Jorge Pullin: Our speaker today is for the person who's going to speak about Gregory progress and relational quantum
Dynamics.
Philipp Hoehn: Yeah, thank you very much and also thanks to those worth invited me to give this talk. Yeah. So my aim here is
Philipp Hoehn: To give an overview over progress in the field of relation quantum dynamics in recent years and this talk is based on various works with various collaborators.
Philipp Hoehn: As far as questions or concerns, please feel free to interrupt for any clarifying questions if there's anything larger to discuss me. Let's try to keep that whether the discussion.
Philipp Hoehn: Um, yeah, so
Philipp Hoehn: Some of the slides, don't move on. No. Okay. Yeah, so the starting point for this whole talk as as ever. So often the problem of time which in its most basic form.
Philipp Hoehn: Many of you know of course is rooted in the fact that the Hamiltonian of January covariance serious
Philipp Hoehn: Constraint or the inner combination of constraints and that leads to states or theory in the quantum case that is often called timeless.
Philipp Hoehn: But this is really quite a misnomer us why it is true that is background timeless, because after all, we're
Philipp Hoehn: Quantization space time and the space time isn't really evolving with respect to any external reference
Philipp Hoehn: It is by no means internally timeless and that is, of course, I'm really at the heart of the relational approach to finding quantum dynamics, namely to the idea in to extract the notion of dynamics from within the dynamic of degrees of freedom encoded in that physical state. And of course,
Philipp Hoehn: Exist many different approaches.
Philipp Hoehn: To implement this and
Philipp Hoehn: If one tries to adopt that point of view one still will face a whole host
Philipp Hoehn: Of other sub problems which are actually nicely reviewed by chemical corrosion press item in a seminar reviews and not going to mention all these problems that one is facing. I will just mention a few of them that I will touch upon in this talk.
Philipp Hoehn: So every off the topics that I mentioned here in the slide, I will review in the coming slides bit more carefully so that we're all on the same page.
Philipp Hoehn: And then I will tell you about what can be done about each of them are has been done in recent years.
Philipp Hoehn: So one of the issues that is there, as already mentioned exists, many different ways in which one could try to implement
Philipp Hoehn: A relational approach to the problem of time and, in particular, they exist three major canonical approaches such as relational server bolts. The privatisations
Philipp Hoehn: And the page, whether it's conditional state formulation and one may as well of course ask what's the relation or which one is the right one.
Philipp Hoehn: Then equal cash.
Philipp Hoehn: Has lavish three series arguments against the viability of the page. What does formalism some challenges that have to be met.
Philipp Hoehn: As another problem. That's one could summarize under the slogan that realistic clocks always or may run backwards probabilistic Lee, which might lead to some issues in
Philipp Hoehn: The relation between evolving and clock degrees of freedom and there's the multiple choice problem that you don't have distinguish choice of an internal time and the global time problem that's
Philipp Hoehn: You may be dealing with clocks that don't monotonic. We ran forward and there's of course many other problems. So yeah, I will tell you a review of these different topics here. Now the coming slides. And as I mentioned, and I will tell you what we found in recent years ago.
Philipp Hoehn: So let me start with the three canonical approaches to three major economic approaches to the
Philipp Hoehn: Termination quantum dynamics. The first one that's most of you of course quite familiar with relational records on both which was really started by Carlo.
Philipp Hoehn: Well, when I was 30 years ago. And the basic idea being that you ask you pick some dynamic degree of freedom that you declared to be a time function that he asked for.
Philipp Hoehn: The evolution of some other observer balls relative to that time function and these observable or encoding the question, really, what is the value of that function when the clock T reads tall.
Philipp Hoehn: Now during our PhD thesis Bianca has given us beautiful policies expansion for such
Philipp Hoehn: Relational goals which here. I'm only giving you for the case of a single constraints for most of the target will be focusing on the single constraints.
Philipp Hoehn: And what they really encode us the gauge invariant evolution of some faith based function F relative to that cloth function.
Philipp Hoehn: T and the way you can think about it, the pictorial as you have your constraints or force.
Philipp Hoehn: With the gauge or bits in it. And these clock functions are they equal time surfaces defined by that redefining something like a gauge server.
Philipp Hoehn: Gay surface that cuts the gauge orbits and then the gauge invariant evolution corresponds to basically scanning through your
Philipp Hoehn: Through your constraints surface with equal time surfaces and then you always ask what the value of that function f is on that time surface.
Philipp Hoehn: But now, of course, they are gauging variance or their commute.
Philipp Hoehn: With the constraints. And so what you really can consider those objects to be as they are, what is often called a gauge invariant extension of a gauge fixed quantity gauge fixed because you
Philipp Hoehn: You asked for the function f on the hyper surface equals constant, but then you extend that engagement during manner.
Philipp Hoehn: And now what is the quantum dynamics on would like to do here is fun in some way of performing a direct monetization, or would like to promote these relational durables two operators.
Philipp Hoehn: Okay, so now let's come to the second approach the penetrations
Philipp Hoehn: So that's usually done in the sense of renewed faith based conversation and exits many different ways in which one could do that.
Philipp Hoehn: So here we'll pick a specific way, namely the penetrating through cemetery reduction relative to a chosen clock particular
Philipp Hoehn: surrogates fixing and because that will be useful later on. So what's the first step to do one performs a canonical transformation on the constraint surface.
Philipp Hoehn: That splits the degrees of freedom into pure gauge and pure gauging varying degrees of freedom. So if you have some canonical variables like here on the left hand side and you can
Philipp Hoehn: map them. Let's say to some clock function t. Let's say, for simplicity and only the first two and then we can find a
Philipp Hoehn: canonical transformation in such a way that the momentum conjugate to that clock is actually proportional to the constraint. So that's the pure gauge and constraint part
Philipp Hoehn: And then you can map these other canonical Teresa of freedom to the relational observable. It's a relative to that clock in at least locally that will work.
Philipp Hoehn: So then basically what that gives us on the right hand side is just the repayment translation of the constraints surface then, as a second step, what we can do.
Philipp Hoehn: Is again fixing. We can just
Philipp Hoehn: Fix it to t equal some constant, for instance, equals zero.
Philipp Hoehn: And what that does is that it really just removes the redundancy in the description of the gate and bad degrees of freedom.
Philipp Hoehn: And we don't lose any dynamic information because we still have that evolution promise at Tom corresponding to the clock readings.
Philipp Hoehn: And what it does is it really removes the reference degrees of freedom, eight o'clock degrees of freedom from among the dynamical variables. But again, no loss of information.
Philipp Hoehn: And we end up with a gauge fixed reduced face space, which only has the remaining degrees of freedom on it. And those dynamics of those are still encoded in these
Philipp Hoehn: Reduced relationships over bolts, which in fact satisfy some equations of motion generated by some true Hamiltonian on this real estate space.
Philipp Hoehn: And this we can interpret then really as the dynamics seem are relative to the perspective of that clock. And then when kind of course, try not to contest that reduce face face to get some motivational company.
Philipp Hoehn: Now the third approach is so called page. What does formalism. So it says follow. So these guys started straight away with a splitting of the constraint and two o'clock part and the system partner interaction.
Philipp Hoehn: And then they defined some clock states and they behave as follows. With respect to that clock Hamiltonian. So they behave
Philipp Hoehn: Monotonic Lee under that clock Hamiltonian and then they defined some conditional probabilities for some dynamical
Philipp Hoehn: A variable of the system. And you asked her, What's the probability that some auto FS some given outcome get another four weeks talk
Philipp Hoehn: And so you can you find that an extension of the bond rule using physical states in this manner here.
Philipp Hoehn: And what is interesting in this case as you can define a conditional state of the system, given that the corporates tall and defined in this manner. So you conditional physical states on oxalate
Philipp Hoehn: And that gives you a states and Hubert space on only the system degrees of freedom and it's not very difficult to convince oneself that that state actually satisfies the equation with respect to that system.
Philipp Hoehn: Okay, so this is a review of the three
Philipp Hoehn: Now let me summarize the three criticisms that could crash and actually raised against the page with this form of autism and they are actually as follows.
Philipp Hoehn: So the first criticism was that could argue that these conditional probabilities are actually inconsistent with the constraints.
Philipp Hoehn: And his reasoning was as follows. You have these kinematic operators that you throw into this in a product of physical states which don't come up with a constraint. So when you act with
Philipp Hoehn: Those kind of magic operators on the physical Hubert's on the physical states it throws you out of the physical Herbert space and, as such, you're violating the constraints.
Philipp Hoehn: Then building up on that argument cool cash.
Philipp Hoehn: Wanted to show that this approach actually leads to completely inconsistent results and he then wanted to show that
Philipp Hoehn: That it leads to the wrong transition probabilities for a non relativistic particle. So, when you ask, what's the probability that
Philipp Hoehn: The project has positioned to prime time top prime, given that earlier was a cure time talk. And then he extended this condition probability in this manner here. So the first. What do you see other on the other sides are just the conditioning on the
Philipp Hoehn: Initial conditioning. And then the final conditioning in the middle and then you can easily see that the result you get us just copy the wrong results here mainly no dynamics at all. And so he argued, it's copied on results.
