Jorge Pullin: Okay, so it's bigger than this Marcin keselowski who will speak about a new approach to homogeneous isotopic sector quantum gravity.
Marcin Kisielowski: Thank you, thank you.
Marcin Kisielowski: In my today's talk i'll talk about my my paper from 2020 where I propose the new approach to studying extracting homogeneous is a Tropic sector of quantum gravity.
Marcin Kisielowski: Let me start with a brief motivation and my motivation comes basically from the two papers.
Marcin Kisielowski: Were promising candidates for the models of quantum gravity have been proposed in the canonical quantum gravity one is a model of quantum gravity coupled to rotational dust and the second one is about quantum gravity coupled to.
Marcin Kisielowski: Muslim scholar field and.
Marcin Kisielowski: My motivation is.
Marcin Kisielowski: I think well described by the following quote from one of the papers, with a.
Marcin Kisielowski: Strong title gravity enticed.
Marcin Kisielowski: In in the loop quantum cosmology models of the homogeneous mass the scholar field coupled to gravity.
Marcin Kisielowski: Big Bang turnouts turns out to be replaced by a big bands, as a result of the quantum gravity affects now.
Marcin Kisielowski: With our model we can consider the same system of fields, from the point of view of a default theory without the symmetry reduction.
Marcin Kisielowski: Similarly, we can also consider the quantum gravitational collapse quantum black holes and theory entropy all those cases are manageable within our model and the only difficulties of a technical nature.
Marcin Kisielowski: and
Marcin Kisielowski: So I wanted to talk today about this technical nature, simplifying this technical problems which arise when we when we try to study the symmetry reduced systems in the full theory.
Marcin Kisielowski: I I focus on on homogeneous is a Tropic sector, but I believe my ideas can be applied to also.
Marcin Kisielowski: quantum black holes and.
Marcin Kisielowski: what's the difficulty.
Marcin Kisielowski: In those two models, the matter fields play they're all of.
Marcin Kisielowski: dynamically coupled observer with respect to which the dynamics happens and the dynamics is formulated and then.
Marcin Kisielowski: add a panorama authorization technique is used to map.
Marcin Kisielowski: Fully constraints system to system with a true non vanishing hamiltonian and, in fact, what is.
Marcin Kisielowski: What needs to be calculated is an evolution operator and I view this problem as a diagonal ization problem.
Marcin Kisielowski: Therefore, I will focus on the on the on the problem of day they're gonna lies in.
Marcin Kisielowski: The operators and I wanted to talk about.
Marcin Kisielowski: Symmetry reduction today as kind of technical tool at the quantum level.
Marcin Kisielowski: which can be used to simplify the problem.
Marcin Kisielowski: And I will focus on a class of physical hamiltonian so which, for me, seem to be technically simpler aware by technically, I mean.
Marcin Kisielowski: When when when you want to diagonal is some mattresses after introducing some kind of on a computer.
Marcin Kisielowski: It seems to me that this class of hamiltonian of this.
Marcin Kisielowski: is best for start.
Marcin Kisielowski: After after this short motivation i'll go to symmetric states on the lattice I think my ideas are best described by studying.
Marcin Kisielowski: A case of a cubic lattice i'll consider this cubic lattice on a three terrorists to make to make it more regular.
Marcin Kisielowski: Then.
Marcin Kisielowski: I will pass to symmetric states on the lattice with loops it's because there's color constraint operator and physical hamiltonian is a function of his color is constrained operator and add and subtract loops.
Marcin Kisielowski: These loops are tangential to do two different links and.
Marcin Kisielowski: In fact, I will focus on in various spaces of this operator in various spaces, which in my opinion can be considered to be homogeneous isotopic.
Marcin Kisielowski: This scholar comes the southern conference spaces is is important is useful in the diagonal ization problem, because you can restrict this operator to the environment, space in particular and eigenvector is an element of.
Marcin Kisielowski: of an off some invariance space.
Marcin Kisielowski: We this operator is defined on the vertex kilobits a space.
Marcin Kisielowski: which I will introduce.
Marcin Kisielowski: In the third section.
Marcin Kisielowski: that's why I chose this ordering first i'll introduce the hilbert space, then the operator and then the metrics stage homogeneous is a Tropic states on a lattice with loops and, at the end i'll summarize.
Marcin Kisielowski: As I said, I consider the spaceman default to be the three terrorists.
Marcin Kisielowski: And the three terrorists are, if you as a caution space, I take our three and quotient it by the group subgroup of the group of translations this sub group is labeled by a parameter real parameter a real positive parameter a.
Marcin Kisielowski: And it is formed of translations of this for far more P P Q R ar ar integers.
Marcin Kisielowski: So this three terrorists can be viewed, for example as as a cube.
Marcin Kisielowski: let's have a look at the the biggest cube here in this picture, with appropriate faces I identified so the Left face and the right face that up and bottom and the front and back.
Marcin Kisielowski: Now.
Marcin Kisielowski: we introduce a lattice.
Marcin Kisielowski: here on this picture we either pick that a lattice where I take this epsilon equal one half.
Marcin Kisielowski: So I split each segment of the lightest into two parts.
Marcin Kisielowski: And when we quotient, so this is a part of our three, but this lattice can be extended to how are three when we questioned it and we obtain a lattice on a three tours.
Marcin Kisielowski: And this lattice as a symmetry grew up.
Marcin Kisielowski: This symmetry group is a semi direct product of have finished subgroup of a group of rotations, this is a group of orientation preserving symmetries of acute.
Marcin Kisielowski: And times kind of translations, but this translations are in a sense, cyclic they are adjusted to the terrorists, so we consider translations in our three and question did.
Marcin Kisielowski: To get translations on at RS.
Marcin Kisielowski: Now, an important concept is what I call homogeneous is a Tropic spin network.
Marcin Kisielowski: So I call a spin at work homogeneous is a Tropic if it has the furrowing transformation property with respect to the default morpheus.
Marcin Kisielowski: corresponding to the symmetry of the lattice so when acting on the spin network it.
Marcin Kisielowski: It leaves this state invariant possibly changing the face of the way function.
Marcin Kisielowski: And I will assume that this face is one dimensional unitary representation of.
Marcin Kisielowski: The symmetry group.
Marcin Kisielowski: Far homogeneous is the tropics been networks in follows in particular that all spins are set to be equal to some number J.
Marcin Kisielowski: So this is a kind of assumption.
Marcin Kisielowski: starting point as assumption, this is a restriction of the States.
Marcin Kisielowski: We see that it already puts the oldest spins equal to some given spin.
Marcin Kisielowski: But that's not the end of the story.
Abhay Vasant Ashtekar: Can I just ask a question for the last one.
Marcin Kisielowski: Yes, please.
yeah.
Abhay Vasant Ashtekar: So the face depends on the state as well fi some s.
Marcin Kisielowski: Yes.
Marcin Kisielowski: Yes, at the end of the day, I will need just dependence on on this number Jay.
Abhay Vasant Ashtekar: I say.
Abhay Vasant Ashtekar: But, to begin with you're saying that it depends on this allowed to it.
Abhay Vasant Ashtekar: So it's not a mean yes or something i'm in the sense of if I if I add as well.
Abhay Vasant Ashtekar: Right okay it's just not a legal representation good.
Yes, yes you're right Thank you.
Marcin Kisielowski: and
Marcin Kisielowski: When you look at the action of the default murphy's miss on on an intertwined.
Marcin Kisielowski: And this default murphy's may have already orientation preserving symmetries of a queue up and.
Marcin Kisielowski: let's look at the links meeting at the origin of our coordinate system.
Marcin Kisielowski: And it permeates the links.
Marcin Kisielowski: And this leads to a permutation of the indices of an entertainer because an index is assigned to Ellen.
Marcin Kisielowski: And I make a projection by appropriate group averaging.
Marcin Kisielowski: Where.
Marcin Kisielowski: I use this face to make the proper dependence, I stated before, and the symmetric states will be the image of this projection operator.
Marcin Kisielowski: and homogeneous is a Tropic spin networks will have all notes labeled with the same intertwine or which is symmetric.
Marcin Kisielowski: This symmetry reduction.
Marcin Kisielowski: is based on this choice of the face honestly speaking first of all, I chose it to be equal to one.
Marcin Kisielowski: But as soon, I discovered that.
