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Home , Abhay Ashtekar, Ashtekar variables, Induced gravity, Problem of time, Quantum gravity, Theoretical physics

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UM-P-95/2^ gr-qc/9503xx March 1995

The Quest for Quantum

G.Au* Research Centre for High Physic* School of The University of Melbourne ParkviUe 3052 Victoria Australia

One of the greatest challenges facing lies in reconciling Einstein's dassical of gravity - general rdativity - with quantum Add theory. Although both have been experimentally supported in their respective regimes, they are as compatible as a square peg and a round hole. This artide summarises the current status of the superstring approach to the problem, the status of the Ashtekar program, and addresses the problem of in . Theoretical physics cannot tell us the fundamental reason why apples fall down from apple trees!

* gauOtanon.ph.unimelb.edu.an. "It is very important that we do not all follow the same the possibility of changing itself in the fashion... It's necessary to increase the amount of context of the physics of the early and the way variety... and the only way to do this is to implore you in which it came into being. few guys to take a risk with your lives that you will not The basic problem lies in the incompatible way in be heard of again, and go off in the wild blue yonder to which the concept of time is treated in QM and general see if you can figure it out." (1965), relativity (GR). One of the main researchers studying in physics award address. this deep problem is Chris Isham at Imperial College, London. In QM time evolution is described by the Schrodinger Introduction equation. However the time parameter that comes into the equation is a background parameter. This is un­ The road to quantum gravity (QG) is a long and wind­ like other quantities which can be represented by Her- ing one, littered with pitfalls, tortuous inclines, neck mitian operators. So from the very beginning QM at­ breaking descents, and the skeletal remains of failed at­ taches a special significance to the idea of time. It's like tempts along the roadside. Although we arc unlikely to the steady ticking of an external clock hand in the back­ see the day of a direct test of quantum gravity, such a the­ ground, which we have no control over. Can we physically ory is needed to describe the early stages of the . measure this quantity? Is it possible to reformulate QM Around this time the universe would be about the of so that the idea of a background time doesn't appear? the Planck volume. Quantum gravity effects, ignored in According to Chris: "It is possible to do this in a formal because of their weakness, would have a sort of way but it does not really get you far. The ba­ pivotal influence on the subsequent evolution of the uni­ sic fact is that conventional quantum theory presupposes verse. Also a QG theory is desperately needed to make an external time whose ontological* status is the same the a consistent physical framework. as it possesses in classical Newtoni?n physics. In rela- Why are we spending so much time on such a lofty tivistic quantum theories this is generalised to become a goal? Richard Feynman commented on the aims of sci­ background Minkowskian - so it is the causal ence: "If you expect to give you all the answers structure that is fixed. There are some intriguing mathe­ to the wonderful questions about what we are, where we matical (going back to Pauli I believe) showing are going, and what the meaning of the universe is, then that you cannot construct a that would I think you could easily become disillusioned and look for exactly measure this background time. In that sense, a mystic answer to these problems. some people might want to consider 'time' as part of the "The way I think of what we're doing is, we're explor­ external classical realm that Bohr postulated was neces­ ing - we're trying to find out as much as we can about the sary to interpret the equations of quantum theory. I have world. People say to me, 'Are you looking for the ulti­ always suspected that Bohr himself would have regarded mate laws of physics?' No, I'm not. If it turns out there the subject of 'quantum gravity' as a non starter!" .s a simple, ultimate law which explains , so be it; that would be very nice to discover. If it turns out Why does time deserve a special status in QM? Well, if it's like an onion, with millions of layers, and we're sick you could raise it to the status of an T then its and tired of looking at the layers, then that's the way it conjugate variable would be the energy H. These would is. But whatever way it comes out, it's , and she's need satisfy the commutation relations [T,H] = ih. going to come out the way she is! Therefore when we go Mathematically it can be shown that it is impossible to to investigate it we shouldn't predecide what it is we're have a pair of Hermitian operators satisfying these condi­ going to find, except to try and find out more.*" tions, in which one of them has only positive eigenvalues. This relates back to Chris' remark about the problem of The quantum clocks. In other words, we have the problem of a system with negative energy. A key question which faces any approach to QG is In QM, measurements of are taken at fixed the 'problem of time': how does a time variable, with , a complete set of commuting operators (defining its special properties in relation to the probability in­ the maximum information you can obtain from a sys­ terpretation of quantum (QM), emerge? A tem) is required a fixed times, and the scalar product related topic which has been attracting increasing atten­ on the HUbert of states has to be conserved under tion, especially in the context of '' time evolution of the Schrodinger equation. Time plays a is the 'consistent-histories' approach to quantum theory. foundational role in the technical and conceptual aspects One of the most exciting features of the programme is of quantum theory. However it is an idea grounded in the

