000 --> 01:19:20.330
lsmolin: that lead to the further effects held.

497
01:19:20.440 --> 01:19:26.149
lsmolin: But you, you seem to leave it open that there might be very different behavior in other cases.

498
01:19:28.920 --> 01:19:36.200
Wieland, Wolfgang Martin: Okay, that may have been a misunderstanding. Indeed, the discreteness in the spectrum

499
01:19:36.540 --> 01:19:44.180
Wieland, Wolfgang Martin: is not the result of any gauges, but really comes directly from the quantization of the fundamental algebra.

500
01:19:44.370 --> 01:19:49.690
Wieland, Wolfgang Martin: In fact, what you see here all this. So L.

501
01:19:49.900 --> 01:19:53.560
Wieland, Wolfgang Martin: A. A. And a dagger. This is just a number operator B.

502
01:19:53.560 --> 01:20:02.530
Wieland, Wolfgang Martin: Diego's just a number operator, and these have a discrete spectrum in the quantum theory. This is, in fact.

503
01:20:03.500 --> 01:20:13.930
Wieland, Wolfgang Martin: reminiscent of the spectrum that we get the area spectrum that we get in loop quantum gravity. The only difference is that here it is equidistant. But

504
01:20:15.420 --> 01:20:17.420
Wieland, Wolfgang Martin: this may have other.

505
01:20:18.530 --> 01:20:24.180
Wieland, Wolfgang Martin: So oh, it's

506
01:20:28.330 --> 01:20:39.500
Wieland, Wolfgang Martin: there's data for. But the important is that this organization of a classical algebra very simple algebra, in fact.

507
01:20:40.410 --> 01:20:45.199
lsmolin: Maybe we should discuss this. You got cut off halfway through.

508
01:20:48.750 --> 01:20:54.729
Hal Haggard: Before we return to Deepak and Abai. Were there any other questions?

509
01:20:57.330 --> 01:21:00.770
Hal Haggard: Okay, Deepak. If you had another short one.

510
01:21:01.700 --> 01:21:05.147
Deepak Vaid: Yeah, yeah. Well, I mean, that's relative.

511
01:21:06.555 --> 01:21:07.180
Deepak Vaid: So

512
01:21:07.780 --> 01:21:18.299
Deepak Vaid: I mean, I mean, I guess I guess again, this is probably a naive question. But I mean what you're doing. Wolfgang is basically you're you're starting with.

513
01:21:18.948 --> 01:21:21.879
Deepak Vaid: The I can inhibit action with the

514
01:21:22.040 --> 01:21:24.250
Deepak Vaid: this thing. What do you call it?

515
01:21:24.600 --> 01:21:25.679
Wieland, Wolfgang Martin: Hello! It's Tom!

516
01:21:25.680 --> 01:21:33.599
Deepak Vaid: Holds term holds term right and then and then pulling that back to the null surface and then quantizing it is that more or less.

517
01:21:34.530 --> 01:21:34.900
Wieland, Wolfgang Martin: Yes.

518
01:21:34.900 --> 01:21:44.849
Deepak Vaid: So my my question is that you know, we do that in for spatial hyper spatial surfaces right? And we end up with the

519
01:21:44.960 --> 01:21:51.468
Deepak Vaid: spin network representation, right? So shouldn't it be possible to.

520
01:21:52.320 --> 01:21:59.250
Deepak Vaid: you know, a extend the spin network representation directly to null surfaces.

521
01:21:59.360 --> 01:22:02.109
Deepak Vaid: or or or or or or B,

522
01:22:02.400 --> 01:22:05.480
Deepak Vaid: you know, I mean, because, because, like you know, I'm

523
01:22:05.740 --> 01:22:09.540
Deepak Vaid: I love formalism. But like I also like to see

524
01:22:09.680 --> 01:22:12.159
Deepak Vaid: simpler ways to do things so.

525
01:22:12.160 --> 01:22:14.350
Wieland, Wolfgang Martin: Yes, yeah, thank you for this.

526
01:22:14.350 --> 01:22:15.340
Deepak Vaid: Is, that is.

527
01:22:17.040 --> 01:22:17.780
Wieland, Wolfgang Martin: Hey?

528
01:22:18.240 --> 01:22:29.750
Wieland, Wolfgang Martin: Yeah, thank you for this remark. It's a very interesting point. And indeed, I think Simona would have perhaps something to say about that there are

529
01:22:29.990 --> 01:22:53.610
Wieland, Wolfgang Martin: proposals for putting spin networks on a on a light cone, or or on a null hypersurfaces, or on a null hypersurface, and the same the same criticism, or or however you would call it what was mentioned. Of course those applies there, but but still one can ask.