Marcin Kisielowski: Although intuitively living spiritually coherent entertainers at first sight, should be in the space, they were not is because of.
Marcin Kisielowski: transformation properties of the paradigm of coherence states with when a rotation is applied, because when you rotate.
Marcin Kisielowski: The state with as you do metrics.
Marcin Kisielowski: it's not the same as rotating the normals with SL three metrics, but there is also a face, so when you multiply the phases in this particular case, you get the following a choice in order to include living speciality coherent intertwined enters.
Marcin Kisielowski: The group of orientation preserving symmetries is a morphic to the group of permutations of the four diagonals of the cube.
Marcin Kisielowski: And so we then out by side, no sign of G the design of the corresponding permutation of as far.
Marcin Kisielowski: And we take the face to be equal sign to the power to Jay and with this choice, there is always.
Marcin Kisielowski: A living spiritually coherent intertwine are in this space and it corresponds to a cube there is actually a second one which corresponds to a cube with the opposite orientation.
Marcin Kisielowski: If you had you chose.
Marcin Kisielowski: The face to be called one year would be a in the art in orthogonal space and you would explode delivery and speciality of your entertainers.
Marcin Kisielowski: let's have a look at the dimension of their reduced space I calculated it up to.
Marcin Kisielowski: spin equal 45 half's.
Marcin Kisielowski: And we see that the two girls polynomial Lee with the spin.
Marcin Kisielowski: At least in the large spin limit another interesting point in the slot is that for J equal to one half the dimension of the space is equal to, and.
Marcin Kisielowski: The tool, leaving speciality coherent intertwined there's I talked here about here are linearly independent and then they span the space.
Marcin Kisielowski: So first been equal to one half the space is completely described by the living, especially for your entertainers.
Marcin Kisielowski: And in order to study how much this cemetery reduction reduces the dimension of the space of impact whiners I I studied.
Marcin Kisielowski: This ratio, which tells us how many times there's the symmetric space, the space of homogeneous is a Tropic intertwined nurse is smaller than the full space.
Marcin Kisielowski: We see that, with increasing spin this reduction right gross.
Marcin Kisielowski: and
Marcin Kisielowski: I believe.
Marcin Kisielowski: There is a chance that it stabilizes at the devalue 24 I calculated only up to such spins because.
Marcin Kisielowski: I did calculation on a single note of it, of the computer cluster and.
Marcin Kisielowski: This is how far I could get in in in a week or two.
Marcin Kisielowski: But this is not this, this was not the focus of my research, so I didn't.
Marcin Kisielowski: See what's happening later.
Marcin Kisielowski: Let me now apply this ideas.
Marcin Kisielowski: To the to the scholar constraint operators, first let me recall some definitions of the vertex Gilbert space.
Marcin Kisielowski: And it is.
Marcin Kisielowski: well known here what is kinematic hilbert space of the quantum gravity, it is the akashic completion of the space of cylindrical functions.
Marcin Kisielowski: And the by so gamma I will consider the space of function cylindrical with respect to graph gamma and the advice he tell the gamma it's a cushy completion.
Marcin Kisielowski: The economical hilbert space has an orthogonal the composition.
Marcin Kisielowski: notice that there are no till this here, because you need to consider and not all not all.
Marcin Kisielowski: functions cylindrical with respect to graph gamma but just the proper ones and the.
Marcin Kisielowski: euro can handle proposed such definition is.
Marcin Kisielowski: It that you need to do it because of cylindrical consistency, if you, for example, divide divide a link and then one of the spaces is is the subspace, of the other so.
Marcin Kisielowski: In this, the composition, they will be not orthogonal So this is the proper definition and my and this, this will be the definition of edge camera.
Marcin Kisielowski: Now.
Marcin Kisielowski: Now, in the vertex Gilbert space, one does and are very gene with the respect to the group of different morpheus fixing the notes of the graph gamma that is.
Marcin Kisielowski: Fixing I mean they act three valley on the set of the notes of the graph and this a very drink is like the standard one standard, I mean when you impose a vector constraint just one limits to this sub group this group of the former business fixing the notes of the graph.
Marcin Kisielowski: I will consider only Su to gauge invite or Yun cylindrical functions.
Marcin Kisielowski: And so the vertex Gilbert space in my case will be the kashi completion of the image of the gate of the Su to gauge and variances in the recall functions.
Marcin Kisielowski: This.
Marcin Kisielowski: This space has a composition into orthogonal spaces.
Marcin Kisielowski: And, first let me notice that when two graphs gamma gamma prime I related by the former fees fixing each element in the set of notes of the two graphs, then the images of this spaces are the same, with respect to this leveraging map.
Marcin Kisielowski: So it is justified to introduce a notation where.
Marcin Kisielowski: He got mine, the brackets were examined the brackets is diff equivalence class of graphs with representative camera now this the composition.
Marcin Kisielowski: Of the following form, where the direct some here is.
Marcin Kisielowski: is all very the affinity subsets of.
Marcin Kisielowski: Of the space manifold Sigma.
Marcin Kisielowski: And each of these spaces each other it's basis invariant under the action of the group de de de de forma sufism data trivially understand V.
Marcin Kisielowski: And it has the the compass, the following the composition in terms of the spaces h.
Marcin Kisielowski: gamma in the brackets and I impose the gauss constraint, this is just an addition.
Marcin Kisielowski: And Sam here is over all equivalence classes of graphs which have the and that which notes are precisely the said V.
Marcin Kisielowski: This color constraint is up is defined in this space, and I will assume.
Marcin Kisielowski: Some general properties which coincides.
Marcin Kisielowski: to match extent with their property stated by Hannah and bulik.
Marcin Kisielowski: First of all, following them I assume it is locked out that it has the the this.
Marcin Kisielowski: This following farm, it is a some of the points of the manifold of operators the cx multiplied by values of the labs faction function at the point X.
Marcin Kisielowski: And each operator cx.
Marcin Kisielowski: In their space age V is invariant space of the of the each operator cx and.
Marcin Kisielowski: When X is.
Marcin Kisielowski: Not a point of V then cx vanishes in this space HIV.
Marcin Kisielowski: And I will assume also that each operator cx does not change the interest whiners associated to the notes different from X.
Marcin Kisielowski: Following here, I can assume that it is covariance that it has the following transformation property with respect to all.
Marcin Kisielowski: analytic the former fierceness in particular if the if the former fizzing fix the notes of the graph the this, it will be invariant.
Marcin Kisielowski: And I assume that it has the following splitting where the sum is all very Paris of links and epsilon is zero if the tangent vectors to the links at the at the node X are cleaner cleaner and one other wise.
Marcin Kisielowski: And each of the operators.
Marcin Kisielowski: maps to this page is gamma into the following some where.
Marcin Kisielowski: He gamma is again here and age gamma plus a minus gamma plus is obtained from gamma by adding a lot tangential to links and other prime according to the prescription which i'll show me the scribe and gamma minus is obtained by removing such low.
Marcin Kisielowski: And the operator does not change.
Marcin Kisielowski: The operator source I consider do not change representation labels of links and.
Marcin Kisielowski: Then you Lopes will be labeled with some fixed representation label l.
Marcin Kisielowski: There is one subtle point which, in my presentation is non standard.
Marcin Kisielowski: it's big I choose my non standard approach, because at some point I introduce a cutoff and wants to diagonal is mattresses.
Marcin Kisielowski: and
Marcin Kisielowski: In order to get this mattresses as.
Marcin Kisielowski: Smaller I believe.
Marcin Kisielowski: In order to get this mattresses smaller I use a different regularization from the standard one proposed by Ls yes and use dilemma of skin makinen.
Marcin Kisielowski: In their paper they carefully describe how the new look.
Marcin Kisielowski: added by this color constraint operator is added.
Marcin Kisielowski: This slope is tangential to do different links of the of the graph.
Marcin Kisielowski: and
Marcin Kisielowski: An important concept here is that tangentially the order of links.
Marcin Kisielowski: And I will then out by T ln J the order of tangent Charlotte, you have the link I would think lj and following the outdoors i'll introduce the notation K I J is just equal to the tangent reality order between links ally and the DJ.
Marcin Kisielowski: And the authors introduce the order of tangential ality of the link I.
Marcin Kisielowski: am not an.
Marcin Kisielowski: As them maximal order of tangential quality of this link with any other link.
