Prof. Neil Turok | Path integrals and the universe

hello Anna hi nice to meet you you could use a co-organizer you're organizing this okay oh good okay so please please please be rude just stop I'm ready good morning everyone so it is a pleasure to introduce Neil Turok who's going to talk about paratheticals and the universe so the universe is yours thank you okay thanks very much and thanks to Gerald for the invitation um I'm going to tell you about an application of path integrals to a very big problem um namely the universe uh maybe the biggest problem but actually rather small compared to the consensus view as I'll explain um a few papers recently we've put out five short papers on this topic with Lathan Boyle based on some earlier work we did and just to advertise something perhaps more directly relevant to this Workshop we've recently put out a long paper on Picard left shit's Theory as applied to quantum mechanics um and and their references in that so I kind of had a choice today as to whether or not to talk about this very technical and difficult work namely defining path integrals in real time uh rigorously we think we've made some steps towards that or I was going to talk about something very exciting and ambitious and brand new which is about the application of path integrals to the universe so I chose to do the the latter um and so this will mostly be a pictorial talk um to motivate the study of path integrals and the universe rather than dwelling on the technical details so starting point is why do we need a rethink well this is the current consensus about cosmology which is that the singularity is incomprehensible um the it was succeeded by a period of inflation which is a made-up Epoch in cosmology which preceded the standard hot big bang and the only role of this Epoch is to prepare the universe for its simple and uh pristine Evolution subsequently so basically the idea was that the Universe was somehow in a rather messy random state you introduce this dynamite called inflation to blow it all up and make it very smooth and simple and that's what we see and the consequences that on very large scales the universe is crazily complicated and even more incomprehensible and so a surprising number of people uh are interested in this concept of a Multiverse where our universe is just what we see is a small piece of this infinite Ensemble of different universes and the universe is very wild and chaotic on large scales it's absolutely not what we observe okay so personally I think this is uh likely to be a theoretical or dead end I think we need to take the observations much more seriously and and build theories that are much more firmly grounded in the basics and that's what this talk is all about in particular inflation predicts gravitational waves produced from this early phase of exponential expansion and these are not seen and the experimental bounds are coming down and down and down this is the current state of play um I had a bet with Stephen Hawking while he was alive and we were both working next door that um the signal would fall below five percent um and and Stephen agreed to that bet well now it has fallen below five percent but he's not around to pay what he owes me um and it's ever sinking so one can make inflation models in which this signal is arbitrarily low that's one of the things I don't like about inflation it's extremely adjustable but the simple potentials like fight to the fourth was ruled out a long time ago fire is not even on this plot uh fire it's up at 0.3 if I squared is now ruled out and people are playing with potentials like this which a little bit strange uh this has no minimum well neither these have a minimum but uh anyway this is where the consensus has led to rather strange and contrived models was crazy Multiverse on large scales so what's my point of view well I think let's start from what we know this is the physics we know is valid at least in some regime and and it works pretty well in fact it works amazingly well uh here is gravity gauge Theory particles described by by dirac's equation and the Higgs field and as far as we know this describes everything we haven't seen anything clearly inconsistent with this description so and notice I've written it as a path integral all amplitudes are as far as we know are described by this formula and let's just let's start from this and look for deeper look deeper at this formula I.E one immediate question is do path integrals like this make sense they're still not nobody has succeeded in defining them properly pick out lepshit's theory is our best best bet but basically this is telling us that the every process in the every physical process is nothing but an interference pattern this is the sum of phases you add up all the phases and and you get the the amplitude and that's the probability so let's think more deeply about this formula and let's start looking for simpler explanations of the universe we see which is stunningly simple okay so in small scales particle physics the surprise is how simple things are A Large Hadron Collider has not discovered any new particles beyond what we already knew or strongly suspected were there all the particles we know of we the simplest addition to the standard model is three right-handed neutrinos very good evidence for these from neutrino oscillation measurements solar and Atmospheric neutrinos and as far as we know this is it there's no evidence currently for anything beyond this surprisingly simple well the large-scale universe is even simpler even simpler it's amazingly simple all the observations we have currently are consistent with just five parameters describing the large-scale universe three for the matter or energy content the baryons per photon it's just one number the dark matter to baryon density ratio to another number and the cosmologic constant or sometimes