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Hal Haggard: Welcome everyone. It's my pleasure to introduce Renata Ferrero.

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Hal Haggard: who will be telling us about asymptotically safe and canonical quantum gravity. Thank you, Renata.

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Renata Ferrero: Thanks, Hal, for the nice introduction, and, thanks to the organizer of the Seminar series to give him the opportunity to present his work.

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Renata Ferrero: This work was based on a common effort together joint effort, together with Thomas Thomas Thiemann, to try to communicate between different approaches to quantum gravity. I come. I did my Phd in asymptotic safety. And yeah, I started getting also interested in canonical quantum gravity. And yeah, that's what I'm going to present. So this is based on these 2 papers. The 1st one is a paper by Thomas.

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Renata Ferrero: where he sets the basis on like to in order to bring to the asymptotic safety community, the basis of canonical quantum gravity and vice versa. And the second paper is a paper we had together, where we apply, like our joint methods to a more concrete example, and it would turn out to be an example of a relational Lorentz and asymptotic, safe model.

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Renata Ferrero: So let's start really from from basics, and try to like.

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Renata Ferrero: Make clear which are the common principles which these 2 different approaches to quantum gravity share, and also which are which are the differences

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Renata Ferrero: the main requirement I will make for my theory of quantum gravity is the requirement that the theory has to be background, independent and like explicitly said, this means that I will never fix at any stage of my computations a fixed metric. So I will make, never make an assumption about an input of a metric which I will use to do my computation. But instead

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Renata Ferrero: I will. I will leave my background completely arbitrary.

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Renata Ferrero: So there are 2 realizations of background independence from from my point of view, and one is the one loop quantum gravity satisfies, which is, if you want, very clear one, because, in fact, no background structure is needed at all.

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Renata Ferrero: But there is also a second realization of background independent, which is the one in which asymptotic safety lives. And it's the one which is based on the background field method a la Devite, where one does a splitting between a background and a fluctuation of our metric field.

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Renata Ferrero: but without specifying what our background our G Bar is. So this

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Renata Ferrero: it's just a tool in order to split between these 2 objects. But this G bar is kept completely arbitrary.

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Renata Ferrero: Then, if one wants, one can ask ourselves afterwards whether there is a preferred background which my dynamical system, my quantum gravity. Dynamical system prefers to live in, but this won't be set

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Renata Ferrero: by hand, but will be something which is emerges after one lets the dynamics

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Renata Ferrero: drive

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Renata Ferrero: this principle of background. Independence, as we know, is a principle which is not realized in perturbative approaches. If one looks at the seminal work by Gaurav and Saniotti of the eighties, where they find out that gravity is perturbatively non-renormalizable at 2 loops. The computation which they perform is done around a flat fixed background, so they fix this G bar to be exactly Minkowski metric.

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Renata Ferrero: also a second principle, a second characteristics that the 2 theories of quantum gravity, both canonical formalism and asymptotic safety try to insist on is a principle of non-perturb activity, so I will never perform an expansion in powers of the carrier fields in metric fluctuations, and nor an expansion in small couplings.

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Renata Ferrero: So, having set the stages about, the 2 main shared principles, which is 2 approaches share, let us also see whether we have a purple intersection zone between the blue canonical quantum gravity and the red asymptotic safety quantum gravity.

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Renata Ferrero: So

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Renata Ferrero: 3 main points which characterize canonical quantum gravity is are the fact that it naturally leaves in Lorenz's signature

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Renata Ferrero: as we said, it is manifestly background independent. We we never specify a background background structure at all.

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Renata Ferrero: and there is no need to do any truncation of our theory.

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Renata Ferrero: Asymptotic safety is instead for those who are not familiar with it. I will also give an introduction introduction to it, but

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Renata Ferrero: in a few words is a quantum field theoretical based approach which seeks for an UV completion of quantum gravity by means of an ultraviolet, non-trivial, fixed point which is found using methods of non-perturbative normalization group.

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Renata Ferrero: However, one of the main critique, or like 1 point which is different from canonical quantum gravity, is that

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Renata Ferrero: the majority of the computations up to now have been done in Euclidean signature.

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Renata Ferrero: Then, also the questions of background. Independence arises, as we said, because we need this tool to split between a background and a fluctuation.

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Renata Ferrero: and also in order to to solve our renormalization group flow equations. We need to perform truncations.

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Renata Ferrero: However, I will show you how there are

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Renata Ferrero: points in this purple zone such that make this apparent contrasts to be overcome, namely.

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Renata Ferrero: there. In the last years some Lorentzian version of the flow equation have been developed and computed, and we also do the same together with Thomas, we will use our version of a Lorentzian flow equation

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Renata Ferrero: background. Independence is indeed enforced via this background film method. So a little bit, really, this, in the spirit of David.

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Renata Ferrero: and also truncations, is something which is natural to consider when wants to do computations in practice, using renormalization group methods. And here I'm citing a paper by Thomas, where he is also like argumenting about necessity of truncation in Hamiltonian renormalization.

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Renata Ferrero: So what is the idea.

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Abhay Vasant Ashtekar: But I make sure that I mean maybe you'll discuss it later. But I think most of us don't understand how you can do the Lorenzian flow, just because, you know, there's no such thing as

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Abhay Vasant Ashtekar: small K, or because of the it's indefinite signature. So the whole idea about ultra wallet infrared, etc. In the Lorenz's signature is not obvious.

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Abhay Vasant Ashtekar: because you cannot just say K is larger. K. Is small, etc. Because of indifferent signatures. So are you going to explain that sometime, because that has been a real building really

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Abhay Vasant Ashtekar: block for us.

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Renata Ferrero: Yes, I'm going to explain it. And, in fact, what I'm going to advocate is that in Lorentzian signature, when we consider the flow we lose this intuition of

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Renata Ferrero: high energies, low energy, my scale K, which I will introduce. My renormalization group scale, won't have any association to any energy fluctuations anymore. But it will be merely a tool in order to like, manipulate my path integral. And in the end, what I will be interested in to

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Renata Ferrero: is really the limit of a vanishing case. So when I remove completely my tool, which is a regulator, right in, in like the strict sense of the word. So I agree with you. We lose the Euclidean like interpretation of integrating out energy, degrees of freedom or momentum scale. Or, yeah.

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Renata Ferrero: So this physical intuition along the floor is is lost.

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Renata Ferrero: Thank you.

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Renata Ferrero: So where does this purple intersection zone leave? So it is natural to realize it via the relational formulation. So the idea is to use the framework from canonical quantum gravity, namely, the reduced face-pace formulation, where we are going to distinguish between the physical degrees of freedom and the gauge degrees of freedom and to construct

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Renata Ferrero: a time, order, correlation functions as a path integral

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Renata Ferrero: now and then. After that we have this object. So this time order, correlation function, we will give it onto the machinery of asymptotic safety

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Renata Ferrero: which furnishes some tool to manipulate this path integral in a in a straightforward way.

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Renata Ferrero: And the 1st application I'm going to show you is simple Einstein-hilbert action. So Ricci and cosmological, constant, coupled to 4 massless Karloff Fields and yeah, I will show you how the Lorentzian flow is realized, and the tool which we are going to use is Lorentzian heat kernel and a suitable cutoff function.

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Renata Ferrero: So that's the plan of my talk. I will start from the canonical quantum gravity framework and introduce reduced phase space. I know that people in this community are familiar with it, but and also like, make the transition to the path integral formulation.

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Renata Ferrero: And then I will introduce asynthetic safety. What's the renormalization group? How the flow equation looks like. And then this new tool which we used in our Lorentzian version, the Lorentzian hit kernel, and finally, I will apply to my showcase model.

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Renata Ferrero: So let's start from like we have. Gr, so it's a constrained Hamiltonian system. And I will work in the reduced phase space approach, which, yeah, can be seen in a 2 different perspective, either like by constructing relational direct observables and deriving the physical Hamiltonian, or also from a gauge fixing point of view by selecting the true degrees of freedom, and then

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Renata Ferrero: constructing the reduced Hamiltonian, which is a function only of the true degrees of freedom, and generates the equation of motion as the primary Hamiltonian. When when my system is reduced, when the phase space is reduced only to the physical degrees of freedom.

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Renata Ferrero: so the interpretation of it is that the 2 degrees of freedom, and hence the reducing Newtonian. They depend really on the

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Renata Ferrero: my gauge fixing choice. And so, if I have my through degrees of freedom. For instance, Q. And P. They are the observables which are realized on the gauge cat, G of J, equal to 0,

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Renata Ferrero: and they coincide with the Dirac Observables, OP. And Oq. At the time 0, and their evolution is generated by the reduce in Miltonian, generated by the following gauge condition gauge fixing condition.