Marcin Kisielowski: Now, when I look is that, then it has a tangent reality orders, with the links ally and lj equal to K I plus one and kj plus one.
Marcin Kisielowski: In.
Abhay Vasant Ashtekar: But I couldn't stop here because I think some of those little bit lucky.
Abhay Vasant Ashtekar: I think what would be good to be that even the previous transparency where you are summarized the work of Hanoi Europe.
Abhay Vasant Ashtekar: So there's a whole bunch of conditions.
Abhay Vasant Ashtekar: On the constraint operating.
Abhay Vasant Ashtekar: And the question is really are these conditions just motivated because you want to have simple simplify your life in some ways and calc simplify the calculations, or is there some physical.
Abhay Vasant Ashtekar: meaning behind this these restrictions there's a place for my first question.
Marcin Kisielowski: Yes.
Abhay Vasant Ashtekar: Oh, you can just answer this one, then I can.
Abhay Vasant Ashtekar: answer that.
Marcin Kisielowski: So I I just gathered some properties of the operators proposed in the in in in the papers which I listed at the beginning, so I kind of summary of of their properties So these are some operators proposed.
Marcin Kisielowski: proposed by.
Marcin Kisielowski: by your leg know.
Marcin Kisielowski: Luca make D and m n O Allah.
Marcin Kisielowski: And, and these are just their common properties, and this is not a kind of isolated as an assumption, but it's actually just a summary of the properties which they have in common, so I don't want to focus on one specific operator from those papers, although I have this.
Marcin Kisielowski: Muslim scholar field case and.
Marcin Kisielowski: From from the paper of Indica make the end and direct.
Marcin Kisielowski: in mind about, but actually the class is a broad there.
Marcin Kisielowski: and
Abhay Vasant Ashtekar: I understand that, but mathematically I understand it, and I understand that you're summarizing their work, so the question was just to remind us.
Abhay Vasant Ashtekar: Whether this restrictions came about.
Abhay Vasant Ashtekar: purely for mathematical convenience or was there some kind of physical more reason that you know if you use this constraint operator that dynamic so so nice property like a physical expected property and such such take so that is my first question.
Marcin Kisielowski: To have.
Abhay Vasant Ashtekar: What is the motivation behind I mean.
Abhay Vasant Ashtekar: Just I know it's.
Abhay Vasant Ashtekar: Just somebody so what.
So.
Marcin Kisielowski: My understanding of this, but maybe the outdoors will be the shed some.
Marcin Kisielowski: Some other lights is that.
Marcin Kisielowski: This operators work well with.
Marcin Kisielowski: With.
Marcin Kisielowski: With with the proposal from the paper of gravity quantization I mentioned as a motivation.
Marcin Kisielowski: Is.
Abhay Vasant Ashtekar: OK OK.
Abhay Vasant Ashtekar: The next question was that you said that well there's a regularization so what happens if you didn't recognize yoga just a State operate written down up here if I did not realize it then what happens when, why do you need a question.
Marcin Kisielowski: So it's actually yes, I call this slide.
Marcin Kisielowski: regularization but so.
Marcin Kisielowski: But this is actually should be it's about some technical differences in the regularization.
Marcin Kisielowski: I use with from the regularization the outers use and and but as far as I understand the outers regularize the operator with this operator are adds.
Marcin Kisielowski: or subtracts infinitesimally small look.
Marcin Kisielowski: Where on the space of solutions of this vector constraint this infinitesimal parameter drops out.
Marcin Kisielowski: And therefore, it is useful to use this vortex Gilbert space because it's enough to.
Marcin Kisielowski: use.
Marcin Kisielowski: Not the full space of solutions of the vectors constraint, but this one step before where you are very just with respect to the former physical fixing each node often of the graph.
Abhay Vasant Ashtekar: Right, but why do you like to regularize is the operator, not very defined if you don't quote unquote regular eyes or is a word regularization and little bit of a misnomer, or what.
Marcin Kisielowski: So, my understanding is that the regularization is used to be, because he you take an operator with infinitesimally small look, and so you have this regularization upside down, which controls the coordinate size of the loop and the.
Marcin Kisielowski: and
Marcin Kisielowski: By appropriately choosing this this look.
Marcin Kisielowski: This this regularization parameter.
Marcin Kisielowski: is irrelevant The limit is just a limit of of the same the same numbers.
Abhay Vasant Ashtekar: So that these loops are not lying in the legacy that you had.
Marcin Kisielowski: They are not.
Abhay Vasant Ashtekar: They are not good, so that was the point that are not completely clear to.
Abhay Vasant Ashtekar: me okay good okay.
Marcin Kisielowski: Thank you, thank you so i'll draw in a second.
Marcin Kisielowski: Some some loops, so I hope this will become.
Marcin Kisielowski: This will become clear.
Marcin Kisielowski: So I wanted to mention that I use a slightly different regularization and in this regularization I.
Marcin Kisielowski: I choose first I consider so maybe i'll show first how that looks look like, because this was really missing in my transparency.
Marcin Kisielowski: Previously, so I look at it by the scholar constraints operator.
Marcin Kisielowski: Is tangential to two different links of the graph and the.
Marcin Kisielowski: And the size of the loop coordinate size of the loop is not important because you're in the vertex hilbert space and there are different, more systems which can resize resize this loop and.
Marcin Kisielowski: And then they look needs to be checked tangential to two different links of of the graph and it's important How did the order of tangible reality of their look to the graph this is.
Marcin Kisielowski: The, this is the former fees in invariant concept, so this is an information left after the very thing is the the order of tangential ality of this look to to the link and.
Marcin Kisielowski: The differences between the original regularization and my regularization is in what way, to what extent the loop is tangents to to the link and the outers.
Marcin Kisielowski: previously said that you just take the maximal order of tangible reality of the other link with another link and add the new look, is it increases the order by one.
Marcin Kisielowski: and
Marcin Kisielowski: In in my case, it is slightly more complicated still.
Marcin Kisielowski: leads to smaller sub spaces.
Marcin Kisielowski: First, I consider a sub graph of graph gamma formed by links which do not form loops and I, I mean they do not form loops even if you remove the redundant to violent notes.
Marcin Kisielowski: I will call them regular links.
Marcin Kisielowski: And I will make this simplifying assumption, because I focused on the lattice.
Marcin Kisielowski: I will ask him that the regular links are pedophiles non tangential and any look in gamma tangential to precisely two different irregular links.
Marcin Kisielowski: bye Kappa I will denounce the order of the wedge of regular links which is just the maximal order of tangible reality of this look with with any of their links in the wedge.
Marcin Kisielowski: Then I will especial look at it, between the regular links.
Marcin Kisielowski: don't usually the order couple plus one, so it increases.
Marcin Kisielowski: The order of the wedge, and then a loop, which is.
Marcin Kisielowski: that there can be not only regular links in the graph but also loops.
Marcin Kisielowski: And by the conditions I imposed, there is also always a unique pair of regular links to which this look.
Marcin Kisielowski: Is tangent so now, the special.
Marcin Kisielowski: added by the scholar constraint operator.
Marcin Kisielowski: tangent to do this, look alpha coincides with the loop I described previously.
Marcin Kisielowski: and
Marcin Kisielowski: The difference between the two regularization boils down to can be described in this example that consider to sequences first the scholar constraint operator out at a look tangential to one and three than one and and cool and in the other sequence in in another.
Marcin Kisielowski: In another order, then in in my regularization in both cases the sequences leads to the same time generality orders of loops and links and the in the regularization by last year sanusi 11 dusky McKinnon.
Marcin Kisielowski: They lead to different tangent reality orders, so when I introduce a cut off.
Marcin Kisielowski: Then the spaces of the original proposal are bigger than the one I study, but the ideas, if you don't like my regularization for some reason.
Marcin Kisielowski: The ideas go through a while to the original regularization as well, just when you try to calculate something on a computer you try to make the mattresses as as small as possible.
Marcin Kisielowski: What.
Marcin Kisielowski: A he so in in my regularization at the end of the day, what what's important is what I call a loop configuration.
Marcin Kisielowski: At.
Marcin Kisielowski: At a given out and it tells me how many loops are there between link.
Marcin Kisielowski: between a segment of the lattice I and J.
Marcin Kisielowski: Well, I.
Marcin Kisielowski: I use for simplification, I use this condition that the they cannot be that tangent vectors to the links cannot be colina are.