called dark energy so three numbers for the matter and energy two numbers for the geometry these are conveniently parametrized by the gravitational potential the Newtonian potential as far as we can observe it's a random gaussian field with an RMS which is approximately scale independent so it's amazing clue just looking at the microwave sky or the Galaxy distribution we see a scale invariant spectrum of Newtonian potential there's a small there's a certain amplitude which is three times ten to the minus five and in this talk I'm going to tell you how to calculate that amplitude from the standard model okay it's kind of crazy but you'll see this is a new new result we believe we can explain this number so essentially a fine structure constant squared is is this number and so that's the amplitude and then there's a small red tilt meaning that the RMS fluctuation amplitude grows slightly at longer wavelengths only two percent effect and uh so these are two numbers and they suffice to explain everything that we see and they're now many many many observations which are completely consistent with this many quantities consistent with zero which a priority would be allowed like the spatial curvature of the universe the gravitational wave amplitude um deviations from what's called adiabaticity of the perturbations and so on and so on so let's try to explain this simplicity with the simple formula I showed you before here's the cosmic microwave sky um here's the fit to a theory which was postulated long before scale invariant spectrum of Newtonian potential perturbations fits beautifully with just the parameters which I've showed you and then in 94 I was new in this game and we calculated the correlation between the polarization and the temperature of the microwave Sky previous authors had claimed this was Zero we calculated it it's not zero this curve is has no free parameters if you fit the model to this curve so you fit those numbers I showed you to this curve this is a complete predict this is a prediction with no freedom and it fits perfectly okay so this convinced me something really simple is going on in the universe it it obeys the laws we know the laws that go into this are plasma physics and the Einstein equations plus the assumptions I listed on the previous slide so it's really amazing my point of view is that the basic puzzles in cosmology are our best Clues let's think about these puzzles let's not ignore them and in fact my basic principle is let's not introduce any new parameters or particles okay we've been doing that for 40 years it's led to the Multiverse let's just tie our hands and say no more okay no arbitrary particles we're not allowed to add any more there's no evidence for them let's try to solve these problems without adding anything more and Occam's razor should be our overriding principle okay so start with the big bang Singularity completely crazy I'm going to explain our point of view that the resolution of it is is conformal symmetry and analyticity okay and you'll see you'll see why the large-scale geometry of the universe why is it so symmetrical the universe is almost the same in all directions and it's spatially flat Euclid could have described the geometry of our universe okay you don't need Einstein to there's no spatial curvature it's just 3D flat space um and why and I'm going to tell you a new resolution which is that there is a measure on four geometries provided by the gravitational entropy we've calculated it using the standard model of of what we know and uh and it predicts that the most probable universes are flat like like the one we see the vacuum energy I'm going to explain if I have time a new cancellation mechanism this is really something that hits you in the face if you do Quantum field theory in a curved space-time you immediately discover these anomalies that doesn't actually make any sense the theory and as I'll explain we found a new way of canceling the anomalies and making at least a free field level this much more sensible and uh and and and and so on so I'm going to talk a little bit about CPT CPT symmetry once you open your mind to this point of view the solution of the Dark Matter puzzle becomes kind of obvious right dark matter is one of the things which have motivated all this model building but I'm going to explain that the Dark Matter was already on one of the slides I showed you okay and this is the new work has not yet written up but I will give you a flavor of it so let's talk about conformal Symmetry and analyticity so when we study cosmology we deal with line elements like this these are so-called Friedman Robertson Walker universes in which space is taken to be maximally symmetric and the the whole dynamics of the universe the the the if you like the zero mode of the universe is described by the scale factor okay so this is a convenient way to write the line element uh and T here is called conformal time because the metric is conformal to a static metric so then there's a rather uh immediate consequence which is that if we go back to the Big Bang Singularity right very high temperatures and let's imagine that the matter is conformally symmetric okay first approximation it is because it's radiation hot radiation which has P equals one third row traceless stress tensor but um it's not much of a extrapolation to imagine that conformal symmetry actually became exact at T equals zero okay it's a hypothesis um and in fact one could say that the only or almost the only conceivable explanation of the Big Bang Singularity would be in the theory that's conformally symmetric why because the scale factor goes to zero and the only way to describe physics when the size of the Universe goes to zero is if the size doesn't actually matter the physics is independent of the size so who cares that it goes to zero it's uh you have conformal symmetry so if we