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Renata Ferrero: So after we've done, we've reduced our space space. We can.

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Renata Ferrero: and we have derived the reduced Hamiltonian. We can canonically quantize. So we can construct the vial elements and seeking for a representation. We are looking for the Gns data. So a Hilbert space, a state and a vector in this Hilbert space and a representation of a state on on this vial, algebra

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Renata Ferrero: and and like, equipped with these tools, we can, we can pass toward a path integral formulation. So a canonical path, integral.

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Renata Ferrero: and our main aim is to construct time, ordered correlation functions and its generating functional under the assumption that our Hamiltonian is banded from below below and has a unique vacuum state.

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Renata Ferrero: Omega 0. So this is the time order, correlation, function with sandwich on our Hilber space, our vacuum state with a time order.

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Renata Ferrero: Vile elements where U is nothing else as the time, evolution, the typical time evolution under this Hamiltonian

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Renata Ferrero: and Wt. Is the time evolved vial element.

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Renata Ferrero: So we can then construct a generating functional where we exponentiate

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Renata Ferrero: our 2 degrees of freedom, multiplying with this is the F, which is the so-called source term.

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Renata Ferrero: And then we want to manipulate this, this generating functional. And in order to do that, if we suppose that F only has compact support on a given interval, minus toe to toe. Then we can consider a time which is much larger than

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Renata Ferrero: this this interval, and we can discretize in Delta N being tau over N, where in the end I'm going to send N to infinity, and hence we can write down our partition function as the limit over this object Z.

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Renata Ferrero: Which is nothing else. As the time sliced, generating, functional, and I recover my full, generating functional by sending both n. To infinity and T to infinity, and this limit can be done by means of Feynman Cac. Arguments, and in the end one arrives

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Renata Ferrero: at like a symbolic structure for our generating functional, which looks like that. So let me analyze it a little bit. We have an integration over phase space. So both momenta and configuration space. We have, like this these 2 factors which are reminiscent of our State dependence.

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Renata Ferrero: Omega. And then we have the typical exponential which also appears like in covariant derivation. So the source term coupled to my variables. And then here we have what would be the Lagrangian? And of course it's legend transform. So we have Pq. Dot minus our Hamiltonian.

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Renata Ferrero: It's how you.

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Abhay Vasant Ashtekar: Just one quick question here about

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Abhay Vasant Ashtekar: the preview

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Abhay Vasant Ashtekar: up to now, when you're going to reduce phase space, you have this capital P and capital. Q,

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Abhay Vasant Ashtekar: and now

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Abhay Vasant Ashtekar: I mean, typically the reduced space space is going to be manifold. It's not going to be a vector, space. I mean, almost never a vector space right?

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Abhay Vasant Ashtekar: Because of the

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Abhay Vasant Ashtekar: yeah reduction that got done.

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Abhay Vasant Ashtekar: And so what is P and QI mean, because you choose one pair and I choose another pair

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Abhay Vasant Ashtekar: when you define the while operators. Normally, if it's a vector space, it doesn't matter

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Abhay Vasant Ashtekar: what polarization I choose.

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Abhay Vasant Ashtekar: But if it's not a vector, space. It does matter. So what is a prescription for choosing a preferred polarization for preferred P's and Q's capital, P's and capital, Q's.

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Renata Ferrero: I mean.

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Renata Ferrero: this is, I guess this really depends on the choice of the gauge. You're doing right. I mean, this is really the the main basics of the reduced phase space quantization. I am choosing a base. So this, if you want this path integral, is only it's dependent on on my choice of the gauge, and then really.

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Abhay Vasant Ashtekar: The gauge is just a gauge fixing. It's a surface in the phase space. Yeah, such that each constraint vector, field, each each gauge vector field. Each gauge orbit intersects once and only once.

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Abhay Vasant Ashtekar: That does not give you a preferred canonical

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Abhay Vasant Ashtekar: coordinates. P. And Q. So you are choosing some canonical coordinates. And it seemed to me that your answers are going to depend on the choice. It would not be independent of those choices in the vector space case they are independent, but in the manifold case I don't see why they could. They could be independent at all.

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Renata Ferrero: I seem.

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Renata Ferrero: I mean in

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Renata Ferrero: in the end, the way the way I'm going to the way I'm going to to implement my gate. Choice is really like to

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Renata Ferrero: the way I'm going to the parameterize this object. And so in this, this P. And I see your point, this P. And Q, they, they live in vector spaces.

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Renata Ferrero: yeah, I think I have to think more about that.

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Abhay Vasant Ashtekar: Go ahead!

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thiemann: Maybe I can interject here. Yeah. Yeah. So the the little P and little Q are really the the 2 degrees of freedom that you caught

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thiemann: capital P. And Q.

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thiemann: So that's maybe just a misprint. And at this point.

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Abhay Vasant Ashtekar: No, that's not. That's not a problem with me.

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thiemann: I know. I know. I just want to make sure that you understand that this these are already the the 2 degrees of freedom.

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thiemann: and of course you are correct, and for a given system one will

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thiemann: look for such canonical coordinates which have as much linear structure as possible.

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thiemann: Okay.

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Abhay Vasant Ashtekar: Please go ahead.

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Renata Ferrero: Thank you.

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Renata Ferrero: So 2 in observations are in order right now, if we want to compare this generating functional with a standard canonical one, let's say, the Q standard Qft based one.

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Renata Ferrero: and namely, usually we just have an integration over configuration space. So

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Renata Ferrero: yes. Thanks for for the observation. This this would be just my physical configuration space capital Q. As I introduced before. But now we also have an integration over my physical momenta. So

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Renata Ferrero: for some simple forms of Hamiltonian. I can perform the integration over momenta, and then, of course, I will have some reminiscence of this integration, which I can like consider as like a a modified measure for my Dq.

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Renata Ferrero: and also, of course, a difference. An important difference is the dependence on the cyclic vector Omega, which is associated to our state, so a dependence on the state.

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Renata Ferrero: So what can asymptotic safety learn from from this construction? On one side, is like this from the reduced space quantization in particular, to learn how to extract physical degrees of freedom from our theory.

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Renata Ferrero: and then to explicitly quantize the theory which is a procedure which, in asymptotic safety is not done in an explicit way. Of course the main challenge is that as we saw it, it would be very nice to to go, Lorentzian, and this is indeed

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Renata Ferrero: what we are going to do.

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Renata Ferrero: So let's see instead, what are the asymptotic safety tools? We learned a little bit in our Qft courses, how to perturbatively renormalize theories which looks a little bit little bit like this cartoon, like

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Renata Ferrero: putting infinities under the carpet. But as we saw this breaks down in for quantum gravity.

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Renata Ferrero: and so maybe there is some other notion of randomization or some tools which are intrinsically non-perturbative how this, which can realize a randomization in a more clever way. And this was a little bit based on Weinberg's conjecture, and I think in the 79. He introduced that, namely, when he introduced the word asymptotics, the objective, asymptotic safe. He was

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Renata Ferrero: like conjecturing that it can be existing. There could there could exist. Non-perturbative mechanism, such that really physical objects, such a

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Renata Ferrero: scattering amplitude, could be finite at energy scales which are greater than the Planck scale. And this is actually realized in a more technical language by the presence of an ultraviolet, non-trivial, fixed point. So interacting fixed point.

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Renata Ferrero: This conjecture by Weinberg was investigated over the years, both in the continuum and in the discrete, in the continuum, which was, I'm going to show you right now by means of this functional Renormalization group and the discrete.

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Renata Ferrero: For instance, in Edt. So in Euclidean dynamical triangulations or in Cdt, there is this nice, very recent paper by Jan Ambirn when he tries to connect asymptotic safety with Cdt.

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Renata Ferrero: so let's try to embed a little bit the Frg. Into the Rg. Like history we learned

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Renata Ferrero: in our Qft courses about the our gene perturbation theory, and namely, the Kalan-symanski equation, which is an equation arising by requiring that our effective action doesn't depend on my Rg. Scale introduced. But this is an equation which furnishes only a finite number of beta functions which are those related to the relevant couplings.

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Renata Ferrero: Some years afterwards, on

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Renata Ferrero: almost like. Simultaneously. Also, Wilson introduced a different philosophy for the normalization group, namely, the Vinsonian Renormalization Group, where he was trying to integrate out the quantum fluctuations in a path integral in a progressive way. And this was inspired by Kadanov spin transformations, where I essentially average over a larger and larger number of spins.

