What Every Physicist Should Know About String Theory: Edward Witten

welcome to the second half of this special session on the centenary of general relativity our next speaker is Professor Edward Witten from the Institute for Advanced Study in Princeton a professor which is a major figure in the development of string theory he's also found unexpected connections with mathematics is the only physicist to ever win the Fields Medal in my eyes he's going to talk to us today about what every physicist to know about string theory thank you so much so for those of you from other areas of physics of it and also for students who aren't yet string theorists I'll be telling you what you might want to know about string theory if you're not working in that field of course if you do want to work in string theory there are lots of more specialized things you'd want to learn for those who are string theorists I won't be saying anything that's remotely new as will become clear one suck at one modesty nasaw after mine will perhaps feel a little more contemporary but possibly I'll help remind us to appreciate some of the wonderful things that we sometimes come to take for granted so as I said I'll be trying to explain today the minimum that any physicist might want to know about string theory I'll try to give answers to a couple of basic questions how to string theory generalized standard quantum field theory why does string theory force us to unify general relativity with the other forces of nature well as David goes from archit at the end of his talk standard quantum field theory makes it so difficult to incorporate general relativity why are there no ultraviolet divergences in string theory and what happens to einstein's conception of space-time I thought explaining these matters even though what I'll be telling you about them is not extremely contemporary is possibly suitable for a session devoted to the centenary of general relativity anyone who studied physics is familiar with the fact but while physics like history does not precisely repeat itself it rhymes with similar structures at different scales of lengths and energies for example the gravitational waves we heard about in the second talk are now August two light waves or a waterway and many other kinds of waves that are studied in many different systems many different length skills will begin today with one of those rhymes an analogy between the problem of quantum gravity and the theory of a single particle now in Einstein's theory of gravity the geometry of space-time is dynamical rather than space-time being something that exists beforehand in which physics unfolds how space-time unfolds is part of the theory and what that means is that although we don't really understand it quantum gravity sunl should be some sort of theory in which at least from the macroscopic point of view we average in a quantum mechanical sense over all possible geometries of space-time in the classical theory there is a definite space-time geometry that unfolds quantum mechanics involves fluctuations in which in a quantum mechanical sense we would have to average over all possible geometries we don't know to what extent this description is valid microscopically but at least macroscopically that's what it oughta mean to quantize the geometry of space-time now in the simplest case the averaging is done with the weight factor which is the exponential of minus the einstein hilbert action so I've written the Einstein Hilbert action and in the traditional way R is the curvature scalar and lambda is the cosmological constant which as David ghost remarked was introduced by Einstein in his first attempt at a cosmological model nowadays we know it's important in the real world now we could add matter fields but general relativity in four dimensions is a perfectly interesting theory without matter fields so we could think of this as the purest version of Einstein's theory will you consider only gravity and not matter fields so the physical dimension of space I'm at least microscopically is four so we're interested in Einstein's theory in four space-time dimensions and that's hard and we don't understand it quantum mechanically what would happen to if we practiced by making a theory like this in which the space-time dimension was one and set a for well one thing that would happen is that there are not many options for a one manifold or I've more or less drawn the only possible one dimensional manifolds and in contrast to the four dimensional case there is no Riemann curvature tensor in one dimension so there's no close analog of the alling sine Hilbert action but Einstein's fundamental idea was really not the Einstein hilb reaction but the principle of equivalence which he developed into the idea that the geometry of space-time is fluctuating that the fundamental laws are our laws in which the geometry has to be determined by solving equations and we can do that in one dimension even though there isn't an Einstein help reduction and but to do it in an interesting way now we have to include matter fields so we can still make a non-trivial theory of quantum gravity by which I mean a theory where the geometry fluctuates but will have to include matter to make an interesting theory so in the simplest case we take the matter to consist of some scalar fields which i've called x sub i where i ones from 1 to d d matter fields and well write down the most obvious action we follow i'm sons general rules that we're supposed to write generally covariant action so we have a 1 by 1 metric tensor that we use to write the action I've included a cosmological constant but I've renamed it as M squared over 2 and then the scalar fields for the scalar fields I've written the most obvious generally covariant action functional which is just the square of the time derivative of a scalar field but waited the way Einstein would tell us to wait it now if we introduce the canonical momentum just as in classical mechanics the momentum is X dot the time derivative of X then the Einstein field equation which is supposed to be the field equation that we get by varying the metric the action