make this our starting hypothesis The Straits of the stress tensor was Zero at the Big Bang then of course the Einstein equations imply that the Richie scalar was Zero and then it's very easy to see that this function a of T is analytic at T equals zero okay just write out the equation and you find the solutions if you look in more detail this is the zero zero component of the Einstein equations it's the Friedman equation or hamiltonian constraint and it looks like this for a realistic universe in these coordinates okay so the um the the Friedman equation is uh only involves velocities no accelerations so a DOT squared rate of change of the scale factor squared is given by these formulas so you see for radiation you have a constant R describing the density of the radiation in a conformal frame which where you've removed the expansion and obviously the solution to this equation is that a is linear in t a is square root of R times T but now add everything in that we know of in the universe this matter this space curvature and there's a cosmological constant or dark energy I haven't put any numerical factors just so that you can see how simple the equation is and actually just looking at this equation you realize it has an analytic solution which is an elliptic function a particle moving in aquatic potential General solution is a is a Jacobi elliptic function and amazingly enough before our work nobody realized that that's not not a known fact most cosmologists would very happily just solve this on a computer so who cares that there's an analytic solution maybe people in this room would care but so as far as I know we're the first people to realize this is analytically solvable now that doesn't really matter as I said you could saw this on a computer but what does matter is to understand its complex analytic properties because as soon as you realize this is analytically solvable you realize that in the complex plane it has remarkable properties namely it's doubly periodic it's periodic in in a if you like in a real Direction periodic in an imaginary direction as soon as you have periodicity and imaginary time you have a temperature and then you have an entropy okay so it all follows from this more or less trivial observation so here is a typical solution in fact here all the solutions the one we'll focus on is this blue one which seems to be the real universe the scale factor came out of a singularity at which a was linear in T and the scale factor diverges to the Future as 1 over T it has a simple Pole and that's the solution to a universe with a cosmological constant is that a this conformal this conformal factor or scale factor has a simple pole in t so we live here in our past was the Big Bang singularity in our future is a simple poll that's a future future asymptotic Infinity of this base time um and so here it is a of T is now then the importance of knowing the analytic solution is you can see its structure in the complex t-plane it's single valued in the t-plane and doubly periodic it's only Singularity is a simple pulse periodicity and imaginary time implies the Hawking temperature Okay so that's what I'm going to use now I just want to mention in passing what initially motivated us to study this space-time was actually solving the Dark Matter problem okay what's the dark matter and there are tens of thousands of models of the dark matter but as I've explained we already we could already tell what the dark matter is based on what we know and this fact why well let's take this seriously let's take this analytic structure seriously it's telling us that the Universe has a mirror image Universe before the Big Bang where a of T is negative but of course a squared is positive line element is fine and uh and and and and and that's what the universe looks like okay and what this means is that the universe the analytic extension of the space-time we know this to negative T is actually at CPT symmetric image okay so see this space time has CPT symmetry T goes to minus t uh is time reversal symmetry with c and p uh you had CPT symmetry of the laws of physics and the background so our hypothesis was extremely simple the universe does not break CPT okay that's it from this we could and knowing the scale factor a of t we solved the Dirac equation for right-handed neutrinos that equation um essentially because neutrinos are conformally symmetric apart from the mass term and the mass term just turns up ends up going like t squared so it's analytic and so you just solve the equation for right-handed neutrino on this space-time impose CPT symmetry on the vacuum state and that automatically predicts how many right-handed neutrinos there are in the future in the asymptotes of this universe okay so there's no adjustability in this aside from the right-handed neutrino Mass there's one parameter they're created as a type of Hawking radiation from The Big Bang itself and we fit the Dark Matter observations if the right-handed neutrino mass is 4.8 times 10 to the 8GB so that's a prediction of the theory turns out that so this is a really simple explanation for the dark matter it's just one of the three right-handed neutrinos happens to be stable um which is consistent with what we know of of the standard model and there's a very clear prediction it turns out which will be tested in the next three to five years which is that if and I'll show you where the prediction comes from comes from me so here's the right-handed neutrino this one is the dark matter prediction comes from this so this is the Seesaw mechanism which explains the left-handed neutrino masses the left-handed neutrino can oscillate into a right-handed neutrino virtual right-handed neutrino for a brief instance and then oscillate back into a left-handed neutrino and as you turn up the mass of the right-handed neutrino the oscillation lasts shorter and shorter