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Renata Ferrero: This was then like this. This notion of integrating our degrees of freedom can be encoded in

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Renata Ferrero: an effective action which depends on a given scale. K. And one can imagine that this effective action, by integrating our degrees of freedom, passes from the deep ultraviolet where no degrees of freedom is integrated out.

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Renata Ferrero: where ideally travel fixed point lies and goes down to K equal to 0, where I recover my full fledged, effective action. So where all the fluctuations have been integrated out.

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Renata Ferrero: And finally, in the nineties there was a different, a slightly different point of view inspired by Wilson, but not implementing the Woodson idea in a straight, straightforward way.

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Renata Ferrero: Was introduced by Betterisch and used by Reuter shortly afterwards the functional normalization group. And this is really based on the introduction of a scale dependent version of the effective action. So this sort of Gamma K

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Renata Ferrero: in order to better manipulate our path integral.

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Renata Ferrero: So the functional randomization group is a tool which helps us to manipulate the path integral and implements the underlying randomization idea already at the level of a scale dependent action, and how this like is a tool in order to prove asymptotic safety

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Renata Ferrero: is because it's a tool which can can like helps us help us to study the flow and see whether in the ultraviolet limit there is a fixed point, and where the trajectories do flow into this this fixed point.

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Renata Ferrero: So let's see how this scale dependent action can be derived in a more like formal way. This is really, very, very goes parallelly as how you derive the effective action in a standard Qft course. So you have your generating functional. This was the same Z. We had before, also the canonical quantum gravity section, and then you perform a legend, transform and you define your effective action.

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Renata Ferrero: Now for the scale dependent version of the effective action which I will call effective average action.

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Renata Ferrero: The only modification is that you add such a cutoff function

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Renata Ferrero: to your generating functional, which is dependent on our Rg scale K, and has a structural function, a functional form which is like sandwiching 2 times our fluctuations of the metric, and this is reminiscent of like a mass term for our metric.

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Renata Ferrero: This is why we historically, this form was was chosen. Then what one does is to perform the Legendre, transform again and to subtract again this this cutoff function, for from my effective average action, and one can prove

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Renata Ferrero: that by taking this this function to be 0 when I send K. To 0, I recover. Indeed, my full fledged, effective action.

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Renata Ferrero: So what is K. What is this R of K. This is

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Renata Ferrero: this cutoff function is a parameter, one parameter family of integral kernels which depends on the background version, even though, I said, the background is, is arbitrary, but it will depend on the Eigenvalues of this general

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Renata Ferrero: background, and as Abai was pointing out in the Euclidean setting. This has nice interpretation of integrating out aggressive freedom. We have a unique ordering in the signature, but in Lorentzian instead, this interpretation is lost

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Renata Ferrero: also because there is an ambiguity. What I'm going to integrate out time like space like modes. So we like on a more conservative footing. We are just saying, Okay, we are taking an oscillating function, which is

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Renata Ferrero: which we are required to have the property to be vanishing when we send K. To 0 in such a way that when we are going to be interested in this limit, K to 0, we can study our effective action, which is the generator of our one pi and endpoint functions.

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Abhay Vasant Ashtekar: All these integrals are completely formal, right? And because it's the infinite dimensional integrals with oscillating functions and so on.

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Abhay Vasant Ashtekar: So I mean, you're not doing approximate evaluation by side point, approximation, or anything like that, or anything like that.

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Abhay Vasant Ashtekar: So what is the meaning of these things? The mathematical meaning of these things rather than just formal expressions

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Abhay Vasant Ashtekar: that once in the last last

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Abhay Vasant Ashtekar: yeah slide.

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Abhay Vasant Ashtekar: yeah. So like the, for example, Z. Bar prime.

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Abhay Vasant Ashtekar: you know, on the right hand side. I just.

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Renata Ferrero: Obviously.

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Abhay Vasant Ashtekar: 3 dimensional, integral. There are oscillating factors.

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Renata Ferrero: Yeah, you are asking me the mathematical.

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Abhay Vasant Ashtekar: Meaning right? I mean, it doesn't a mathematical meaning, I mean, so.

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Renata Ferrero: They are going to regularize my, for instance, my propagator, because, of course, if you think about the one loop effective action, then I will have divergences. I need a regulator right? This, otherwise it is divergent. For instance, in the infrared in that case.

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Renata Ferrero: So I it's it's a it's a needed regulator. Then, of course, you can usually install.

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Abhay Vasant Ashtekar: Again, I mean in the Lawrence and case, I mean I don't know what infrared means, so.

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thiemann: Can I say something to the

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thiemann: the derivation of the wetelish equation from this very formal objects, which are prior or very ill-defined

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thiemann: is.

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thiemann: gives you an equation which the people in asymptotic safety, then use as a starting point.

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thiemann: and solutions of that betterch equation would, by the relations that Renato shows in the middle, then give you a well-defined, generating function, from which you can find

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thiemann: all the endpoint functions that you might be interested in.

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thiemann: So the derivation is starting from this formal object is, of course, only formal.

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thiemann: but the final equation is is well defined, and this is the starting point. If you want.

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Abhay Vasant Ashtekar: So the final equation is well defined in the infinite dimensional space with an arbitrary G bar, and infinitely many H. Because even the Euclidean signature. For example, you know we have this usual problem, that the action is not unbounded below. So even E to the minus is even in the Euclidean signature doesn't make any sense, and so on. So. But you're saying that none of those problems are there in the vetterich equation.

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thiemann: And is a function of these smearing functions which were not the notes by G hat and G. Bar.

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thiemann: and they're supposed to be

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thiemann: well. The the assumption of the vectorization is that the the solution of the vectorization is a well-defined

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thiemann: generating function of of distributions in

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thiemann: by taking functional derivatives with respect to those

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thiemann: that's the assumption.

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Abhay Vasant Ashtekar: Equation is a equation on on some

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Abhay Vasant Ashtekar: distribution.

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thiemann: Is, is an equation which which involves this. What Renato denotes by

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thiemann: Gamma, prime bar K

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thiemann: at the at the end of the day.

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thiemann: So they they are manipulating that

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thiemann: object and all the equations that lead to this vector equation. You can call them formal if you want. But

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thiemann: the assumption is that you start from there and then try to construct the the 3 vice versa, not going from the path, integral, formal, a formal expression, but

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thiemann: working out backwards the consequences that the well-defined Gamma Bar

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thiemann: Prime K.

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thiemann: Would give for your generating functional Feynman distributions.

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Abhay Vasant Ashtekar: Sorry. But just last question, what is G. Hat? I mean? G. Bar was a background. What was G hat.

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thiemann: So G hat is is the is the where. So if you if you think in terms of the genre transformations

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thiemann: you have, you have if you look at this this object, Z. Bark, prime K,

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thiemann: you have a smearing function. F.

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thiemann: And now you do a later on the transformation with respect to F

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thiemann: and the new variable after the round, the transformation is V hat.

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thiemann: Okay. Thank you. From from velocities to moment, momenta, and vice versa.

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Abhay Vasant Ashtekar: She had depends on R.

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thiemann: She had was just a smearing function.

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Abhay Vasant Ashtekar: It depends on r, right? Because of the

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Abhay Vasant Ashtekar: yeah.

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thiemann: No, no, G had to stress. The smearing function does not depend on R.

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simone: So should I not think of Jihat as the dynamical metric on top of the background metric Gbar.

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Abhay Vasant Ashtekar: No.

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thiemann: It's what people ineffective.

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thiemann: If you think of the effective

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thiemann: action in usual quantum theory, this is what people call J, the the source term.

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Abhay Vasant Ashtekar: So jihad is like the source term. Is that what you're saying? Yeah.

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thiemann: That's a very

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thiemann: okay.

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Renata Ferrero: Okay? Yeah.

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Renata Ferrero: So by by starting like, from this Gamma K, from this scale, dependent, effective action.

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Renata Ferrero: one can prove that this Gamma K satisfies functional.

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Renata Ferrero: an integral differential equation, which is what we call what was already mentioned, the vectoric equation. And this is the wetteric equation in its Lorentzian version. So we have, as you see, this one over 2, I appearing in front of that. So what is that? This is on the left hand side of this equation, we have the scale derivative.

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Renata Ferrero: So the derivative with respect to K of my gamma K of this effective average action, which in the end is going to give me the Beta functions on the left hand side.

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Renata Ferrero: while on the right hand side we have a trace over an inverse of what is the Hessian. So the second derivative of my Gamma K.

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Renata Ferrero: With respect, for instance, with respect to my Dg hat

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Renata Ferrero: plus my cutoff function. So this is what this is this is the term which in the Euclidean version, will regularize the Ir modes, because it will add a small mass to my one over box operator.

