Michael Freedman - The Universe from a Single Particle - IPAM at UCLA

so one of the things i like about physics is it gives one an opportunity to comment on why you know why things are the way they are sort of foundational questions and you know one of the themes in the last 50 years in that direction has been emergence uh you know some theory is supposed to emerge from other physics at higher energy emerge at lower energy and all the theories that we discuss foundational theories [Music] of course are interacting theories and what i want to talk to you about today is a mathematical possibility for understanding the emergence of interaction itself you know how could uh interacting physics emerge from something simpler well what's something simpler we'd be like uh freshmen quantum mechanics single particle quantum mechanics actually as much as i enjoy these masks it's sort of fogging my glasses and then i can't see you the masks are great for high altitude training i think trying to suck in the air but other than that i'm not sure uh so this emergence uh that i want to discuss will be a a kind of a novel application of a familiar idea of spontaneity spontaneous symmetry breaking uh and basically it'll be single particle quantum mechanics will break to interacting physics so let me first review symmetry breaking in its sort of more familiar context which is symmetry breaking of states an example let me give three examples uh the something called the curie temperature if you for instance uh start cooling iron below about a thousand degrees kelvin at some point the magnetic moments of the iron atoms align and you you get magnetic domains which might be a few thousand atoms across or a few million atoms across depending on how quickly you cross that transition so you know basically this is sort of like a freezing or crystallization so here symmetry breaking leads to a crystal uh it's called spontaneous because you haven't prejudiced by an external field necessarily which way they were going to line before they all freeze together they kind of point any which way so it has full rotational symmetry but then that's vastly reduced when they all pick one direction uh another example is um the uh ginsburg landau theory which led to an understanding of superfluids and superconductors before there was a known microscopic mechanism before you know the pairing mechanism so this led to superconductors and an example from high energy which is a little bit different sort of symmetry breaking the gauge symmetry higgs leads to you know mass of other particles so this is this is just meant to convince that interesting things happen when you break symmetry and i want to talk about breaking symmetry but today or tonight the symmetry breaking will be on the level of operators rather than states and as you'll see when i get to this point of the talk it's actually even not really breaking on symmetry breaking of operators it's symmetry breaking of a certain of a metric on a leigh algebra of a symmetry group which leads to a breaking of a distribution which leads to breaking of operators so i'll explain that sequence in due course so the starting point is the starting point for us will be the gue the gaussian unitary ensemble of hermitian matrices which is a probability distribution it's a density which is uh the probability of a hamiltonian drawing a hamiltonian is some normalization constant e to the minus l2 norm of the hamiltonian squared some people may recognize this may write it as a trace of h squared that's the same thing so this is a um i would call this the single particle distribution on hamiltonians there's it's sort of completely undifferentiated any in dimension n any n by n hermitian matrix is pulled out of this distribution with its entries iid identically independently distributed according to a gaussian its diagonal entries are pulled out that way and it's real and imaginary parts of its off-diagonal entries are pulled out from the same distribution it does not know anything about interacting physics it's it's i would say it's the undifferentiated uh template for hamiltonians and you notice it's based on a a metric i called it the l2 metric it's based on a metric on all hamiltonians which is a very symmetrical metric you could think of it as the round metric it's just the metric which assigns uh a length to the hamiltonian which is the sum of the square root of the sum of the squares of its entries so that's the geometry which leads to the gue but there are other geometries that are possible and other other metrics other metrics i'll just call the metric gij on the space of hamiltonians so let's just call them the hermitian matrices in dimension n so n by n hermitian matrices other metrics on this um vector space will lead to other distributions they'll lead to a new probability distribution which is based on g which would be up to normalizations e to the minus well the length of the hamiltonian in that metric the length squared in that metric which would be gij contracted with the components of the hermitian matrix in its basis so the index there represents the basis that i write down the matrix in so if i change the metric on the space of hermitian matrices i change the notion of a random matrix and uh i want to give an example of a second choice of metric just to kind of you know put us as they say on the same page by the way i hate that corporate terminology i think we should all be on different pages so uh this example actually motivated my interest in this subject and it's the last reference i put on the whiteboard the brown suskin paper the second law of quantum complexity uh and they talk about what they call the penalty metric sorry its indices