and the effective mass of the left-handed reader goes to zero that's why it's called a seesaw mechanism now if you switch off this coupling then the right-handed the corresponding right-handed neutrino is stable because this is the only decay mechanism so then it becomes a dark matter but switching off this coupling means that the corresponding left-handed neutrino is massless so the prediction if the if one of the right hand neutrinos is the Dark Matter then one of the left hand neutrinos is massless and that prediction is going to be tested in the next three to five years uh we already know two Mass differences there are three light neutrinos we know two of the mass differences from atmospheric and solar oscillations we don't know the absolute scale of the masses so our prediction is that the lightest neutrino is massless okay and uh and basically here is the current data the prediction essentially there are two predictions depending on the normal or inverse hierarchy the inverse hierarchy is already almost ruled out um and so most likely it's the normal hierarchy uh here are the current data pushing downwards and the prediction is 0.0.06 so as the data get more and more powerful which they will very quickly with new surveys the Prospectors of the this gaussian curve essentially focusing around 0.06 and this should be possible with 0.06 representing about five Sigma in in the measurements okay so if this works the way we predict then I think the Dark Matter puzzle is essentially solved this is easily the simplest model anybody ever made of the dark matter uh here's here's the five Sigma I mentioned okay now the main thing I want to talk about in this talk is this problem emphasized by Roger Penrose that the large-scale geometry of the universe is amazingly simple it's homogeneous isotropic and spatially flat it's consistent with euclidean literal euclidean geometry and so Penrose imagine the Creator you know picking this needle out of a haystack and saying that's the Universe I want how to explain flatness okay so I'm going to use an analogy that you know when I walked to the Newton Institute this morning I didn't have to worry about the curvature of the Earth because I didn't walk very far and so here here we are in the UK as long as we travel tens of kilometers something like that we really don't know about the flatness of the universe of the Earth surface but let's try and explain why it's flat anyway why why is the surface of the Earth so flat locally okay and and this is a real picture from space and you can see it's really this polished almost perfectly smooth rolling ball rolling ball so one explanation is somebody made the universe it was a mechanism and somebody hammered it into shape and made it and this is essentially the inflationary mechanism let's cook up a mechanism for flattening and smoothing the universe it's not a very good explanation there's a much better explanation which is that the Earth is large it's a big number 10 to the 50 atoms in the Earth and then we have gravity dissipation and entropy gravity pulls things in dissipation means they don't rebound or oscillate a very complicated geometry here with spikes and so on if one of the spikes collapses due to dissipation its potential energy is converted into heat and and there are vastly more ways of Distributing that energy among the heat among the Motions of the molecules and atoms of the earth than there are for putting very spiky creating very spiky geometries and so uh entropy explains why the unit why the Earth is so round and flat um I have to mention there's some beautiful recent supporting evidence for this explanation which is that the large mountain ranges on Earth are surprisingly enough they were formed shortly after life filled the oceans uh what happened is that there was a explosion of Plankton in the oceans and they ate up all the carbon dioxide and the Plankton then died and fell to the seabed and formed a layer of graphite they were squashed by sedimentation and converted into graphite and as you probably know graphite is an excellent lubricant so now what happened is as the continental plates on the surface of the Earth were floating around and when they collided there was now lubricant and so adjacent plates slid one above the other and that's how all the large mountain ranges were formed okay the Andes the Rockies the Himalayas and this recent work uh just last year studied I think 15 or 20 of the largest mountain ranges on earth in every case there's a thick layer of graphite which is correlated with the existence of these Peaks so basically this says when you remove the dissipation indeed you form but more random geometries gravitational entropy okay so this is what we're going to use to explain the geometry of the large-scale universe we won't have atoms instead we'll have gravitational microstates okay the number of quanta if you like of the geometry um that will be analog of the entropy of the uh of the Earth and so this topic is still a bit mysterious but there is by now a lot of evidence that one can associate an entropy with the space-time and this is the topic of black hole thermodynamics it started in the 70s and recently has had a Renaissance associated with a holography and ideas and similar ideas so we're going to use this concept we're going to calculate the the entropy of a cosmological space-time and this is the first time it's been calculated and what we'll discover is that the largest entropy space times are those which are flat spatially flat just like we see okay so um just as we don't need initial conditions from the a for the air in this room right the air in this room is very homogeneous and isotropic we don't need the initial theory of initial condition to explain it it's just the maximum energy state and likewise for the universe the maximum energy state turns out to be flat