Random Matrices in Unexpected Places: Atomic Nuclei, Chaotic Billiards, Riemann Zeta #SoME2

what do atomic nuclei chaotic billiards and a million dollar math question all have in common well nothing really at least at face value atomic nuclei are physical systems comprised of protons and neutrons that interact with each other meanwhile billiards are a common example of a system studied in chaos theory a field which deals with dynamical systems that are sensitive to small changes and the million dollar math question we are referring to is the riemann hypothesis a now 163 year old conjecture about the properties of a particular complex function known as the zeta function aside from relating to technical subjects these three examples appear to have very little in common but looks can be deceiving and it turns out these three examples all take part in a bizarre and remarkable coincidence they are all surprisingly well described by random matrices that is to say these three very deterministic objects atomic nuclei with a definite composition of particles billiards that evolve under deterministic physical laws and a well-defined mathematical function all share a common thread of looking random [Music] this video is the first installment in what will hopefully be a whole series on the subject of random matrix theory otherwise known as rmt in this video we'll be assuming a basic familiarity with linear algebra and probability theory but that being said this video is meant to be introductory and we encourage you to continue watching even if you're somewhat rusty on those subjects we'll also be linking some great resources in the description box below so that you can reference them if need be before we begin it's helpful to know what kinds of context random matrix theory shows up in shown here are some examples of the fields that rmt makes contact with the topics are loosely color-coded according to their proximity to the four subjects listed in the corners really this connectivity diagram only contains a fraction of the topics related to random matrix theory mostly drawn from examples that are most familiar to the creators of this channel but hopefully it still conveys how rmt can play the role as the bridge between very different fields sometimes even making surprising connections between topics throughout the mini-series we hope to highlight many more examples of connections between rmt and other fields such as physics machine learning and number theory so what is a random matrix theory in short it's a model for matrices whose entries are random numbers usually they're constructed by specifying the probability distribution each entry is drawn from some examples are the gaussian distribution of mean mu and variance sigma squared or the uniform distribution from a to b the entries may be real complex quaternionic piatic or whatever type of number you please throughout this series we will usually be considering square matrices with real or complex entries as an example here's a 5x5 random matrix with real entries which are each drawn independently from identical uniform distributions between -10 to 10. we call the collection of all possible matrices that can be constructed from the random entries an ensemble an analogy with the ensembles one encounters in statistical mechanics as a quick aside in probability theory when we have multiple random variables drawn independently from identical distributions we call them iid random variables you may be wondering how this is any different from considering a collection of 25 random variables what difference does it make if we organize the elements into a rectangular or square object and call it a matrix indeed if all the entries of an n by n matrix are drawn independently then there really is no difference between regarding them as matrix entries or a list of n-squared independent random variables however most matrices when encounters in the real world have additional structure which places a constraint on the entries of the matrix so they are no longer all independent for example when working with real valued matrices one often requires that the matrix be symmetric meaning the entry in the i throw and the jth column matches the entry in the jth row and the ith column for any i and j you choose the consequence of this is that when choosing a random symmetric matrix we have less freedom in choosing the entries of your matrix or fewer free parameters once you've picked out a little over half the entries specifically n times n plus one over two of them you've determined the rest of your matrix entries but why should you choose to impose such requirements in short it's because the problem you're working with dictates it for example symmetric matrices emerge naturally in statistics where they assume the avatar of covariance matrices such matrices tabulate the covariance between all possible pairs of variables from a list or vector these variables may represent the prices of different stocks the firing rates of different neurons or whatever else you choose to measure in general given a vector x of variables x1 x2 and so on we can construct the covariance matrix for that vector by placing the covariance between x i and xj in the ijth entry of the matrix so the entry in say row 1 and column 3 would represent the covariance between x1 and x3 since the covariance between a pair of variables doesn't care about the order of the arguments the jith entry of the matrix is necessarily equal to the ijth entry which implies the covariance matrix is indeed symmetric one encounters similarly structured matrices in physics for example in quantum mechanics one is primarily interested in the hamiltonian which is a complex valued matrix that encodes the dynamics of the system the possible energies it can have and many other important features a physicist might care about you can think of