*No Ordinary Genius - The Illustrated Richard Feynman, 'Ontology is the part of metaphysics concerned with the Christopher Sykes, W.W.Norton and Company, 1994, p251. nature of existence. world of continuum . If it breaks down at When we try to add GR to the QM picture, the intrica­ the Planck scale, where some believe a discrete structure cies are much more difficult to handle. will emerge, what are we left with? Indeed, how much of With a curved spacetime equipped with a Lorentzian QM will hold? metric tensor, how many ways are there to foliate space- According to John Wheeler: The word 'time' was not time as a one parameter family of space-like surfaces? handed down from heaven as a gift from on high; the Chris continues: "In some there is a topolog­ idea of time is a word invented by man, and if it has ical obstruction to performing any global foliations. But puzzlements connected with it, whose fault is it? It's our if such foliations are possible then there will be an un­ fault for having invented and used the word.*" Taking countable number of them, each corresponding (in gen­ this into mind, could we redefine time so that it reduces eral relativity) to an allowed definition of time." In other to the familiar clock ticking concept in some classical words, unlike in SR, we cannot define a unique time ac­ limit? cording to which we evolve a system. This is the main Chris Isham is very sympathetic to this view: "I would difference between the concept of time in special and in bet strongly that whatever the 'final' theory of quantum (classical) . It is an important aspect gravity will be, we shall see our standard notion of time to the problem of time in QG. On one hand we have a only emerging in some semi-classical sense." Hamiltonian l'ormulation of QM for privileged SR space­ The role of time in relativity is very different to its like surfaces. On the other we have role in QM. Relativity treats time on equal par with the in GR which states that no one set of space-like surfaces other three spatial variables. Indeed, time and space are has privilege as a reference system over others. unified into a four dimensional spacetime manifold. When we get to Planck scale physics, spacetime geom­ On this Chris explains: "The notion of a sin­ etry could be subject to Heisenberg's uncertainty princi­ gle, 4-dimensional, 'spacetime' (rather than separated 3- ple, varying about quantum mechanically. In other words dimensional space plus 1-dimensional time) first appears the GR metric tensor g,,¥ could have fluctuating compo­ in (SR) because of Einstein's realisa­ nents! How could one define a cone and hence a tion that any absolute split of 3+1 is observer-dependent. space-like, light-like, or time-like separation under such General relativity works with a curved version of the fiat circumstances? Without such a definition, the idea of a space-time of special relativity. space-like surface seems invalid. Perhaps the idea of a "In both special and general relativity a 'moment of spacetime (a continuous manifold of points) doesn't have time' corresponds to a single space-like hypersurface in physical meaning in the QG regime, where the the space-time. Time itself appears as the parameter that could be smeared out. labels the elements of a one-parameter foliation* of space- To this Chris says: "You cannot defined a light cone time by such surfaces. In the case of special relativity you etc. except in some 'background' sense, and this perhaps restrict your attention to the case where the space-like is the essence of the problem of time in QG: there is no surfaces are (i) hypcrplancs"; and which (ii) are mapped fixed microcausal structure which can be used to con­ into each other by actions of the Poincare group. The struct a relativistic quantum theory in any of the stan­ main challenge in the quantum theory is to show that all dard ways. such admissible families give the same physical answers. "However, this does not rule out a priori a continuous The "tuation in general relativity is more complex. manifold: the ideas of causal structure and continuous One says that a hypersurface is space-like if the vectors manifold are in no way synonymous. You can easily have tangent to each point of the surface are space-like. The the latter without the former (although I myself am very idea here is that the Lorentzian metric on spacetime in­ sympathetic to the idea that the idea of a 'continuum duces on each tangent space a copy of the Minkowski spacetime' is not something that will carry across to the metric of special relativity, and it is with respect to this 'final' QG theory)." latter metric that the tangent vector has to be space-like. One possibility at the Planck scale is for topology Put slightly less rigourously, a hypersurface is space-like changes in spacetime . Fixing a background if any 'infinitesimal'transformation in the surfac always topology and differential structure to that of Minkowski points in a space-like direction." space, as is done in quantum theory, seems to pre­ The crucial point here is that the problems encoun­ suppose a lot at the Planck scale. tered when incorporating SR into QM can be overcome. If we stick with a continuum formulation of QM based on complex numbers, could we be missing out on reveal­ ing some sort of discrete structure at the Planck scale? One can see how a discrete theory could reduce to a con­ ''* A Brief History of Time: A Reader's tinuum one in the large scale limit, but to shei! light on Companion, prepared by Gene Stone, Bantam Books, 1992, a discrete theory while working from the perspective of pl25. a continuum one seems difficult to achieve. 'A foliation it a one-parameter family of hyperplanes which, Chris has felt this for a long time. One of the many taken all together, fill out the entire spacetime. reasons why he is so interested in the new decoherent- **A hyperplane is a flat surface of 3. histories approach to quantum theory, and is developing