Philipp Hoehn: And then the third criticism that he raised was that when you actually apply this conditional probability to relativistic systems in the category systems condition on the makovsky time, then you get a completely incorrect.
Philipp Hoehn: Localization probability, namely given by the shooting a type probability amplitude with respect to the solution to the client or any questions.
Philipp Hoehn: So these were pretty serious criticisms against that approach and effectively that ended research on this approach, at least for a while was recently asked up again for a while there was a quietness.
Philipp Hoehn: So the next problem or challenge that I that I just want to briefly review is one that actually applies to any of these approaches.
Philipp Hoehn: To relational quantum dynamics. That's the so called multiple choice problem meds basically a problem as they exist. Many equivalent choices for relational clocks.
Philipp Hoehn: It may indeed need to equivalent quantum dynamics. And then the question arises, which one would be the right one, or is there any way in which one could relate this different
Philipp Hoehn: Kind of emotions and note that, of course, this problem does not arise, classically, because when you have to clock variables, the relations between them obviously fixed at least locally on a constraint surface.
Philipp Hoehn: Now then.
There is another
Philipp Hoehn: Challenge that has been raised against relational notions of dynamics.
Philipp Hoehn: And that is basically stated like this so perfect clocks for bounded Hamiltonian. The first one is an observation made by Polly
Philipp Hoehn: In a footnote actually has book. Many of you know of course familiar with it. So if you have a hat bonded Hamiltonian and exits know south of joined operate it. That is conveniently punter get to it.
Philipp Hoehn: Fair enough. But then later under involved developed a sort of strengthening of that result.
Philipp Hoehn: And again for the case study of a bounded Hamiltonian and they argue that it does not exist any SAFA joint the joint operating at that satisfies of certain wanted to this city property, which roughly summarizes as follows. There. No, I get states have that operator.
Philipp Hoehn: That are ordered in this manner. So it was increasing clock readings, which are such that the transition attitude with respect to shorting a time
Philipp Hoehn: And for going from a smaller O'clock reading to a higher talk reading is non zero for at least some positivity.
Philipp Hoehn: And that is always zero for all positive T for transition of the tooth going from a larger cooperating to a smaller cooperating. So the clock can never run backwards.
Philipp Hoehn: But it can run forward. That was the inputs and then additional at this cannot exist.
Philipp Hoehn: For itself to join to that conclusion, less than anywhere in the state clock which can run forward in time must have a non banishing probability to run backwards in time.
Philipp Hoehn: And so, and then they argued that this is a serious problem for relational dynamics, because other variables that you want to describe
Philipp Hoehn: Loosely want to describe may actually be multi valued at a given the clock reading. And so to counter that day and actually developed quantization of, you know, margin on the gravity that
Further
Philipp Hoehn: And finally, there's the focus of global problem of time and that's basically just the problem. What if you don't have any clocks, or you're just dealing with clocks.
Philipp Hoehn: That are not monotonic, and you have a multi valued in this off the relations between the evolving degrees of freedom and the clock and that can lead to some serious problems.
Philipp Hoehn: Challenges, at least in the conference here.
Philipp Hoehn: Okay, so that's the again the various topics that I want to address here in this talk. And here's the menu for the rest of this talk. I'm not entirely sure I will be able to cover all these topics but I will try.
Philipp Hoehn: So one of the things that we have now shown us that these three canonical approaches are actually exactly equivalent for my class of systems you can consider it a really as three phases of one in the same relational dynamics.
Philipp Hoehn: In that's construction and we have also clarified what quantum end of August of symmetry reduction and also have agent or the extensions of gauge fix quantities using that equivalents. We have actually completely resolved. Also, who cares three criticisms against the waters formalism.
Philipp Hoehn: This issue of realistic clocks may run backwards. I can address using so called quarter and copy of the EMS multiple choice problem we're trying to dress.
Philipp Hoehn: Developing a covariance and that's it via code change us. And lastly, maybe I will still have time to talk about the global problem, the proposal is to
Philipp Hoehn: Think about the dynamics in terms of what might want to call an estimate protection.
Philipp Hoehn: So I'm in the remainder of this talk, I hope I can
Philipp Hoehn: cover these different topics. So let me begin with this issue of realistic clocks may run backwards. So what can we do about this one thing that we
Philipp Hoehn: Want to do here is, in fact, this give up somewhat orthodox idea that or observable in quantum theory have to be selfish on operators and in fact
Philipp Hoehn: There's no by no a huge body of literature on so called generalized measurements.
Philipp Hoehn: Of operating measures that are completely standards in the literature and quantum metrology kind of information on the foundations and that's what we want to use here and using such a few of the EMS will actually still inherit some of these nice features that you would want to have
Philipp Hoehn: For a nice clock. But of course, we have a price to pay for that.
Philipp Hoehn: So we're going to use Colburn copy of the items to model our quantum clocks. So what are these covariance POV.
Philipp Hoehn: So first of all, they define a probability measure for the clock reading. So, suppose the clock readings tag value and all the results, then you can integrate, what are called effect densities over some interval of the real it so that it is an interval here, it's not
Philipp Hoehn: A number. And so then you associate to each interval some positive operator and you require that they define a resolution of the identity. So they add up to now, in contrast to park or to project the value of measures.
Philipp Hoehn: These objects here, the so called effect operators don't have to satisfy the property that if you have effect operators associated to non overlapping time intervals that they have to be orthogonal. So in fact, in general, they want
Philipp Hoehn: And now we want to require a so called covariance property of these effect operators and that's the following, they should behave monotonic Lee under the evolution generated by O'clock Hamiltonian HC
Philipp Hoehn: And so in this manner here. So then you can ask the question, how can one construct something like that. And I will just give you a schematic construction recipe here.
Philipp Hoehn: The details will always depend on what your clock Hamiltonian is but basically you define your effect densities here that go into these POV EMS
Philipp Hoehn: The segments are just degeneracy labeled for the energies of the clock Hamiltonian
Philipp Hoehn: And then you have these clock states here and you can find them schematically as follows. You just take Eigen vectors of the clock Hamiltonian and with such a pre factor here and then it's very easy to see that when you act with the
Philipp Hoehn: Unitary off the clock Hamiltonian on it. Then they behave in this way, methodically and you can really think of these clocks states in some sense of coherent states of the group generated by. It was called kind of Tonia
Philipp Hoehn: And using these these these effect operators are these effect entities, we can then actually define a generalization of
Philipp Hoehn: Spectral the composition and can define so called and small and operators of the pure VM in this form here. So we just take it to the end and then take the next density and integrate that of the reels.
Philipp Hoehn: And these objects actually satisfied generalization of the canonical controversy relations in this form. So it's something that you would want to have a nice clock satisfied.
Philipp Hoehn: Now, so we see they satisfy some nice properties that, of course, we don't get a free lunch, there's a price to pay for that and
Philipp Hoehn: Well, we have a consistent probabilistic interpretation, but we have to accept that typically these ends moment operators will not be salvaged operators.
Philipp Hoehn: These clock states T, they will generally not be orthogonal. So they will not be perfectly distinguishable. And generally, they will also not be eigen states of the first moment operator.
Philipp Hoehn: Okay, I'm just as an aside here when can use these programs also to construct conjugate clock constraint pairs. So in the classical theory and the construction of relational deliverables young cast, for instance, use a lot
Philipp Hoehn: This relation year that given some classical clock function t
Philipp Hoehn: You can always find a constraint that's conjugate to it by rescaling the constraints. So effectively what we're doing here in the quantum theory is
Philipp Hoehn: Kind of the jewel procedure, given some fixed Amazonian constraint or the clock Hamiltonian, you know, try to find a contract gets the contract moments that satisfy this condition. And in fact, this can be done for fairly wide cost constraints of all kind of
Philipp Hoehn: Okay. If you have never seen these objects. Let me just give you an example and particular supposed to clock Hamiltonian is given in this form here is of course appears in relativistic systems.
Philipp Hoehn: But then we can define this clock states, we have to split them into positive and negative frequency modes and they can be defined in this form here.
Philipp Hoehn: And it turns out, then they are not orthogonal. That's not a problem. But they of course satisfied is covariance property here. So they're monotonic under that clock evolution.
Philipp Hoehn: And for instance, if redefined the first moment operator off that pure VM in this manner here or if it finds us that it's actually equivalent
Philipp Hoehn: To the symmetric quantization of that clock function year which classic me is, is of course conjugate to that clock Hamiltonian and the price we pay us that object is clearly not have a joint, but we can still work mostly
Philipp Hoehn: Okay.
Abhay Ashtekar: This transparency, what is sigma
Abhay Ashtekar: Can you tell us
Philipp Hoehn: Sorry. Oh.