each entry of the hamiltonian as encoding the probability that a system will transition from one state to another in a small time step with the ijth entry capturing the probability of transitioning from state j to state i since the hamiltonian is complex we can't directly think of the entries as transition probabilities since probabilities have to be real nevertheless it's not too hard to find the actual transition probabilities from the entries so you can interpret each entry as containing the information needed to find the actual transition probabilities meanwhile the eigenvalues of the hamiltonian represent the energies of the different states or configurations of your system for physical reasons we won't detail in this video the hamiltonian matrix is required to be hermitian which is essentially a generalization of the symmetry requirement for complex valued matrices essentially instead of the off-diagonal entries being equal to each other the off diagonal entries are complex conjugates of each other meaning each entry h i j is the complex conjugate of h j i meanwhile the diagonal entries which have no partners so to speak are required to be real the point of offering these two examples is to show that most matrices when encounters in the real world which we will henceforth refer to as physical matrices are required to have additional structure unlike a list of n-squared random numbers as we will show in this video and throughout the rest of the series these constraints are precisely what defines the salient features of a chosen random matrix ensemble so far we've defined the main players in a random matrix theory namely matrices with random entries which we will assume to have an additional structure like symmetry but what can we do with such objects as with any other study of random variables we will primarily be interested in computing ensemble average quantities that is for any given m we can compute some quantity which is a function of m like the entry mij or some function of all its entries like the trace or determinant and then average that quantity over all possible random realizations of m the process of computing this average can be written schematically as shown in this equation the integral over dm refers to an integral over all possible realizations of m or alternatively all possible realizations of its independent entries you can think of this as shorthand for a multi-integral over all independent entries mij meanwhile f m is the quantity we wish to average and p of m represents the probability density of encountering a particular matrix m there is a huge number of possible functions of a matrix that we can compute the averages of but which ones will be of the most interest or will come up the most frequently it turns out in many real world applications one is primarily interested in rotationally invariant quantities that is functions of a matrix that don't change even if you rotate the coordinate axes used to express the matrix let's look at an example suppose you have a system where the matrix of interest m governs transformations on real vectors for instance m may act on the pair of vectors here and transform each of them into new vectors such that the parallelogram they define has a new area this change in area represents an example of a physical transformation to objects in the vector space and in this case is governed by the determinant of n however if you were to say rotate your coordinate axes the actual coordinate of the vectors will be different and so the matrix that performs the equivalent transformation on the rotated vectors should also have different entries said another way the coordinates that the matrix m is written in should change in lock step with the coordinates used to express the vector space it acts on and in fact this is what it means to rotate a matrix in this example we denote by r the matrix that represents the change of coordinates on the vectors which in this case is simply a rotation it turns out the way we correspondibly transform the matrix m is to sandwich it between an r and an r transpose which produces a transformed version of m which we denote by m tilde this process of multiplying m by r and r transpose on either side is called conjugation by r and is precisely the recipe we need to ensure that m tilde acts on the rotated vectors in an analogous fashion to m on the original ones we can see this by rotating the transformed vectors and their corresponding parallelogram back to the original location note that throughout this process of transforming m although the matrix entries of m changed its determinant did not since the determinant of m tilde is the same as that of m so the determinant is an example of a rotationally invariant quantity or more mathematically an orthogonally invariant quantity since a rotation is an example of an orthogonal transformation the greater implication of this is that the determinant is a good candidate for a physical quantity we might want to take an average of when dealing with complex vectors as we often do in physics we use the complex generalization of orthogonal transformations known as unitary transformations correspondingly when we transform m we conjugate it by a unitary matrix instead of an orthogonal one in most of our examples we will demand that we have this more general unitary invariance by the way there isn't anything very special about the determinant since there are many more unitarily invariant quantities available the trace of m the trace of m dagger m and the trace of its powers the underlying reason that these quantities are unitarily invariant is that they are all functions of the eigenvalues and using some basic machinery from linear algebra it's not too hard to show that the eigenvalues themselves are unitarily invariant physicists often refer to the collection of eigenvalues as the spectrum