Marcin Kisielowski: It it's because it works faster on the computer and.
Marcin Kisielowski: and
Marcin Kisielowski: I will also so when when you look at the number and I J it tells you that there are.
Marcin Kisielowski: There are this there is this number of loops between link I and J in in in the lattice with loops so gamma l.
Marcin Kisielowski: Is a lattice with loops where the loop configuration is and called it an L and.
Marcin Kisielowski: Allah just gives the number.
Marcin Kisielowski: Of loops that are between the links.
Marcin Kisielowski: I introduce also a total number of loops at the note.
Marcin Kisielowski: and
Marcin Kisielowski: An inquiry on space a space invariant under the action of the scholar constraint operator and looks one of the spaces looks like this.
Marcin Kisielowski: We have direct some are very possible look configurations.
Marcin Kisielowski: And off, and here we have hilbert spaces associated two graphs with a.
Marcin Kisielowski: With a different look configurations, so this is a space we, in our case, this will be a lattice and there will be loops tangential to different links and possibly many loops tangential to.
Marcin Kisielowski: The two different segments, and this is encoded in this expression.
Marcin Kisielowski: This is in very unsafe space because, adding that the the operator adds or subtracts loops so starting from the space it ends in in the space again.
Marcin Kisielowski: And it is convenient in our case on the largest we can think of this space directly in terms of spin networks, the embedded spin networks.
Marcin Kisielowski: And because the embedding is an isometric between between the spaces.
Marcin Kisielowski: And now, so.
Marcin Kisielowski: As I said, use Peter way the composition DRM to make the composition of the space, this is just a standard spin network, the composition.
Marcin Kisielowski: not letting me notice that the inquiry on spaces.
Marcin Kisielowski: Look in the following way So these are just the levels of the segments of the lattice l is the fixed representation label of the look and the loop starts and ends at this note, so there is one index up one index down in this interview in each intertwined.
Marcin Kisielowski: and
Marcin Kisielowski: And and.
Marcin Kisielowski: And now we apply this composition display network, the composition, to our environment space.
Marcin Kisielowski: And we get.
Marcin Kisielowski: In summary, what I said we get this expression, so this is the network, the composition and these are intertwined our spaces here and we have a direct some are very possible look configurations, because the graph we consider different graphs with different configurations of knobs.
Marcin Kisielowski: An important point which can be somehow.
Marcin Kisielowski: and an important step, but it seems trivial, from the point of view of an eyesore murphy's law is that this space is is a morphic further to this space.
Marcin Kisielowski: Just this is an mathematical transformation, but conceptually it leads to something.
Marcin Kisielowski: Different objects before we had just spin that works, and here we have 10s or products over the notes of the lattice of some towers of spaces of invariance sensors.
Marcin Kisielowski: And each.
Marcin Kisielowski: scholar constraint operator at the node and can be viewed as acting in a in a such tower of invariance basis.
Marcin Kisielowski: Therefore.
simone: question but is because it's been several the same right.
simone: No, or you can use with the swap.
Marcin Kisielowski: No, no, sluggish vertex will have an independence been on a link coming out of it.
simone: Then the nearby vertex to each attached don't believe this permutation the general case of plus and times you did.
Marcin Kisielowski: Why the spirits are fixed here.
To be all the same I suppose.
simone: So they don't in this space you don't have operators that can change the spins.
Marcin Kisielowski: Yes, they cannot change the spin.
simone: Okay okay yeah okay good and then another question, so what is the real issue you know.
Abhay Vasant Ashtekar: When you said that the same I don't do you mean that they're not all equal.
simone: I meant I meant that they're not looking at operators that can change this pins and because otherwise.
Marcin Kisielowski: yeah uh huh yes, yes, yes, so this is.
Marcin Kisielowski: The just that this construction is adjusted.
Marcin Kisielowski: To this class of operators so.
Marcin Kisielowski: I believe, if you if you can see that.
Marcin Kisielowski: Operators which change the spin.
Marcin Kisielowski: You could, in principle, think of similar construction, I think, but but yes, this construction will not work with, for example, female.
Marcin Kisielowski: scholar constraint operator from algebraic.
Okay.
simone: And I had another clarification these spaces that you're introducing now loop, how do they relate to the initial librispeech to introduce the one of the symmetry reduction, so you you propose a response which was associated with these regular lattice and met the whiners.
simone: Yes, and now, if you are the with a generic operator.
simone: Okay, you said.
simone: he's not gonna change the law is fine.
Marcin Kisielowski: But it will change.
Marcin Kisielowski: You also want to.
simone: pursue this image of the entertainers.
Marcin Kisielowski: So i'll come to this in a few moments, yes, but this is a you catch the important technical problem because first of all I wanted to.
Marcin Kisielowski: which I needed to solve in this project, so first first.
Marcin Kisielowski: I gave you general my first ideas, how the symmetric states should look like, but then the operators, I study they change the graph they are the subtract loops so.
Marcin Kisielowski: So I will show you in a few slides how big how I will consume what what states, I will consider to become a genius eyes, a Tropic on such spaces on such lattice with loops.
Marcin Kisielowski: And and just did this there, they will have the component which has no loops which will coincide with with what I previously described.
Marcin Kisielowski: But the spaces will loops with look differently out the average Inc will move loops and this the states will be.
Marcin Kisielowski: sam's of such states with loops in different positions.
Okay.
simone: Thank you.
Marcin Kisielowski: Thank you.
Marcin Kisielowski: Yes, so this This brings me to what what semana raised.
Marcin Kisielowski: As now consider a lattice lattice with the with loops.
Marcin Kisielowski: First of all.
Marcin Kisielowski: Now we restrict to lattices such that each.
Marcin Kisielowski: So lattice liquid loop will be gamma L and our lattice is just gamma began.
Marcin Kisielowski: So each lattice with loops can contains.
Marcin Kisielowski: Our lattice and we assume that the links of the lattice are labeled with the same same spin J.
Marcin Kisielowski: And the loops are labeled all with spinner some is given spin.
Marcin Kisielowski: Now we still have a default Murphy isms corresponding to.
Marcin Kisielowski: Our group of orientation preserving symmetries of the cube.
Marcin Kisielowski: And as before we can we can make this leveraging but now this operator does not change is not only an intertwined but also.
Marcin Kisielowski: This.
Marcin Kisielowski: Look configuration.
Marcin Kisielowski: So.
Marcin Kisielowski: If there were five links between five labs between.
Marcin Kisielowski: Between link one and toe and you apply a permutation then after acting with after acting with this transformation, you will have five links five loops between different links this will be i'll just tell it precisely in a second and.
Marcin Kisielowski: Here we again have this phase factor.
Marcin Kisielowski: And again, due to the covariance property of this operator.
Marcin Kisielowski: scholar constraint operator it commutes with each of the which each of the default morpheus and therefore it can commute with this projection operator.
Marcin Kisielowski: So now we we look at the image of this projection operator this image is no longer.
Marcin Kisielowski: A single intertwined in a in a single intertwined our space it's in some power of intertwined intertwined.
Marcin Kisielowski: And now we take such a symmetric state and.
Marcin Kisielowski: We multiply we assign this state to each of the nodes of of the lattice and we tend, we tend to multiply we tensor multiplied the same number.
Marcin Kisielowski: The number of times there are not in in the lattice gamma so this way you obtain.
Marcin Kisielowski: You obtain a.
Marcin Kisielowski: kind of homogeneous homogeneous isotopic states.
Marcin Kisielowski: The states will be invariant under the group.
Marcin Kisielowski: Of.
Marcin Kisielowski: Of cemeteries of of the lattice I I talked before.
Marcin Kisielowski: and
Marcin Kisielowski: So if you, for example, wants to buy a gun allies, the full scholar constraint operator you don't like the diagonal is it at a single node find an Eigen value.
Marcin Kisielowski: Eigen factor added this note and.
Marcin Kisielowski: multiply this the same I I again factor for for each note of of the lattice.
simone: Can you give us an example like you pointed out, these simplest case with the regular Q, but that you are getting in the initial part of your talk now suppose that you act to by just adding a single loop on a pair of links in these six violent vertex.
Yes.
simone: Can you describe it would be the Tower of intertwines Starting from that symmetric one and what would be the resulting in intertwine or after the projection like in a simple example, so that we get some intuition.
Marcin Kisielowski: So, first of all.