2 his own quantum-logical version, :

3 normalised, and they must sum to unity. Complex proba­ Jim Hartle made the important observation that one bility amplitudes can be assigned to histories, but a prob­ can talk about 'generalised histories' in QG in which the ability is defined as the modulus squared of an amplitude. word 'history' need not presuppose a conventional type of This operation introduces cross terms which violate the time parameter. According to Chris: "The basic entity probability axioms. in the scheme is a 'possible universe'. A simple, but in­ To handle this problem 'consistency conditions' are in­ formative example would be a Lorentsian geometry that troduced. These determine sets of histories which neg­ is not globally hyperbolic. Such a spacetime does not ligibly in'erfere, and to which probabilities may be as­ admit a global foliation by space-like surfaces, and hence signed. These conditions can be combined into a deco- there is no global time parameter. Nevertheless, it is a herence functional, which is the main calculational tool viable classical possibility since time still makes sense lo­ of the QM of history. It must satisfy the properties of cally (i.e. the local along a ). This hermiticity, positivity, normalisation, and the superposi­ was one of the original motivating examples suggested by tion principle. The functional is computed using stan­ Hartle. dard . Griffiths, Omnes, Hartle and "However, my own interest in this scheme is when the Gell-Mann's contribution is the new probability interpre­ basic entity is something more basic that does not neces­ tation. Here the notion of 'superposition' does not mean sarily include any manifest time-like features at all. This quite the same as it does in standard way of superposing would be in line with the general idea that time appears wave-functions. It is more a question that the decoher- only in some semi-. Of course, this leaves ence functional must be additive with respect to a disjoint quite open what types of object one would actually use sum of histories. as 'possible ' since this depends entirely on the In order to decouple interfering histories by coarse actual theory that is to be developed (the decoherent- graining we require the idea of decoherence. Consistent histories approach is not a theory per se, it is more a the­ (decoherent) histories are then sets which obey the con­ oretical framework in which new types of theory can be sistency conditions. Consistency is a property of families developed). For example, I am currently thinking about of histories, of which there may be many. the possibility of constructing such a theory in which the But what is the mechanism of decoherence? It has basic entities are simply point-set topologies. been postulated** that the environment decoheres by act­ "In theories of this generalised type Hilbert do ing like a sink, by draining off - similar to a not appear as basic mathematical structures in the same piece of blotting paper soaking up excess ink. But if way that they do in normal quantum theory; indeed, in the system is a closed QM universe how do you define the scheme that I am developing the basic mathematical an environment? What other methods could there be to structure is much closer to that of (i.e. an decohere closed systems? algebraic approach) than it is to Hilbert space method­ To this problem Chris remarks: "The mechanism for ology. On the other hand, in any theory in which time decoherence in a truly closed universe is, to my mind, still could be shown to 'emerge' in some coarse-grained limit I rather problematic. There have been attempts to, for would expect the usual Hilbert space structure to emerge example, identify part of the gravitational-field complex too at the same level. However, as no one has yet written as the environment, and part as 'really quantum' freedom down an explicit example of this type, it is hard to be in this environment. But I am doubtful whether this is more specific!" really a universal notion. The examples I have seen have all been for very simplified models." The Ashtekar Program In a closed QM universe, would it be possible to use black holes as decoherence devices? When matter goes The QG problem is a subtle one. People have ap­ into a , the only properties of the hole which proached it from many different directions, along many are via its distant electromagnetic and grav­ roads. All roads might lead to Rome, but some are ity fields are , , and angular . All more easily traversed than others. Most attempts suf­ other information is lost. Would this loss include 'inter­ fer from seemingly insurmountable problems. A short ference' information between histories? list of avenues includes higher derivative Lagrangians, twistors, , Kaluza-Klein theories, Eu­ Unfortunately Chris says: "One could do that when clidean quantum gravity, covariant , studying quantum theory in a fixed background, since discrete gravity, non-linear quantum mechanics, net­ then one could consider a background containing black works, asymptotic quantisation, quantum cosmology, the holes. However, this is not too meaningful a priori in a decoherent histories just covered, full theory of quantum gravity itself since the black holes in curved spacetime, superstrings, and canonical gravity. will also have to be quantised in some sense." The Ashtekar program belongs to the last category. is at the Centre for Gravitational Physics and Geometry, Pennsylvania. Although his pro­ gram involves esoteric mathematics, some of its key fea­ "Physic* Today 44, 36, (1991), Decoherence, Wojicek Zuiek. tures can be described qualitatively.