together if you like i is a double index and j is a double index and the normalization is that the each very the variance of each entry is one uh yes i i the usual thing to do is to draw them independently distributed gaussian variables of there we go yeah so let me let me give you um an example of a second metric which i'll call the brown suskin penalty metric it's called penalty because it penalizes high body interactions so in order to explain this metric on hamiltonians uh let me first say what the herme the hermesian matrices are so the hermitian two by two matrices are the span of the identity matrix and poly x paulie y and paulie z i'm guessing at this point those letters are familiar this conference is okay and if i look at the hermitian matrices in a power of two this is simply uh the tensor product of n copies of the hermitian matrices which are two by two and whereas these were poly matrices x y and z a nice convenient basis using this tensor decomposition would be poly words of length n so the basis here is the span of the pali words so a poly word might look something like identity identity x identity z y more identities a total of n of these and i'm i'm not writing but i'm thinking tensor product between each of these symbols and that's a orthonormal basis with respect to the usual l2 or killing inner product uh and we'll just we can think in terms of that basis and then this brown suskin metric uh g i j is uh delta i j so it's diagonal on this basis but then it's some exponential i'll just say e to the minus uh the weight of the ith word so what is the weight the weight is the number of non one number of xyz letters in the polyword so this example i wrote for you had weight three so what does this metric do oh actually sorry i often think of the metrics inverse but for the metric i want a plus sign here not a minus sign so i want the metric to charge more to say you're longer in the metric you measure longer if you have three letters in there instead of two letters and this metric is remarkably very congenial to people who interest are interested in quantum compiling how to write a program for a quantum computer and for studying black hole physics the authors were more interested in black hole physics when they wrote this paper in 2017 but in the last four years the distinction between quantum compiling and black hole physics seems to have completely disappeared they seem to be the same subject so the idea of this metric from a uh compiling sense is that we think it's harder to build gates that involve more inputs and outputs maybe exponentially harder with the number of inputs and outputs so a metric like this might represent an appropriate measure of distance to get to a certain unitary which does the job you want like factors numbers you know like the shore algorithm unitary how long does it take to make that unitary as a composition of gates this would be a good measure but of course it's infinitesimal it's not discrete so this is more like the mike nielsen point of view on quantum compiling that you should understand it instead of discrete steps you should think of this hamiltonian evolution and this is telling you how expensive it is to evolve in certain directions and you might go to even a more extreme kind of cliff metric where it's infinitely hard to have three qubit gates and you have to build everything up by bracketing two qubit gates and that's that's kind of an extreme version of this metric and the connection with black hole physics is it seems that um in order to understand various time scales of black hole evolution you should think of information propagating you you should ask how does the state of the black hole evolve in time and if you let it evolve with the original killing or add invariant metric the l2 metric that i started with the time scales would be much too short things would happen too fast but if you look at the lead group the symmetries of the hilbert space s u n and you give it this penalty metric it has exponentially large diameter and n whereas the standard symmetrical metric it's like two pi so a metric like this is appropriate if it's going to take a long time for complexity to evolve in time yeah well it it no no not neces either way either way you can ask what's the diameter of the group if you're allowed to navigate but you just pay according to integrated arc length so every time you choose a hamiltonian it's a time-dependent hamiltonian it's like driving your car in the lead group and then you have a bill at the end that you have to pay which is sort of how far you went so in order to get the time scales to work out with various other predictions this seemed like a very attractive metric to them but it got me thinking about you know where qubits come from in the first place uh in this black hole discussion and i what i want to point out about a metric like this is let's just go through the steps we start with the brown suskin metric and what then we can make from that a a gaussian distribution based on that metric and i think i wrote the formula earlier it's just you measure length squared in the new metric instead of the l2 metric so that gives you a new probability distribution on the space of hamiltonians and then using that probability distribution you draw from it a hamiltonian and what you'll see when you draw this hamiltonian at random is it won't look like a hamiltonian you drew from the gue it will look like it's interacting physics this hamiltonian will look interacting and what that means is that with respect to the tensor structure in which we wrote this basis most of the mass of this hamiltonian must have its l2 weight will be represented in small body interactions it'll be