Marcin Kisielowski: First of all, you would.
Marcin Kisielowski: yeah first of all, you would add.
Marcin Kisielowski: You would consider.
Marcin Kisielowski: First start with the lattice then add the.
Marcin Kisielowski: then add, for example, one loop between a pair of links.
Marcin Kisielowski: Then you apply this are very aging procedure, which tells you.
Marcin Kisielowski: Why.
Marcin Kisielowski: Which which tells you that you should.
Marcin Kisielowski: You should.
Marcin Kisielowski: One second because i'm thinking about i'm going further and back.
Marcin Kisielowski: Because I.
Marcin Kisielowski: will describe it in.
Marcin Kisielowski: To some extent i'll describe it later, so it will be easier for me to.
Marcin Kisielowski: Talk about the example after two slides.
Okay.
Marcin Kisielowski: So let me just finish finish this one point, and then i'll talk about this example and.
Marcin Kisielowski: First, first of all i'll introduce a cut off in the number of loops so at each note the there will be, I will not allow for more than ella number of loops.
Marcin Kisielowski: scholar constraint operator is.
Marcin Kisielowski: is acting between some is acting in some infinite them dimensional hilbert spaces so after I introduce a cat have to get the metrics which can be a diagonal line on a computer, for example, and this is when you introduce a limitation of a number of loops at each node.
Marcin Kisielowski: And then.
Marcin Kisielowski: Then.
Marcin Kisielowski: Then, then this is one of possible cut offs then.
Abhay Vasant Ashtekar: Why then face value and the academic scale constraint them because scale because.
Marcin Kisielowski: It is no no.
Marcin Kisielowski: No, no, it is not.
Marcin Kisielowski: It is not this cat have brakes the brakes day invariance just.
Marcin Kisielowski: it's, for example in.
Marcin Kisielowski: I learned that, for example in nuclear physics one introduces a cut off and then.
Marcin Kisielowski: One star studies, Mr three mattresses.
Marcin Kisielowski: After this kind of looks at the Eigen vectors of this matrix and the moves that cut off looks how this.
Marcin Kisielowski: If there are eigenvalues which converts to some value when you move this kind of this is a numerical study and then, if you see that some of the eigenvalues one stabilizes.
Marcin Kisielowski: let's say the lowest or one.
Marcin Kisielowski: or one from the lowest.
Marcin Kisielowski: Then you believe that numerically you have found and eigenvalue and eigenvector.
Abhay Vasant Ashtekar: But so that metrics right so here.
Abhay Vasant Ashtekar: metaphor, in fact, the scale a constraint is going to map me from edge loops to help us one dupes.
Abhay Vasant Ashtekar: Then.
Abhay Vasant Ashtekar: Then, then I don't have a matrix right, so what What do you do we just.
Marcin Kisielowski: So, so in look quantum cosmology, for example, you could introduce a cat of in the volume.
Marcin Kisielowski: And, and then calculate data analyze the matrix on the computer and.
Marcin Kisielowski: And it will turn up turn out that given a cut of in the volume.
Marcin Kisielowski: gives you well estimation of some lowest energy eigenvalues of of the physical hamiltonian.
Marcin Kisielowski: So, because.
Marcin Kisielowski: Because in this Eigen States after this volume there.
Marcin Kisielowski: There isn't really not.
Marcin Kisielowski: Not much happening at larger volumes so for large energies, you could introduce a cut of in the volume and for for low energies, you could take this approximately approximate eigenvalues and obtain.
Marcin Kisielowski: obtain a good approximation of the of the of the full system.
Abhay Vasant Ashtekar: Right, but here we are interested in the solutions to the hamiltonian constraint right.
Abhay Vasant Ashtekar: So when we just wanted isn't that what you mean I i'm not sure about I understand what the goal is isn't the goal to find the solutions to the hamiltonian constraint, or something some others something so the goal.
Marcin Kisielowski: Yes.
Marcin Kisielowski: It is that it is.
Abhay Vasant Ashtekar: But in presence of some scale up with something not to scale, a field is not included here at the moment.
Marcin Kisielowski: So this killer field works is included in a in this sense that.
Marcin Kisielowski: Once you calculate.
Marcin Kisielowski: Once you calculate an evolution operator, then.
Marcin Kisielowski: Then.
Marcin Kisielowski: Technically, you can construct.
Marcin Kisielowski: You can construct the direct observed bubbles and.
Marcin Kisielowski: Practically you solved the system.
Marcin Kisielowski: So here technical problem is to.
Marcin Kisielowski: The technical problem is to.
Marcin Kisielowski: exponentially eight and operator.
Marcin Kisielowski: Right so.
Marcin Kisielowski: So my idea is, for example, that maybe it will turn out that.
Marcin Kisielowski: When we restrict to this to this sector, we will find some.
Marcin Kisielowski: Eigen vectors which can approximate.
Marcin Kisielowski: which can approximate.
Marcin Kisielowski: The.
Marcin Kisielowski: Full.
Abhay Vasant Ashtekar: it's much clearer now okay so you're thinking of this constraint, this only the geometrical part of the constraints, so this will be the.
Marcin Kisielowski: hamiltonian yes.
Abhay Vasant Ashtekar: And then the statement is that you're looking for kind of low lying I.
Abhay Vasant Ashtekar: Just don't understand it okay.
Abhay Vasant Ashtekar: Now, thank you.
Marcin Kisielowski: Yes, thank you.
Marcin Kisielowski: So i'm coming to see Manas question.
Marcin Kisielowski: And let me first describe how how this, the former fees from from the orientation preserving cemeteries have a cube act.
Marcin Kisielowski: On.
Marcin Kisielowski: In this space in this Tower of spaces, so this Tower has a label which tells us.
Marcin Kisielowski: How many loops there are between links, so we take segment one and and far, for example, and take the number as evaluated at one and four and we get the number five This means there are five loops.
Marcin Kisielowski: And this, the farmer fizzle moves the loops because, for example.
Marcin Kisielowski: Link one and farm is moved to link to and five, so now we have different configuration, we have five loops between link of two and five.
Marcin Kisielowski: But we also have a different intertwined there.
Marcin Kisielowski: and
Marcin Kisielowski: And this this.
Marcin Kisielowski: This transformation.
Marcin Kisielowski: Of look configuration is.
Marcin Kisielowski: Quite standard one is I as I described you in words it's just a very mutes the indices I J, but it can, it can also flip flip a loop.
Marcin Kisielowski: So, because I assume I is smaller than Jay so this G can break this condition, then I will call it a flip and this flips have a have important have consequence on the action on the Internet whiners when I flip happens, it introduces a phase phase of minus one to the power to.
Marcin Kisielowski: So the action on the intertwined or space is some permutation of the innocence of the intertwining multiplied by the number of times flips of loops or cure multiplied here by sorry invite to twice the.
Marcin Kisielowski: Fixed spin level.
Marcin Kisielowski: and
Marcin Kisielowski: Now this averaging operator looks.
Marcin Kisielowski: Has the following farm, so my idea so on on the example see mana asks asks.
Marcin Kisielowski: So when you consider, for example, one look at it between link.
Marcin Kisielowski: link.
Marcin Kisielowski: Say one and far then.
Marcin Kisielowski: What kind of state, you would consider symmetric I mean what kinds of let's go back what kind of state with one loop, you would consider symmetric.
Marcin Kisielowski: And my idea is that okay take this loop between one link then transform it with this.
Marcin Kisielowski: With this.
Marcin Kisielowski: group element from our cube.
Marcin Kisielowski: And added to the previous one, so this is a kind of.
Marcin Kisielowski: This is a kind of some.
Marcin Kisielowski: Some of states with with different with loops in different positions so.
Marcin Kisielowski: let's go to for violence case, maybe for an example because in for violent case when when you have this could be.
Marcin Kisielowski: transferred to for violence case when this group will be just.
Marcin Kisielowski: will be just also permutations but so they will permeate all the links, so the symmetric state, the first idea for symmetric state is.
Marcin Kisielowski: Take a look, between a pair of links plus.
Marcin Kisielowski: A loop between another pair of things, plus.
Marcin Kisielowski: Another plus and.
Marcin Kisielowski: Plus any configuration of a little bit when elling so and this this state would be kind of.
Marcin Kisielowski: What would be.
simone: semi Matilda.
Marcin Kisielowski: Sorry.
simone: What is.