with respect to the metric tensor the Einstein field equations in this example is going to be a very equation it just says that P squared plus M squared is zero but quantum mechanically P is a derivative with respect to X and we would interpret P squared plus M squared as minus d squared by DX squared plus M squared and the statement that at zero quantum-mechanically really means that this operator should annihilate the quantum mechanical wave function so the content of our general relativity theory in one dimension tells us that this operator should annihilate the quantum wave function well that's an operator that you may have seen before if you've studied classical general relativity the equation we found is the relativistic klein-gordon equation in d dimensions but in Euclidean signature if we want to give a sensible physical interpretation we probably should reverse the sign of the action for one of the scalar fields so the action will look more like this where one of the scalars which I'm calling x0 has an unusual sign of the action and now the equation abate by the wave function is a klein-gordon equation in learn signature so we found an exactly soluble model of quantum gravity in one dimension and what it describes is a spin zero particle of mass M propagating in d dimensional Minkowski space-time so what we found is certainly something that textbooks and classical general relativity describe but they don't necessarily describe it in the way of explains us coming from one dimensional quantum gravity actually we could replace Minkowski space-time by any D dimensional space-time em with the Lorentz signature metric gij and then the action looks like so and the equation abate by the wave function will now be a klein-gordon equation on our spacetime m so we could get not just the klein-gordon equation in flat space time but the general relativistic version of the klein-gordon equation where here general relativistic means in four dimensions or in d dimensions the a general relativistic equation we get from our baby version of quantum gravity in one dimension so we got the mass of klein-gordon equation in a curved space-time now to continue though I want to make the equations a little simpler and more familiar so i'll abbreviate g IJ p IP j SP squared and also write formulas in Euclidean signature just to avoid having to explain a few details now let's calculate the amplitude for a particle to start at a point X in space-time and ended another point Y so here's a picture the particle started at X and ended at Y and it followed some path in classical physics we would solve Newton's laws to determine what should have been the path but in quantum mechanics any path is allowed but a path has to be weighted with the exponential of minus the action so but what is the action well but in quantum gravity there's not only a path in space line but there's also a one dimensional geometry so part of the process of evaluating the the amplitude to go from X to Y in quantum gravity is to integrate on them over the metric on the one manifold modulator if you morphisms in other words modular what Einstein called general coordinate transformations but up to a few more phys a more general coordinate transformation a one mount of photos only one invariant the total length town no matter what the metric is on this path the only invariant of one dimensional geometry is the total length tell which will interpret as the elapsed proper time so it has to calculate the amplitude for the particle to get from X to Y where an arbitrary proper time tail might elapse along the path and it might get from X to Y along an arbitrary path well first we'll do this keeping tau fixed and if tau is going to be fixed we can take any one metric that has length tau for example the one metric could be one we're the one manifold is parametrized by a time variable T that goes from zero to tell so and then keeping tail fixed we simply integrate over all paths the particle might follow in a way that was described 50 or 60 years ago by richard fineman when he introduced fine integrals and then the precise calculation I'll describe is the first example in the book of fine men and here besson path integrals so all the one manifold parameterize by time that goes from zero to tell we have to integrate over all pairs that start at X at time zero and ends at Y a time town and the action started as being a general relativistic action but once we set GTT to one it's simply a standard non relativistic action for a point particle so it's literally the same action that you would read about in Feynman and hips where they explain how to do this basic path integral so according to Fineman the integral over all paths so the Hamiltonian of the nonrelativistic quantum mechanics is P squared plus M Squared and according to Fineman the result of the path integral is the matrix element of the exponential of minus tau H where H is the Hamiltonian so H can be diagonalized by going to momentum space so to get the amplitude from to propagate from Y to X we integrate over all momenta if the momentum is P the amplitude is e to the IP dot Y minus X and then we include the matrix element in a plane wave sight of the exponential of minus teletypes the Hamiltonian so this would be the amplitude to get from X to Y in a proper time town what we have to remember to do the gravitational part of the path integral which here just means to integrate over the elapse proper time so the complete path integral for our problem integrating over all metrics in all paths modulo de few morphisms gives the integral from 0 to infinity over the proper time of what we had for fixed proper time which was this so we have to integrate this expression over the Overtown and that just turns the exponential of minus Talon times P squared plus M squared into 1 over P squared plus M Squared so finally our answer is what's written here that's the amplitude to propagate from X to Y and it's the standard Fineman propagator in Euclidean signature a similar derivation and Lorentz signature would give the correct Lorenz signature Fineman propagator with the I epsilon so we've