of a matrix so we'll be focusing on the so-called spectral data of a matrix which are quantities related to the eigenvalues so to recap in our investigation of rmt we will primarily be interested in computing the average of unitarily invariant quantities and specifically quantities that are functions of the spectrum now that we've defined the quantities of interest things like the eigenvalues of a matrix or functions of the eigenvalues we can now use the quantities to compare different matrix ensembles when we say ensemble a looks like ensemble b what we mean is one or several rotationally invariant quantities computed by averaging over ensemble a looks the same as when you average over ensemble b we would like to say that two matrices whose entries are drawn from the same ensemble indeed look the same for example if both m1 and m2 have gaussian random entries then we would expect them to have a similar eigenvalue spectrum but what if m1 and m2 were different sizes but with the entries still drawn from the same ensemble if m1 is a 500 by 500 matrix and m2 is 1500 by 1500 then we would still get a similar spectral profile in both cases they both look like semicircles but one is simply bigger than the other it turns out this is just an artifact of the fact that the magnitude of the eigenvalues scales with the square root of the size of the matrix we can actually compensate for this by cleverly scaling down the variance by a factor of n so while the semicircles are different when we don't scale the variances appropriately when we have the variances scale like one over n the two semicircles have the same width even though they came from matrices of different sizes in general scaling the variance of the entries in this manner will allow us to do comparisons like both of these spectra look like semi-circles without having to worry about what size matrix they come from so far we've discussed what a random matrix theory is but now we want to study how it actually works and to do this we first have to actually choose around the matrix theory remember that any random matrix theory essentially has two ingredients number one is a set of matrices we've already specified those to be either real symmetric or complex hermitian because those are the matrices that we work with in physics or care about in real life scenarios number two and this is a more interesting point is a probability distribution on those matrices which we denote by p of m and the reason this is an interesting choice is that we're essentially asking the question what is the probability distribution for these matrices that accurately captures the physics whilst maintaining sufficient simplicity to be amenable to calculation while in physics the canonical example is the harmonic oscillator which is simple enough to be solved easily but is also powerful enough to capture the physics of almost all systems near equilibrium and this is actually the same for random matrix theory what we call the hydrogen atom or harmonic oscillator of the subject is the gaussian ensemble or really gaussian ensembles we should say because there are several ones of them and that choice actually depends on point one which is whether we choose those matrices to be real symmetric or complex hermitian now we usually write the matrix distribution in the form of a partition function denoted by z if you've seen statistical physics before you know that the motivation is that we can calculate essentially all physical quantities of interest once we know this partition function but if you haven't seen it before don't worry at all you can really think of it as just a normalization factor for the probability distribution notice also how we have two things that we've seen before in this expression number one is this unitarily invariant or rotationally invariant trace of mm-dagger and number two is the scaling factor of n over two the one-half is essentially there to make the gaussian look a little nicer but the n is really important because it ensures that our eigenvalues lie within some finite range the new ingredient however in this expression is the measure dm we've currently chosen it to be flat meaning that each matrix entry mij is integrated with an equal and flat or uniform weight dmij the reason this choice of random matrix theory is easy and amenable to calculations is twofold number one is that the entries are uncorrelated and what that means is that the probability distribution p of m factorizes into a product of individual probability distributions for the matrix entries number two is that both the weight and the measure are unitarily invariant so they depend only on the spectrum we haven't actually shown that the measure is unitarily invariant and it's really not obvious why this should be true for the weighting factor it's pretty clear because the trace itself when you conjugate a matrix in there you can bicyclicity of the trace cancel that effect out and so it's invariant but the measure is a little more complicated to understand and the true reason is actually quite deep and it's difficult to show but we can get a good sense of it by looking at a much simpler model which is just a two-dimensional gaussian integral we immediately see that the probability distribution can be factorized so we can sample the random variables x and y independently furthermore both the weighting factor and the measure are invariant under rotations the form is just because rotations have unit determinant and the ladder because the weighting factor x squared plus y squared is just the radius squared now to make the connection with our matrix partition function let us write the trace and the exponent out you can clearly see how this big matrix gaussian is really just a product of gaussians for the individual matrix entries we've motivated the rotational invariance but we haven't motivated