Marcin Kisielowski: This Sigma tilda is a permutation of the.
Marcin Kisielowski: Of the indices of of the intertwine are because.
Marcin Kisielowski: Because this.
Marcin Kisielowski: This sort.
Marcin Kisielowski: of it.
simone: So, because you started with the semantic.
simone: So if the stage we're not semantically and now you start normals you make a mess, of course, but I guess the fact that you were starting from something that was already very symmetric is what.
simone: may make the procedure work so is the intuition that basically you're just.
simone: and put on furthermore you're probably flipping every time to normal so so that there should be no problem with the closure conditions as well right.
simone: So basically the intuition, is that you're just intertwined is always the same name it's always in the six volume case is always 100010 and so on, as you pointed out, is just that now.
simone: Which one is 100 which one is 010 you are paired mutating them.
simone: And when you bear mute them at your commute every time to so that the closure condition is always satisfied, is it something like this that is going on.
Marcin Kisielowski: I didn't.
See I.
Marcin Kisielowski: didn't but so did.
simone: The final result you can write it as a sample intertwines right.
Marcin Kisielowski: refineries, yes, yes.
Marcin Kisielowski: Yes, this is some of the intertwined there's but notice that when you change and put a look, when you have already a loop in some position.
Marcin Kisielowski: And you move it with that the former ISM to another with with this G with G here when you move it to another position you most probably end up in orthogonal space.
simone: Maybe was a very naive question usually like in demon's regularization when you add the loop, but you also change this been here, this is not happening right.
Marcin Kisielowski: Yes, this.
simone: Is preserving the spins.
Marcin Kisielowski: Yes.
simone: Okay, so when you see now state you mean just in terms of the intertwining or.
Marcin Kisielowski: No, no, no.
Marcin Kisielowski: No Okay, because it's a.
Marcin Kisielowski: I i'm in this space, I make a very jink with respect to the form morpheus which do not move nodes of the graph so this G is not such in the form of racism and so when I have a loop between link one and link to.
Marcin Kisielowski: In my state and in a spin networks, they take a spin network state with the loop between link one and one link three and then consider SP network state with a loop between link one and link five.
Okay.
Marcin Kisielowski: Then.
Marcin Kisielowski: Then they are, they are in different different spaces in this vortex Gilbert space.
simone: I see, yes, these will do defined earlier on, yes okay.
Marcin Kisielowski: So now.
Marcin Kisielowski: So now, I need to take into account that this out cube is is, can I can take me from one space into orthogonal space or can leave me in the same space so.
Marcin Kisielowski: But it's important it's only important that it.
Marcin Kisielowski: It it moves along in this Tower changes this look configuration.
Marcin Kisielowski: And States with different look configuration or a toggle now.
Marcin Kisielowski: So they're in different.
Marcin Kisielowski: different components in.
Marcin Kisielowski: In this, the composition, although the invariance basis are the same, but then look configuration is different and the spaces are orthogonal.
simone: I see, so I cannot think of the result of this operation in terms of the.
simone: In terms of our linear combination of intertwined there's on the regional graph without any loop added, I cannot do this.
Marcin Kisielowski: Yes.
Marcin Kisielowski: Okay, just because it's they have another label they.
Marcin Kisielowski: have this look configuration label which.
Marcin Kisielowski: Which.
Marcin Kisielowski: understates with different labels are orthogonal.
simone: And just to understand what do you have in mind here is to realize the full Armenian constraint or adjust the induced scale or curvature part of it.
Marcin Kisielowski: fall for.
simone: The fool so also the part that contains extrinsic curvature he would like to realize within this framework.
Marcin Kisielowski: Yes.
Marcin Kisielowski: I mean.
simone: I would expect that to change the spins maybe we can come back to this at the end at least now I also better how these construction workers.
Marcin Kisielowski: So in the approach I followed.
Marcin Kisielowski: It from from the from the papers.
Marcin Kisielowski: I described before.
Marcin Kisielowski: They this beans are are are not changed just.
Marcin Kisielowski: Just the.
Marcin Kisielowski: loops are added in.
Marcin Kisielowski: English.
simone: Thank you.
Marcin Kisielowski: in different ways, thank you.
Marcin Kisielowski: What I looked at.
Marcin Kisielowski: is how how the dimension of of the space after the cut off.
Marcin Kisielowski: looks like when we vary the cut off it's because.
Marcin Kisielowski: i'm thinking about some calculation on on a computer and.
Marcin Kisielowski: 10 to the power six is the rank of the metrics you can typically diagonal lights on on a supercomputer.
Marcin Kisielowski: The restriction is.
Marcin Kisielowski: mainly due to memory, because if you have 10 to the power six and you need to start a matrix, then you have 10 to the power 12.
Marcin Kisielowski: Bytes roughly speaking and.
Marcin Kisielowski: This is a around the capacity, you have and we see that would spin one half we could.
Marcin Kisielowski: We could calculate up to five loops would spin one up to four three half's and tool, the same up to four loops and the rest, I in my study five have up to five, we could do only diagonal is up to three loops.
Western: it's you know very quick question.
Western: gamma the Barbarians parameter or and the do you have any difference in the competition or whether you use a different one.
Marcin Kisielowski: So right now.
Marcin Kisielowski: Right now, I I.
Marcin Kisielowski: Fixing gamma will be at the level of data analyzing the operator so once you figure gamma, then you get some operator and operator matrix which.
Marcin Kisielowski: Which.
Marcin Kisielowski: So this slide I did in this work, I do not fix gamma because i'm just talking about some general properties, but in calculating the eigenvalues yes, I will need to fix gamma it, it appears explicitly in the in this physical hamiltonian.
Marcin Kisielowski: and
Marcin Kisielowski: I could, in principle, look for different values of gum our fix it to the standard one.
Marcin Kisielowski: This is in front of me.
Western: Thank you.
Marcin Kisielowski: Thank you.
Marcin Kisielowski: What I observed, is that the symmetry reduction rate that is the dimension of the reduced space to the full space with the cut of L.
Marcin Kisielowski: stabilizers seems to stabilize quite fast, at the very value 24, which is the number of elements of the group of symmetry of the cube.
Marcin Kisielowski: and
Marcin Kisielowski: And we see the larger spin this, the faster it stabilizes at the edit this at this value 24.
Marcin Kisielowski: And i'll discuss.
Marcin Kisielowski: Let me know summarize and discuss.
Marcin Kisielowski: In particular i'll discuss this value 24 in the summary second discussion section.
Marcin Kisielowski: The restriction.
Marcin Kisielowski: To homogeneous is a Tropic sector of loop quantum gravity that I proposed leads to substantial reduction of the degrees of freedom, first of all.
Marcin Kisielowski: We limited to a lattice with with all equal spins on the sides.
Marcin Kisielowski: And, moreover.
Marcin Kisielowski: We took one Eigen vector of on operator CN at some fixed node and we tensor multiply it for each node of the lattice.
Marcin Kisielowski: And so, this is also a huge reduction of degrees of freedom, and this is something you could.
Marcin Kisielowski: You could, I think, think of.
Marcin Kisielowski: But what what I bring here more is that there is also a symmetry reduction at the level of the intertwined.
Marcin Kisielowski: And this symmetry reduction.
Marcin Kisielowski: is useful here in the canonical of quantum gravity, but I believe it can get to be interesting, also, for example, for.
Marcin Kisielowski: group field theory where they consider.
Marcin Kisielowski: some kind of comedy in psychotropic states which are quite similar.
Marcin Kisielowski: I noticed that diverging should include non trivial face factor to accommodate leaving speciality coherent intertwined.
Marcin Kisielowski: And I I found that after truncation APP to Allah loops we get roughly 24 times smaller sub spaces of intertwined hours, so there is a further reduction of the degrees of freedom.
Marcin Kisielowski: And is it is that 24 small or large, probably it's small compared to the restrictions we did.
Marcin Kisielowski: We did here but.
Marcin Kisielowski: For computers this factor 24.
Marcin Kisielowski: turns out to be also important because most diagonal ization algorithm algorithms have complexity close to our end cube where and is the rank of the metrics.
Marcin Kisielowski: So this means that we get around 25 to the power three, that is, two to the power force speed up compared to the approach, then, if let's say approach, where we would not make this a very junk in the space of entertainers and practically this speed up means that instead of diagnosing.