4 An obvious question that people might ask about QG quantum mechanics. In quantum field theory, however, research is: Why are we spending so much time and ef­ there is a key difference. Whereas in quantum mechanics, fort trying to obtain a theory which is most probably we know from general principles that the exact answer (intestable, and which won't have any practical applica­ exists and is finite and perturbative techniques are used tion? Abhay doesn't quite agree with the assumptions only as a computational tool, in realistic quantum field here. He says: "We are seeking a pkysicml theory and theories in 4-dimensk>ns, we do not have an assurance therefore it will definitely have testable predictions. The that there is an underlying well-defined theory; the per­ initial tests will be probably indirect. For example, we turbation series itself is being used to define the theory. have this great pussk for over 50 years: quantum field To make worse, each term in the perturbative ex­ theory provides incredibly accurate predictions in par­ pansion diverges because one is allowing virtual processes ticle physics yet it is, in a sense, only a set of calcula­ of arbitrarily large momenta. In renormalisable theories, tion^ recipes, without a coherent, complete mathemat­ the infinities that arise are of a special nature; they can ical framework. Again, the recipes work incredibly well be absorbed into a finite number of parameters associ­ and should be taken seriously. But surely, they are in­ ated with the theory such as charges, and cou­ complete in some essential way. The ultraviolet diver­ pling constants. Once these parameters are renormalised gences arise largely because we assume that space-time by the appropriate infinite factors, all individual terms is a continuum at all scales. If a quantum theory of grav­ in the perturbative expansion become finite. The actual ity provides an alternate picture of space-time, it should renormalised parameters can be determined by a finite lead to an alternate way of doing quantum field theories number of . Further experiments then pro­ which is coherent and mathematically consistent. So, in vide tests of the theory. In a non-renormalisable theory, a sense, the vast experimental data we have accumulated this can not be achieved and therefore the perturbation in particle physics could provide tests of the space-time expansion has no predictive power. models that come from quantum gravity. "Another way of stating the difference is that, in a "Furthermore, it is possible that quait-um gravity renormalisable theory, predictions for phenomena at a will also lead to 'practical applications'. Recall that, given length scale are insensitive to what is happening at when Einstein discovered special relativity, most people much smaller scales. So, for example, since QED is renor­ thought that it was an esoteric theory which would have malisable, its predictions for processes at, say, a GeV little impact, if any, on daily life. However, the theory scale are insensitive to all the goings on at the Planck then led to E = mc2 which has had profound ramifica­ scale (1019 GeV). In non-renormalisable theories, this is tions on the entire world-order of this century. Quantum not the case; the predictions for one length scale are sen­ gravity will also change our understanding of space and sitive to what is happening at much smaller scales. It is time radically and may therefore have some really deep thus harder to extract the physical content. practical implications. I would hope that they would be "If perturbative general relativity had turned out to be more peaceful in nature! renormalisable, on one hand, life would have been sim­ "But of course we don't have the faintest clue of such ple. One could have calculated scattering cross-sections applications today and, as you indicate, it is beyond in gravity using familiar methods. On the conceptual today's technology to test the quantum gravity effects side, however, at least some people including me, would dinctlj. The primary motivation is really conceptual. have been disappointed. It would have meant that most Again, there is a similarity with special relativity. Just as of physics is insensitive to the 'true' small scale struc­ the primary motivation for special relativity came from ture of space-time and hence it would have been harder the incompatibility between Newtonian mechanics and to 'probe' this structure. I have expressed my unease Maxwell's electrodynamics, the two pillars of the 19th with the current status of quantum field theory. It is the century physics, the primary motivation from quantum non-renormalizability of gravity that us to seek the gravity comes from the tension between general relativ­ true micro-structure of space-time and holds clues for ob­ ity and quantum mechanics. There should be a deeper taining something which goes beyond (the calculational theory which unifies the principles of both in such a way recipes of) quantum field theory." that these two theories emerge in suitable approxima­ Perturbation theories assume spacetime is a continuum tions. That is the theory that workers in quantum grav­ at all scales. Could this be a reason why they fail for QG, ity are trying to construct." where the short distance behaviour of spacetime might Abhay's program was inspired by the failure to find not be a continuum? An analogy has been made that a renormalisable perturbative theory of quantum GR spacetime is a 'foam'of sorts at Planck scales, subject to (QGR). What this is, and what would have happened Heisenberg's . if this failed program had been successful? "That's right! The key infinity in perturbative treat­ He explains in depth: "Perturbation theory assumes ments is the ultraviolet one where processes involve arbi­ that answers to physical questions can be obtained by trarily large . Surely, when the energy becomes a method of successive approximations, generally ex­ bigger, gravity should become more and more important, pressed in terms of a power series in the coupling con­ and the space-time geometry, more and more non-trivial. stant. We routinely use such approximations already in Yet, in the perturbative treatments, one uses a fixed,