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Renata Ferrero: And this is further also multiplied by the scale derivative of our K, and this term in the numerator is instead what is going to regularize the large K limit.

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Renata Ferrero: So they ultra valid in a Euclidean signature.

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Renata Ferrero: So this equation is, and the derivation of this equation is exact and nonpertormative.

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Renata Ferrero: and it can be used to construct a well-defined gamma. So in the limit, for instance, where K. Is sent to 0 also in the Lorentzian signature rather than starting from the generating functional. So this is what this is the object that we are going to to use.

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Renata Ferrero: Of course, I mentioned in the introduction that in order to solve it.

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Renata Ferrero: we need to do truncations, because every time in principle this is like an infinite order equation. So the systematics in which truncation is performed is to tailor, expand both the left hand side and the right hand side

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Renata Ferrero: in power of my metric field, g mu nu, and then to compare the coefficients, which are multiplying the same operators, the same order in the expansion, both on the left hand side and on the right hand side.

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Renata Ferrero: And also.

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Abhay Vasant Ashtekar: There is no G menu in the top equation at all. So how how are we doing this expansion?

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Renata Ferrero: I mean, if you think about okay. For instance, if you think about Einstein Hilbert, I can split g mu nu in g hat plus h mu nu.

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Renata Ferrero: and then I can expand in powers of I can do an expansion of this object, right of this of this, of like of this.

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Renata Ferrero: Split it action in powers of in in this case, in powers of H. Mu. Of course.

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Abhay Vasant Ashtekar: I agree, but the equation has no G in it at all. Right. There's only a G Bar and G hat. G hat is an external source. G bar is a background metric.

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Abhay Vasant Ashtekar: And so where is G in that equation? There is no G at all. So how suddenly am I getting a G, or expansion, or ice.

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Renata Ferrero: I mean.

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Renata Ferrero: I think in the end, I mean, I think it's it's an expansion in terms of G bar in this side. But right? Yeah, because I'm taking the derivative of Gamma Gamma K. With respect to Dg hat.

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Renata Ferrero: But do you have.

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Abhay Vasant Ashtekar: Is like an external current. It is not a background. Metric

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Abhay Vasant Ashtekar: is not a dynamical metric.

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Abhay Vasant Ashtekar: It's not G menu. It's not.

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Renata Ferrero: Yeah. But

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Renata Ferrero: let me just look a second at this

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Renata Ferrero: this object.

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Renata Ferrero: I think, Thomas. Then here, here I have to ask you. I think that then the the external source is not. Is not F here? Or is G hat.

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Abhay Vasant Ashtekar: It's not correct here.

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Renata Ferrero: Yes, the external current. I think we we do. The Legendre transform with respect of to with respect to

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Renata Ferrero: F is the external corner and jihad would be like

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Renata Ferrero: my my expectation value

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Renata Ferrero: with respect to the derivative with respect to my external current.

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Renata Ferrero: so FF is the external current. And then, when I do the Legendre transform with respect to FI get my expectation value, which is G hat.

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thiemann: So the the thing is that, of course, the better equation. Once you have written it down, you have to. You want to solve it, and you have to make an answer.

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thiemann: and that, and that starts from some

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thiemann: some formula that you guess for Gamma

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thiemann: Bach, prime K,

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thiemann: which depends on G hat and G bar, and then you make A,

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thiemann: you consider this as a function of both variables.

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thiemann: and you can take the functional derivatives with respect to G hat.

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thiemann: and one of the common

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thiemann: answers that the people make is that it's it's it's exactly the Einstein Hilbert action

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thiemann: where G. Is given by the sum of G bar and G hat, or G hat is.

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thiemann: or the little h that she wrote there up up there in the 1st equation is related to the difference between G. Hat and G. Bar.

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Abhay Vasant Ashtekar: Okay. But in that case, then, I think what Ranata is saying is makes sense to me, which is that

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Abhay Vasant Ashtekar: jihad should be really thought of as a

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Abhay Vasant Ashtekar: dynamical metric or expectation, variable dynamical metric. This is, I think, what Simonia was asking, and

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Abhay Vasant Ashtekar: rather than external current, and that F may be an external current. Is that correct, or is that not? Are we mixed up.

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thiemann: So I'm just saying external current. So maybe this is just a problem of language. So

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thiemann: J. What I, what I mean by the external current is is the variable J, which is

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thiemann: used in order to compute the legionne transform, ie. The the effective action.

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thiemann: So if you're familiar with Euclidean quantum theory. Then people write out, Gamma of J,

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thiemann: okay. And JJ is, is the variable that you use to compute the Laronda transformation of the

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thiemann: generating function of connected Feynman Feynman functions.

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thiemann: and that role of J is played by G. Hat. Here.

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Abhay Vasant Ashtekar: Okay, I think we should go ahead because we can discuss this at the end, since it's taking too much time.

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Renata Ferrero: Yeah, we can discuss this later.

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Renata Ferrero: So yeah, this is like, the in order to do truncations, we would just expand in power of if yeah of G mu nu, which then turns out to be powers of of G hat. So, for instance, the idea is like that in gravity you start. The simplestruncation is Einstein, Hilbert, and you can consider

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Renata Ferrero: quadratic gravity and higher order, and people have computed, till R. With rich scalar to the power. 72. In order to test the presence of this fixed point, and of course also other contractions like R. Mu nu r mu nu, and so on and so forth. So it is a powerful tool in order to. And yeah, depending on the truncation, you can study the presence, whether there is a fixed point or not in the ultraviolet.

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Renata Ferrero: So how I told you this is this can be performed in a background, independent way, and I want to show you why. So we saw that in our flow equation we have the trace over this operator. So this like

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Renata Ferrero: propagate modified propagator. So this would be the propagator by. And we added this R. Of K.

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Renata Ferrero: And how can we perform this trace in a general background? We are going to use the following combination. So both the background film method. But what I introduced before the splitting between a background and fluctuation

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Renata Ferrero: and the Lorentzian heat kernel.

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Renata Ferrero: So, for instance, if you want to compute for for Einsen Hilbert gravity this object. So the Hessian of our effective average action, then you will get operators of such a form. So you have Laplace operator plus non-minimal derivatives acting on our fluctuation, and we want to compute the trace over this object.

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Renata Ferrero: How to do that. So the first, st

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Renata Ferrero: like. The 1st identity we will use is the Schwinger proper time integral? So if you have, this is also introduced in Qft books. If you have one over box here. We will also add a sort of master, I mean hit. Then you can write it in this proper time. Integral representation as an integration over Dt.

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Renata Ferrero: And then you exponentiate your box with E to the E to the it, and then you have to add this minus T. Epsilon, where, after you perform the integration, you can analytically continue X epsilon to minus. I say, ck, which was the so-called mass term in here.

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Renata Ferrero: So now now that we've done this rewriting in terms of the proper time integral. We can write the traces of this this object, and in particular.

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Renata Ferrero: if you, if we have functions of our box operator, which we saw is always appearing in the in the Hessian of our effective action. Then we can write down

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Renata Ferrero: the general general function. Ok, as the transformation. Ok. Hat of T times this operator. So the same operator which was appearing in here, E to the IT. Box.

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Renata Ferrero: and we want to compute traces over that we have. We can exploit the so-called heat kernel expansion for a general manifold.

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Renata Ferrero: So, for instance, H. Of T.

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Renata Ferrero: Can expand it in proper time. So in this variable T in a given dimension, also Lorentzian one, and it has the following analytic form, where we have an exponential of this sigma, which is a so-called cinders, word function, and this omega t are the coefficients of our expansion in small proper time. T.

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Renata Ferrero: So, for instance, if one has the, if one considers the 1st the leading orders in this expansion, the trace over this box, then one has the following expansion, where the leading order, for instance, in D equal 4 dimensions in one over t squared. So

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Renata Ferrero: this will be, of course, the the TE to 0 limit is the limit we want to regulate.

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Renata Ferrero: and then you see that we will have an expansion in powers of T, which multiply some curvature monomials. So this is why it's general in a general background, because we are not specifying the forms of these curvature monomials, and the expansion will come in the Lorentz sense acting with alternating signs of eyes.

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Renata Ferrero: So now the last step that we miss is how to choose a suitable cutoff function in order to regularize these traces. For instance, I told you in the limit, T. Equal to 0 in 4 dimensions, this will be divergent.

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Renata Ferrero: And so we made a choice of a regulator such that we get convergent traces. And, as said, this regulator is really a tool which has to be thought, as in order to then compute the limiting of vanishing regulator. This is like the proper sense of it, so K, equal to 0. And it has this function. So it's

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Renata Ferrero: it has a wave function normalization. This is customary in our computations, then, is a K squared times, a dimensionless function which has this form and also is a proper time, integral, and is essentially the same transformation with respect to

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Renata Ferrero: sorry.