Philipp Hoehn: Sigma is really positive negative frequency mode so
Philipp Hoehn: It's the degeneracy sector of
Philipp Hoehn: Of he says
Abhay Ashtekar: Is this plus or minus
Philipp Hoehn: Yeah, just plus or minus
Abhay Ashtekar: Thank you.
Philipp Hoehn: OK, so now we want to use these clock POV comes in most of the cycle of this talk.
Philipp Hoehn: And so now I will come to this equivalence of these three different approaches that I will try to explain. Well, at least give you an overview of why they're equivalent. We call that the trinity of relation quantum
Philipp Hoehn: So that starts. So since we're going to work also with equivalent to page voters. We're going to adopt a little restriction that they also use that we assumed as a splitting of the constraints to cook and system.
Philipp Hoehn: So there's no interaction between clock and evolving degrees of freedom.
Philipp Hoehn: And for the time being, and later relax that but for the time being we assume that the Kakuma Tony has a continuous spectrum in January. It's really a group that is
Philipp Hoehn: Basically our so the clock is monotonic at the system. Hello, Tony, can be anything. And that's still covers a wide range of models that are interested in these for instance in cosmology.
Philipp Hoehn: Okay, so the first step is that we want to contact these relations rocket variables.
Philipp Hoehn: So, I mean, first of all say what we do, classically, so we've also constantly at least locally. It's always possible to some time function that is conjugate to to that constraint.
Philipp Hoehn: And we will restrict to observable off the system. And so we will ask for the value of FS one to talk real talk. And then this
Philipp Hoehn: House here is construction by Bianca actually simplifies in this manner here and then now what we want to do is a contest that object. And basically, what we see is
Philipp Hoehn: Really one of the crucial things we need us quantization of this T T n. And now we know how to do it. We have these covert O'clock POV. I'm so we'll just use the moment operator off the field.
Philipp Hoehn: And so here's just in a nutshell. How you do that in the quantum theory.
Philipp Hoehn: So the colonization of that power series just looks like this, as in this line. I hope you can actually see
Philipp Hoehn: My pointer. And so this year is really just a moment operator. So what I'm pointing at here of the pure VM.
Philipp Hoehn: The top is really the inherited from the classical expression. And then we have the policies here in terms of commentators
Philipp Hoehn: And then it's not very hard to convince herself. They can rewrite that whole expression in terms of some over degeneracy sectors.
Philipp Hoehn: And then you have a coherent group leveraging and so called G twirl off this operates in brackets here. So you have a conditioning or a sort of a projector on to the clock time tall.
Philipp Hoehn: And while the sector sigma is the frequency sector sigma and then times the observable you're interested in. And so that's called the detail.
Philipp Hoehn: And so what you see also here that classical parameter talk has actually also become a quantum degree of freedom.
Philipp Hoehn: And now it's also not very hard to show that these objects actually are.
Philipp Hoehn: A gauge it better. And they're stronger October, those that come up with the constraints and have a bunch of other nice algebraic properties. For instance, they also define homomorphic comes from these little FS to the observable.
Philipp Hoehn: Go further into that.
Abhay Ashtekar: you emphasize that the moments.
Abhay Ashtekar: To to the end. I'm not self a giant. So what happens here with the F.
Philipp Hoehn: Yeah, capital A few mean yes let me, let me try to address that a bit later. So
Abhay Ashtekar: Thank you. Thank you.
Philipp Hoehn: Um, yeah, if I forget it, you can come back to me.
Philipp Hoehn: So yeah, so that's so that was basically as much as I wanted to say about direct causation here know that see why it is equivalent to so called quantum the parameter ization
Philipp Hoehn: So what we want to do now is we want to emulate what I explained before, classically for classically, we had the splitting.
Philipp Hoehn: Of the pace pace are the constraints surface and pure engage engage in varying degrees of freedom. And then again, fixing and now we want to do the same in the quantum theory. So how do we do that.
Philipp Hoehn: So the first thing to do is we also perform a transformation that is actually generally also maybe find on the physical space.
Philipp Hoehn: That splits the degrees of freedom into pure gauge and drop observer bolts and it does so actually cross the tents of the kinematic intensive factorization to clock in system.
Philipp Hoehn: So this transformation has that form here. You can call it in a way it doesn't angler it's a kinematic of isn't angler
Philipp Hoehn: To be the capital without and what does it do well. For instance, the end moment operators, it actually really. But basically, he was an invariant, and the constraints of sorts. It's transformed purely into the clock factor. You might wonder why it has an epsilon here that's just because
Philipp Hoehn: So that's zero lies in the spectrum of HC minus epsilon because it may be. That's why the total constraint. He has he has their own spectrum HC by itself does not necessarily have to have it. So, epsilon, there's just some eigenvalue off HC
Philipp Hoehn: But what you see is the constraint. Now, and the gauge degrees of freedom only live on the seat attentive factor. And if you look at the clubs are those the relationship circles. They transform. I know that.
Philipp Hoehn: Transformation to purely him to have the system factor. And that's actually a week relations only holds on the physical Herbert space. And what you get is really that the right hand side here. It's just the highs and back operator.
Philipp Hoehn: Now the second step is then the analog of gauge fixing what we do is we just condition on declare class with gauge and on sigma and
Philipp Hoehn: On the frequency sector. So that's as in the page with this formulation. When we get to reduce server space on your system degrees of freedom.
Philipp Hoehn: So when you look at the intermediate states what we're already doing us. We're just getting rid of that pre factor, but there's no physical information inside all the information is only in the system toxin and that's all that survives.
Philipp Hoehn: Now,
Philipp Hoehn: So what. So basically, the total reduction ID parameter ization not this, then the concatenation of that trivialization disentangle our tea with that conditioning here.
Philipp Hoehn: And you can show that it isn't is on the tree at least her sigma sector, and it turns out it's actually, it's it's convertible and
Philipp Hoehn: When you invert it and you conjugate the shock absorber balls with that's a parameter ization map, you just get the
Philipp Hoehn: Heisenberg operators. So you see the relational dribbles just becomes a standard relational Heisenberg picture operators and the states of course they transformed from the physical space to these highs and back state. So we really good. I get a relational picture.
Philipp Hoehn: Now, and let me just give you an overview comparing the classical versus quantum symmetry reduction.
Philipp Hoehn: So the classical structures on left quantum and outs and the rights over the first two, I guess that fairly clear chromatic face base maps is
Philipp Hoehn: A bird space constraints surface for physical space. Now the class. Well, we can do a gauge fixed reduced face basis per frequency sector relative to clock.
Philipp Hoehn: The quantum equivalent to that reduced hybrid space and that I just gave you on the previous slide, the analog. The bottom analog of this canonical
Philipp Hoehn: Transformation splits engage in two degrees of freedom to engage engage in varying degrees of freedom is that
Philipp Hoehn: trivialization or disentangling operation, the gauge fixing condition t equals top prime is given by the conditioning on this clock PM states.
Philipp Hoehn: The gauge fixed of the observer both classically that satisfy these equations of motion or just given by the relation Heisenberg operators.
Philipp Hoehn: And now if you want to do the inverse. So not the quantum analog of gauge fixing
Philipp Hoehn: At the quantum analog of engagement by an extension of gauge fix quantities. Remember, classically, that was just these relational observer goals, but we now also know what the quantum matter lockers
Philipp Hoehn: These FS here in the middle, they're just they're like the quantum gauge fixed observer bolts and now the congregation with these
Philipp Hoehn: Reduction maps, they embed those operators back into the physical here but space and I can convince myself.
Philipp Hoehn: That there are indeed weekly equivalent to the relational deliverables and so does tie sigma in front and the Tita sigma here. They're just projected onto the onto the different frequency sectors.
Philipp Hoehn: OK, so now that we have these different structures that hadn't. We might ask about the relation between direct and produce quantization.
Philipp Hoehn: So he has all these analogous structures in one diagram. So we have the kingdom article face space.
Philipp Hoehn: Constraint and gives the constraints surface than doing these classical reductions a symmetry reduction gives us then a reduced face to face. And then we may do some reduce quantization. By contrast, we can also do a direct monetization of the kinetic hybrid space. And you can imagine.
Philipp Hoehn: Using it came out of a bright space, then there was some group averaging project. I got a physical a workspace and then do our part of symmetry reduction procedure and get some reduce to every
Philipp Hoehn: Now the question is, is a does that diagram commute. And of course the answer is sometimes it does, but not in general. And that is because obviously in general constraining you're contacting don't commute as many people have shown
Philipp Hoehn: So here what you can say, more precisely, is actually or the slogan. I think would be more precisely phrased as, since we don't know what's mature reductionism the quantum theory that cemetery production organization don't
Philipp Hoehn: Um, I think there was a question or if I'm not mistaken.
Okay, maybe not.
Philipp Hoehn: OK, let me move on.
Philipp Hoehn: So what in particular. So in general the reduced quantum series of pain and the two different ways, who are generally the equivalent. And that's mostly expressed in the fact that spectra observer bowls us reduce quantum series may not be the same.