Marcin Kisielowski: compute and metrics on on all the resources of my computing Center I could diagonal is a matrix on on my laptop.
Marcin Kisielowski: And I hope that this will allow us to study the Eigen states of the operator of this scholar constraint operator in the reduced space.
Marcin Kisielowski: In the reduced spaces.
Marcin Kisielowski: And I we introduced a cut off and my other result from another paper about a volume operator suggest that this truncation has an interpretation in terms of.
Marcin Kisielowski: truncation in the volume.
Marcin Kisielowski: At least in the spin one half case I showed that.
Marcin Kisielowski: The truncation is precise is a truncation in the volume.
Marcin Kisielowski: Yes.
simone: I need I need to leave, can I ask a question now.
Marcin Kisielowski: Yes.
Yes, of course.
simone: curiosity about this 24 like what, why is it so big in fact that reduced, you may have expected you know, like thinking of the quantum reduced.
simone: quantum gravity that intertwine is basically just one regardless of what is the spin.
simone: And so that the So what are because it's the only day like the regular cube.
simone: So what are these other shapes that you are allowing when you increase the speed, but stealing system is on virginity and so there'll be I think it will be very interesting to understand the gentleman this do you know it already.
All right.
Marcin Kisielowski: Well, I don't know, but I have some idea So yes, and also when I mentioned gfp, let me just mentioned that in JFK they also choose the highest highest volume eigenvalues.
Marcin Kisielowski: So they also make a bigger restriction in a sentence and and notice that this factor it when.
Marcin Kisielowski: This 2024 up.
Marcin Kisielowski: appears for larger spins and spins equals one half we have two dimensional space described by the tool.
Marcin Kisielowski: Living special a coherent intertwined there's corresponding to the Tokyo one with one orientation and the other with opposite orientation so.
simone: It seems to me to exist for any value of this pain right.
Marcin Kisielowski: Yes, that's right.
simone: that's right.
simone: And what are the other states, now that appear when you increase the speeds, do you know.
Marcin Kisielowski: I don't know I expect that they may be some kind of.
Marcin Kisielowski: some kind of a mixture some kind of combination of the spin spin spin one half.
Marcin Kisielowski: cubes with opposite orientations I don't know this is my earthly idea that since two orientations appear here, I suppose, they may be responsible for for for for for this this many configurations I ended when I started this project I expected a bigger reduction.
Marcin Kisielowski: So this was disappointing.
simone: For me it's very interesting.
simone: I think it's very easy in a sentence promising that you're not getting at these bigger reaction that we naval expected, it means that you're hinting at other degrees of freedom other one can.
simone: study, even though is restricting to homogeneity and so there'll be means that is not just simply.
simone: There are certain positions orientations, maybe do something like this that is also occurring that.
Marcin Kisielowski: Might not notice that I do not do not restrict.
Marcin Kisielowski: The spatial curvature in in my model so so kind of there is this parameter the spatial clarify chair.
Marcin Kisielowski: I can be can be different it's not set to one or minus one or zero so, so there are different it's comma genius is a Tropic but not fixed fixed by the spatial curvature is not fixed so.
simone: In these extra degrees of freedom, you have a maybe what tells you what the core.
simone: gives you room before non trivial curvature and yes that's right.
Marcin Kisielowski: that's right.
simone: Sorry, I had to leave, thank you very much.
Marcin Kisielowski: Thank you, thank you.
simone: So.
Marcin Kisielowski: As I said before.
Marcin Kisielowski: My goal is still to look for the.
Marcin Kisielowski: For the Eigen states and actually.
Marcin Kisielowski: My my my dream, if everything goes well that, for example, I can it turns out that the lowest energy eigenvalues are.
Marcin Kisielowski: Are are well described, I mean I get some converging of the eigenvalues when I increase the cutoff.
Marcin Kisielowski: Then I could consider kind of similar states coherence states another states analogous to the one in accuracy, for example, peaked at.
Marcin Kisielowski: at low values of the gravity scholar gravitational part of this color constraint.
Marcin Kisielowski: operator and.
Marcin Kisielowski: And one could in principle in this region study study, if there is a big banks or investigate the stability of the States under the quantum dynamics generated by the physical hamiltonian and Let me close and now with just short remark that this.
Marcin Kisielowski: technical simplification does not apply only to the.
Marcin Kisielowski: To the canonical theory, it also has some application to the covariance theory at least in the formulation, we proposed in 2019 in our spin phones a history of symmetry reduced state is described by states in the symmetry reduced space and.
Marcin Kisielowski: This means that.
Marcin Kisielowski: That in order of the vertical expansion, the complexity of the problem gets roughly 21st to the power and minus one times smaller due to symmetry reduction, thank you for your attention.
Marcin Kisielowski: Yes.
Cong Zhang: Okay, so, so I have a I have a comment on this on this on this algorithm you want to use this final cut off to to to get this, I can state, and I can value.
Cong Zhang: So, as my understanding, so what you want to do here is to write this again stated as the real function of the numbers of the loose right.
Cong Zhang: Something like that.
Marcin Kisielowski: I wave function of volume.
Cong Zhang: Of whoa Okay, a volume of the loop okay of volume but anyway, so this is something like a discrete the value of something and as my understanding, if you want to do this finance.
Cong Zhang: at all, so the precondition should be that the against data of this hamiltonian operator should be exponentially decay, something that is potentially the key for large value of the volume.
Cong Zhang: So if the against data is exponentially decay, then the icon status will be concentrated add some finite value of this volume So if you just choose a very large volume.
Cong Zhang: Such that this value of this volume is added to it as the exponential, the key is exponentially, the key features of this against data, then the against data of the hamiltonian operator can be well approximated by the against data of this finite matrix.
Marcin Kisielowski: Just this would be ideal.
Cong Zhang: yeah so actually I did I actually use the dis master the to do diagonal lines that this hamiltonian operator of the loop quantum black hole modal and I get this experience in intuitively.
Marcin Kisielowski: Great Thank you.
Cong Zhang: And also, I have a question so so so so for for your symmetry reduction so as my understanding, so you'll have like a to.
Cong Zhang: You have to group arrogant or something like that the first is that so when you do this simply reduction, so you do some averaging procedure with respect to the.
Cong Zhang: To the cemetery group of the cubicle graph and the second is that the venue do that when you do this when you do this work hats here, but the space, there is also a group averaging weather is back to the symmetry of the graph right.
Marcin Kisielowski: So that's the largest the lattice.
Cong Zhang: yeah a lot.
Cong Zhang: yeah so I just I didn't get the difference between this to.
Cong Zhang: This region So what is this too.
Marcin Kisielowski: So maybe my presentation was confusing there's no to everything there is only one just I somehow thought that it would be best to present first the idea on some simple example some first.
Marcin Kisielowski: It was a kind of example of.
Marcin Kisielowski: When you take the cut of an equal.
Marcin Kisielowski: Equal zero.
Cong Zhang: mm hmm.
Marcin Kisielowski: So this is the first is how it works in the space where you take zero loops so the cut of zero.
Marcin Kisielowski: But this is.
Marcin Kisielowski: The average income is one.
Marcin Kisielowski: And in the full space, it was somehow.
Marcin Kisielowski: There are many technical ideas there and I thought that, first, if I start first with the with the with this simpler case and equal to zero, I can explain better what's happening in the intertwined nurse because I found its particular interesting.
Marcin Kisielowski: So directly recently watched.
Cong Zhang: him to reduce the.
Cong Zhang: status is a is a is in this work exhale bit of space right yes.
Marcin Kisielowski: Yes, yes, but he noticed that.
Marcin Kisielowski: Since the diverging map in this hour simple case is an isometric so.
Marcin Kisielowski: So you can just work in this space of spin networks.
Marcin Kisielowski: So when you introduce a cut off from the calculation out point of view, you can work.
Marcin Kisielowski: I mean.
Marcin Kisielowski: I mean when you said equal to zero and use this map into.
Cong Zhang: This is is.
Cong Zhang: This either is the is to define this work, I see with a space right.
Marcin Kisielowski: Yes, yes, yes.
Cong Zhang: yeah yeah.
Marcin Kisielowski: yeah there's only one my.
Marcin Kisielowski: answer is basic, my answer is that there is only one leveraging.
Marcin Kisielowski: And this is a.
Marcin Kisielowski: This was just a particular case.