interpreted a free particle in d dimensional space-time in terms of one dimensional quantum gravity how it may include interactions there's actually a perfectly natural way to do this there are not a lot of smooth one manifolds but there's a large supply of singular one manifolds in the form of graphs there is a graph for example our quantum gravity action makes sense on such a graph we just take the same action views before summed over all the line segments that make up the graph so we integrate the action over this segment and this segment in this segment and so on so now to do the quantum gravity path integral well the quantum gravity part of it is that we integrate over all metrics on the graph off to general coordinate transformations in other words up to de few more physics but the only invariance or the total lengths or proper times of each of the segments so I've labeled some of the line segments by a proper time variable tau 1 tau 2 or Tel 3 I didn't label all of them the natural amplitude to compute is one in which we hold fix the positions of the external particles and integrate over all the proper time variables and all the paths of the particles follow on the line segments do it now for the moment we keep fixed the proper times and just integrate over the paths here we observe that if we specify the positions at the vertices and therefore on each end of each line segment so this line segment goes from y1 to y2 this one goes from y1 to x1 but now all vertices are and external on points are completely labeled so each end of each line segment is labeled and we specify at what point in space-time that should be so once this is done the computation we have to do in each line segment is the same as before and gives the Fineman propagator so we have a graph and to each segment in the graph we have a Fineman propagator but we still have to integrate over the internal positions y1 up to Y for in this example the integration over the Y's will just impose momentum conservation at vertices and we arrive at fineman's recipe to compute the amplitude associated to a fine than graph we take a Fineman propagator for each line and an integration over all momenta subject to momentum conservation so if you're familiar with Feynman graphs will recognize you'll recognize that what we've arrived at is what Feynman said to do and otherwise just accept that somehow this quantum gravity procedure applied to a one-dimensional world with the space-time that was a graph has materialized fineman's recipe for computing a scattering amplitude in standard quantum field theory so we've arrived at one of nature's rhymes if we imitate in one dimension what we would expect to do in four dimensions to describe quantum gravity we arrive not at quantum gravity in space-time but if something that's certainly important in physics we have arrived at ordinary quantum field theory in a possibly curved space-time we've not arrived at quantum gravity but we've arrived at quantum field theory in other words at the mathematical framework that's used in the standard model of particle physics in a given space time which could have been fired or it could have been curved in the example I gave the ordinary quantum field theory we arrived at is scalar Phi cube Theory because of the particular matter system we started with and assuming we take the graphs to have cubic vertices quartic vertices for instance would have given 5 for theory and a different matter system would give fields of different spins so many or maybe all quantum field theories in d dimensions can be derived in this sense from quantum gravity in one dimension so that's one of nature's rhymes but there's actually a much more perfect rhyme if we repeat this in two dimensions that is for a string instead of a particle so one thing we run into immediately is that a two manifold Sigma can be curved I've drawn one and it's curved so you see in general gravity in two dimensions will not be nearly as simple as it is in one dimension there will be a Ricci scalar and an Einstein Hilbert action so the integral over two dimensional metrics promises to not be trivial this is related to the following fact a two dimensional metric in general is a two by two symmetric matrix so it's a two by two matrix where G 2 1 equals G 1 2 but it has three independent matrix elements this one this one and this one now of the three matrix elements in the matrix you can remove two of them by a few morphism that locally depends on two arbitrary functions but you only can remove two of the three functions so you really will be left with one arbitrary function in the metric tensor to integrate over and if that's the answer you will not get a close analogue of what I explained for the particle but now we notice that the obvious analog of the action function that we used for the particle which is this one well now I simply integrate in two dimensions instead of one but I write the obvious analog of the action we used for the particle is conformally invariant that is it's invariant under multiplying the metric tensor by any positive function each of the volume if we require conformal invariance as well as if you morphism invariance then this is enough to make any metric on sigma locally trivial locally equivalent to the Euclidean metric which is analogous to what we had when we were studying the point particle and some very pretty 19th century mathematics now comes into play here I've drawn an example of a Fineman diagram I've picked a simple one with only three propagators and therefore the one-dimensional geometry only depends on three proper time parameters now below I've drawn the string theory analog of this Feynman graph which I've achieved by slightly thickening every line into a tube that represents the propagation not of a point particle but of a closed string and then every vertex I've smoothed out so that the tubes join smoothly and now it turns out but as in the one-dimensional case the metric can be parameterized by finally many parameters it can be parameterize up to general coordinate transformations and also rescaling of the metric which are called vinyl