Philipp Hoehn: Okay. And so I just wanted to make that point in order to emphasize that the equivalency of this Trinity is really in terms of quantum the parameter ization and not in general in terms of the retail space based compensation.
Philipp Hoehn: Okay, so now let's also see why the page. What is formulation is actually fully equivalent
Philipp Hoehn: To the direct quantization of relationships variables. And so let's recall we had seen before the conditional state of the system, given that the clock reads Tom as defined in this manner and remember that satisfies a shorthand equation.
Philipp Hoehn: So we can now use that analogy to that caught him cemetery reduction that I told you about before to define another reduction map that we defined in this way so conditioning on the clock states now taking into account that we also have degeneracy sector sigma
Philipp Hoehn: And so
Philipp Hoehn: Basically what we're doing is we're doing the same.
Philipp Hoehn: Procedure. As for the relational highs and back picture. So I'm doing a cemetery reduction but without the intermediate step of this trivialization or this disentangling operation. So we're just skipping that step.
Philipp Hoehn: And then, of course, we get this reduced to a workspace and the state satisfy the shooting and equation. And what you can convince yourself that actually this reduction, not this, what we call the patient with us reduction lap is actually weekly equivalent to the parameter ization
Philipp Hoehn: map that I gave you before this other symmetry reduction of up to a unitary that unitary is just a unitary on the system with that time of Russian
Philipp Hoehn: Parameter top and so then you should not be surprised that if we know also conjugate is relational observable from the physical Hibbert space with us page, what size reduction back there. We just get the shorting a picture observable so
Philipp Hoehn: Here in this picture states evolve and tall and observable don't evolve so we just get the relation shooting I picture should be obvious that it is equivalent to that relational hasn't back picture that we had before.
Philipp Hoehn: So this year is done for the class of systems that we've considered the equivalence
Philipp Hoehn: Given by these mobs page. What does reduction map quantum the parameter ization map and then here between the relationship, getting a picture of it back, which is just a unitary and then while that relational hyphen back picture sometimes equivalent to reduce content creation.
Philipp Hoehn: Okay, just as a side note, so here that Trinity was showing for a while just a single constraint and certain restrictions.
Philipp Hoehn: And it actually turns out that the structures and equipment. This whole way more generally, and can be generalized. In fact, under certain conditions.
Philipp Hoehn: Required reference system so arbitrary locally, contact me groups and so yeah these are much more general structures and hopefully in the near future. You can read more about that.
Philipp Hoehn: Okay, now I'm coming again to the criticisms that could cause raised against the page. What does formalism that I summarized before so we can reap the fruits of that equivalents to actually
Philipp Hoehn: resolve these criticisms completely. So remember the first one was the argue that the conditional probabilities would be incompatible with the constraints, because you have these conditional. I'm sorry. These can imagine
Philipp Hoehn: You know, and they don't come up with the constraints. So they would throw the physical state out of the physical Herbert's
Philipp Hoehn: Now a corollary from the Trinity. Remember that the revelation observer boats are completely equivalent to the reduced observer balls in the shooting a picture. And so here
Philipp Hoehn: Who cares was considering the non degenerate cases here. We can also consider that on the general case. And a corollary from the equivalence of the Trinity is
Philipp Hoehn: That if you evaluate the relational directives are both in the physical enough product. So this is a manifesto engagement very an object if I'm on the physical space. It was exactly equal to the inner product. Sorry that expectation value of the corresponding
Philipp Hoehn: System observable on the reduced Herbert space evaluate and they're shooting on a product relative to these states evolving with respect to the shorting equation.
Philipp Hoehn: And now we can come back to that conditional probability you so here the first line is just what I had before, but now using that observation. This camaraderie. Here you can just rewrite the first line in the following way. So the sorry the
Philipp Hoehn: Numerator and so the numerator has just given by the direct observable corresponding to
Philipp Hoehn: The projection operators on certain outcomes of that observable, but these are also relational backups are both evaluated in the physical product.
Philipp Hoehn: And then you divide again by the physical product. So, this denominator here is just the gauge fixed expression of the physical product.
Philipp Hoehn: So in other words, these conditional probabilities are manifested engagement variant. They're absolutely not in in conflict with the constraints, they're perfectly compatible with the constraints.
Philipp Hoehn: And so this is actually also nice because these conditional probabilities. They provide us know with a nice conditional probability interpretation of relational observable that we didn't have in that form before
Philipp Hoehn: So the upshot from this observation is or this Trinity is really that you should regard the page. What does formalism as quantum analog of a gauge fixed formulation of a manifesto engage in very relational dynamics defined on the physical Herbert space. It really is equivalent
Philipp Hoehn: Now, coming back to the second criticism that could cause race against the page. What does formulas, namely that would lead to wrong propagates us for non relativistic systems.
Philipp Hoehn: So when can now see that actually the way kuqa
Philipp Hoehn: Conditions was actually wrong. So what we can do is now, given that we have this equivalence of the page waters for relational relational Draco observer boats.
Philipp Hoehn: We can just go back to the physical hybrid space define a correct to time conditioning for any observer bolts. A and D on that physical Hibbert space.
Philipp Hoehn: They are encoded in these projection operators on the physical here good space that I'm pointing at here.
Philipp Hoehn: So the pie. A and pie be there, just the projectors, or even keel VM elements, if you wanted on certain outcomes, corresponding to any of the articles and be
Philipp Hoehn: And why you throw them into the physical in our product and you normalize by this factor here. So this is the right conditional
Philipp Hoehn: Probability on the physical Herbert space and that gives you the well what you have on the right hand side. What's the probability that be has outcome little be at cooperating top prime, given that the observable a outcome at a time talk. And now, given our equivalence with
Philipp Hoehn: Page. What is we can translate that whole stuff into the reduced hybrid space. And what we find is this result here. And that is really the correct transition probability in the shooting a picture for general observable.
Philipp Hoehn: But if that's not enough for you. If we can we can restrict that whole
Philipp Hoehn: Construction here now to the non relativistic particle and just set a and b equals to the position operator.
Philipp Hoehn: And set the outcomes be to Q prime and a to queue. And then it just reduces to exactly the right transition probability that we should get. And I just emphasize this is really exact know approximations needed here and it works really for a wide class of 20 constraints.
Philipp Hoehn: At this point, I should mention that they have been previous proposals for getting the transition probabilities right and
Philipp Hoehn: In the page. What does formulas in order to address the glucose criticism most notable one year is really by Rodolfo forehead and have a porter Sebastian Ronan
Philipp Hoehn: And they also work with drops over balls, but they don't use the equivalence that we use, but rather they use a combination of the page waters ideas and dropped on a physical space.
Philipp Hoehn: In contrast, us to integrate all the clock ratings tall and then then you find some relations rockets always been a new clock and then that leads to some coherence and you get the right proper gators
Philipp Hoehn: Only approximately approximately
Philipp Hoehn: And there is another proposal for a resolution that was developed by these guys here on the left and I should emphasize, so they get the rights.
Philipp Hoehn: Propagate us, however, it's strictly to idea clocks and what they do is they have to add additional Insula degrees of freedom for each new sequential measurement for each conditioning.
Philipp Hoehn: And in order to do so, they also have to modify the Hamiltonian constraints and add interactions to the Hamiltonian constraint and your degrees of freedom.
Philipp Hoehn: So it is a very interesting proposal, however, I would argue that by actually modifying the original problem. They can argue that they have actually
Philipp Hoehn: Found a resolution to the original problem. So overall I would still stand by their by our resolution as being the lights resolution of criticism.
Philipp Hoehn: OK, now let's come to the last criticism of cash.
Philipp Hoehn: Flow cash, as I mentioned, showed that if you condition. If you do the paperwork does conditioning and with respect to makovsky time for Gordon systems.
Philipp Hoehn: And you get this expression here website is just a solution to the category equation, no separation into positive negative frequency modes, of course, that does not correspond to any accepted notion of localization for relativistic particle systems.
Philipp Hoehn: And so the resolution of this issue is somewhat more subtle than the other ones. And that's because relativistic particle mechanics away from fear theory as you might know this know exact notion of localization probabilities
Philipp Hoehn: Due to the theorems of elements and Perez Wilder and so the best one can really hope for some approximate notion of localization probability
Philipp Hoehn: And that is, for instance, done in terms of the Wagner localization and now it turns out, so this is somewhat peculiar
Philipp Hoehn: We have shown us that if you condition, instead of on the mccroskey time operator condition on these cold air and clock POV AMS
Philipp Hoehn: Then you actually get a Wagner type localization probability. We're now at separates into positive and negative frequency modes as it should.
Philipp Hoehn: And what you have here the shorting equipment solution to the shorthand equation which is going away function.
Philipp Hoehn: And here I should also emphasize that, in this case, the physical enough product for the sigma modes of corresponds, of course, to this.