very rare to draw something there that looks like it's million body interactions so now i want to think of this whole process as drawing h but h is now going to be the hamiltonian of our universe so something like the uh hurdle hawking hamiltonian or something like that i want i want to ask you know we live in an interacting universe how did it come how how might that have come to be what could cause like the most symmetrical possibility for selecting hamiltonians to be rejected and instead hamiltonians are drawn from a different distribution and that's where the symmetry breaking idea will come in that that will i'll explain how symmetry can be broken and lead to something like the brown suskin metric on hermitian operators and hence a probability distribution on hamiltonians okay so uh in order to do this it's now time to sort of start the math and just give you a definition yeah so your first metric yep the killing factor do you want to think of those hamiltonians as non-interactive yeah actually there's a historical irony here which maybe is i'm guessing is your question that wigner introduced the gue because he wanted to model complex interaction and he had no way of disentangling it except on the basis of symmetry class so in 1956 gue was introduced to represent interacting physics but it only represents interacting physics on a very superficial level in terms of the statistics of the spectrum you get the semi-circle law if you look more deeply if you look at entanglement measures and so on it's not representing interacting physics at all did i guess your question yeah okay yeah so uh yeah so what i wanted to find just first is the notion of a qubit structure on a hilbert space that initially has none but let's assume for convenience and we'll step away from this later that the space that doesn't have qubit structures has to mention a power of two so it could have a qubit structure we'll talk about what happens in other cases soon so suppose we have a space that happens to be a power of two dimension and then what a cubit structure is is an isomorphism j from qubits from n qubits so choosing an isomorphism from left to right here is what i'm going to call a qubit structure and it determines um an isomorphism which we'll call little j from the hermitian matrices which are two by two tensored up end times to the remission matrices on the large scale so before i mentioned that these two were isomorphic but the choice of the isomorphism is given by the choice of the qubit structure on the large space so with that definition i can come to the fundamental definition which is we say a metric gij on hermitian sorry let me use the dimension 2 to the n and 2 to the n by 2 to the n we say that such a metric is cac and this is a acronym for nose about qubits okay so the metric knows about qubits if there is a basis uh uh of principal uh principal axes for the metric i'll call that basis h k because of course they are hamiltonians they're vectors but they're they're in the space of emission matrices um if there's a basis such that for some fixed little j is above each one of these h k is j of the tensor product of 1 k nk okay so let's let's first absorb the definition um what does a metric look like you know how do you picture a metroid a metric a metric should be thought of as some kind of ellipsoid tells you how long it is to go in different directions that's what a quadratic form is of course in order to draw it as a ellipsoid you really need a background notion of geometry and that's where i'm using the l2 norm otherwise you just draw it around you'd have nothing to compare it to but when i talk about the principal axes i'm thinking the principal axes of this ellipsoid drawn in the with respect to the l2 geometry now if you're not comfortable with principal axes and like eigenvalues better if i just take the metric gij and i raise one of the indices to make it an operator i raise it using the killing form the add variant metric then the principal axes just become the eigenvectors so what i'm asking for i'm saying a metric knows about qubits if all its special directions all the principal directions of this metric actually are tensor products in some fixed qubit um structure so you don't get to change the qubit structure for instance if we did this with if we thought about the lead algebra of su4 we'd be looking at hermitian four by four matrices and i put a little zero there for the s we're taking out the identity matrix so it turns out this is 16 minus one this is 15 dimensional so there'd be 15 principal axes if there was no degeneracy and i'm only saying that a metric on su4 is cac knows about qubits if by some miracle all 15 of its principal uh axes factor as tensor products it's kind of like a one in a million thing it's not going to happen at random and in fact if you do dimension counting you see asymptotically as the dimension or increases um calc metrics are about square root uh common in the sense of dimension so i call that square root common that is if i have a space of metrics which is about a million dimensional the variety of cache metrics inside it is up to log factors like thousand dimensional i mean even a code to mention one variety you have measure zero of hitting at random but this is very thin variety so it'll be quite astonishing that we i'll present numerical results that symmetry breaking lands you know up to numerical precision like three decimal places into cac metrics it's not it's not kind of an accident okay so uh that's the one definition you have to remember [Music] so now symmetry breaking is always associated with some kind of potential or functional so we have to define a potential and i'll call it delta for a