transformations and so there's a fairly close analogue with the classical case but there are a few important differences one difference is that the parameters are complex rather than real but a second difference which will be more important today is that the range of these parameters is restricted in a way that allows no possibility for an ultraviolet divergence so we'll make a theory that in some ways is analogous to the traditional one but it is a theory in which there can't be an ultraviolet divergence so I've drawn another picture to underscore how to manifold is understood as a generalization of a Fineman graph so I drew one example here but here's another example in this case this is a one loop Fineman graph for two particle scattering two particles came in and to go out and on the right I've thickened it slightly replacing the particles by tubes so the line here is the world line of a particle the tube is the word tube of a string and now the tubes joined smoothly so the singular one manifold with vertices is replaced by the smooth two manifold but here is a one loop Feynman diagram for two going to two scattering in the one-dimensional or two-dimensional case now I come to a deeper rhyme than the one we've told about before with one-dimensional quantum gravity we arrived at quantum field theory in a spacetime that might be curved but we did not arrive at quantum gravity in space-time the reason we did not get quantum gravity in space-time to state it in a very succinct but slightly perhaps abstract way is that in quantum mechanics there is no correspondence between operators and States if you've read a quantum mechanical a book on quantum mechanics or if you take a course on quantum mechanics you'll learn about the quantum wave function which is a state in Hilbert space you also learn about operators like the position or the momentum that act on the quantum states but nobody ever tells you that there's a map between operators and states because in general in quantum mechanics there isn't so what we did with our 1 dimensional quantum gravity was we considered the 1 dimensional theory with a dynamical metric tensor and then we arrived at Fineman diagrams and the external states in the Fineman diagrams were just the states in this quantum mechanics remember the first thing we did was to quantize the quantum mechanics and get the klein-gordon equation and the quantum mechanical States were the solutions of the klein-gordon equation a deformation of the space-time metric would appear where well the space-time metric appears in the action so if we change the metric a little bit that would represent a change in the action and the action would change by the integral over time of the change in the Lagrangian density so if I change the space hi metric gij a little bit the Lagrangian density will change by this operator curly o so that operator would represent a change in the space-time metric but not a state in the quantum mechanics so as I said our operators in quantum mechanics do not correspond to States if you ever studied quantum mechanics you weren't told that no one ever told you of a correspondence between operators and States and that's why the one-dimensional Theory did not describe quantum gravity in space-time in fact the one-dimensional theories I presented it led to Phi Cube theory in space time rather than quantum gravity an operator Oh such as the one describing a change in the space-time metric appears on an internal line in the Fineman diagram not an external line so you see to calculate the effects of the perturbation we insert the integral of the perturbation well we integrated with the position on the graph where the operator is inserted I just drew one possible insertion points that could be inserted anywhere but it's not particularly at the external points that represent the particles in the quantum mechanics or the states in the quantum mechanics so because there is no correspondence between operators and states in quantum mechanics the objects which are scattered in our Fineman diagrams do not have to do not generically describe fluctuations in the geometry of space-time so we arrived at quantum field theory in a fixed space-time but not at quantum gravity in space line but in conformal field theory and remember to make the theory work in two dimensions we had to assume conformal invariance in conformal field theory there is a correspondence between operators and States which actually is important not just for string theorists but also in statistical mechanics and certain areas of condensed matter physics so the operator in the two dimensional case the operator that represents a fluctuation in the space-time metric automatically does represent a state in the quantum mechanics and therefore the theory those describe quantum gravity in space-time so since the correspondence between operators and states that exists in two dimensions were not in one is the reason that string theory describes quantum gravity in space-time I want to take a few moments to explain it it comes from a nineteenth-century relation between two pictures that are conformally equivalence so here I've drawn a two manifold a 2-dimensional world sheet as we call it in the jargon and in that red dot is a point at which an operator might be inserted but 19th century mathematicians discovered that if you remove the point from that red point from the two manifold the metric on the two manifold is conformal equivalent to this one with on tube going off now if a long tube goes off we'd say that the what is at the end of long tube would the estate in the quantum mechanics well in this picture we'd say that the points on the two manifold is inserted an operator in the quantum mechanics that represents a fluctuation in the action perhaps coming from a change in the metric of space-time so the conformal equivalence between these two pictures is the reason that while in one dimension we got quantum field theory in space-time in two dimensions we get quantum gravity and based on I therefore will take a moment to explain in more detail this conformal mapping for the basic idea we write the