Cong Zhang: The fund is simply reduce the state.
Marcin Kisielowski: Sorry.
Cong Zhang: It is this this this for Jackson P here is used to define the same to reduce the stadium.
Marcin Kisielowski: Yes, you get a tower here, and the one the lowest component of the Tower is the space I described before so.
Marcin Kisielowski: So this is also reflected in in in mind notation so I plot here DEM dimension of a cube jl.
Marcin Kisielowski: and
Marcin Kisielowski: At the beginning, I plot kill dimension H cube G zero.
Western: Hello Marcin.
Western: Okay, can you ask the question.
Cong Zhang: Okay, so so can can.
Cong Zhang: Can can.
Cong Zhang: Ask my question further okay so so so let's so let's meet describe my my question in another way, so the point is like is this simply use the State to do more with them in the environment.
Marcin Kisielowski: that's a good question, so the answer is.
Marcin Kisielowski: As a state is it is as a wave function, it is not.
Marcin Kisielowski: Because of this, face factor here, so if you want to have living speciality coherent intertwined there's you need to choose this face.
Marcin Kisielowski: And with this face your state at the end of the day, will be different murphy's invariant but the way function with the way function will be not because you will have this transformation property.
With.
Marcin Kisielowski: With with with this face factor here so.
Marcin Kisielowski: So the vertex hilbert space, does the average in to the respect to the former physics, which you do not.
Marcin Kisielowski: move the notes what you're left with are.
Marcin Kisielowski: What you are left with our just our the default murphy's miss which which move the notes.
Cong Zhang: yeah I know.
Marcin Kisielowski: So.
Marcin Kisielowski: So if you set this face factor equal to one you get.
Marcin Kisielowski: You get the former policeman very young boy function and if not, you get the not default murphy's mean variant one way function but you get.
Marcin Kisielowski: The form Orpheus mean variance state.
Marcin Kisielowski: So there is this subtle point.
here.
Marcin Kisielowski: which I didn't race because of.
Marcin Kisielowski: Time so.
Cong Zhang: If this G is is just a number rather than this happy is does the indigo rather than is happy bigger.
Cong Zhang: Right, this is this, then this is the form of a dummy environment.
Marcin Kisielowski: Yes, yes.
OK OK OK.
Western: alomar senior just a very nice question.
Marcin Kisielowski: uh huh.
Western: Okay, so can we go back to the first block that you showed us, yes, that one okay so here, I understand what you're doing but it's not entirely clear to me how you're doing it so.
Western: I already see as the computational complexity.
Marcin Kisielowski: that's me think.
Marcin Kisielowski: What i'm doing here is first is the.
Marcin Kisielowski: First, I calculate the dimension of the space okay.
Marcin Kisielowski: So for this, you need to calculate.
Marcin Kisielowski: You need to calculate this this object here.
Marcin Kisielowski: So you need to pair mute the indices of of this intertwined are here.
Marcin Kisielowski: and add up for different.
Marcin Kisielowski: cemeteries of the queue.
Marcin Kisielowski: Some.
Marcin Kisielowski: So the first try is.
Marcin Kisielowski: The first try is to do it natively.
Marcin Kisielowski: Then you need to construct the standard source as.
Marcin Kisielowski: As using the standard three cotton three construction, as you construct the intertwine or as a three.
Marcin Kisielowski: And then.
Marcin Kisielowski: And then you.
Marcin Kisielowski: You need to mute.
Marcin Kisielowski: and
Marcin Kisielowski: and add up, so you get 24 summations and the difficulty is in constructing all the.
Marcin Kisielowski: All the intertwining of this farm, but this is not how I did it.
Marcin Kisielowski: Because it was going to slow.
Marcin Kisielowski: instead.
Marcin Kisielowski: Transposition I used to.
Marcin Kisielowski: Use the following observation that transposition of.
Marcin Kisielowski: Has a representation matrix which can be written in terms of a six day symbol right simple one, it is a sparse matrix.
Marcin Kisielowski: and
Marcin Kisielowski: which I calculated for different path transposition, then I wrote down.
Marcin Kisielowski: A permutation as a product of trans positions neighboring trans positions that is transposing neighboring numbers.
Marcin Kisielowski: And then.
Marcin Kisielowski: It was a matter of multiplication of sparse matrices which I did a music into mk sparse matrix or routines.
Marcin Kisielowski: And what you end up with is you want a dimension of this space, so you want a trace of this operator.
Marcin Kisielowski: So it's a matter of strike through calculating a trace of a product of some mattresses which are sparse.
Marcin Kisielowski: and
Marcin Kisielowski: I found it fastest to be done.
Marcin Kisielowski: To be done.
Marcin Kisielowski: like this, so.
Marcin Kisielowski: So.
Marcin Kisielowski: it's a.
Marcin Kisielowski: In order it's hard for me to estimate the complexity right now I I just looked for the fastest method and calculate I didn't plan to.
Marcin Kisielowski: To to go further, just my my point was just.
Marcin Kisielowski: You see here I reach I reach the limit of.
Marcin Kisielowski: Of of size of the mattresses which I could calculate and I could find all their points here.
Marcin Kisielowski: and
Marcin Kisielowski: So this was enough for me, so it was a part which I needed to optimize to get to spins 45 half's but I didn't study it really how I see that.
Marcin Kisielowski: I don't remember, but I think the complexity grows grows.
Marcin Kisielowski: Like.
Marcin Kisielowski: exponentially, first with with increasing spin even.
Western: Okay now it's medically you Thank you.
Marcin Kisielowski: Thank you, thank you.
Jorge Pullin: Is there another question in western.
Abhay Vasant Ashtekar: classical quick question, this time, if not, I can.
Jorge Pullin: Go ahead.
Abhay Vasant Ashtekar: So the question was really sad end of the day, there are several you know, like.
Abhay Vasant Ashtekar: approaches to also coming from Warsaw in many ways to kind of bridging call them know quantum Dr Lupo and cosmology in which one then writes down the expectation value of the the.
Abhay Vasant Ashtekar: Execution hamiltonian and then sort of tries to see some effective dynamics so on so forth, so what happens in this approach and how is this related to what other people have said.
Marcin Kisielowski: So.
Marcin Kisielowski: In this approach.
Marcin Kisielowski: In this approach, what is what is different here is that.
Marcin Kisielowski: The.
Marcin Kisielowski: The homogeneous isotopic space is an invariant space of the scholar constraint operator exactly.
Marcin Kisielowski: So, for example, it is preserved by the evolution.
Marcin Kisielowski: and
Marcin Kisielowski: So there isn't, it is not an.
Marcin Kisielowski: That the approximation here is in using cut off.
Marcin Kisielowski: and not in the in varying the parameter of a coherent state.
Marcin Kisielowski: And maybe my hope is.
Marcin Kisielowski: To construct something.
Marcin Kisielowski: which looks like like the elk you see.
Marcin Kisielowski: Coherent state from.
Marcin Kisielowski: The paper out.
Marcin Kisielowski: By by you and and tomek.
Marcin Kisielowski: For example.
Marcin Kisielowski: and
Marcin Kisielowski: I hope to get a better picture of the.
Marcin Kisielowski: Of the low low volumes.
Marcin Kisielowski: If I think about this application.
Marcin Kisielowski: But, am I.
and
Abhay Vasant Ashtekar: So this is still working progress, so you can compare it with what other people have that in this year.
Abhay Vasant Ashtekar: terms of in terms of this.
Abhay Vasant Ashtekar: In terms of dynamics yet.
Marcin Kisielowski: Yes, so I want to just say this thing that.
Marcin Kisielowski: That in in.
Marcin Kisielowski: In this symmetry reduction it's not kind of approximate because it seems, but because it's it's kind of exact in the sense.
That.
Marcin Kisielowski: When you read make this reduction, you still your operator does not bring you out of this space your gravitational of the scholar constraint operator.
Marcin Kisielowski: Acting on homogeneous is a Tropic state is again homogeneous isotopic state.
Marcin Kisielowski: And for coherence states.
Marcin Kisielowski: In other approaches this.
Marcin Kisielowski: seems to me more like approximate.
Marcin Kisielowski: that the States are approximately homogeneous psychotropic.
Okay.
Abhay Vasant Ashtekar: Thank you all, thank you.
Marcin Kisielowski: Thank you.
Jorge Pullin: Any other questions.
Thank you.