5 continuum background space-time. Because space-time at face value, and try to modify GR in some way, or take geometry is a dynamical entity in general relativity, it GR as a weak field limit. This is the case for superstring seems clear that in quantum gravity we can not prcsap- theory, where GR appears as a low energy limit. The fost what the micro-structure of geometry must be; we dimensional nature of the basic - the Planck should let the theory itself tdl us. Of course, one might length, time, and energy - lends credence to the idea that have been lucky and made the right guess at the start. a QG theory could reproduce GR in regimes well away But the failure of perturbative methods tells us that this from the Planck scale. If QM is only an approximate was not the case with the continuum hypothesis." theory itself, which fails to hold at Planck scales, then Might quantum GR exist non-perturbatively, i.e. as in the spirit of 's suggestion, QM will also an exact theory? have to undergo a revision. Abhay feels that this is in "Yes, it might. We don't have any clear evidence to part why a//attempts at quantum gravity that are being the contrary. Incidentally, there do exist theories in 3 pursued are likely to be incomplete. "However", he says, space-time which are non-renormali-- ble but "it is very difficult to provide concrete models of how exactly soluble. Note that I am not saying that quan­ QM should be revised. I think the best bet is to push all tum GR must exist non-perturbatively. However, since viable approaches to the end and if cracks develop, they there are qualitative reasons to expect that the non- will suggest how we should change QM." perturbative theory will be very different from the pertur­ Loop variables gravity seems like a conservative ap­ bative one, the question is well worth investigating. More proach compared to superstring theory. One major dif­ importantly, even if the answer turns out to be in the neg­ ference is that according to your program, quantum grav­ ative, the exercise will have been extremely worthwhile ity can be obtained without a of all forces. In because it has already provided many new mathematical the superstring theory, such a unification is vital for con­ and conceptual tools to construct quantum field theories sistency. To be consistent, superstrings also need super- without a background space-time. Like many others, I (a mathematical relationship between firmly believe that, at a fundamental level, the finalsolu ­ and ) and higher spacetime dimensions (nine tion will have to be background independent; space-time space and one time). The extra six spatial dimensions will emerge only as an approximation. And we have very are supposed to be compactified into a very small re­ little with quantum field theories which have gion. Do you see these differences as complementary or an infinite number of degrees of freedom and which do conflicting issues? "On the whole, I think the differ­ not use a background space-time." ences are complementary. theory provides pow­ Do you see the need for a fundamental revision of the erful constraints on allowable interactions. On the other present concepts on which the standard model is based? hand, it has proved difficult to analyse non- In other words, are we in need of a conceptual revolution perturbatively, without assuming a background space- to solve this problem? I once spoke to Bill Unruh, and time. As I indicated above, our approach provides con­ he didn't think QG could be obtained through 'tinkering cepts and tools for a background independent approach. with mathematics'. He said we needed to feed in new Over the last year or so, more concrete calculations have conceptual ideas. been proposed which may pull the two approaches closer. According to Roger Penrose "...if there is to be a final "I should also point out that we don't have a proof theory, it could only be a scheme of a very different na­ that quantum GR exists non-perturbatively. However, ture. Rather than being a physical theory in the ordinary the tools required to analyse this issue finally exist. It sense, it would have to be a principle - a mathematical is possible that we will find, e.g., that principle whose implementation might itself involve non- or some specific couplings are necessary. This year is an mechanical subtlety." Is this your point of view as well? exciting one for our program because we have begun to "I believe that, at a fundamental level, the continuum analyse such issues." picture has to go. This would be a profound change, both I spoke to Peter Bergmann (one of the founders of the in terms of physics and mathematics since the picture is geometrodynamical route to QG, and a co-worker of Ein­ embedded so deeply in the conceptual fabric of physics. stein) while I was an undergraduate and he was very So, I agree with Bill that 'tinkering' with mathematics skeptical of superstrings. Is it the same with you, con­ alone will not suffice. However, I also feel that new math­ sidering most of the work in this field has been grounded ematics will be needed, (i mean mathematics that was in perturbation theory? not previously used in physics.) This has happened with "I am perhaps more impressed than Peter by the math­ most big breakthroughs in physics. Newtonian physics ematical successes of string theory. I agree that the use needed ; general relativity, differential geometry of a background space-time is a severe weakness. How­ and quantum mechanics, Hilbert space* and operator al­ ever, the very recent results due to Ashoke Sen*** and gebras. Indeed, without access to new mathematics, it would not have been possible to formulate new questions, let alone analyse them! Certainly, what Roger suggests is a possibility." Black Hole Solutions In On A Most approaches to the QG problem seem to take QM