homogeneous and isotropic just like what we see so uh uh the starting point of these studies were the Hawkings observation of uh made just near here that black hole has a temperature soon afterwards this was realized that it corresponds to an entropy the entropy is proportional to the area of the Horizon and the mass squared of the black hole um and so these are very famous formulas moving one step closer to cosmology the simplest possible cosmology is deciduous base time where you have a cosmological constant an Einstein gravity and the solution is just this four hyperboloid embedded in 5D minkowski space and that's the city of space time so Hawking and collaborators Gary Gibbons and Malcolm Perry figured out how to calculate the entropy or the temperature and entropy of the city space time and the calculation is almost trivial you take the city space time you write it in global time coordinates continue the time to imaginary values usual Wick rotation and it becomes a forcephere obvious solution of the euclidean Einstein equations the Hawking temperature is nothing but the length of a great circle on the four sphere and uh and and so here it is and then as I'll explain uh slightly more formally the gravitational entropy is just the exponent in the path integral is I've put H bar to one but it should be is over H bar it's the exponent in the path integral as calculated on this classical solution so it's a saddle point calculation of the entropy so is of course the negative of euclidean action euclidean action looks like this the usual Einstein Hilbert term the dark energy or cosmological constant energy density and using the trace of the Einstein equations this just becomes rho Lambda times the volume of a four sphere put in the volume and you get the radius is H inverse put in the volume and you get this formula for the gravitational entropy today we have a cosmological constant if you work out in magnitude the value of the entropy it's 10 to the 122. so there's huge entropy associated with the geometry corresponding to an empty Universe with nothing but cosmological constant so at face value if you believe this calculation which I will okay uh it counts the number of microstates number of microstates is just e to the s but what so decided space time is not particularly interesting because it doesn't have any matter or radiation in it but uh and and essentially yeah so but we're just going to generalize this calculation and calculate the entropy of a realistic universe as a function of the cosmological parameters and then we'll understand how uh you know what type of space times it prefers so a few subtleties about the calculation a realistic cosmology is not close to equilibrium obviously came out of the Big Bang it's going towards future decider space um it's not perfect to sitter the sitter is uh completely um symmetrical has maximal symmetry so every point is the same as every other point so this is space time is really an equilibrium the real cosmology is not it's not asymptotically flat like a black hole so you don't have the mass of the black hole as a parameter to play with there's no asymptote in a realistic cosmology no asymptotic space spatial you know asymptotic boundary to space at the very least we have two very different temperatures we have the radiation temperature three degrees Kelvin today hotter in the past and we have the DeSoto Hawking temperature which is about 10 to the minus 39 of the radiation temperature today so it's tiny so you know there's two different temperatures so obviously it's not in equilibrium third third subtlety is that for spatially compact space times and that's what I'll assume either um a spherical geometry for positive curvature or a compact hyperbolic geometry for negative curvature so it's basically a Subspace of a three hyperboloid so I'll take space to be compact and then there's no boundary and so the total hamiltonian turns out to be zero how do you do thermodynamics when the hamiltonian is zero on all states okay uh and and that's what we'll do so one can still Define a statistical ensemble um and the key is to realize that the expansion rate of the space time or the rate at which things change is much smaller than the radiation temperature and so the typical time scale of the radiation is minuscule compared to the time scale of the change in the space time so the space-time evolution is adiabatic as far as the matter is concerned so that says first Trace out the radiation and matter and then perform the gravitational path integral so I want to make um yeah there's something very interesting property of part so we're going to use the path integral we're going to trace over the radiation and then do the gravitational path integral something may be of interest to people in this Workshop is that for path integral amplitudes transition amplitudes one can quote prove that the exponent is always have to be has to be has to have a real part which is negative okay this is essentially the Picard lefchets theorem says that any relevant subtle Point has to be has to have a height uh lower than the height of the integrand on the real axis which is so integrand is one on the real axis so moddy to the is has to be less than one and there's a series of papers with felberger we use this to disprove the Huddle Hawking proposal for the quantum wave function of the universe um so that's transition amplitudes what about statistical ensembles so here it's quite different you're not calculating a transition between one pure State and another you're calculating a Trace can also write that as a path integral with imaginary time so the amplitude is periodic in imaginary time beta and we all know the partition function is e to the S minus E over t not very surprising that this is exponentially large especially if you think about polymers or strings they have exponentially