metric of a plane which I've written a two dimensional plane a flat plane I've written the metric of a plane in polar coordinates where think of inserting an operator at the origin of polar coordinates now if the origin of polar coordinates is the point we're going to remove and after we remove it we make a conformal transformation multiplying the metric by 1 over R squared here's the new metric and now I introduce a new valuable U which is log R it goes from minus infinity to infinity and now the new metric is just a five metric again but now on a cylinder because Phi is an angular variable so I have described a conformal mapping from the plane minus the origin to the cylinder so this conformal mapping gives an equivalence between an operator inserted at this point and in initial state and the quantum mechanics that comes in from minus infinity on the tube that conformal mapping is the reason but unlike in the one dimensional case string theory leads to a description of quantum gravity in space-time so we arrived at quantum gravity in space-time but we could have tried to arrive at quantum gravity and space-time by taking the einstein hilbert action reading drock's book about the passage from classical mechanics to quantum mechanics and formally following the recipe in fact people did that starting in the 30s but it doesn't work well because one runs into intractable ultraviolet divergences which in the 30s seems only a little bit worse than the analogous divergences in autumn agonism but by nineteen by the 1950s when people had learned to tame the divergences in quantum electrodynamics and even more later when the full standard model of particle physics was developed it became clear that the methods the tame the ultraviolet divergences in quantum field theory without gravity do not work for gravity so now we should ask if this alternative approach to quantum gravity and space-time eliminates the ultraviolet divergences which plague a naive attempt to quantize Einstein's gravity according to tech support rules so I want to spend a few minutes to explain why this type of theory does not have ultraviolet divergences well first of all where do ultraviolet divergences come from and field theory they come from the case that all the proper time valuables in a loop go to zero so I've drawn a 1 loop diagram that we discussed before and now I've labeled all of the internal lines by proper time valuables t12 l1 and l2 and so on up to tell for and to evaluate the Fineman amplitude or if you wish to evaluate the one-dimensional quantum gravity path integral as we interpreted it we have to integrate over these proper time valuables in the example shown ultraviolet divergences can come on all the proper time valuables going around the loop simultaneously go to zero so that the total proper time elapsed around the loop vanishes now it's true as I said that a Riemann surface can be a riemann surface is just a fancy nineteenth-century name for the two dimensional manifolds that we are using instead of Fineman graphs it's true that a riemann surface can be described by parameters but roughly mirror the proper time valuables of Fineman graph so here's a picture I drew before with the proper time parameters and there are analogs in two dimensions but as I said before there are a couple of crucial differences and one very important difference which is the reason that there are no ultraviolet divergences in string theory is that although the proper time valuables tale of a Fineman graph covered the whole range from 0 to infinity the corresponding two dimensional parameters towel hat are bounded away from 0 so the reason sorry the region integration that leads to the infinities of a Fineman graph are simply absent when we replace conventional Fineman graphs that our eyes from one dimensional quantum gravity with their string theory analogues literalized from two dimensional quantum gravity so what I've just said is a matter of some more 19th century mathematics and I won't give a complete explanation but I will give a restrictive example let's say but first let's just say again given a Fineman diagram we can make a corresponding riemann surface but the mapping only works well if the proper time valuables tell all I are not too small so the read the reason sorry the region that leads to trouble in conventional quantum field theory is lost when we go to the string theory while the physical phenomena that are important in quantum field theory and that we observe really have to do with large tell and those persist in string theory so I won't give a general explanation but I'll explain how it works for the wom loop caused logical constant so the Fineman diagram is simply a circle with a single proper time parameter tail which is the circumference of the circle and the resulting expression for the one loop causal multiple constant and is what I've drawn here written here where we integrate over all town from 0 to infinity H is the particle Hamiltonian and the trace is a trace over all want all states of the particle hilbert space and the integral diverges at small town because of the momentum integration that's part of the trace so technically when you perform this trace you get a badly behaved function of town or more exactly Paves badly for small town tail going to zero and that's the famous ultraviolet divergence of quantum field theory now going to string theory root means replacing the classical 1 loop diagram by its string G counterpart which is a two dimensional manifold that's a torus 19th century mathematicians showed that every Taurus is conformally invariant to a parallelogram in the plane with opposite sides identified you glue the top to the bottom and the left to the right but to explain the main idea without extraneous details and keep our chairman happy regarding the time I'll consider only rectangles instead of parallelograms so we'll label the height and base of the rectangle as SN T so we have a rectangle in the plane whose height is s in whose base is T but you see only the ratio of T over s which I'll call you is conformally invariant so in a conformally invariant theory we