6 others on black boles within string theory are providing classical. When the background starts curving noticeably some glimpses of the possible relation between perturba­ itself, what reference n there for these ripples? The prob­ tive results and non-trivial ." lem seems to be that the arena and performers can't be Unlike the other forces, the gravitational is non­ distinguished anymore. linear. In contrast, the electrical forces are linear because " are indeed ripples on a space-time back­ the total force due to a number of charges is just the sum ground. In the perturbative approach, one does quantise of the forces due to each charge acting alone. This sit­ these undulations leaving the background nntouchcd. In uation corresponds to the vector law of addition. Now our approach, the situation is different. Very roughly in the gravity case, we have a non-linear combination of speaking, by itself is represented as a forces. This n because all forms of energy have mass, quantum loop state - which we call a weave - and certain and all mass acts as a source of gravity and . This nearby states - embroidered weaves - can be thought of as is true for the gravitational potential energy existing be­ gravitons propagating in Minkowski space. However, as tween two separated masses. Thus the total gravitational I indicated above, the picture works only for low energy force involved when two objects interact does not follow gravitons. In this regime, one can distinguish the arena the usual linear vector addition law. Is this non-linearity from the performers." one of the root causes of all the problems in quantising gravity? This profound non-linearity is part of the difficulty. But this part is technical. The the most important dif­ Abhay's program takes conventional GR coupled to ficulty, in my view, is conceptual: it lies in the fact that matter as the leaping point. It is based on the 'canon­ gravity is encoded in the very geometry of space-time. ical' approach to QG whereby the full four dimensional To quantise gravity, one has to quantise geometry. One spacetime is split up into an observer dependent 3+1. has to learn to do physics and mathematics in absence of The first step in any canonical approach is the Hamilto- space-time.'' nian formulation of the theory. This was achieved in the early sixties by Dirac, Bergmann, Amowitt, Deser, Mis- I guess one of the things readers want to know is how ner and others. In their formulation, the basic variable everything reduces down to the picture of an apple falling is the metric on the three dimensional space; it captures from a tree. Newton's idea of a 'force' pulling the apple the intrinsic geometry of the hypersurface. The Hamilto- down is intuitive. In GR, this picture was replaced by nian framework was therefore baptised 'geometrodynam­ the concept of the of a four dimensional space- ics' by Wheeler. Unfortunately, in this framework, the time continuum. The apple's mass curves spacetime, and equations of the theory are complicated and therefore, the 's mass curves spacetime. This curvature tells as far as the full theory is concerned, the quantisation how the masses should move. The two masses feel the program itself did not really take off. combined curvature, and fall towards each. In QFI, this 'attractive' picture is replaced by the concept of the ex­ In geometrodynamics the canonical variables are the change of particles called gravitons. The gravitons medi­ intrinsic geometry and the extrinsic curvature of the 3D ate the gravitational pull. Does the loop variables theory spatial hypersurface (which is embedded in a 4D Rieman- preserve this idea? nian spacetime). The subject has close links with Gauss' "Yes. Although in our approach gravitons don't exist Egregium. at a fundamental level, in the low energy approximation Abhay explains: "The intrinsic geometry of a 3- one can identify certain loop states with gravitons and dimensional hypersurface is the geometry of space that then the picture is the one you indicate. we can 'experience directly'. It tells us, for example, how to measure lengths of curves lying in the surface. The "Incidentally, the loop variables also have a direct ex­ extrinsic curvature on the other hand, tells us how the perimental significance. If you move an electron around surface is embedded in the 4-dimensional space-time. To a closed loop in a , its spin undergoes a see the difference, let us go one dimension down and con­ rotation. The basic loop variable just measures this rota­ sider a cylinder embedded in a 3-dimensional Euclidean tion. Furthermore, if you know the rotation for all closed r space. Since the cylinder can be obtained by rolling up a loops, you know the gravitational field completely. piece of paper, its intrinsic geometry is the same as that One of the problems with gravitons is that they seem of the paper, i.e., a 2-dimensional plane. The difference to make sense only when the gravitational field of GR lies in how the paper and the cylinder sit in 3-dimensional is linearised, i.e. split up into a flat space metric, and a space. Technically, their extrinsic curvature is different. perturbation. Gravitons are like ripples in a flat space- time background. Conventional particle theory tries to "Coming back to GR, the extrinsic curvature measures quantise these undulations, while leaving the background how the intrinsic geometry is changing in time. So, if we think of the intrinsic geometry as the configuration vari­ able of general relativity, as one does in geometrodynam­ ics, then extrinsic curvature can be thought of as the conjugate momentum. In quantum geometrodynamics, Torn*, Athoke Sen, T1FR-TH-94-47, 20 pages. one would subject the two to the Heisenberg uncertainty relations." theory. I won't b* surprised, however, if there is another By splitting spacetune into (3-t-l)D, are you treating way to ensure in the semi-classical the spatial and time co-ordinates on a different footing? regime." Physically, this would correspond to choosing a special By casting GR into Hamiltonian form, it b found that observer, whose wristwatch keeps track of the distortions the theory b a dynamically constrained system. These in the 3D spatial hypersurface. This method is very dif­ restrictions go by the name of the Hamiltonian and dif- ferent to the 'covariant' schemes of quantisation like the feomorphbm constraints. What physical significance do path integral approach. they have? "That b quite right. From aesthetic considerations, it "Constraints of thb type signal the existence of a large would be nice to use, say, path integrals. However, no gauge group. In electrodynamics, for example, there b one has seen a way to make mathematical sense of path an analogous constraint, the Gauss law, which says that integrals in GR beyond certain simple minisuperspaces the electric fields should be divergence-free. It ensures (which result when you freeze all but a finite number of that the classical (and quantum) theory b gauge invari­ degrees of freedom of the gravitational field and replace a ant. Higgs pointed out, quite early in the game that the quantum field theory problem by a quantum mechanics diffeornorphbrn constraint of general relativity b rather problem). The mathematical framework that my col­ similar. It ensures that the spatial difteomorpiusms - leagues and I have developed over the last two years is, active coordinate transformations on the 