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Renata Ferrero: Sorry. I don't know what is going on. Yeah, it's the same transformation. With E to the it box, and

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Renata Ferrero: it has a kernel E to the minus t squared, minus one over T squared. So essentially, the insertion of this kernel is the thing that is going to regulate both the infrared, and so both the T equal to 0 and the T equal to infinity limit.

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thiemann: Can I also just say something just.

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Renata Ferrero: Sure.

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thiemann: To make to answer one of the questions that Abbay was saying.

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thiemann: Raising K. Squared is really positive here, so it has nothing to do with the Euclidean norm of A vector and the Minkowski norm of a vector or something like that.

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thiemann: So it's it's just a number, if you want.

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thiemann: But number that is positive.

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Abhay Vasant Ashtekar: What is it? How is it related to

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Abhay Vasant Ashtekar: the Lawrence.

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thiemann: Nothing. It has nothing to do with with the Minkowski square of a vector

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thiemann: absolutely nothing.

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Abhay Vasant Ashtekar: But does it have to do with

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Abhay Vasant Ashtekar: eigenvalue of some operator? I mean some like block or whatever. What.

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thiemann: No, it's it's it's completely independent of that.

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thiemann: So like, Renata was saying that that kind of interpretation is missing here.

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Abhay Vasant Ashtekar: So K. Is just the external parameter that.

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thiemann: K is just an external parameter. Yeah. And K squared is always non-negative.

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thiemann: Okay?

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Abhay Vasant Ashtekar: Okay, it's just an external parameter. You're just defining some. Okay.

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Hal Haggard: Timo Ne.

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simone: Yeah. So this was a long time ago for me. So sorry. The question is stupid, but

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simone: when you do like it, kernel of non elliptic operators like in your case. Don't you have like issues with your 0 modes that you will have to treat separately or something?

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simone: Usually the formulas. I remember they were only defined in the Euclidean.

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simone: so is there some subtlety, or actually not really

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simone: thanks.

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Renata Ferrero: So there are. If you talk to mathematicians, they will tell you that we still haven't proved some nice convergence properties. So it's we have to

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Renata Ferrero: pay attention to deal with this object, but I mean in the in the, in the 1st iterations of this, if you go to the papers and also the seminar books by fooling, and, like all the Qft in curved spacetime books, they they also introduce it in Lorentzian signature.

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Renata Ferrero: There are also some nice paper where they relate the Lorentzian hit curler coefficients with the Hadamard expansion. But still, yeah, you're right. There are some convergence properties which have not been tested yet

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Renata Ferrero: that said, we are using that. So this is yeah.

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Renata Ferrero: Of course, it's it's something we which we

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Renata Ferrero: would like to understand better.

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simone: Can you remind us? Why is it important for you to do, Lorentzian? Couldn't you, as a 1st attempt to connect the theories, use Euclidean, and or there's some important reason why

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simone: comparing the theories in the ingredient would not be meaningful.

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Renata Ferrero: Yes, that's a good question. It depends on the on the risk, on what the theory, the model we are considering. So, for instance, I will show you that with these 4 scalar fields.

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Renata Ferrero: when we deparamatrize, it would be necessary to use Lorentzian signature, because otherwise we have to deal with the square root of a Hamiltonian, and that we don't want. But there is some work in progress where we can consider some other matter fields in such a way that we can work in Euclidean. So yeah, you had the right intuition.

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simone: Thank you.

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Renata Ferrero: So this, yeah, this function has this very nice properties. That is a decrease rapidly when T equal to 0, and it becomes 0 for negative T proper times. It will always give us convergent traces in the flow equation because of this numerator factor which we had in the batterych equation, which

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Renata Ferrero: essentially, we'll always give an insertion of this kernel, such that the traces are convergent. And then, yeah, also important to notice is that it has a positive support in order to like regularize. For instance, this denominator

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Renata Ferrero: in in the heat kernel expansion.

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Renata Ferrero: There is a price to pay, however, for the use of this Lorentz and heat kernel, namely, we saw before that the coefficients come with the I factors. So with imaginary factors, and hence the flow will become complex valued. So we will have complex valued coupling constants.

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Renata Ferrero: And this, yeah, this is something we would like to better understand. But as a 1st investigation, what we asked ourselves, is whether there is, exists also in a complex values flow, whether there exists on trajectories

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Renata Ferrero: which have dimensionful couplings when I send my K. To 0. So when I want to study my effective action in when the K equal to 0 limit, I'm going to study whether my theory admits some trajectories which have real valued, dimensionful coupling constants.

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Renata Ferrero: So this is, for instance, I here I just plotted in order to show you how this R. Of K. Looks like for fixed K, and this is the P squared. But again I said, this is merely a tool and

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Renata Ferrero: but yeah, just to get you some intuition. How this Fourier transform looks like.

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Renata Ferrero: So what can canonical quantum gravity learn from from this asymptotic safety tool.

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Renata Ferrero: how to manipulate the path integral which we have to write before, and in the end the final aim is really to maybe have an object from which we can compute correlators. Starting from an Hamiltonian theory.

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Renata Ferrero: Now I will finally apply all these equations, which I've been deriving in a very simple model, namely, Einstein-hilbert action. So Ricci, minus 2 lambda

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Renata Ferrero: with 4 scalars, 4 scalar fields coupled

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Renata Ferrero: so I will before start up with a reduced face space approach, and I can impose the gauge fixing on the configuration and solve the constraints for the d momentum.

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Renata Ferrero: For instance, I can choose such a choice of a gauge fixing, and then I am able to separate between the gauge degrees of freedom, the Phi I. And the pi i. And identify the true degrees of freedom, the Q's. And the P. And then, by imposing the gauge stability condition.

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Renata Ferrero: I can afterwards write down the reduced Hamiltonian and quantize it. So this is just the application of what what I've been saying before to this specific model.

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Renata Ferrero: So I I find I can. I can write down the cyclic representation which corresponds to my state omega with respect to which to construct the time, order, correlation, function. And finally.

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Renata Ferrero: we arrive at this object, which we derived also before the generating functional in a general signature. S,

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Renata Ferrero: and this j of me, guys, what before was this? Vector

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Renata Ferrero: omega? So this is the term which is reminiscent of my state dependence of the path integral.

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thiemann: Yes, just for clarification, so that

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thiemann: the signature is always Lorenz in here.

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thiemann: But the the S labels 2 possibilities. Either you want to derive

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thiemann: the generating functional of stringer functions

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thiemann: which would be S equal to 0,

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thiemann: or you want to derive the generating function of time ordered functions. With respect to the notion of time that is dictated by the choice of gauge fixing.

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thiemann: That would be this choice. S. Equal to one.

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Abhay Vasant Ashtekar: Yeah. And in this case little Q and little P are really what was previously capital. Q. Capital. P.

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thiemann: That's correct. That's correct.

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Abhay Vasant Ashtekar: And but then that they're really nonlinear right? And they're not linear. Is it right? Or am I missing something here.

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thiemann: This. This is something that people in in asymptotic safety always ignore, that

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thiemann: that metrics of certain signature have no vector space structure.

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thiemann: And we also adopting that point of view here, if you want to avoid that, you need to, would need to go to A

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thiemann: and bind formulation.

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thiemann: So so.

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Abhay Vasant Ashtekar: Just to understand. So to understand that that

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Abhay Vasant Ashtekar: the the gauge fixed surface that you're looking at actually is is nonlinear, and therefore

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Abhay Vasant Ashtekar: the Q's and P's are subject to some constraints, and therefore it's not.

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thiemann: So what we are doing in loop quantum gravity. We always ignore the fact that we are supposed to quantize a system in which, for example, the metric is positive, definite.

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thiemann: but we are also the we are considering flips in in signature of the triad. And all this kind of things.

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thiemann: So that's quite similar here.

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Abhay Vasant Ashtekar: It's different, right? Because there are also constraints.

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Abhay Vasant Ashtekar: Am I missing something? I'm I'm missing.

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thiemann: Constraints are already solved at this point.

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Abhay Vasant Ashtekar: Yeah. But therefore, because of constraint, then Qnp would be nonlinear. I mean, it's not.

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thiemann: No, no, but we have 4 4 scalar fields, so the the Qab, so the the adm variables Qab and Pab are no longer constrained because of the 4 scale.

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Abhay Vasant Ashtekar: That's that's what I was missing. Thank you. I'm with you. Okay.