the independent and identical entries so far and it turns out that this is actually just a choice that allows us to do calculations analytically if you think about it it's actually very unphysical because if you visualize this matrix m as having matrix entries mij that denotes some transition amplitude between input state j and output state i it suggests that there are absolutely no correlations between any such processes finally you may wonder if the gaussian is really the only ensemble that satisfies both one and two for instance we've seen before that any higher trace moments or even the determinant are also unitarily invariant so why would exponentiating those not give quote-unquote good random matrix theories well the reason is that precisely they would correlate the entries and as we've said before this would cause us some great trouble in computing analytical averages or carrying out the integrals so the answer is yes as shown rigorously in a theorem by two mathematicians potter and rosenstein the gaussian is the only ensemble that fulfills these two requirements this section is going to be a little longer so we'll give a quick breakdown the kind of setup or teaser of the problem is to find the distribution of the spacings of the eigenvalues we've just now built a theory where the matrix entries are uncorrelated so what do you think will happen to the eigenvalues and especially their spacings will they also be uncorrelated first thing we're going to look at what the distribution of eigenvalues would look like if they were all sampled independently next we're going to give an intuitive argument on how this should be different for eigenvalues that are correlated after that we'll get to the main meat namely we'll look at the distribution for the random matrix theory we've just introduced above the gaussian we'll do this for the simplest or smallest possible matrix a 2x2 matrix the goal is to find the distribution of spacings between uncorrelated events or eigenvalues to do this we'll actually look at the cumulative distribution function of the spacings so define q of s which is the probability that the spacing t is greater than s you can also think of q of s as the probability that there is no eigenvalue a distance s from any given one we can write out a differential equation for this quantity by considering what happens in a small interval d s what this equation means is just that the probability that there is no eigenvalue in the interval s plus ds is the conditional probability that there is no eigenvalue in the interval ds given that there was no eigenvalue in the interval s in the case of uncorrelated events this factorizes and we can write q of s plus ds is the product of q of s times q of ds next we take advantage of the interval ds being small so that we can expand q of ds around zero we know that q of zero is equal to p of t greater than or equal to zero which is equal to one simply because the probability density is normalized or another way to think about it the probability that there is a spacing greater than zero has to be one so taylor expanding this to first order we find that q of ds is 1 minus c times ds where we've defined c to be the probability of having an eigenvalue in the unit interval and notice that this is a positive fraction because we have a finite number of events or eigenvalues in a finite range so our equation now looks as q of s plus ds is equal to q of s times 1 minus c times ds we can just rearrange this to get a more familiar-looking differential equation which in the limit of ds going to zero which in the limit of ds going to zero gives us an ordinary differential equation we can solve it to find that q of s is a decaying exponential as a function of s now the last step to finding the pdf from the cdf is that we have to take the derivative one minor detail is that there is a minus sign because we took the cdf to be the probability that there is a spacing greater than s so we're integrating from a lower bound to infinity all of this gives us that p of s is equal to c eta minus cs what this tells us is that for uncorrelated events or eigenvalues the distribution of spacings is exponential now what happens if the events or eigenvalues are correlated we can still set up the equation as before q of s plus ds is equal to the conditional probability of ds given s however now the conditional probability doesn't exactly factorize anymore and that's because q of s is no longer translation invariant in s namely if these eigenvalues are correlated then whether there is a spacing or not will depend on how large the spacing already is so to account for this we add a functional dependent on s to q of ds which we denote by subscript i know that this is not mathematically rigorous but just bear with me we're just trying to get an intuition at this point and we'll get the rigorous derivation soon enough so what should this functional dependence be since the eigenvalues are correlated we can guess that if the spacing s is already large the probability of having an eigenvalue in their interval d s should be large whereas if the interval s is small it should be small the simplest model for this is a linear behavior in s as before we can set this up as a regular differential equation this time it has a solution of a gaussian again to find p of s we need to take the derivative with a minus sign where we've absorbed the constants into a proportionality symbol granted we made some rather rough assumptions along the way but for now we found that the distribution of spacings for correlated events or eigenvalues is a gaussian multiplied by a linear and s term now that we've built up some intuition let's work out the simplest example explicitly the simplest example is just a two by two matrix where each matrix entry is sampled from this simple harmonic oscillator potential or the gaussian