3-dimensional in principle, also applicable to the Euclidean path inte­ hypcrsurface - don't change physics. Thb constraint b gral approach. However, there are two key difficulties. universal in the sense that its form and meaning b the The first is conceptual. Even if one did have a co..iplete same for all gravity theories in which there b no back­ Euclidean theory, it seems very difficult to extract physi­ ground metric. cal predictions from it since, in quantum gravity, there is "The b more tricky. It b re­ no analog of the 'Wick rotation' which replaces the time sponsible for time evolution, i.e. . If you treat it coordinate f by if in Minkowskian quantum field theories in the same fashion as the diffeomorphism constraint, one thereby enabling one to pass from the Euclidean to the b led to what people call 'the frozen [time] formalism' in Lorentsian regime (i.e. from Schwinger functions to the which nothing evolves and physical states are represented Wightman functions). The second difficulty is technical, by entire space-times. There b nothing wrong with thb associated with the specific form of Einstein's ." picture. It b like doing Hamiltonian dynamics in terms One of the main criticisms of the approach is that it of constants of motion; it b therefore inconvenient from breaks the general covariance of GR. Would the general from technical and interpretational perspectives." covariance concept have any meaning if the continuum I have heard that the constraint equations are (high picture of spacetime is lost? One of the ideas brought order) polynomials in terms of the basic canonical ge- up by your program was the possibility of spacetime be­ ometrodynamical variables - the 3-metrics and extrinsic ing combinatoric instead of geometric in nature, at the . What effect does thb have on trying to quan­ Planck scale. tise the theory? "For technical clarity, I should first point out that, "It simplifies the technical problems of quantisation while canonical approaches do lack manifest covariance, enormously. In quantum field theories, what one might they are, nonetheless covariant; for example, the full naively think of as operators are really operator valued Poincare group has a well-defined action on the phase distributions. Just as products of the Dirac delta dis­ space of electromagnetic fields in Minkowski space. With tributions are ill-defined, products of these 'operators' this side remark out out the way, let me answer your are also ill-defined. This b the origin of the infinities. question. If anything resembling our approach is correct, So, if you have an expression which w non-polynomial then, in the quantum theory, there will be no such thing in your basic operators, in general, there b very little as a space-time geometry at a fundamental level. The chance that you would be able to regulate it and extract picture, as you say, will be combinatorial and therefore a meaningful operator. If the expression w a low order the diffeomorphism group will not have a preferred role. polynomial, the chances of success are much greater. At Space-times will arise only semi-classically. In this ap­ various levels of ,thi s simplicity has been exploited proximation, one can ask if there is 4-dimensional diffeo­ by Rovelli, Smolin, Briigmann, Gambini, Pullin, Morales, morphism invariance. The answer will be in the affirma­ Nicolai and Matschull to solve the quantum constraints. tive. However, it will be arrived at from the Hamiltonian No such solutions exist in quantum geometrodynamics. picture; the 4-dimensional covariance will not be mani­ "A amplification came in the mid eighties when it fest. This is a drawback of the approach but only from was realized that there is another way of obtaining the an aesthetic viewpoint. Hamiltonian formulation, where the basic variable is a "Incidentally, for these reasons, I had first tried to use connection which enables one to parallel transport chiral the covariant phase space for general relativity and avoid fermions. Thus, general relativity can also be regarded the 3+1 splitting. That approach faced technical prob­ as connection-dynamics. This formulation has two key lems and I now believe that, even if the technical prob­ advantages. The first is conceptual: general relativity is lems could be resolved, it would not lead to an interesting brought closer to gauge theories which govern all other

8 bask forces of Nature. The second simplification is tech­ differences between the coanectwos that Sen was using nical: equations of the theory become low order polyro- and the ones I was led to use, it was not obvious to me that all of Einstein's equations would be low order poly­ nomials in the triads and connections." The The next step is the the transition from classical GR to quantum GR. By using these Ashtekar variables, you At this stage, you came up with the idea of the are trying to quantise a theory of connections, instead Ashtekar variables. Where did the inspiration of these of a geometrodynamk theory based on spatial 3-metrks. new variables come from? Was Amitabha Sen's work a In terms of connections, the constraint equations of GR major influence? bear a great resemblance to those of ara-Atciica Yang- "For me, personally, a lot of the inspiration came from Milb theory. results on chiral solutions to the Einstein equations by 'Progress came about because there already existed Newman, Penrose and Pkbanski in the mid seventies. a rich machinery to deal with quantum gauge theories. In a certain sense, this sector of Einstein's theory is We did have to make some important changes eventually completely integrable. Therefore, it seemed natural to because in Yang-Milk theories, one makes a heavy use try to base quantum theory on chiral variables. Sen's of the background Minkowski metric and our framework work was definitely an important influence up to a cer­ had to exist without reference to any background struc­ tain point. About the same time that Sen published his tures. But a number of key idea came from Yang-Mills papers, Horowitz and I also had some results on positiv- ity of the Hamiltonian of GR (for open universes) which used similar techniques. The two together suggested the "Incidentally, because all other basic interactions can direction that I finally took." be formulatedi n terms of connections, the geometrody- namic formulationdi d create a distance between GR and What physical picture can you give us of them? You other gauge theories. For example, Weinberg, in the