530
01:22:53.730 --> 01:22:55.380
Wieland, Wolfgang Martin: is it possible

531
01:22:55.830 --> 01:23:07.980
Wieland, Wolfgang Martin: to quantize initial data on a null hypersurface and find a representation of the canonical commutation Heisenberg commutation relations. And this is one approach

532
01:23:08.090 --> 01:23:09.360
Wieland, Wolfgang Martin: for doing that.

533
01:23:09.550 --> 01:23:22.230
Wieland, Wolfgang Martin: What about so? But let's dive a bit deeper. So what about the similarities and differences to a from a usual spin network representation in here?

534
01:23:23.250 --> 01:23:37.639
Wieland, Wolfgang Martin: What we see is is a kind of hybrid or a mixture of of the 2 of of a of the 2, I mean, of a speed network representation and a more standard fork representation. In fact.

535
01:23:39.790 --> 01:23:48.369
Wieland, Wolfgang Martin: in what I'm treating the in a way, I'm treating differently the angular directions on the on the Spatial Cross section

536
01:23:48.840 --> 01:23:56.569
Wieland, Wolfgang Martin: from the null directions along the light rays. It's like using a spin network representation on the

537
01:23:57.020 --> 01:24:04.770
Wieland, Wolfgang Martin: in the angular directions in which I only end up with punctures, on the, on, the.

538
01:24:05.010 --> 01:24:09.370
Wieland, Wolfgang Martin: on, the, on the celestial sphere, if you wish. So, instead of

539
01:24:09.900 --> 01:24:20.740
Wieland, Wolfgang Martin: infinitely many degrees of freedom in the angular directions, I only have punctures where these so n harmonic oscillators, let's say.

540
01:24:21.040 --> 01:24:30.900
Wieland, Wolfgang Martin: corresponding to end punctures, while in in the null directions I use a different representation. I in here

541
01:24:31.060 --> 01:24:32.690
Wieland, Wolfgang Martin: I chose.

542
01:24:33.980 --> 01:24:55.050
Wieland, Wolfgang Martin: I mean, what I'm doing here is investigate. No, I investigate a physical model, and the simplest way to start out is to make some simplifying assumptions, for instance, using a very rough truncation, in which I only work with piecewise constant functions. But of course one could use

543
01:24:55.530 --> 01:25:10.619
Wieland, Wolfgang Martin: other truncations, for instance, using a mode expansion in E to the INU. Or something, and then then the difficulty would be that the constraints would couple all these modes.

544
01:25:11.030 --> 01:25:14.930
Wieland, Wolfgang Martin: But okay, one could also try that. But then.

545
01:25:15.600 --> 01:25:22.520
Wieland, Wolfgang Martin: what I'm suggesting here is that we can investigate the resulting physics

546
01:25:22.760 --> 01:25:46.299
Wieland, Wolfgang Martin: way more directly by using a local truncation, not like a cut off in in momentum space, but in position, space, so to say. So, using only pulses and then quantizing each pulse separately. And

547
01:25:46.340 --> 01:25:56.379
Wieland, Wolfgang Martin: what I would argue for or and and I hope I convinced you, or some of you, that that indeed something interesting is happening

548
01:25:56.510 --> 01:26:01.950
Wieland, Wolfgang Martin: in this form theory that deserves to be explored.

549
01:26:03.340 --> 01:26:05.540
Hal Haggard: Carlo, did you have a question.

550
01:26:07.190 --> 01:26:12.455
Francesca Vidotto: So yeah, just a comment about that. And

551
01:26:13.542 --> 01:26:34.859
Francesca Vidotto: and relate to the previous questioning. So if I understand correctly what you're saying, and also the answer you gave to buy so just wanted to see if I'm in. In a simple, in simple words, following what you're saying, you're saying, let's let's we're studying the quantization of finite

552
01:26:35.090 --> 01:26:38.229
Francesca Vidotto: patch of space type. Right? That's a key point.

553
01:26:39.343 --> 01:26:45.956
Francesca Vidotto: And and let's do things consistently in in this context. Now, this makes sense, of course.

554
01:26:46.730 --> 01:26:54.036
Francesca Vidotto: it. It might generate terminological confusions, because, you then attempted to use

555
01:26:55.481 --> 01:27:17.288
Francesca Vidotto: expression and notions that make sense at infinity. But and and this might be a source of confusion, because one of my racks say, Wait a minute. I mean this. This is defined infinity out there. But you're defining things locally and and and consistently located. Now, once you do that and you do that consistently so, in a sense, that's arena you're considering. Once you have a result

556
01:27:17.700 --> 01:27:24.399
Francesca Vidotto: it makes sense to take this result and ask what does imply

557
01:27:24.530 --> 01:27:31.410
Francesca Vidotto: if I see the luminous or something at a large distance, nevertheless, is that is that the logic.