reason you'll see in a moment which will go from m and m will be my notation for the space this will be uh metrics this will be gijs or metrics on hermitian traceless hermitian and by n matrices so the sp so by the way that's fourth power then the hermitian matrices are like n squared many and squared over two and the um metrics on it of course is another square so this is fourth degree in m and this will be a potential function to r and what i'll be looking at is i draw this as a single axis here but this is this very high dimensional space of matrix m and what we'll find is some well some kind of it's not that wiggly it's a function oh you can't erase like that sorry oh all right we'll find some function that has a lot of it'll have a local maximum the functions i talked about usually will have a local maximum at the add-in variant um or the l2 i sometimes call it l2 and i sometimes call it killing i call it killing when i think of it as symmetric on the skew hermitian matrices on the lea algebra they call it l2 and i think of it matrices but you can always put an i and change back and forth but what we'll find is that many of these local minima that we find when we consider the an energy function i'll think of it as sort of a pseudo-energy delta many of these are cac you know when i say many like after hundreds and hundreds of finding local minima by extensive computer searches probably maybe even thousands of local minimum probably 20 or 30 percent or cac it's a order one fraction yeah let me give you i didn't say so let me give my favorite example of delta favorite example delta but it's not the best one for computation because there's no expansion parameter but the best example of delta theoretically is the ricci scalar so let me just ask does everyone see immediately why the richie scalars are functional on metrics so the point is that a metric on hermesian matrices is equivalently as i just mentioned a metric on the lee algebra of of s-u-n and a metric on the lee algebra is the same as a left-invariant metric on the entire group and in romanian geometry as soon as you put a romanian metric on the group you can start talking about curvature and then the scalar curvature is richie scalar curve which is a number that you get by contracting up all the indices of the riemann tensor and it's very simply expressed in terms of the two input tensors there's only two tensors in the game we have the structure constants c i j k of the lee algebra and we have the metric which in might be the brown suskin metric or some other metric gij and then of course we have the inverse metric g upper ij and the ricci scalar is actually represented by this diagram these diagrams the ricci scalar is i have just room to write it here it's minus one quarter this diagram minus one half of this diagram and i don't know if this notation is this diagrammatic notation is completely familiar to everyone it's sort of finement notation by the way i didn't find this in the literature but we computed this from milner's article on left invariant metrics on league groups but what this means in terms of you know written out in coordinates is each one of these y-shaped things is of course a copy of the three tensor and the curvy lines whether they're up or down like this or copies of the metric or the inverse so for instance this diagram writes out to c i j k c i prime j prime k prime now we have to contract the indices so g i i prime g j j prime and then lower indices to contract the k's that's what that diagram is explicitly in this diagram let's just do that one too i j k k is the index on top so any guesses how to get rid of this just touch it okay um so the top index went to the right index so that means i should put a k here this j index went to the top so this is a j here and this index here would be an i prime and that would have to get contracted with a copy of the metric inverse okay so this is just what the richie scalar is it's some number and of course it's the same curvature at every point because it's a homogeneous space so here's the amazing thing if you say well we're going to kind of consider this as an energy functional on left invariant metrics it has an addition to the maximum at the most symmetrical metric which is kind of a roundish metric uh it's a positive non-negative curve metric on the lead group but the metric gets all floppy and has negative sections becomes quite long and stretched out in different places as you vary this metric and very quickly you get all kinds of minima some of which we were able to look up in the differential geometry literature and then find numerically after spotting them in the literature and some we found by just searching with random seeds using a lot of machine learning and not me but emoji did the machine learning so now this is just representative i said the actual functionals i'll show you um i'll show you uh in the transparency here yes they know that i guess that's how it is but my question is i would think that you're further interested in pac metrics that are sort of like the penalty metric well yes but the point is our numerics i mentioned this n to the fourth we have to search metrics over a space that's about end of the four dimensional so the limits of our successful computer work has been three qubits n equals eight so the decay with high body is something that's only barely visible to us yes yes that is our hope so i'm going to go now to some data slides but before i do that the last thing i want to write uh is just to remind you of the the sequence of uh events so the first event is we pick a um a delta which is like our functional on metrics and then from that we get to a minimum we find a gij which is a minimum