Philipp Hoehn: Client Gordon, our product with sigma in front of it and that is actually equal to this kind of shorting and our product for these women are away functions. So this is not an invariant
Philipp Hoehn: It's an approximate notion of localization. It's not actually Lawrence invariant as as well known, but it's often considered sort of the best possible localization notion and relativistic practical dynamics.
Philipp Hoehn: And it's peculiar that one gets that here. So you might wonder, why should you care about this. Well, you should care if you're interested also in in relation observable. It's because
Philipp Hoehn: Relational observable through a trinity equivalence their equivalent to this page with us formulation and if you have some issues for interpretation issues in the in the page. What is formulas and then they would also extrapolate to
Philipp Hoehn: To relational deliverables. But here we see that we get some sort of an acceptable notion of localization.
Philipp Hoehn: Through this this attack organization, also in Dubai for for relational tentacles.
Philipp Hoehn: Alright.
Philipp Hoehn: So,
Philipp Hoehn: Let me now come to the multiple choice problem and so
Philipp Hoehn: You know the problem that we, there may be many different internal time choices and I will just tell you how I might want to address this in this may go under the name of covariance covariance
Philipp Hoehn: And the me just summarize the very basic ideas. So the basic idea is, and it goes back to what I said in the very beginning is to not view the physical state satisfying
Philipp Hoehn: Satisfying Hamiltonian constraint as being some timeless object, but rather as being a clock neutral state. So what I mean by that.
Philipp Hoehn: But it is a description of the physics prior to having chosen temper reference system relative to which we described to remain the dynamics of the remaining degrees of freedom.
Philipp Hoehn: Now you might ask, why is that possible.
Philipp Hoehn: And that interpretation is possible here because there's of course a symmetry constraints and use redundancy in the description of the physical Herbert space.
Philipp Hoehn: In terms of the kinematics data in which you write down physical states, but there's also fact redundancy among the physical degrees of freedom, sort of constraints. They're not all independent
Philipp Hoehn: And so that gives us many different ways in which we can describe the same invariant physical states in terms of the Prince of kinematic operators.
Philipp Hoehn: And the basic idea is that we associate these different ways of describing the same invariance states as being related to different clock choices.
Philipp Hoehn: And as you have seen earlier in the talk. We had these quantum symmetry reduction maps that removed redundancy.
Philipp Hoehn: And then, well we could interpret the results in quantum see or the reduce quantum serious being basically the dynamics relative to a particular choice. So the
Philipp Hoehn: Idea is now that we basically interpret these reduction maps in some sense as quantum Gordon apps that sort of map from this clock neutral description to a given clock perspective.
Philipp Hoehn: Now let me
Philipp Hoehn: just summarize that schematically how that works in practice. So they exist by know various papers.
Philipp Hoehn: Was references I give there in the upper right corner, how that might be done. So suppose you are given a Tony constraint as here on the left. So again, support, we have no interactions and as a clock Hamiltonian wanted another call Hamiltonian to and Sam system.
Philipp Hoehn: And so the basic idea is now that
Philipp Hoehn: Well, we have the clock neutral physical here workspace. The physical hybrid space might split them to different frequency sectors for each clock and
Philipp Hoehn: And so in that case, we actually have to find these reduction maps.
Philipp Hoehn: On each of these frequencies sectors and what we can do is actually map. So for instance, what we see here support them. So, this we can interpret as the reduced
Philipp Hoehn: Herbert space relative to clock see one, so sort of the perspective of klutzy one and then we have to reduce Herbert space relative to the two clocks in tune. So sort of the perspective of proxy to in certain frequency sectors.
Philipp Hoehn: And then the way to map from the perspective of proxy one to proxy to would just be inverting the reduction map sort of as a quantum Gordon map.
Philipp Hoehn: embedding it back into the physical here but space into the relevant sector of it and then concatenate it with a foreword map into the new perspective.
Philipp Hoehn: And so that happens frequently sector why schematically these transformations, they will have that form here on the left.
Philipp Hoehn: And you can of course apply those those two states so states, which is really transforming this manner. So, so these would be states, of course, to and the system relative to clock one they would map and to certain states.
Philipp Hoehn: Relative to cop to but now become the system of one system.
Philipp Hoehn: And you can do the same thing for observable so you can transform observable in this way. So, of course, what I'm getting at is really just a schematic summary of it.
Philipp Hoehn: But the basic idea is that we always describe the same physics that relative to different perspectives and sort of structures.
Philipp Hoehn: On the physical here big space. So this clock neutral here but space are ready to invest in structure. So you can think of them and some analogy to the
Philipp Hoehn: Like Lawrence invariance scanner and special relativity. But then there's this many different ways in which we can write that invariant information relative to different frame choices and this year is sort of quantum analog of that.
Philipp Hoehn: Now this leads to very interesting physical consequences, and I'm just going to throw them here at you if you're interested in knowing more about it. And you can ask me or consult these references to indicate here.
Philipp Hoehn: One thing is so overall, you get
Philipp Hoehn: Certain tempo of frame dependence of clock dependence of the physics.
Philipp Hoehn: You get something that is loosely speaking, something like a quantum relativity of comparing readings of different quantum clocks or synchronizing different quantum clocks.
Philipp Hoehn: So you might say, well, that's maybe not a surprise, it is actually interesting because it's a pure quantum attack that does not arise classically because classically
Philipp Hoehn: The relations between two different clocks, also in terms of relational observer boats, they would correspond to two different kinds of relationships over bolts, but of course the pairings of the different clock readings that will always be the same. It doesn't matter which
Philipp Hoehn: Which clock perspective you choose. And it turns out that this is no longer true and the quantum theory and then has to do with the fact that, of course, now we're dealing with operators and also quantum states have mantra goes on turtle spreads.
Philipp Hoehn: So another interesting feature that appears as that's actually
Philipp Hoehn: An evolution that appears as being temporarily local relative to one clock may appear as a superposition of time revolutions relative to another. And so that's an interesting feature.
Philipp Hoehn: And you can also get some funny Cox alpha inference effects. And then here, there's two papers that I just want to mention they don't use the same framework as I'm summarizing here. But what was shown in this in this
Philipp Hoehn: Reference by Stephen given and then this is that's singularity singularity resolution quantum cosmology may depend on the clock, there was also this recent paper by Christina geezer. The and
Philipp Hoehn: For a show in reduce quantization the enemy confrontation or nomics on different choices or box You could ask other one could maybe embed that into Scott change framework.
Philipp Hoehn: Now let me also emphasize that these clock changes I've just described at four o'clock changes.
Philipp Hoehn: But that this whole quantum covariance scheme is actually much more general you can define that also for spatial quantum reference frames, you can control these references here and let me also emphasize that I know it is possible to be fine or extend that covariance of
Philipp Hoehn: Quantum frame perspectives. So for also spatial ones to arbitrary liquid compact eagles under certain conditions.
Philipp Hoehn: But the basic idea is always that you have sort of quantum reference frame perspective changes as the form of what you might want to call it. Quantum coordinate changes.
Philipp Hoehn: You have what we then call more generally perspective neutral description, that's really just the physical here but space.
Philipp Hoehn: And then you have two different continents symmetry reduction maps sort of as quite accorded maps in different reduce carbon series and then you go from one description to the other always I'm in this composite manner and just analogous to coordinate changes.
Philipp Hoehn: Okay. And that also has some repercussions for the way function of the universe.
Philipp Hoehn: So why might want to develop a new perspective on the way function, the universe.
Philipp Hoehn: Using these insights
Philipp Hoehn: So everyone wants to quantifies gravity. In the end, let's say economically.
Philipp Hoehn: Whatever the physical states are they will be patterns of redundancy in the states and the proposal then has really to to view that way function of the universe.
Philipp Hoehn: As being a perspective neutral quantum state of the universal global description.
Philipp Hoehn: Sort of a global description prior to having made a choice of quantum reference frame relative to you which you want to describe the physics of the remaining degrees of freedom.
Philipp Hoehn: And you could also view that way function of the universe then sort of as a link between all the different internal reference frame perspectives on the universe. And so, that is, for me, it's sort of my personal proposal to render
Philipp Hoehn: One of this concept of a wave function of the universe compatible with colors relational quantum mechanics that maybe some of you might be familiar with.
Philipp Hoehn: So they are the ideas that states are always really relative to some observer and which at first sight seems to be somewhat in conflict with the notion of a function of the universe. To me, this year would be my best answer that I could give to
To to
Philipp Hoehn: Make the two consistent with one another.
Philipp Hoehn: Okay. And here is the final topic so seems maybe so I'm like three minutes overtime, but maybe I will need like five minutes and then I'm finished. That's okay. And so, so far we had already considered
Philipp Hoehn: Global clock. So monotonic clocks and now I just want to give you some hot outlook on what happens when you have something like periodic clocks are non monotonic clocks and was actually a lot of funny stuff happens.