558
01:27:32.821 --> 01:27:34.680
Wieland, Wolfgang Martin: Yes, that is the logic.

559
01:27:34.680 --> 01:27:37.009
Francesca Vidotto: Thank you. Thank you. So I I think I get it.

560
01:27:39.260 --> 01:27:42.119
Hal Haggard: By? Did you have any? Follow up.

561
01:27:42.120 --> 01:27:48.360
Abhay Vasant Ashtekar: Well, I think that you know. I mean, I feel that this is like very interesting work, and

562
01:27:49.180 --> 01:27:54.459
Abhay Vasant Ashtekar: but I think there are gaps, and I think that if those gaps are filled, then it will have much more impact.

563
01:27:54.730 --> 01:27:59.680
Abhay Vasant Ashtekar: For example, this, this thing that we have on the the board on the screen right now

564
01:27:59.830 --> 01:28:03.140
Abhay Vasant Ashtekar: of using the the pulses.

565
01:28:03.270 --> 01:28:10.479
Abhay Vasant Ashtekar: and to what extent, for example, I mean some all over the Place D. By du of Sigma, was assumed to be 0,

566
01:28:10.860 --> 01:28:17.720
Abhay Vasant Ashtekar: and and the bounds are obtained only in that case, what happens if I do have d by Du is that Sigma or Sigma.

567
01:28:18.320 --> 01:28:19.130
Abhay Vasant Ashtekar: that this part.

568
01:28:19.130 --> 01:28:20.500
Wieland, Wolfgang Martin: This signal, Sigma. Right?

569
01:28:20.770 --> 01:28:28.239
Wieland, Wolfgang Martin: Yes, so you're saying there would be a Delta chronicle delta in Sigma dot.

570
01:28:28.420 --> 01:28:31.140
Abhay Vasant Ashtekar: Yeah, they'll be called chronic Delta and Sigma dot.

571
01:28:31.490 --> 01:28:32.669
Wieland, Wolfgang Martin: So that will be in.

572
01:28:32.670 --> 01:28:35.769
Abhay Vasant Ashtekar: Numirosity. Distance is really not well defined.

573
01:28:36.340 --> 01:28:37.000
Wieland, Wolfgang Martin: 4 or something.

574
01:28:37.000 --> 01:28:40.160
Abhay Vasant Ashtekar: Not. Yeah. 6. It will be delta squared.

575
01:28:40.370 --> 01:28:42.300
Abhay Vasant Ashtekar: So I kind of feel that

576
01:28:42.610 --> 01:28:44.220
Abhay Vasant Ashtekar: it will be much better to.

577
01:28:44.930 --> 01:28:47.599
Abhay Vasant Ashtekar: you know. Not. Have this assumption

578
01:28:47.800 --> 01:28:51.310
Abhay Vasant Ashtekar: and see what one can do. Right? I mean, in the sense of

579
01:28:51.640 --> 01:28:58.140
Abhay Vasant Ashtekar: I mean to make even distant things at finite distance possible at the moment.

580
01:28:58.460 --> 01:29:04.379
Abhay Vasant Ashtekar: There are too many unsettled elements, in my view to make it sort of a you know

581
01:29:05.260 --> 01:29:14.599
Abhay Vasant Ashtekar: that to make the result have big impact, and I'm my suggestions might seem like criticisms, but they are constructive suggestions as to what needs to be filled

582
01:29:14.850 --> 01:29:16.659
Abhay Vasant Ashtekar: rather than you know.

583
01:29:17.120 --> 01:29:25.289
Abhay Vasant Ashtekar: I also sent you an email message. Oh, sorry chat message, old gang, and if you are available for discussion we can continue with you, Junior.

584
01:29:26.860 --> 01:29:27.190
Abhay Vasant Ashtekar: But.

585
01:29:27.429 --> 01:29:28.149
Wieland, Wolfgang Martin: Yes, of course.

586
01:29:28.150 --> 01:29:29.920
Abhay Vasant Ashtekar: Junior can send you a link.

587
01:29:32.100 --> 01:29:41.399
Abhay Vasant Ashtekar: Yeah, I think I should stop because I you know, it's only half an hour past and we can continue. But I think there are many open issues, so we can continue.

588
01:29:41.400 --> 01:30:07.679
Wieland, Wolfgang Martin: No, I agree this this will not be the end of this story, neither for me, and I hope neither for other people that are interested in this research to follow up and see whether they can confirm this conjecture from from a different direction. That would be certainly wonderful.

589
01:30:09.410 --> 01:30:15.249
Hal Haggard: Excellent. In that case. Let's thank Wolfgang again, and thank you all for a very nice discussion.

590
01:30:22.010 --> 01:30:22.840
Wieland, Wolfgang Martin: Thank you.