or a local minimum and then from that we create a probability distribution which is the gaussian based on that metric and then we sample we draw from that probability distribution and that's how we get our hamiltonian so that's how that's the proposal for you know in this toy model for the universe's hamiltonian and now um what i want to uh show you uh next is actually you know what are these um you know what are the deltas that are most useful to us what do we find so for that let me uh oops let me pull up my powerpoint good oh it worked okay so um this shows the is it too small to read okay the person in the front row said he could read it uh so this [Music] this functional if you can read it it's um it's kind of faux physics you know what we did is um we took like the first few pages of a standard text and field theory where they teach you how to write perturbative expansions for uh perturbed gaussians so what we thought is the g the metric that looks like a quadratic term and the c i j k that looks like a cubic term so you know these aren't scalars these are tensors but what we asked ourselves is what is the simplest finite dimensional integral of that form that we can write down um which we could then use as a functional on on on the space of metrics since you know g is in there and if you just write down the very first thing you think of it turns out that the cubic correction just vanishes uh identically because of the skew symmetry of the first two indices c i j and c j i give you the same value so to prevent that you have to do some kind of tripling of the state where you throw in a vector three times so that that's this formalism where x is three copies of y so i won't belabor this i'm just saying that i'm i believe this is the simplest perturbed gaussian that you can make out of those two tensors to get a number and um it gives you we can take look at its real imaginary part as possible functionals and we gave them nicknames we called i think one of them you'll see in the later notes is like the two four functional and the others the two six and that is just to remind us how many vertices were in the feynman diagrams up to where we truncated them so when we actually threw this on the computer of course we only used the beginning of the perturbative expansion and uh hidden in that formula is an expansion parameter k and by making um by by varying k we get more flexibility than we had with scalar curvature and we used that to do a little better job of searching out local minima but that's this was kind of the workhorse functional and then a very quick summary of what we found in su4 is this slide uh the ellipses on the left are labeled as cac and the ellipses on the right are labeled as sub because one thing we realized early on is that we might get some false positives we might get some the symmetry might break not break to something that particularly is concerned about qubits but it might be concerned about some subtly algebra that it likes to point hamiltonian terms in you know for instance s u n might break down to s o n or s u 2 n might break down to spn and we did find examples of that so it isn't that it doesn't happen and very often because you can write generators for these special of algebras with poly representatives it turns out that they can be interpreted as knowing about qubits but we thought they were a little bit illegitimate because they had another interpretation as having found us different subway algebra so we in order to be honest we highlighted the things we found according to the suspicious ones that might have a different explanation we put those left ellipses so in in this dimension um i guess everything we found by scalar curvature was in the intersection of the two but when we used 2 6 functional we found well actually in both cases we found some things that were strictly cac we found a symplectic thing that was both uh the two four functional we found many we also have the interesting thing which we're starting to explore now in much more detail we found the majorana pattern you know which it looks like a list of binomials that is if you take four majoranas that'll generate a 16-dimensional space and if you throw away the identity of the 15-dimensional space so you might imagine that you might get degeneracies of the principal axes of the um eigenspaces according to some pattern based on whether you were seeing operators that could be written as one two three or four majoranas and we actually did see that that was one of our patterns so parallel to the qubit discussion there's a majorana discussion which involves a different kind of decomposition the what the asterisk means i agree with you it says slightly delicate here and what that means is that this local minimum that had the majorana pattern was not persistent as we varied the expansion parameter we varied it between like 100 and 400 sometimes it came in and sometimes it went out then we went to su-8 because we're looking for cubit structures and again we found a lot of patterns and actually this slide is not updated we have about six more patterns that we have found in the last month oh i should tell you what these long strings of numbers means i didn't explain that so let me just go back um if you see a pattern like this this cac pattern three one one eight two if you add up those numbers you'll get fifteen yeah yeah uh that worked yeah okay good so fifteen is the dimension of the algebra of su4 and what this means is that we didn't have uh it broke into i'll call these principal directions eigenspaces i think you know it's raising the index the eigenspaces were not all one dimensional that would be a one one one we found up to numerical precision some