that we've introduced before and it is very easy because in the case of a two by two matrix we only have two eigenvalues and hence only one spacing we're also going to be working with the matrix m being real symmetric just to make our life a little easier because then we only have one real value on the off diagonals instead of two real values if the entries were complex now we want to get the eigenvalues of this matrix so we have to solve the characteristic equation and in this case it's just the quadratic polynomial and eigenvalues lambda so it's not too hard to solve we obtain two eigenvalues lambda plus and lambda minus and to get the spacing we just take their difference and we always sort them in increasing order so that the spacing is always positive now let's look at the spacing and pause for a second you can see that the spacing is a square root of two positive terms so this is really begging to be interpreted as a distance and you can actually really think of the difference between the two energy values as the distance of a point from the origin with coordinates h11 minus h22 and h12 and we see that this distance can only be 0 when h11 equals h22 and h12 is 0. furthermore if you make this analogy with a distance from the origin you can kind of figure out that for small values of s or small values of the radius in this case the distribution itself should be proportional to s because the volume element in polar coordinates is proportional to the radius we usually write it as r dr d theta in our case r is just s but at the same time we obviously know that p of s can't just be proportional to s for all s because that would suggest that the eigenvalues are completely unconstrained and they can really just fly off from one another if their distance can be arbitrarily large okay so that's just a little bit of intuition looking at s but we can actually find it because we have all the ingredients so we set up this integral here where we specify the spacing to be equal to what we found above and now we have these three probability distribution of these matrix entries and we've already specified those to be the gaussians the only subtlety here is that the variance of the diagonal matrix entries is twice that of the off diagonal entries and that's just because we've sort of constrained the matrix to be real symmetric so that is sort of you can think of it as a constraint on the off diagonals so their variance is a little smaller okay this is a good place to pause the video and try to figure out how to solve this integral and if you're not in the mood for it then don't worry we'll do it together now as a reminder we're letting ourselves be guided by the interpretation of s as a distance from the origin but before we can actually convert to polar coordinates we need to do a small linear transformation to make our coordinates a little nicer or a little more symmetric since this is a linear transformation again we can just absorb it into the proportionality constant because the jacobian is just going to be some constant we immediately see that the v variable disappeared from the delta function so we can integrate it out and again absorb it finally we can do the promised conversion to polar coordinates and again what happens is that we have an integral over theta that is just a volume element so we can absorb it and again we get an integral over theta that factorizes so we can just absorb it into the proportionality constant and we're left with a simple delta function integral for the radial variable from this we find that p of s is proportional to s e to the minus s squared and this is exactly what we sort of guessed based on our intuition of having correlated eigenvalues so it's really really nice to see that this intuition and the simplest example already match and for those of you who figured it out congrats you've essentially derived the wigner surmise for the smallest possible random matrix as a small comment this distribution is not yet normalized but you can easily do this by hand or in mathematica to conclude we've derived two distributions one for uncorrelated random variables and one for correlated random variables we found that the distribution for the uncorrelated random variables is just a poisson distribution which is an exponential decay in the spacing on the other hand for the eigenvalues that are correlated that came from the random matrix ensemble we looked at we found that the limiting distribution is a linear function of s multiplying a gaussian this distribution is referred to as the vigner dyson distribution and the important distinguishing factor between the poissonian and the vigner dyson is in the limit as the spacing s goes to zero for the wigner-dyson distribution the limit as s goes to zero is zero which means that the probability of two eigenvalues lying on top of one another is zero on the other hand the poisson distribution goes to one which means that there is actually a very high probability of the spacing being 0 or the eigenvalues lying on top of one another this is the important distinguishing factor between correlated and uncorrelated eigenvalues and why we have come to refer to the latter as experiencing repulsion note that this level repulsion came out of doing a change of variables from matrix entries to eigenvalues before we move on let's do a little recap of what we've learned so far we started by asking what is a random matrix and we said well it's just a matrix with entries sampled from a probability distribution next we asked what is a random matrix theory and we say well it's when we start studying ensembles of such matrices specifically we're looking at quantities that are averaged over the matrix distribution next we asked what is the simplest random matrix theory we can think of and the answer was the gaussian it's the only random matrix theory that has