large density of states and so we expect something exponentially large and as we saw 10 to the 122 is a huge huge number for Gravity so here's the subtlety the hamiltonian annihilates all physical States so this partition function is just the trace of one okay it just counts the number of states um likewise the energy is zero for all physical States and and so on the right hand side we just have e to the s but the number of states has to be bigger than one I guess it could be equal to one I should put a equality there but for a universe we don't expect one physical state so the entropy the entropy has to be positive and and for the saddle point approximation to be valid the exponent has to be positive so in this case it has to be negative in this case has to be positive it's the opposite Behavior okay so now maybe it's not too surprising that the euclidean action for the uh giving the entropy of a black hole or decided space-time euclidean action is negative so that the exponent is positive so let's go to cosmology I've put in this lapse function here yes uh I'm just going to yeah it's time Reaper amateurization invariant so any system in which is invariant and the time re-parameterization as a Vanishing hamiltonian okay that's a theorem so String Theory uh any geometrical theory always has Vanishing hamiltonian as you see here all I've done is introduce a lapse multiplying the time so n could be any function of t and the theory is invariant under re-parametrizations and then basically it's a one-line argument to show that implies the hamiltonian must be zero okay so the way it works is that the path integral over this lapse enforces the constraint that the total hamiltonian is zero okay essentially it's set you see if the hamiltonian was not zero then time Evolution would mean something but in in gr there's no preferred time the time coordinate is just an arbitrary choice so you can't really have a hamiltonian otherwise that would define a time a particular time and there is no particular time so the resolution is the hamiltonian is zero so the way we'll do this is first Trace over the radiation okay so we have the gravitational hamiltonian this governs the scale factor of the universe and the radiation hamiltonian the usual um e squared plus b squared hamiltonian and we'll treat the radiation as the conformal fluid at a certain temperature we just Trace over the radiation by the way it's very important in this calculation that the radiation is conformally invariant so it doesn't see the expansion of the universe it completely decouples from Gravity the total hamiltonian is just a sum of two terms so when we do the radiation Trace we get the normal formula Sr is the radiation entropy u r is the internal energy of the radiation and and then we continue this formula by saying beta's i n to calculate this trace of e to the minus I hrn and voila so we've traced over the radiation we're left with the trace over the metric okay so now we have to do this gravitational path integral uh this this we do just write down the action these two terms come from the Einstein Hilbert term here's the cosmological constant here's the radiation I forgot to write the matter there should also be Mata term down here and we just have to do this path integral and there's some definition of these constants and the way we'll do this path integral is just by finding saddle point finding the classical solution plug it into the action and that gives gives us the gravitational entropy so I think I'll skip this this is a side point on Quant appeal theory in curved space time um so here we have uh the metric again Friedman equation I forgot to put the matter in here as well um and basically they're all the classical Solutions you can think about them all just just by realizing this is a particle in aquatic potential here the different cases so you have a universe coming out of the big bang and recollapsing or Universe bouncing like decider space time but it can have some matter and radiation in it so that changes its Evolution a little and then there are various uh analytic continuations of such space times to imaginary time and I won't dwell on them but we analyzed all of them um here is the picture of of the case of a flat universe in the complex T plane so the magnitude of a of T in the complex T plane so a of T has zeros like the big bang and it has poles like the future the sitter space-time it's a kind of curious thing that the fundamental domain in the complex T plane is square for a flat universe okay there's a special symmetry for a flat universe which is the the uh if there's spatial curvature then this fundamental domain is still rectangular but uh has different lengths in the two directions so uh so here's the modulus of a of t and then because of this double periodicity you see actually this represents a Taurus this is a function defined on a Taurus and whereas the real time evolution is along here the imaginary time Evolution being periodic can be thought of as defining an imaginary time Contour which goes wraps around a Taurus and the action is just the action along a non-contractable loop around that cycle of the Taurus so it's a topological invariant here it is so again these are some particular cases in the complex t-plane here's the Taurus the is we calculate is the action or exponent is calculated along a non-contractable loop around this Taurus and that gives the formula for the entropy uh so last month there was this article in quanta magazine about our work and they made a rather nice picture for us and basically this says that imagine I've got an ensemble of universes it might just be some notional Ensemble they don't have to exist and that Ensemble gives universes with different cosmological parameters we now have a formula that tells us the weight the statistical weight you should associate with each such