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Renata Ferrero: Thank you. So in this case, for instance. And this is the question that Simone, arised in the case of the 4 Scalar fields, H involves a square root, and in the hands, in order to, we would like to get rid of the integration of a momenta or physical momenta Dp.

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Renata Ferrero: In order to do that, we have to unfold the reduced phase space to the unreduced phase space, by also adding this smeared delta over the constraints and gauge fixing, and to integrate over Phi, i and Pi, i. And so what we see that if we do that, then we find that if we would have, like P. And Q dot

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Renata Ferrero: with different I factors. If we would work in in Euclidean signature, we would have P. And Q. Dot with different I factor with respect to Phi, Dot and Pi.

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Renata Ferrero: And so, in order to to have, like a consistent and proper expression for the generating functional. In this case we are forced to work with S equal to one. So in the Lorentzian signature, but with other forms of the like, with different choices of reference, frames of, or and so to like, perform the reduced face space

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Renata Ferrero: quantization such that H doesn't involve a square root. We could, in principle, of course, also work with S equal to 0. So in the Euclidean case.

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Renata Ferrero: So now I'm writing down my z. 1. So where I put S to one in Lorentzian, and in the end we end up with such an object. So because we can integrate out the piece, we said, and we have following action, which is the Einstein Hilbert action

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Renata Ferrero: plus this term coming from the scalar field, and as a curiosity one can also write down the state dependence. J. Of Omega as a ghost matrix. And yeah, and then also consider ghosts in our fields, or one can also actually use it like, perform a field redefinition of our G and absorb it in a redefined field.

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Renata Ferrero: So now we we are going to give this object into the machinery of the asynthetic safety. frg machinery. And so in the 1st investigation we did, we

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Renata Ferrero: did not consider the state dependent. So I will ignore this J. Of Omega. But this is, yeah. Currently a work in progress. So inspired by this action, we will write down our effective average action. So our K dependent gamma

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Renata Ferrero: in a very like parallel fashion. So we have the Einstein-hilbert term, with, of course, K dependent coupling constants, so my Newton Constant would be k dependent, and also my cosmological constant.

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Renata Ferrero: and the term reminiscent from the Scalar fields, where we actually specialize to the gauge in which Phi i is equal to Chi i.

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Renata Ferrero: And so this, like Chi mu Nu. This matrix of Kaplan, constant in front of it, just turns out to be proportional to a delta function.

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Renata Ferrero: With a with a K dependent coupling constant Kappa, a unique one.

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Renata Ferrero: So we will go to second order of the expansion of In in of our geometric series, in in order to solve the flow equation.

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Renata Ferrero: And what we find

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Renata Ferrero: is that 1st of all, at this level of the truncation in T equal 4, the coupling constant in front of the scalar fields. It's not flowing. And hence, yeah, it's due essentially to the fact that at this order this traces are all equal to 0 in 4 dimensions, and hence the scalar fields they just contribute in order to disentangle the physical degrees of freedom from the gauge degrees of freedom. But they are not dynamical. They do not contribute

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Renata Ferrero: to the running of the gravitational camp links and do not run itself.

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Renata Ferrero: Instead, we find here do not get scared. This is just like some mathematical object. But these are the equations of the Beta function of my dimensionless, cosmological, constant, and dimensionless Newton Constant, which are expressed, as you see

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Renata Ferrero: in terms of this proper time integrals with our insertions of our regulator functions in hit. So this is, yeah, it looks pretty complicated. But it's really nothing else is really the resulting expressions for our traces by means of this heat, kernel expansion.

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Renata Ferrero: and what we find this is indeed very nice is that this 2 beta functions. They have a fixed point, so they both vanish when K. Is sent to infinity

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Renata Ferrero: for these particular complex values. So we have a complex fixed point. The real part is very close to the right and fixed point. This is quite nice, and also we can compute the critical exponents which are the exponents which describe how

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Renata Ferrero: the lambda and G. So the coupling constants scale around the fixed point, and we find out that they have to both positive real part. And this would what people would associate to relevant directions. Of course, here, here the interpretation is a bit less clear than in the Euclidean case. But yeah, still, we find it nice to perform this analysis.

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Renata Ferrero: And then we wanted to know whether the trajectories did flow in the limit. K. To infinity to the fixed point.

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Renata Ferrero: But of course, to study that we have to project our flow into the real or imaginary part, or like mixed real for one coupling, constant and imaginary for the other one.

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Renata Ferrero: And so, for instance, this is the G real G. Newton dimensionless, real lambda, dimensionless, real, and we find that the trajectories flow into the fixed point, and this is the same for the imaginary part

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Renata Ferrero: and other projectors projections. But what you see is that all the arrows. For K to infinity do go into this purple point.

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Renata Ferrero: And finally, and this is really the last bit. We investigated the possibility, or, like the existence of admissible trajectories. So to see whether we had dimensionful coupling constants when K is sent to 0 for some trajectories.

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Renata Ferrero: And this is actually like this, investigation is performed in 3 steps. We have to integrate down to K to 0, which we could do, and then study how the real valued dimensioning couplings in the infrared looks like, whether imposed a condition that they have to be real.

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Renata Ferrero: and then also test that these trajectories do indeed flow into the ultraviolet fixed point in the dimensionless version of the coupling constant. So how this is done numerically, this is a numerical investigation. One fixes an arbitrary initial condition. K. Bar equal to one for the real part, then one integrates down to K to 0,

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Renata Ferrero: and then one studies, which is the like, associate the imaginary part initial condition in such a way that my lambda at K equal to 0 and my G at K. Equal to 0 are both real.

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Renata Ferrero: So what is important here is that

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Renata Ferrero: it is no agreement for granted that they exist. And also it's also not given for granted that they are unique. Right? There could be several solutions for a given initial initial condition. So we investigated a little bit in order to see unicity and existence.

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Renata Ferrero: And, for instance, we started with this initial condition, and we were able also, by means of this graphical method, to see that it exists an admissimal trajectory. Because what I'm plotting here essentially is, the 2 surfaces are my dimension for lambda and G, when K is equal to 0. So effective action.

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Renata Ferrero: And on this plane they are the initial condition on the imaginary

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Renata Ferrero: subsector, because I fixed the real one for K equal to one. And what they see is that the 2 surfaces they meet exactly at 1 point

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Renata Ferrero: when both. This is the imaginary part. So I'm interested when they intersect the surface. This gray surface, which is 0 when they take 0 imaginary part, and then I can read which is the imaginary counterpart for G. And lambda for this real initial condition. So there exists an admissimal trajectory for this initial condition.

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Renata Ferrero: And here we plot indeed for this trajectory, with how the coupling constant looked like, you see, that imaginary part goes to 0 when K goes to 0.

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Renata Ferrero: And this is how the real part looks like.

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Renata Ferrero: Okay, this is the dimensionless counterpart. And we were interested in. See whether the flow of this particular trajectory flows into the ultraviolet fixed point, or the K 2 infinity limit into the fixed point we found before.

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Renata Ferrero: and finally we depicted a little bit. We wanted to study the the unicity of these admissible, admissible trajectories.

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Renata Ferrero: And so we, we are plotting here.

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Renata Ferrero: which are the correspondent initial condition imaginary part for any given real part initial condition. So we essentially, we scanned the space of initial condition on the real part, and we saw which was the corresponding imaginary part, such that an admissible trajectory is

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Renata Ferrero: found, and we see that there are like these 2 smooth surfaces. So in this, in this space subspace of this parameters. It's like a regular surface.

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Renata Ferrero: But however, what you see

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Renata Ferrero: is that this surface, like it's only existing for this region, and instead for this other region. It seems that no admissible trajectories are found, and this is also something we have to understand better.

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Renata Ferrero: Yeah.

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Renata Ferrero: So finally, I would like to to wrap up a little bit. Yeah, this was a little bit the 1st attempt in order to communicate, and also joined a force between asymptotic safety and canonical quantum gravity.

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Renata Ferrero: and, namely, the idea was to start from a canonical quantum gravity framework in order to define the true degrees of freedom. The state underlying the Hamiltonian theory, and then, by means of the canonical path integral, write down the partition function only of the true degrees of freedom, and then

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Renata Ferrero: give that into the asymptotic safety machinery to construct a well-defined, effective action. So K. Sent to 0 in order to compute correlators for this Hamiltonian theory.

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Renata Ferrero: we investigate that in a 1st application for an Einstein-klengordon theory. And in order to do that, we have to develop this new or like further develop this tool, which was already introduced many years ago, namely, the Lorentzian heat kernel proper time, and to do that we have to also like, introduce this suitable cutoff function.