of the eigen space is degenerate so this means we've got three dimensional one a one dimensional one eight and two and the reason we write them in that order is that is the order according to the easiness of the direction like the low bodiness so to speak it's the in in this brown suskin terms so this the larger eigenvalues we list first down to the shorter ones so for example if the pattern breaks is 10 and 5 is that that means 10 is easy and 5 is a hard direction to move in and that corresponds to the fact that the symplectic sub algebra is 10 dimensional so it's like su 4 says i'm 15 dimensional but there's 10 easy directions i'm going to be free to move in and five expensive ones which are like kind of the analogy of high body in this brown suskin world so it's actually more interesting that it came out it's 10-5 rather than 510 because you expect it to pick out the easy directions to move in hopefully they'd be associated with the subway algebra okay now uh the the other thing i wanted to say before leaving four and eight is you might we make numerical measurement of whether it's cac or not it's just not a yes or no thing the way i wrote the definition down for you either there wasn't tensor decomposition or not but obviously these are this is numeric so you have to have some tolerance and the really striking thing is we've never found an example where we didn't know if it was a good tensor decomposition or not typically two or three orders of magnitude separate the ones that are tensor factors from the ones that are not tensor factors so we we use some entropy measure to see whether it breaks up as a tensor you know it's the um reduced the entropy of the reduced density functional all right we take partial trace and we see what the entropy is and what we find is that the entropy is either tiny it's really almost zero up to two or three decimal places or it's order one [Music] uh okay now that was sort of the content of our first paper which posted at the end of the year last year but this year we decided to explore the numbers between four and eight yes can i ask why do you think your functionals are picking out yes great that's a good question um i think that the answer is uh if you have a any kind of reasonable natural functional and you minimize it the structure that minimizes it will be rather rich quite often for example if you um if you give some kind of repulsion potential to particles and ask for the minimum you may end up with the e8 lattice or the leech lattice in fact that's a rigorous theorem these days that for a continuous family of energy potentials there's a unique solution in dimension 8 and 24. so i think what i think this is a similar ilk that is we put out we put out a functional that's more or less natural and the structure that minimizes it is going to [Music] have a lot of symmetry internal symmetry just you know like the leech lattice has the monster group symmetry so i think that's what we're finding so i think nature likes these tensor structures and in fact this slide is very much apropos then that this one here because even when we put in things that aren't powers of two what we find is that well for instance when you put in six it's not such a big surprise that it factors as c2 tensor c3 that's just replacing one of the qubits with a trip that's kind of a boring generalization but what i think is more interesting is when you put in five what it will do and maybe i should go to the data slide for five su-5 so now there are 24 vectors that have to line up and i think the ellipsis on the left represented one of the functionals and we found nothing too foreign ellipses on the right represented the two six functional what we find is patterns where it what we call almost cac and what it does is it just ejects a one dimension from the five so what should be five by five hermesian matrices they break up uniformly with the same decomposition as a four by four block and you get kind of garbage on the margin and inside the four by four block you have a a tensor product to high precision of two times two so this is what we call the leaky universe scenario so you can imagine that the universe started with some single particle with a tremendously high number of degrees of freedom or a big hilbert space but it wasn't a power of two or factored in some nice way in terms of small primes so there's sort of a a repertoire number of dimensions that just had to be pushed aside and the rest of the universe could sort of coalesce into interacting degrees of freedom and this is sort of a unnerving thought because if you do a little number theory you turns out that actually here's an interesting little number theory problem you can work on um how common are numbers that are powers of two well they're not that common they get farther and farther apart right but how common are numbers that are their prime factorization only consists of twos and threes turns out those are vastly more common it turns out that you only have to round down by approximately log of the number before you hit something of that kind and if you have three primes in there it's uh one over log squared conjecturally that's that's not known that the log statement is so you can imagine that log many of the dimensions um got left out if things had to break into small primes and the part that broke into primes that turned into kind of our universe and the evolution and it looks um unitary as far as we can tell because there's only this one over log of this huge number error in um you know the the part of the the rows only the rows only leave the tensor product block on a very small fraction of their entries so it looks to all intents and purposes of its unitary evolution