will integrate not over thien hau seperately but only over you and also since it's arbitrary what we mean by the height and what's the base of the rectangle we'll free to exchange s NT by exchanging s and T we can consider only the case well exchange guessin T exchanges you with 1 over u so up to a general coordinate transformation of our two-dimensional world U is the same as 1 over u which means we can restrict to well these statements mean we can restrict to the region where T is no bigger than s surface a no smaller than us or equivalently we take the range of U to be from 1 to infinity now the proper time parameter tau of the particle corresponds to u in string theory and the key difference is that instead of towner which goes from 0 to infinity we have U which goes from 1 to infinity so although the one loop caused multiple constant and field theory is given by this textbook formula in the approximation of considering only rectangles and not parallelograms the string theory analog is almost the same but goes the integral goes from 1 to infinity and set it from 0 to infinity so we simply are missing the region that would've led the trouble in field theory there's no ultraviolet divergence because the lower limit on the integral is one in zero and just for fun I'll say if you include parallelograms the lower limit becomes one half root three instead of one but the important thing is that there still is a positive lower limit so I've explained a special case but this is a general story the string key formulas generalized the field theory formulas but without the region that can give ultraviolet divergences in field theory the infrared region lines up properly between field theory and string theory and that's why a string theory can imitate field theory in its predictions for the behavior at lower energies or at long times and distances and that's why string theory can be compatible with physics as we developed it in the twentieth century but the region that gives the ultraviolet divergences disappears and so there's no difficulty in going to quantum gravity but you can't avoid going to quantum gravity because of the operator state correspondence that I told you about now I want to use last few minutes to explain at least partly in what sense space-time emerges from something deeper if string theory is correct and you'll hear a fuller and probably more contemporary explanation in the next lecture by one Malta Sina will focus on the following fact the space-time M with this metric tensor g IJ of X was encoded as the data that enabled us to define a two dimensional conformal field theory that we used in this construction moreover that's the only way that space-time entered the story we could have used in this construction a different two-dimensional conformal field theory now if the metric is slowly varying compared to the natural length scale on which string theory modifies conventional physics so the radius of curvature is everywhere large then the Lagrangian that we use to describe the two dimensional conformal field theory weakly coupled and illuminating this is the situation in which string theory matches to ordinary physics that we are familiar with we may say that in this situation the theory has a semi classical interpretation in terms of strings in space-time when this description will reduce at low energies to an interpretation in terms of particles and fields in space-time when we get away from the semi classical limit for Ranjan is not so useful and the theory does not have any particular interpretation in terms of strings in space-time in fact the following type of situation very frequently occurs what I've drawn here schematically this region is meant to schematically indicate a family of two dimensional conformal field theories that depends on two parameters that parameterize this region and generically there's no particular interpretation in space-time but there's a limit where you discover strings in a spacetime m1 but there's another limit where you find strings in a different space on M two and in the third limit you discover m three so generically there's no particular space-time at all but in different limits one space-time or another might emerge and lots of other non obstacle things can happen for example in a situation where in classical general relativity the space line develops a singularity the two-dimensional conformal field theory may remain perfectly good meaning that the physical situation in string theory is perfectly sensible so we can say that from this point of view space-time emerges from the seemingly more fundamental concept of two-dimensional conformal field theory from this point of view a more fundamental concept is the two-dimensional conformal field theory that might be described in terms of a spacetime but not necessarily space-time emerges in a limit from a more fundamental concept and in general a string theory comes with no particular space-time interpretation but such an interpretation can emerge in a suitable limit somewhat as classical mechanics sometimes arises as a limit of quantum mechanics though you can study a quantum system that's very far from any quantum mechanic from any classical limit now this is not a complete explanation of the sense in which in the context of string theory space-time emerges from something deeper a completely different side of the story which you'll perhaps hear about in the next lecture involves quantum mechanics and the duality between gauge theory and gravity but what I've described is certainly one important piece of the puzzle it's perhaps the piece that's best understood even today although that's arguable but certainly it's the best understood piece that's been best understood for longest it's at least a partial insight about how space-time as conceived by Einstein can emerge from something deeper and also hope this topic has been suitable in a session devoted to the centennial of general relativity thank you

Channel: International Centre for Theoretical Sciences

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Length: 44min 6sec (2646 seconds)

Published: Wed Jul 01 2015