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Renata Ferrero: and what we found for this model that there is a K to infinity fixed point, that the flow is complex, but that there exists for a given space of parameters, admissible trajectories having real valued coupling constant when K is sent to 0.

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Renata Ferrero: And as a small outlook, yeah, we would further understand how we can classify these Lorentzian cutoff functions, how to incorporate the state dependence which we haven't done yet by means of a ghost matrix term, or redefined by a means of a filler definition.

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Renata Ferrero: Of course, one of the further applications or investigations are over the higher order truncations, in order to see that the result is stable.

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Renata Ferrero: and also to introduce more realistic matter, coupling where, in order to make more like maybe clear connection between the 2 theory, we can investigate our flow in a Euclidean version.

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Renata Ferrero: Thank you.

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Hal Haggard: Thank you, Renata.

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Hal Haggard: and thank you for staying close to time, even with all the discussion.

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Hal Haggard: Questions.

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Hal Haggard: Yeah, Abbay, please.

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Abhay Vasant Ashtekar: Okay, so

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Abhay Vasant Ashtekar: several basic elementary questions really about

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Abhay Vasant Ashtekar: in the beginning, you said that

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Abhay Vasant Ashtekar: you know the background Gbari, chosen to begin with, is arbitrary, but then we fix it by self consistency.

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Abhay Vasant Ashtekar: What is, what are the self consistency that you use to fix it?

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Abhay Vasant Ashtekar: And what are the background here for Gbar in your calculation. And what are the self-consistency requirement that you use to

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01:05:52.400 --> 01:05:53.740
Abhay Vasant Ashtekar: fix it? I'm sorry.

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Renata Ferrero: So I have. I have not used this any self-consistency requirement. Sorry. This was maybe like something I just mentioned to to tell you how this background field method, which is the spirit of the background, fill method. But here we are. We are never like using any notion of self-consistent background at all.

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Abhay Vasant Ashtekar: Okay, so.

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Renata Ferrero: Principle. This can be done if one wants to see like this would be like a an effect like an effective equation of motion for myself, consistent background. But in this framework I am never using that.

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Abhay Vasant Ashtekar: Okay.

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Abhay Vasant Ashtekar: Then I mean as as you. But I could explain to us that G. Hat should could be thought of as an expectation value.

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Abhay Vasant Ashtekar: So I think I would like to understand what is a quantum state in which this is supposed to be expectation of value, or

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Abhay Vasant Ashtekar: I mean, in general.

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Abhay Vasant Ashtekar: I'm rather confused about the relation between quantum states and this, the framework that you have Hilbert space and States, and so on. In the framework that you have developed.

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Renata Ferrero: So you want you. You're asking me which is the state associated to my to my jihad. Sorry.

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Abhay Vasant Ashtekar: Right. But state even what is the space of states in which it can lie? And

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Abhay Vasant Ashtekar: and then.

447
01:07:11.640 --> 01:07:13.960
thiemann: So you mean the you mean Hilbert Space language.

448
01:07:13.960 --> 01:07:15.669
Abhay Vasant Ashtekar: Yeah. Hilbert, space. Like, state, right?

449
01:07:15.670 --> 01:07:17.435
thiemann: So the the thing is that

450
01:07:18.840 --> 01:07:22.690
thiemann: at the end of the day, suppose we can actually solve the the vertex equation.

451
01:07:22.870 --> 01:07:28.339
thiemann: and then you get some some functional which depends on G hat, G bar.

452
01:07:28.850 --> 01:07:29.870
thiemann: and K,

453
01:07:30.000 --> 01:07:30.810
thiemann: okay.

454
01:07:31.270 --> 01:07:39.589
thiemann: and we the the thing that from our point of view, so from the people. From the point of view of canonical quantum gravity, people would be interesting are

455
01:07:39.720 --> 01:07:44.340
thiemann: 2 special cases for this functional, namely, K equal to 0,

456
01:07:44.570 --> 01:07:46.629
thiemann: which gives you the effective action.

457
01:07:46.740 --> 01:07:48.930
thiemann: 1st of all, the background

458
01:07:49.120 --> 01:07:51.820
thiemann: method, effective action.

459
01:07:51.900 --> 01:08:00.440
thiemann: And then you take the following, you set the following values, you set G Hat equal to 0 and J. Bar equal to G.

460
01:08:00.610 --> 01:08:05.080
thiemann: The G is the the metric, and that gives you the effective action.

461
01:08:05.600 --> 01:08:09.049
thiemann: The effective action, then, has to be Legendre transformed

462
01:08:09.640 --> 01:08:13.919
thiemann: back to the generating function of Feynman distributions

463
01:08:13.960 --> 01:08:20.400
thiemann: and the Feynman distributions are hopefully related to expectation, expectation, value functionals of

464
01:08:20.609 --> 01:08:23.050
thiemann: operators in the usual sense.

465
01:08:23.420 --> 01:08:24.219
thiemann: That's well.

466
01:08:24.220 --> 01:08:29.220
Abhay Vasant Ashtekar: It had be 0, because I mean rather explained to us that we can think of it as an expectation value of the

467
01:08:29.649 --> 01:08:32.129
Abhay Vasant Ashtekar: of the metric operator, that dynamical video.

468
01:08:32.130 --> 01:08:41.340
thiemann: So so the the general method to get from the background effective action to the actual background. Independent action

469
01:08:41.540 --> 01:08:46.720
thiemann: is to take this combination of of limits that I was telling you.

470
01:08:47.580 --> 01:08:51.070
thiemann: So that can be proved in in general. Quantum theory.

471
01:08:55.710 --> 01:08:57.620
thiemann: Yeah, exactly. This was the form.

472
01:08:58.970 --> 01:09:02.150
Abhay Vasant Ashtekar: Sorry. Can you repeat again? I didn't understand what you said.

473
01:09:03.069 --> 01:09:08.070
Abhay Vasant Ashtekar: I I thought I mean there was this confusion about whether jihad should be thought of as like external current, or

474
01:09:08.140 --> 01:09:15.019
Abhay Vasant Ashtekar: out of the out of the expectation value of the metric. And then I understood that explained that this should be thought of as Expectation. Value of the Metric.

475
01:09:15.020 --> 01:09:24.410
thiemann: Let let us 1st let us 1st look at, forget about gravity at the moment. And let's just look. Let us look at what is the effective action actually computing

476
01:09:24.840 --> 01:09:29.690
thiemann: the effective action is computing the one pi irreducible

477
01:09:29.770 --> 01:09:31.299
thiemann: endpoint functions

478
01:09:32.729 --> 01:09:36.989
thiemann: and the legendre transform of the effective action gives you back the original

479
01:09:37.029 --> 01:09:39.840
thiemann: generating functions of connected

480
01:09:40.109 --> 01:09:41.800
thiemann: Feynman distributions.

481
01:09:44.890 --> 01:09:53.279
Abhay Vasant Ashtekar: Yeah, but I would like to understand what I was supposed to think of in gravity. Right? I mean, so you're doing some calculation gravity, and I would like to understand purely within gravity.

482
01:09:53.310 --> 01:09:54.909
Abhay Vasant Ashtekar: conceptually, not in.

483
01:09:55.200 --> 01:09:57.379
thiemann: Yeah, conceptually, so.

484
01:09:57.380 --> 01:10:05.899
Abhay Vasant Ashtekar: What is it that am I calculating? The object is depends. This effective action depends on 2 variables, I think, or only one. I don't know.

485
01:10:05.900 --> 01:10:07.920
thiemann: Let's let's have a look at

486
01:10:08.050 --> 01:10:12.149
thiemann: what is called Gamma Bar. Prime K.

487
01:10:13.340 --> 01:10:13.840
Abhay Vasant Ashtekar: Had.

488
01:10:13.840 --> 01:10:15.729
thiemann: Ji hat and G bar. Right?

489
01:10:16.000 --> 01:10:19.919
thiemann: So what we are interested in from the canonical point of view is.

490
01:10:19.990 --> 01:10:22.860
thiemann: 1st of all sending K equal to 0

491
01:10:24.070 --> 01:10:31.490
thiemann: and send setting as, secondly, setting G hat to 0 and interpreting G Bar as as a metric.

492
01:10:32.430 --> 01:10:36.090
thiemann: and that gives you the background independent, effective action

493
01:10:36.230 --> 01:10:41.820
thiemann: if and if it would exist, and if it's well defined, would give you, via legendre, transform

494
01:10:42.100 --> 01:10:47.299
thiemann: the effective, the generating function of connected Feynman

495
01:10:47.330 --> 01:10:48.510
thiemann: distributions.

496
01:10:50.920 --> 01:10:57.019
Abhay Vasant Ashtekar: I guess we would like to understand the same confusion in the beginning that why should jihad be 0?