both iid entries and is unitarily or rotationally invariant finally we just answered the question of how are the spacings in this gaussian random matrix theory distributed and we found that the answer is a probability distribution that goes to zero as the spacing goes to zero what this means is that the eigenvalues don't want to lie on top of one another or we just colloquially say they repel remember how we said that the hamiltonian encodes the dynamics of a system but in order to predict those we need to extract the spectrum of the hamiltonian and to find a spectrum in a quantum mechanical model we need to solve the schrodinger equation if you haven't seen quantum mechanics before don't worry you can really just think of this as an eigenvalue problem and since we're now talking about physical systems we'll change notation from lambda to e because we think of the eigenvalues as eigen energies so back to the equation we want to solve it for h representing the hamiltonian of a uranium nuclei at this point in time we're doing this purely out of curiosity rather than trying to construct an atomic bomb but despite our peaceful motives it's not looking good for us uranium has over 200 protons and neutrons in its nucleus so that's going to be one very messy hamiltonian to write down and it's not only messy there are truly interactions in there that we don't know how to model however very many things rushing around and impossible to keep track of should ring a bell the ideal gas think of gas molecules in a box in a room there's order 10 to 23 of them and we can't possibly follow every particle's position and momentum however instead of mapping each particle out we can course grain the model and talk about the probability of a particle occupying a patch in phase space so we associate one probability pi with each such microstate the distribution that maximizes entropy at fixed energy then tells us that this pi is proportional to the exponential weight of this energy multiplied by a factor of inverse temperature note that in the equation above i is the state of the system not the energy otherwise would also have a degeneracy factor so what have we actually done we've replaced the exact state of the system with a probability distribution associated to the energy of set state so we've washed out some microscopic details and are left with the macroscopic variable that we can actually measure instead of phase based coordinates these could also be discrete spin configurations in an ising model but the point is the same we can't possibly keep track of all the spin configurations however we can calculate the energy associated to them if we postulate the interactions between the particles i'm not saying it's easy but at least it's not impossible now the lesson learned is that the large complexity of the system actually helps us perhaps ironically it's much more difficult to say something about a 6 than a 10 to the 23 particle system in the latter we can justifiably replace the deterministic laws of the individual particles with probability distributions that accurately reproduce quantities we care about at the macro scale turning back to our nucleus hamiltonian we can use a similar approach here but at the level of the system itself not just the state namely instead of writing the exact deterministic matrix of the nucleus we model it with a random matrix and instead of obtaining the exact spectrum of h we can say something about its statistical properties for instance we can answer questions like what is the likelihood to measure the eigenvalue lambda is 0.310231 or what is the likelihood that there is a spacing of s is 1.2 for the atomic nuclei it turned out that this approach worked remarkably well the next example where random matrix theory shows up in the spectrum is the canonical example of chaos a single billiard in a square arena with a circular barrier kind of like a pen pole game the hamiltonian consists simply of a kinetic term plus a potential term that's infinity outside of the chosen region setting the potential term to infinity is just how we simulate those barriers the differential operator in the kinetic term is known as the laplacian and since we've imposed this boundary condition it behaves in many ways like an ordinary matrix now if we numerically solve this laplacian in a simple program such as mathematica we find that the high energy part of the spectrum is distributed like the wigner dyson distribution we've seen before and it turns out that we actually understand the reason for this and it is related to the system being classically chaotic but billiards are just one of many deterministic real world physical systems that display random matrix theory statistics there is a whole class of quantum systems which don't have a well-defined semi-classical limit that exhibit random matrix theory statistics in their spectrum one such class of models are one-dimensional spin chains if you haven't encountered such models before don't worry you can just zone out for a minute and appreciate from afar that there is yet another seemingly unrelated system that displays random matrix theory statistics in spin chains the degrees of freedom are binary variables often referred to as qubits the hamiltonian associated with such systems assigns an energy to each spin configuration and these depend on the parameters that we put into the model by hand like how the spins interact with one another or how they react to external forces the main reason we mentioned this model is that it lends itself even better to a face-to-face comparison with a random matrix look at the hamiltonian written out explicitly for two three and four qubits where the letters just refer to some of the interactions we put into the model do you notice something two things should come to mind first of all there is no randomness at all each matrix