Philipp Hoehn: So this part is a bit less or more speculative than the rest that I've told you
Philipp Hoehn: But there's already some interesting observations and here's the rough summary of that. So suppose you are given some
Philipp Hoehn: Some system was periodic clocks and you want to describe in a relational pollution, with respect to that periodical just as an example I can think of the constraint given by free particle and some harmonic Austin later. And you might want to ask
Philipp Hoehn: Every particle given a certain phase reading of that Harmonix also data, for instance, yes, in the cylinder on the right.
Philipp Hoehn: And of course, in general, when you have periodic clocks, you will get a mouth evaluate this off the relations between evolving degrees of freedom.
Philipp Hoehn: And the clock readings. So in this case, that say if you want to know what's the position of the particle when the clock face this reads five, then you will get an infinite tower of solutions for the answer.
Philipp Hoehn: Now, classically, there's no problem and actually still be finding a global family of relational variables. In that case, what do you do well you just pick that phase observable.
Philipp Hoehn: That is your time observable and you just use the classical notion of winding numbers that you can define and you use that in order to sort of unravel the clock. And so you get a new
Philipp Hoehn: Clock function that goes from minus infinity plus infinity. And that really takes into account all the different cycles off the clock.
Philipp Hoehn: And in this manner when can actually construct single valued relationships doubles and in the usual form and this tall here now runs over all of our
Philipp Hoehn: And there's no problem. Classically, they weekly commute with the Hamiltonian constraints. So they are there are couples.
Philipp Hoehn: But now the really funny part happens so you can contest these objects, but it turns out they are not directly job boards and the contents here. They don't really come up with a constraint and
Philipp Hoehn: So now you might want to ask, Why the hell is that the case and well intuitive reasoning here is as follows.
Philipp Hoehn: You have quantum spreads and for quantum theory. So suppose your, your physical state is one that is quite spread over classical orbits in such a way
Philipp Hoehn: That different clock cycles may actually interfere. If that's the case, then you have absolutely no notion of whining numbers and the full quantum theory and he's whining, the numbers weren't necessary in order to define single value of related
Philipp Hoehn: And so you cannot have such an emotion in the full quantum theory.
Philipp Hoehn: However, of course, we have the classical relational dynamics and the classical relational direct observable bowls. And in some sense,
Philipp Hoehn: There is no very non trivial interplay of the classical quantum relational dynamics. And so there's a striking
Philipp Hoehn: qualitative difference. But clearly, there ought to be some semi classical regime in which one talks to the other. And somehow we have to read
Philipp Hoehn: Recover these classical rock observer bolts from these quantum objects. And so this is that completely finished business yet.
Philipp Hoehn: But in some sense these objects. Well, not for quantum direct observable. They have to be in some sense de de Pena directives are balls.
Philipp Hoehn: That is in some semi classical regime, they have to become at least approximate your outcomes and goals. So that, for instance, you might end disaster. That's the expectation value in some semi classical State's office communicator, or to vanish at least at leading order.
Philipp Hoehn: Okay, so that's quite interesting. But now this is actually very much
Philipp Hoehn: Compatible with previous observations that we have made when dealing with quantum in classical relational dynamics.
Philipp Hoehn: In situations when you don't have mobile clocks. And in fact, there's very non trivial features for
Philipp Hoehn: For the semi classical regime and semi classical room does not even always exist, depending on how your contacts.
Philipp Hoehn: So it's just a very rough summary of something that we did various years ago with Bianca Kozlowski and Mike Nelson was a student of mine. So there we considered some semi interoperable models.
Philipp Hoehn: Were basically we're kind of compact to five dynamics on some tourists and in such a way that the momentum our international relations with one another. And that gives you
Philipp Hoehn: Not only monetary value in this off relations between evolving degrees of freedom and whatever you choose either o'clock, so it actually gives you density many
Philipp Hoehn: Places relations. And so in that case we showed that there does not exist any semi classical limit at all if you if you contrast that model using standard quantization techniques, but then
Philipp Hoehn: Actually Bianca came around and said, Well, wait a minute. We actually conversation is really about finding the quantum representation of your observable. So what we should be doing is adapt a matter of quantization.
Philipp Hoehn: To the relational variables that we might want to represent in the in the four columns here and classically at least implicitly these relational problems exist.
Philipp Hoehn: And so then that led to modifying the topology underlying the conversation. And it was really part of my quantization that we are news.
Philipp Hoehn: And as part of my quantization and save the winding numbers in the quantum theory and he might just asked whether polymer quantization might also resolve that issue that I have described on the previous transparency, so maybe Pokemon compositional come to the rescue here as well. Nevertheless,
Philipp Hoehn: In general, even if you might be able to construct such relational observable.
Philipp Hoehn: Even a parliament colonization, they might become horribly complicated objects and intractable to to deal with.
Philipp Hoehn: And that alone. If you have something like not inevitable systems. Maybe they will become even really practically possible to work with. So the question is, what can we do in that case.
Philipp Hoehn: What kind of interpretation of the dynamics, might we haven't in such a circumstance. And so here the best possible proposal that I could come up with. And that also sort of what we analyzed back then is
Philipp Hoehn: We might just adopt something that we might want to call an S matrix interpretation of relational dynamics. So, just like in
Philipp Hoehn: Why is the matrix interesting quantum field theory or we just have to deal with as I'm talking states and we don't have any mathematically tractable formulation of the interactions.
Philipp Hoehn: That happen in the interaction region in terms of observer bolts. So here
Philipp Hoehn: In some rough analogy. We don't have a mathematically tractable formulation in terms of observable or operates us that describe a relational dynamics.
Philipp Hoehn: We could then take a step back and at least work with kinematics will
Philipp Hoehn: States corresponding to certain kinematic observable is reading certain values and then just throw the physical projector on that sort of like the S matrix and then ask for the physical transition aptitudes
Philipp Hoehn: Then say that's our most elementary way of interpreting the dynamics, given some initial clock reading and some observable at that time. What's the probability that we find a certain other
Philipp Hoehn: Observable reading at some other o'clock time
Philipp Hoehn: Okay, so that's just speculation at this point it's not completely clear that it can be done better. But maybe just what we have to deal with. And so, yeah. The question is whether maybe
Philipp Hoehn: That if we want to get an ocean of dynamics from for quantum theory of gravity might have to resort to transition
Philipp Hoehn: So if I speak to people working on spin for models and I might be preaching to the converted, but I still think it's, it's important to appreciate that also from a point of view of relational
Philipp Hoehn: Okay, so that's really all I wanted to say. So I'm so as a conclusion that has by now some some updates on certain facets of the problem of time, some of which are resolved others of which
Philipp Hoehn: Are resolved, but there's maybe some new insights we have gained that might help us.
Philipp Hoehn: To further develop your understanding of dynamics in quantum gravity. Thank you.
Jorge Pullin: Questions.
Carlo Rovelli: I have them question in a comment, if I may.
Carlo Rovelli: Just go ahead
Abhay Ashtekar: Go ahead. God
Philipp Hoehn: I don't know if there was a, um,
Carlo Rovelli: Yeah, so it's a it's a general comment and then a specific question very much connected to your penultimate slide. So the general comment is that
Carlo Rovelli: I think that
Carlo Rovelli: You've done an excellent work, Philip here. I mean, this is a very remarkable because there's all this literature around this old literature, which is a was indeed very confusing and with some statements which were correct. And I think if cleaned up.
Carlo Rovelli: All these values.
Carlo Rovelli: Perspective, how they go together. And I would say I agree with everything you said. I mean, I think it's completely convincing, both in the technical part and in the in the general interpretation part
Carlo Rovelli: Second, also very briefly, I want to mention something because we had a discussion at some point you you told me, but but but you dislike
Carlo Rovelli: Page water you disagree with page water, I agree with everything you said about page water. I think it's a it's a it clarifies entirely the situation.
Carlo Rovelli: When I disagree about is the fact that they in a paper present their solution us ether. The solution progress time requires quantum theory. So there's no time in classical GR and magically appear thanks to this mechanism that's, I think, is wrong.
Carlo Rovelli: What is right is
Philipp Hoehn: That's what
Carlo Rovelli: I did, yeah. What you say is, right, is what you said exactly what you said, so they they have the quantum analog of the classical solution of the problem now that
Philipp Hoehn: I actually have something more to say on exactly that point, I don't know if you have another question.
Carlo Rovelli: Yeah, it's a third comment. And this is just a comment, but I'm interested to know what you think and last point and and this is more of a question is related to your penultimate slide.
Carlo Rovelli: UP THE LIGHT OF EVERYTHING WE HAVE DONE. I think that
Carlo Rovelli: In a sense, there's a lot of junk to throw away. I mean, we have an overall picture.
Carlo Rovelli: And the idea of going to the hubris to kinematic or here with space to have observable Salvadoran observers and they can magically over space and then he has these operators project. There is not a projector, whatever you want to call it.
Carlo Rovelli: That defines
Carlo Rovelli: Position amplitude between those
Carlo Rovelli: Seams. This is the way, by the way, my, my book is written the first one. Now, and the book of Francesca is also written and also this is a way that has been widely generalized and clarified by Urkel Robert lyrical, I don't know if you've looked at his
Carlo Rovelli: Abstract it doesn't go at all in the details of what you do.