497
01:10:57.130 --> 01:11:04.179
Abhay Vasant Ashtekar: I mean, I thought G hat was expectation. Value of of the of the metric G operator in some quantum state.

498
01:11:04.340 --> 01:11:06.719
Abhay Vasant Ashtekar: and I wouldn't expect it to be 0.

499
01:11:06.720 --> 01:11:11.280
thiemann: So JJ. Hat, J. Hat has really has the the same

500
01:11:12.646 --> 01:11:14.560
thiemann: meaning as

501
01:11:16.640 --> 01:11:18.899
thiemann: How do I do? I want to say this so.

502
01:11:18.900 --> 01:11:24.279
Abhay Vasant Ashtekar: Yeah, yeah, you said it was like external current. But then then there was confusion about it a little bit. And then.

503
01:11:24.430 --> 01:11:25.300
Abhay Vasant Ashtekar: okay.

504
01:11:25.300 --> 01:11:34.770
thiemann: Let me let me let me say it in a different way. Let's say what they're doing here is it's it's very parallel to what people do in Europe on the p. 3.

505
01:11:35.360 --> 01:11:47.309
thiemann: Just that you label that you're dealing with the metric as a field rather than some other field. Let's say, a scalar field. So in in, let's say, in in the scalar field you start from the

506
01:11:47.330 --> 01:11:50.109
thiemann: let's say from the generating function

507
01:11:50.250 --> 01:11:52.389
thiemann: of Feynman distributions.

508
01:11:52.980 --> 01:11:57.580
thiemann: And then, if this is well defined, you can take the Ronald transform.

509
01:11:57.620 --> 01:11:58.730
thiemann: And when you.

510
01:11:59.130 --> 01:12:04.639
thiemann: the the generating function depends on some some smearing function, right? So

511
01:12:05.180 --> 01:12:06.480
thiemann: you have to take

512
01:12:06.570 --> 01:12:11.449
thiemann: functional derivatives with respect to that smearing function in order to get the endpoint functions.

513
01:12:12.080 --> 01:12:13.010
thiemann: Now, what

514
01:12:13.200 --> 01:12:28.039
thiemann: what what it turns out is is that's quite nice is that if you want to pass from the connected Feynman functions to the one Pi Feynman functions, all you need to do is to take the ronde transform of the generating function of connection functions, and

515
01:12:28.200 --> 01:12:32.399
thiemann: with respect to some other sphering function which people usually call a J.

516
01:12:33.470 --> 01:12:39.160
thiemann: And if you want, you can compute. You can sometimes. Sometimes people call this a classical

517
01:12:39.180 --> 01:12:53.379
thiemann: field or a classical metric in this case, but it's just another smearing function, just as you want to. This is, if you wanted to pass from the Lagrangian to the Hamiltonian, passing from velocity, phase, space to the momentum phase space.

518
01:12:55.860 --> 01:13:00.110
thiemann: So there's really no mystery about that. This is a common tool in quantum p theory.

519
01:13:02.400 --> 01:13:11.749
Abhay Vasant Ashtekar: Yeah. But the problem is basically that common tools in quantum field theory are just not applicable to general relativity. As we see, you know we're going to Euclidean space looking at K. Looking at these things.

520
01:13:11.820 --> 01:13:13.730
Abhay Vasant Ashtekar: But you are applying it in some way.

521
01:13:13.910 --> 01:13:26.349
Abhay Vasant Ashtekar: and I mean, I have difficulty in understanding which elements of the standard quantum field theory I can still rely on, and which I do not. For example, you talk about fixed points.

522
01:13:26.480 --> 01:13:31.789
Abhay Vasant Ashtekar: On the other hand, you say, K. Doesn't have any meaning by itself except K equal to 0 and K equal to infinity.

523
01:13:32.170 --> 01:13:37.230
Abhay Vasant Ashtekar: but in practice, of course, one is never at K equal to 0 or K equal to infinity. One is close to that.

524
01:13:37.460 --> 01:13:45.239
Abhay Vasant Ashtekar: and the whole point was that it doesn't matter if you're sufficiently close. But here it does matter. So so these are the questions that I, you know, would

525
01:13:45.330 --> 01:13:55.019
Abhay Vasant Ashtekar: is precisely the points at which the standard quantum field theory differs, or general relativity or gravity differs from quantum field theory.

526
01:13:55.450 --> 01:13:58.369
Abhay Vasant Ashtekar: They cause some differences in your framework.

527
01:13:58.520 --> 01:14:03.690
Abhay Vasant Ashtekar: but we still use, or you seem still to use, some intuition from the

528
01:14:03.730 --> 01:14:08.780
Abhay Vasant Ashtekar: standard quantum field theory. I'm not saying it's wrong, but it's difficult for us to grasp

529
01:14:08.980 --> 01:14:09.660
Abhay Vasant Ashtekar: up.

530
01:14:09.820 --> 01:14:16.070
Abhay Vasant Ashtekar: Why, certain things are okay, and certain things are not okay. But I think it's a long discussion, so we can continue. Thank you very much. Both of you.

531
01:14:18.400 --> 01:14:19.390
Hal Haggard: You genuine.

532
01:14:20.990 --> 01:14:28.649
UP: Whitmore 320: Thank you, Renata. This was really interesting. I I have a question not about the dynamics, but about the State.

533
01:14:29.180 --> 01:14:47.460
UP: Whitmore 320: Let me phrase it in quantum field theory the effective action can be used to compute correlation functions in some state, for instance, in informalism or matrix elements in the now formalism. So here, in the gravity, what is the state encoded in this effective action that you are considering.

534
01:14:48.640 --> 01:15:06.869
Renata Ferrero: Yeah. And I mean, this really depends on the, on the, on, the, on the gauge we choose. Right? This is like, yeah, I mean you have to do the entire construction of your Hilber space, and then, like from your Gns, data, and and and so on and so forth. I haven't given any explicit example here.

535
01:15:06.940 --> 01:15:10.466
Renata Ferrero: because we haven't taken into account. But

536
01:15:11.240 --> 01:15:15.510
Renata Ferrero: yeah, this would be that like the procedure to to build. It would be really like

537
01:15:15.590 --> 01:15:16.620
Renata Ferrero: to

538
01:15:16.890 --> 01:15:39.839
Renata Ferrero: like. Give your reduced phase space approach, start your reduced phase, space, approach, and then construct your physical Hamiltonian, quantize it, and then look at the States at the cyclic vectors, and associated to this to the States in in for that specific physical Hamiltonian that said, I have not constructed it yet. This, or like, I haven't given any any explicit expression in here.

539
01:15:40.200 --> 01:15:42.300
Renata Ferrero: But yeah, that would be the recipe.

540
01:15:43.960 --> 01:15:47.250
Abhay Vasant Ashtekar: But then you talk about fixed points and things. Right? So.

541
01:15:47.790 --> 01:15:52.590
Abhay Vasant Ashtekar: But there's some specific calculations. You did do so. It's not just general theory.

542
01:15:52.590 --> 01:16:01.230
Renata Ferrero: Right. Let's say, I computed the fixed point for a constant state. This is what this is, what what we did up to up to now, so we haven't taken into account

543
01:16:01.250 --> 01:16:04.370
Renata Ferrero: any dependence on a on a more.

544
01:16:05.180 --> 01:16:05.920
Renata Ferrero: So it's.

545
01:16:05.920 --> 01:16:08.769
Abhay Vasant Ashtekar: Constant state mean just the words that we don't understand.

546
01:16:08.770 --> 01:16:32.920
Renata Ferrero: Yeah, okay, maybe a constant state is misleading. Let's say I haven't. I haven't. So this object right now. I have discarded up to now, of course, it's what we would like to do as a next step is to consider that as well because it plays a role. I agree it is an element which is in the path integral. And it does play a role.

547
01:16:32.920 --> 01:16:51.349
Renata Ferrero: But this is something which, yeah, also, quite surprisingly, up to now, in the study of this flow equation has not been so much taken into account, but it has to. It's only in the recent. In like recent publications that has been also

548
01:16:52.020 --> 01:17:01.099
Renata Ferrero: said that it has to be included in the form of our of our flow equation. So I agree it's not the complete story, but this is a as a 1st investigation.

549
01:17:01.460 --> 01:17:02.170
Renata Ferrero: Thank you.

550
01:17:03.800 --> 01:17:04.500
UP: Whitmore 320: Thank you.

551
01:17:07.020 --> 01:17:09.029
Hal Haggard: Are there other questions?

552
01:17:15.870 --> 01:17:18.620
Hal Haggard: If not, let's thank Renata one more time.