entry is deterministic and models a well-defined physical interaction in terms of the chosen parameters such as j1 and j2 that we've written into our model secondly the matrices are extremely sparse in fact there are only order n non-zero entries and yet somehow the spectra of these very sparse non-random matrices shows similar statistics to those of a random matrix there are a lot of subtleties and details we we're skipping over here for instance there is a particular energy scale or energy window for which these deterministic systems look like random matrices but the takeaway remains there are many systems for which we put no randomness in by hand and yet they show random matrix theory statistics [Music] what these systems suggest is that the wingnerdyson distribution is truly a universal property for almost any system that is sufficiently complex in fact this empirical observation motivated the definition of quantum chaos as showing random matrix theory statistics for our third and final example we turn to a topic that is very different from the first two and that is complex analysis unlike the first two examples in which the deterministic matrices came from physical systems here we're going to start off with a purely abstract mathematical function the riemann zeta function briefly the riemann zeta function is a particular complex function that maps complex numbers to complex numbers and is often denoted by this funny greek symbol here for zeta for an input with real part greater than one it takes on the following form so zeta of z is the sum of the reciprocals of the natural numbers raised to the power z for inputs on the rest of the complex plane the zeta function is defined in a different manner which is slightly more complicated this is because the expression above diverges for inputs whose real parts are less than one long story short when the expression with the sum diverges one uses a different expression that ensures that the zeta function in a sense has similar behavior on the rest of the complex plane as it does to the right of one by the way this process of extending the domain of the zeta function is called analytic continuation and we'll link some sources about this in the description so why do mathematicians care about the riemann zeta function for starters it encodes information about the distribution of prime numbers you can get an intuition for this by recalling that all natural numbers may be decomposed into their prime factors so in principle you could write the above infinite sum over the natural numbers as a sum over all possible prime factorizations as we've alluded to before there's this long-standing open conjecture about the zeros of the riemann zeta function mathematicians know that the function has a zero at every negative even number which are called trivial zeros since they're more straightforward to find the rest of the zeros called non-trivial zeros are known to lie somewhere within the strip between 0 and 1 also called the critical strip but mathematicians can't prove exactly where they are nevertheless billions of the zeros that have been found all seem to lie on the line at one half riemann's 163 year old hypothesis with a million dollar bounty is that all of the zeros in the critical strip necessarily lie on this line admittedly one could do a whole video on just the riemann zeta function its deep connection with prime number theory how it provides a universal approximator for any complex analytic function and all the beautiful connections it makes with quantum chaos theory and we very well may make a video on this in the future but for now let's skip to the reason why we're mentioning it in the first place and ask how does the random matrix signature we've seen before make an appearance in the riemann zeta function let's take a look at the non-trivial zeros that all seem to be on the line at one half it turns out even though there's no obvious pattern to wear along the imaginary axis the zeros lie if one makes a histogram of the spacings between the zeros one sees the following distribution and there it is our now old friend the wigner dyson distribution even though these zeros are not random and have nothing to do with matrices they're just some zeros of a function they behave like eigenvalues of a random matrix isn't that remarkable this behavior was first noticed due to a chance meeting between mathematician montgomery who had been looking at the distribution of the non-trivial zeros and physicist dyson who recognized the level spacing distribution of the gaussian ensemble this is perhaps one notable example of the very real benefit of interdisciplinary mingling unsurprisingly this remarkable coincidence has led to an explosion of work at the intersection of physics and math one last thing we should mention is that the appearance of random matrix statistics in the spacing distribution of the zeros has led to a second conjecture called the hilbert poila conjecture it surmises that the zeros are determined by some quantum chaotic hamiltonian remember that we said that the eigenvalues of the hamiltonian are interpreted as energies and these are real numbers so one posits that the imaginary parts are given by the energies of some quantum hamiltonian and we've established from before that there's a connection between quantum chaos and random matrix theory so that if such a hamiltonian exists we also expect it to be quantum chaotic some candidates for hamiltonians have been found but there's still a lot of ongoing work in making the arguments rigorous and proving that the candidates can indeed be regarded as hamiltonians of a quantum system [Music] [Music] [Music] you

Channel: ℏistas

Views: 73,123

Keywords: SoME2, Dyson, Wigner, Riemann, Nuclei, Billiard, Chaos, Quantum Spin Chains

Id: Y4mnlIvVJEs

Length: 41min 1sec (2461 seconds)

Published: Tue Aug 16 2022