Carlo Rovelli: It seems to me the right direction where to go, in spite of the fact that the details are not clear yet in a number of things. Because, and here I close
Carlo Rovelli: It seems to me that a lot of the confusion come from mixing up different meanings of what we call time
Carlo Rovelli: We have a psychological sense of time and entropic sense of time and microscopic notion of time which player or
Carlo Rovelli: A summation in macroscopic physics in other contexts. And if we try to interpret fundamental physics in terms of those we get confused. So what fundamental physics gets you is this relational probabilities or less general and
Carlo Rovelli: The values way as you have described the of computing them a clear ended a sufficient seems to me.
Carlo Rovelli: That's it, yeah.
Philipp Hoehn: So, um,
Philipp Hoehn: So concerning
Philipp Hoehn: Concerning your statement on the page. What does formulation and people, they're often what they actually cite the slogan time from entanglement.
Philipp Hoehn: And that is actually misleading that statement. So I actually had a slide prepared for actually debunking that claim. I have thrown it out because I thought I would go over time.
Philipp Hoehn: But now let me just summarize what my answer to that is so, indeed, if you say time from entanglement and you see that really in a lot of the literature and kind of foundations and
Philipp Hoehn: Quantum Information on the problem of time even slogans like the problem of time really disappears. Once you look at it through the lens of of entanglement, which was total nonsense. So, so here
Philipp Hoehn: One thing that one has to be very careful with and that's actually also here. So this is why I pulled up the slide here.
Philipp Hoehn: So the notion of entanglement in the page. What is formalism. So, it is true that the physical state between clock and system is in some sense and entangled states between cocking system, but really only in terms of the kinematic attempts of factorization. So it is not in any way a physical
Philipp Hoehn: Physical or engage in Barrett notion of entanglement and that is because that's kinematic attentive factorization is in fact not inherited by the physical Herbert space.
Philipp Hoehn: So in the physical Hibbert space, you would want to have a notion of entanglement in terms of commuting
Philipp Hoehn: Some algebra. Algebra of relation of drug observer goals which are independent, but you don't find anything like this in terms of
Philipp Hoehn: Of sub algebra. Some of those associated with clock and the system that are invariance, so you don't get any so there's no way in which you could ever test this.
Philipp Hoehn: This notion of entanglement that they talk about in terms of gauging Baron observable. Nevertheless, it is true that, of course, the kinematic level, you could think of it in some sense as a kind of magical entanglement that another
Philipp Hoehn: Point that I wanted to make us. And this is also I call that operation to disentangle I put it in quotation marks, because it is, it isn't angler at the kinematic lever level.
Philipp Hoehn: So with respect to the kinematic attempts and factorization. So what you see here what it really does is
Philipp Hoehn: It does untangles quotation marks the gate pure gauge degrees of freedom from your observer bowls, it puts pure gauge into one terms of factor one canonical attempts to factor the pure gauge in various parts into the into the other types of factor.
Philipp Hoehn: And so, but what you see as the observer boats are then pure product observer bowls and in fact what you also see us these
Philipp Hoehn: disentangled states doesn't take a physical states they contain the same information as the physical states. There was also actually disentangled so there's some over degeneracy sectors, but that's actually super selection. So they are like, like, in fact,
Philipp Hoehn: Equivalent to mix states over the, over the degenerate over the degeneracy sentence.
Philipp Hoehn: But in any case, um, what you can see from this construction and that's actually also explained in our paper, you can get the exact the same relational quantum dynamics, but by starting out from
Philipp Hoehn: Actually quantization a different the same classical system that with respect to a different set of degrees of freedom. And so there you get
Philipp Hoehn: Exact same quantum dynamics without any magical entanglements. And so in that sense, one has to be really careful with the notion of entanglement and that resolution.
Philipp Hoehn: So overall, when has to take it with a grain of salt. And I would say overall it's really fascinating statement now coming to your last comment.
Philipp Hoehn: About these transition aptitudes it's actually true. I should have sides. It's all about a year if all but if you're here, I sincerely apologize. I was in a bit of a rush to finish the slides, but it's obviously true that robot has really developed a whole
Philipp Hoehn: Mathematically extremely well developed framework for talking about a transition amplitude in a very general way so
Philipp Hoehn: That I think is a very, very, very useful framework. So I forgot your other comments in that regards
Carlo Rovelli: So I don't know if there was any girl know all praises praises not not had a comment. So thank you.
Abhay Ashtekar: So I also wanted to say that these are by I also understand that. Yeah. Did this clarity in the whole picture is really very nice and, in particular, I very much have had the same view that, in fact, the the
Abhay Ashtekar: The framework that we have with just our constraints on them. That is not so much timeless framework, but it's really time neutral block neutral or perspective neutral framework. So it's really completely agrees with what but
Abhay Ashtekar: How I think about it, but I still am confused about this issue about the key hack and not being commuting
Abhay Ashtekar: And how much it does not trickle or does not trickle in particular when you had two different clocks and you are you are comparing them and you are lifting
Abhay Ashtekar: The in order compared the description, you're lifting. So there's a questions from one clock frame to the drop neutral frame and then descending again.
Abhay Ashtekar: In that case,
Abhay Ashtekar: I'm confused about things not being self and john not affecting your detailed description.
Abhay Ashtekar: And I can see why I'm asking that question later if you if you want
Philipp Hoehn: Yeah, so I'm sorry you're asking about non commuting
Philipp Hoehn: Copy of the sunset. So it like two different clock PM set on computers that what you're asking for.
Abhay Ashtekar: So the fact that
Abhay Ashtekar: Slide. I think 2526 or something like that us off with nicely pointed out that the way
Philipp Hoehn: I don't know which slides. Those are because there are a number for me.
Abhay Ashtekar: Number for you. Okay, so anyway, when you're talking on this operative and positive a positive attitude measures, you said that, well, but there's a price to pay. Everything seems nice. But there's a price to pay the price to pay, was about was precisely that.
Abhay Ashtekar: That this this this t hat and operators. Yeah, exactly. And that not self a joint and he has not fallen let cetera and I don't know how much of this then affects what you said. Later on, or whether you sort of glossed over this fact and just treat them as if they were Sanford joined
Abhay Ashtekar: In particular, that you're comparing two different clocks.
Philipp Hoehn: Yeah, so I mean that is very good question. So
Philipp Hoehn: So let me come to the so I guess you're asking what's the consequence. I think that was the question you also raised during the talk. So basically what's the consequence for these relational deliverables here, right, or the
Abhay Ashtekar: question I was asking.
Philipp Hoehn: Yeah, sorry. Yeah, sorry, I forgot to answer that. So, um, yeah. So it's a very good question. It's actually one that I tried to prove
Philipp Hoehn: purely physical hybrid space level that these objects at the very least symmetric. Then I was not able to fully do that there's some subtleties that may arise.
Philipp Hoehn: However, there is now something I'm sort of course you know when you have these these these equivalence maps between the physical space and the producers space. They are, of course, at this stage, somewhat formal here. I mean,
Philipp Hoehn: I mean, we made it for for some fairly why class of systems. But in principle, the thing is, since one has
Philipp Hoehn: Since one has these
Philipp Hoehn: These equivalencies. Am I can at least at a formal level say that if these reduced observable actually Salford joint with respect to the reduced in our product.
Philipp Hoehn: That's why the the relational observable should somehow also be south of China near symmetric with respect to the physical product but I agree that at this stage, it's a, you know,
Abhay Ashtekar: To me, that has been a stumbling block and, in particular, this becomes to forefront. When you consider two different clocks and try to prove his equivalence or covariance of the relations. I had exactly the same picture that you had
Abhay Ashtekar: I was thinking in terms of
Abhay Ashtekar: Energy momentum in space it on TV it becoming energy and momentum, you know, in one frame and mixing in the other frame and so on so forth.
Philipp Hoehn: So just come out. Yeah. So I mean, I agree. So, I mean, I agree that what I'm saying here is a bit formal but
Philipp Hoehn: Somehow they should be equivalent. But one other thing I wanted to say about this is
Philipp Hoehn: That, as I mentioned already, before you know the statement that we don't necessarily need to think of observable and Salford drone operators.
Philipp Hoehn: I would argue that in principle or also be fine if we can at least have some interpretation of the relational direct observable, maybe not as alpha john operators, but themselves as being expressed in terms of some notion of pure VM for me, that would still be a
Abhay Ashtekar: Much more foundational quantum mechanics. If you can do that, yes, I agree that that is really what a change in quantum mechanics itself in which one is replacing Selfridge operators, but a
Abhay Ashtekar: Lot of
Abhay Ashtekar: Positive revalued so
Every morning
Philipp Hoehn: I mean this is quite standard in quantum metrology and quantum information so they would not not be at all worried about the fact that you have an observable that's defined in terms of appeal VM, but it's not sell for joining. So it's just a
Abhay Ashtekar: Symmetric right necessarily so.