geometry okay and and again emphasize they don't have to exist this is not a Multiverse this is just the a priori probability of getting a certain geometry so it provides the probability measure on cosmologies and it turns out if you as you read in our paper you get long formally involving Jacobi elliptic functions uh and but you can do all the integrals analytically and then plot the entropy as a function of the curvature of the universe and it turns out the most likely universe is a homogeneous isotropic and flat now this I haven't explained the homogeneous and isotropic so flat we can get from the results I explained homogeneous and isotropic we do small perturbations of the entropy we show those correspond to deak any perturbation decreases the entropy just like in homogeneity and the gas in the this room would decrease the entropy perturbations of the universe decrease the entropy and that that's therefore why the maximum entry state is homogeneous and isotropic you don't need any primordial flattening or smoothing mechanism right like inflation or apparatic universes one of my former sins you don't need anything like that okay just take Einstein gravity and calculate the entropy and it explains why the universe is flat cosmological constant of course is of huge interest massive puzzle called the biggest problem in physics and this calculation actually gives you a very good clue about what the cosmological constant is one interpretation of it which I particularly like is it's nothing but a LaGrange multiplier for euclidean for volume okay so if you think about euclidean action it goes like Lambda times the full volume okay so think of that as a chemical potential for for volume okay so I calculate my euclidean action as a function of Lambda but Lambda is really just a proxy for for volume and you can now actually go backwards and figure out what's the density of States as a function of four volume and uh and it's very interesting large full volume requires small positive Lambda The Ensemble doesn't exist for negative Lambda so people doing ads you're out of luck there's no Ensemble for ads space times if Lambda is positive there is a meaningful statistical ensemble and small positive the most probable space times are one with small positive Lambda and then very recently just in the last couple of days I I have realized this fact that in fact a well-defined Ensemble for the three volume the spatial three volume requires that Lambda is bigger than the matter density and today Lambda is about 0.7 critical density matter density is about 0.3 so the universe seems to be consistent with that this idea that small Lambda is favored by the way is pre-figured hawking wrote a paper about it Sydney Coleman people noticed this the decision result is that the entropy goes like one of a Lambda and so the probability is e to the one over Lambda so Hawking wrote a paper saying the cosmological constant is probably zero because the peak of e to the one over Lambda is that as Lambda goes to zero plus but I think there's a new argument now that actually there's a lower bound on Lambda as well as an upper bound and it seems to be consistent with what we see so I have a final topic but um I think in view of the time I will skip it just to allow time for questions so uh unless anybody asks me I'm willing to explain this cancellation mechanism uh let me see I have one other slide where did it go I seem to have lost a slide okay I won't explain that either I will oh here it is density perturbations okay so it's essentially another topic but I told you that there is a simpler way to get the observed density perturbations let me just say a couple of words about that I told you the trace of the stress tensor was zero in the first approximation in the standard model it's not zero okay it's small and the dominant contribution at high temperature actually comes from hypercharge the U1 gauge coupling that is not asymptotically free it turns out that in the non-abelian gauge Theory plasma the trace of the stress tensor scales as temperature at high temperature whereas the density is going like T to the fourth so it becomes irrelevant at high temperatures it's very very subdominant in the abelian theory the trace scales us t to the fourth and this is a well-known result calculated a long time ago it goes like the fine structure constant for a hypercharge squared and then you have to add up all the different particles in the plasma and their hyper charges so you get some number so then I can go through well I won't have time to go through this but basically you you have some extra feels around these are what we call Dimension zero fields you introduce them in such a way as to cancel this Trace to restore conformal symmetry they don't introduce any new degrees of freedom into the theory all they do is cancel the trace anomaly and it turns out these fields give you density perturbations with a scale invariant Spectrum and the amplitude turns out to be uh to differ from what we see by about a third it's amazingly close to what we see and it turns out there actually are some some dimensionless numbers you have to fiddle to get the right answer but as far as I know this is the first calculation of the primordial density fluctuations from standard model physics so um I will stop there and see if there are any questions [Applause] thank you okay are there any questions or comments do you think there's any um correspondence some of these Monte Carlo methods for yes summing over triangulations of metrics yes could there be some numerical evidence for this also so gravity is of course unusual because the kinetic term is negative kinetic term for the scale factor a is negative so the hamiltonian is zero on physical States and viewed as an operator it's not positive semi-definite it can take both sides off shell um and so so in fact this is an