on the block even though it's truncated it's unitary evolution the big matrix but still you'd have some overlap with these um uncoalesced states that aren't part of interacting physics and it seems rather frightening if your wave function should overlap with those states that aren't even part of our interacting world it seems like it could not be healthy so that's that's sort of universe with a leak and uh and we did the same experiment so here's six i i didn't show you the data but six produced a lot of factorings into two and three and seven actually produced many interesting factorings and some of them seven split is six so two times three plus one and in some of these patterns it's split as two times two plus three so it seems like it was comfortable ejecting uh different amounts of material um see the last line oh this is just some additional data on su-8 i won't go through that so um let me just try to summarize um and you know then take some questions uh so yeah so there i think there are two things you might think about with um numbers that are not powers of two so one thing that we still have to understand better is whether powers of two are really special in some way or not whether clifford algebras for example play a role we'll be looking at su-8 in order to try to understand that you know maybe but i say our preliminary data is that um powers of two are not important and like six works in pretty much the same way as four or eight uh but you might wonder um you might wonder whether if the if the breaking has to go according to prime factors if if if that was the only thing that would happen then it would seem like there might be some very interesting information left over from the prime factorization of the original dimension so in other words if the original dimension of this hypothetical single particle was just some vastly large integer like googleplex you'd really be interested to know what is the statistics of factors of a random number selected in that range you know and it turns out number theorists know all about this there's like a constant it's called the gulab dickman constant which just for your amusement says that if you have a thousand digit number taken at random its largest prime factor will have 628 digits so that number 0.628 is telling you on a logarithmic scale what the expected largest prime is prime factor and a number and that's just like the tip of the iceberg they know all about these things so it's tempting to think if this is some kind of origin story it's tempting to think well what residues of number theoretic facts like that might exist in the low energy world of string theory or something um but then the evidence that i just showed you is maybe prime factorization isn't such an important thing in the story because maybe this leaky universe idea works that maybe the system is happy to eject some inconvenient degrees of freedom and just really really will break into small primes so that's that's a hypothesis uh and then we we want to understand um fermions and majorana fermions whether that somehow coupled with clifford algebras whether that gives interesting decompositions i'd say another thing that we have on our to-do list to look for is one isn't really just interested in isolation in what the initial hamiltonian of the universe h0 is it's the pair the initial hamiltonian and the initial state you should really be looking for h0 comma psi 0 because it's somehow the way they those fit together that must explain what seems extraordinarily extraordinarily special tuned about the way our universe evolves you know that it's constantly producing entropy and not coming to an end too fast uh so that i think it would be very interesting to look at for symmetry breaking producing a pair and that's actually trivial to do in our formalism because remember i showed you that gaussian integral all you have to do is add a source term to that gaussian integral where the source term is a projector onto [Music] a state and then the diagrams instead of being closed diagrams will have that state in it we can still evaluate those integrals and then when we minimize them we'll get pairs of g and and initial state so we're quite interested to do that and see how entropy evolves over intermediate time scales uh under the under the symmetry broken hamiltonian comma state uh and then the last the last project which i wouldn't say is exactly on her to-do list because we don't know how to start it is i haven't talked about spatial organization of these degrees of freedom yet um you know you one would like to conclude from this that the universe is 26 dimensional or something like that right uh you know make contact with some other origin story uh but you know that is hopelessly beyond uh numerical direct numerical study uh because you know we can only deal with a small number three or possibly four at the most qubits we can't possibly look for the leech lattice to start forming or something like that so we would definitely need an effective model for what the symmetry breaking is doing at sort of a larger scale you know a little bit like watson crick and franklin you know studied dna not with the quantum mechanics which was new at the time but by tinkertoys so we need obviously to reduce whatever we learn from quantum mechanical simulation into tinker toys and then use the tinker toys to study large systems we can't study them on a computer unless we build a quantum computer so yeah maybe i'll call it good there thank you

Channel: Institute for Pure & Applied Mathematics (IPAM)

Views: 222

Rating: 5 out of 5

Keywords: ipam, math, mathematics, ucla

Id: JkmC4aG0kEI

Length: 55min 19sec (3319 seconds)

Published: Fri Sep 10 2021