Philipp Hoehn: And while these pure VM is here and they actually symmetric, these, these
Philipp Hoehn: Weights. So these T operators here. These objects are actually symmetric, but they're not necessarily yourself.
Philipp Hoehn: Okay, so, um, but now coming. I don't know if you had another question on blockchain. I
Abhay Ashtekar: Just wanted to say that in the case of this comparison to clocks, a long time ago does
Abhay Ashtekar: Date and we looked at one specific case.
Abhay Ashtekar: Yeah, we could do one one little calculation, which is exactly the same. And we could not do it for others, precisely because of the season that we could not an ambiguous, the lift observable from one clock to though.
Philipp Hoehn: Yeah, so, so, I mean, actually. Yeah, you pointed out that paper to me in Valencia. Last year I had a look at it.
Philipp Hoehn: I would still think that one could actually embed it here in this framework that we have. So the thing is that so what they're what you did is you basically on the
Philipp Hoehn: Clock Hubert space, you use the different partitioning for the different choices of cotton. That's why they didn't come up with. So here in the construction that we have us. We have of course used
Philipp Hoehn: Clock of the two different choices which commute on the magical hybrid space. So that's a different situation and you consider it back then, but I would still think that priority. I don't see a reason why you wouldn't be able to just embed
Philipp Hoehn: This this construction that you made with class and and agitates also in this language.
Abhay Ashtekar: Okay, thank you.
Philipp Hoehn: Okay, can I ask a question for the short term here.
psingh: So, so your whole talk. First of all, it's a very beautiful somebody like but the whole talk is Zoom's essentially that the
psingh: Block and the system degrees of freedom are not interacting. So, do you have some thoughts on this trinity equivalence. When this is not true when the clock and the system are interacting. For example, if you have a massive scale or field you assume the masters one
Philipp Hoehn: So yeah, so that's indeed. I mean, so actually in this. I mean, I mentioned this sort of in passing that problem, the very beginning, but I but indeed you're putting your finger into the ones. So why it's so here
Philipp Hoehn: Yeah, I'm a slide of the global problem of time. So here, this is exactly the case you're referring to. So there's a model. So the clothes clean my universe with a massive scanner theaters model that we had
Philipp Hoehn: Studied with makeup lover and elton john many years ago. And obviously, there's an interaction. And in fact, it's even worse, it's a non editable model against chaos there and so
Philipp Hoehn: Yeah, so here the Trinity, I would have no idea of how to extend it to that. In fact, I would probably argue using standard quantization. There's no way of doing it. Not even principle. And that goes back to, and now should really run back to the other end of the talk to the penultimate slide.
Philipp Hoehn: Sorry about that.
Your
Philipp Hoehn: Slides. So here you have referring to that same paper. So we showed back then. And that's cosmological models we use an effective framework to describe effective relationships eyeballs.
Philipp Hoehn: And so there. And then we also develop the notion of clock changes.
Philipp Hoehn: And what we found there is you can actually only do these changes and sufficiently Pete semi classical states, but really only away from the classical turning region off the off the scale factor.
Philipp Hoehn: So it turns out that the that the turning region off the scale factor or the region of maximum expansion is actually a region with a chaotic scattering
Philipp Hoehn: And so in that region generic classical solutions will have a turning points, both in the scanner field and the scale factor on all scales of the face face.
Philipp Hoehn: And if you then use it just a standard or sort of reduce the width type quantization. There's absolutely no way that you can
Philipp Hoehn: encode these different turning point, all these different
Philipp Hoehn: scales on the face space in your quantum Syria just will be oblivious to it. And I think that's also the reason why there has been no full quantization of that model in terms of the
Philipp Hoehn: Of the related with methods. However, no one could ask, well, what if we do polymer quantization of that model because
Philipp Hoehn: Using a much simpler case, namely the semi interoperable model that I was referring to you in the work was back I did fish Mike Nelson and Tim kozlovsky so there we had something similar.
Philipp Hoehn: Of course, in a much more benign context but there we could sort of rescue the dynamics.
Philipp Hoehn: In
Philipp Hoehn: In terms of polymer quantization.
Philipp Hoehn: Because the polymer colonization, of course.
Philipp Hoehn: It has a much much finer scale. So basically any configuration space point is
Philipp Hoehn: Like an open set. And so that might give you a chance to in the quantum theory. Then also disentangle this this this chaotic scattering that gets in the
Philipp Hoehn: In the, in the, in the call, sweet my universe with a massive scale or field.
Philipp Hoehn: It will, of course, probably horribly complicated and I guess you guys will know much better than email to do that. And now to see, but maybe there is some chance of dealing with that.
Philipp Hoehn: And. However, in practice, I think these observers and that model will probably still be intractable, even if you could write them down maybe implicitly or something like this.
Philipp Hoehn: And that's also especially that model is actually for me. Also, one of the reasons why I'm thinking now in terms of relational dynamics.
Philipp Hoehn: In in situations where you have a very complex dynamics, maybe more in terms of these transition attitudes and this is matrix interpretation.
Philipp Hoehn: But of course, this would be, you know, something that departs from that Trinity picture that we have in this Trinity picture.
Philipp Hoehn: You know this works very nicely. In that case, when the, when you when you have these globally monotonic clocks and maybe it works sort of locally when you
Philipp Hoehn: When you have non global clocks, but they are not too badly behaved. But I think when they're too complex than something like the Trinity may at best work semi classic view and chaotic dynamics, maybe not at all.
Abhay Ashtekar: But but but Philip I mean this paper that is an adage that you are done right in between two turning points.
Abhay Ashtekar: One is the
Abhay Ashtekar: Clock and one could do everything that
Abhay Ashtekar: Memory of your paper. Is that correct, I
psingh: Think in some
psingh: Insight in some projects you're right away, like, one can do but but Philip I had my
Abhay Ashtekar: related question is what what our ships on analysis. I mean, I did not.
Philipp Hoehn: Know, so in that no, sorry, are you referring to the paper with Monica and
Abhay Ashtekar: Without it, right.
Philipp Hoehn: Ya know, so I mean what we showed there is that in some, you know, when the state is nicely peaks in that region. You can do it. But what we also showed us that what much more generic situations.
Philipp Hoehn: Even if the state is initially semi classical it starts spreading out too quickly apart.
Philipp Hoehn: For this change matter to even be pack practically applicable in between various different turning points, of course, the general idea was
Philipp Hoehn: That we try to resolve the global problem of time by just, you know, evolved relative to one clock and just before it starts turning badly switch to another clock and for sort of relatively benign cases of the global time problem. It works. Yeah.
Abhay Ashtekar: But yeah
Philipp Hoehn: But, but in the butt in these really complex dynamics when you have this chaotic scattering in a standard related with quantization. And what we use back then was an effective related conversation then it becomes practically and tractable.
Abhay Ashtekar: So, but, um, you're saying something.
psingh: Yeah but Philip like am I related point was that wasn't there a paper I think recently by am a day in which he showed that if you have, I think he considered probably couple of examples of this.
psingh: Interacting picture and he showed that in fact, you get a very time non local modify shortening that equation and
psingh: Then I'm just for it. What will happen to your s metrics description and transition amplitude picture. Yeah.
Philipp Hoehn: So actually, there was another yeah that's a good point. So it was done was a paper by Alex Smith and media Hammadi so collaborators of mine, indeed. So what they have shown us that if you include interactions in the page. What's ours on
Philipp Hoehn: formula. The formula isn't done that, you get a modified shorting dynamics and it was a non local colonel. That's true. So the question is, however, in that case, um, how would you actually interpret that in terms of the relational dynamics.
Philipp Hoehn: Right. So in that case, I think it is something that works. Formally, and it's something
Philipp Hoehn: You can also. And this is actually part of this project that we're currently doing also on periodic clocks. For instance, you can do page with a square periodic clocks and you can condition.
Philipp Hoehn: Just on some clock reading. But then the question is, how do you define how do you interpret that dynamics and the full quantum theory, it's really a prayer not clear at all because when you when you condition on a on a clock reading some, you know,
Philipp Hoehn: Some angle so that the face variable read some angle and, you know, in the case of the of the
Philipp Hoehn: free particle and the the harmonic oscillator, you would get a horrible a multi valued in this of the of the relation between the system and the clock.
Philipp Hoehn: And so this is something that will formally, you can do that conditioning interpretation of that relational dynamics is actually far from clear and I think
Philipp Hoehn: In this case of the interaction. It's true. You get some some modify trigger dynamics, but to me the interpretation of that in terms of some sort of single value. It's relational dynamics is not clear to me at all.
psingh: Okay, thank you. Phillip
Ivan Agullo: I think the Jorge had to leave and I am the new boss. So is there any, any other question.
Ivan Agullo: If not, let's thank fairly big game for these wonderful docs.
Ivan Agullo: Thank you.
Philipp Hoehn: Thank you.