excuse to show to show something this is the kind of system you want to look at you see if you're doing Quantum field theory in a curved space-time partly motivated by our work on the no boundary proposal we basically said you must take the lorencian path integral seriously and look at these saddle points as complex saddle points and then decide if they're relevant or irrelevant partly motivated by this conservich and Siegel and Whitton considered Quantum field theory in complex space times and they classified those space times according to whether the quantum field Theory makes sense or doesn't make sense and basically it's all about the wick rotation oh you can't see this the allowed region is actually the negative imaginary lapse this n is the lapse leave t as a coordinate but but do the wick rotation as a function of n so this is euclidean quantum field Theory lives down here okay and these people argued Corner field Theory does make sense in complex space times but only within this half plane of the lapse now actually what we're doing is inconsistent with that because what we do is start from the lorenzian path integral we do a weak rotation this way for the matter but that doesn't work for Gravity because it has a negative kinetic term so we integrate out the matter using a standard Wick rotation and then we do the opposite Wick rotation for gravity okay now if you think about that that's that's very strange because that's corresponds to negative temperature and negative temperature sounds like a No-No but in fact negative temperature is fine for systems with finite number of states and this is well known in cadet's matter for example that if you have a spin system for example which uh you know the spins can all be down in the ground state for example classically they're all down in the ground state in the most in the highest energy state they're all up but there's only a there's a finite number of states and the partition function negative temperatures perfectly fine it just means all the spins are up okay so um you get into issues like that so what you need to look for is a system with uh you know a hamiltonian that's not positive but but has a finite number of states and then you will have something that corresponds to gravity so uh in yeah it's not obvious to me what type of systems would be the best models for this yeah yes did the Higgs yeah it's a great question there is no Higgs no it's a great question so uh again if you will allow me to cheat and show show the slide you see we we started off by noticing something very trivial which is that imagine we're allowed to introduce scalar fields with the kinetic term box Phi squared not grad Phi Squad okay yeah that's the that's the canonical view is that theories with higher derivatives are unstable however people don't seem to be aware of the fact that bogaluboff proved this Theory this particular Theory which he called a gauge Theory by the way has only one physical state which is the vacuum it has no particle States I don't want no excited States at all and essentially the argument is that this is uh has an infinite dimensional symmetry if I goes to 5 plus Alpha with oh the Box should be on the alpha with box Alpha equals zero okay so there's an infinite dimensional Symmetry and that symmetry turns out to remove all the physical States apart from the vacuum in the vacuum you get a scale invariant power Spectrum and that's why I got interested in this Theory because we look at the microwave Sky we see scale invariants I I said what's the simplest imaginable explanation well there's some field in the universe which has a scale and variant power Spectrum D3 and it's this Dimension zero scalar okay so there's only one physical state so there's no Pro it's not a pathological Theory at all the best way to understand it is brst symmetry and again brst proves that there's only one physical state it's by the way very large literature in these things none of it seem to be aware of the existence of this brst argument it's a paper by Revell which seems to be virtually unnoticed which shows that so we have some stuff we can add to the standard model without any adding any new particles all it does is change the vacuum and then our observation was here's the stuff that if you add the right number of these Dimension zero scalars turns out to be 36 you simultaneously cancel the vacuum energy and both contributions to the trace anomaly in the standard model and you predict there are three generations of particles okay this cancellation requires three generations of particles in the standard model and no more and and they have to include right-handed neutrinos okay so it's amazingly simple mechanism which makes Quantum field Theory couple better to gravity and then you are but no Higgs the cancellation only works if there is no fundamental case okay and so what's the Higgs so the Higgs must be a composite of these Dimension zero scalars and actually there's a very natural mechanism for forming a composite which is just copied from String Theory because in string theory the world sheet coordinate is a dimension zero field has logarithmic correlations e to the i k dot X the vertex operator has non-zero conformal dimension and so that's my guess that's what the Higgs field is it's a e to the IFI where Phi is a dimension zero field and the Higgs is the composite probably multiplied by fermions which give it the right gauge quantum numbers so it's only a guess but indeed very good point thank you sorry yes and the conversation is really interesting the mindful of time so once again uh thank you uh Neil for your uh wonderful talk thank you so we are now running uh five minutes behind uh well we will take over coffee break so

Channel: INI Seminar Room 1

Views: 16,597

Keywords: INI

Id: TdKYxSvNDQo

Length: 63min 10sec (3790 seconds)

Published: Thu Dec 15 2022