in­ perform a canonical transformation of the jld geometro- m troduction to his book emphasises this point. How­ dynamical variables, obtaining a connection and a triad ever, both Einstein and Schrodinger had presented a for­ as the new variables. A complex, group-valued, self-dual, mulation of general relativity with connections, rather is hard to visualise. In vhat way does a than metrics as the basic variables. The reason the idea triad of ortbonormal vectors on the 3-sIice take over the did not catch on, I believe, is that their equations were role of the extrinsic curvature? even more complicated than those of geometrodynamics. The connection enables us to parallel transport chi­ They used affine connections and simplifications occur ral fermkms along closed loops and therefore has a di­ with chiral ones. I became aware of this piece of history rect physical meaning. Roughly speaking, the connec­ only recently." tion knows both about the (spatial derivatives) of the triad and the extrinsic curvature (the time-derivative of Loop Variables the triad). If you think of the 3-metric as the analog of the position variable q in quantum mechanics and ex­ In QM we have operators and states. Operators act on trinsic curvature as the analog of p, then the connec­ states, and take them from one allowed state to another. tion is analogous to the complex variable z = 9 - ip, The simplest example of a QM system may be the sim­ which one often uses to construct the so-called coherent ple harmonic oscillator. There, the raising and lowering states in quantum mechanics. Clearly, if you know (q,p) operators take a particle subject to a constant restoring you know (z,q) and vice versa. One can do quantum force, to different energy eigenstates. The canonical vari­ mechanics with (z,q) as basic variables, although it is ables, position and momentum, are raised to the status somewhat unconventional. In quantum GR, we are led of operators, and they have to satisfy commutation rela­ to connections and triads because these are the variables tions. What are the corresponding operators and quan­ that simplify the equations." tum states in this approach? The actual canonical transformation is non-linear. The "This is a somewhat technical point and I hope you result casts GR as a dynamical theory of connections. will bear with me. In particle mechanics, we have the The phase space of GR is now seen to be embedded in notion of configuration space - the space of positions, that of Yang-Mills theory (a ). The con­ for example - and quantum states are functions on this straints form surfaces which define restricted on space. In field theories, the situation is more compli­ this phase space. This hidden relation seems remarkable. cated. States are now functions on a quantum configura­ Were you expecting such a relation, or did it just drop tion space which is a genuine enlargement of the classical out of the working? "I knew that if the triads and connections can be shown to be canonically conjugate, the embedding you refer to would be a consequence. But it was far from obvious to '" and Cosmology: principles and application* me that the Poisson brackets between triads and connec­ of the general , , New tions would be so simple. Also, because of some subtle York, Wiley, 1972, xxriii, 657 p., ill, 23 cm. configuration space and the measures which dictate the and Pullin to obtain solutions to the Hamil'ttuaB con­ inner product are typically concentrated on the exotic, straint. Whether such results can be made rigorous is non-classical configurations. an open question; one would have to wait and see if the "The precise domain space of quantum states is a well- current work by mathematicians in this domain leads to defined completion of the space of gauge equivalence useful results." classes of connections. This space can be constructed Are you feeding in what the mkrostructurc of space- without reference to a background metric and we have time should look like beforehand, or docs it come out of also developed integral and on this the equations? I have heard the analogy of Planck space- space to define Hilbert spaces and operators. There is time being like a mesh of mail armour, made up of loops no background metric or connection anywhere. This cal­ weaved together. culus telb us that there is a well-defined 'non-linear du­ "A key point in our approach is that we do not start ality' between connections and loops and hence one can out assuming what should be like but also represent states as suitable functions of loops. The let the theory lead us to it. (Of course, the fram'work connmr ion and the loop representations are 'dual* to one as a whole makes certain assumptions - there should be another in the same sense as the position and the momen­ no background structure and the Wilson loop operators tum representations in quantum mechanics are dual. In should be well-defined - but they are of a general na­ practice, the loop representation is generally more con­ ture). We construct operators that correspond to geom­ venient to use while analysing questions related to the etry and ask for states which, when probed with these op­ Planck regime while the connection representation is bet­ erators, would approximate a classical geometry at scales ter suited to study semi-classical issues. For instance, much larger than the Planck scale. We then obtain spe­ it is the loop representation that has provided the rela­ cific states which have this property but which display a tion between quantum gravity and knot theory as well discrete structure of a definite type at the Planck scale. as glimpses into quantum geometry while the connection These are the states that look like ^-dimensional chain representation has been used to address the issue of time mail. The gravitational field is excited only at the loops in the semi-classical regime." in the chains. However, on coarse graining, the states are "For technical reasons connected with diffeomorphism indistinguishable from classical geometries. More impor­ invariance, quantum gravity states have to be measures tantly, we find that spectra of basic operators such as the on the quantum configuration space, i.e. objects which area of a given 2-dimensionaI surface are quantised in the are slightly more general than functions. The basic quan­ units of the Planck area. tum operators are associated with loops eud strips, i.e. "In the loop representation, all operators act by break­ ribbons. Their commutator algebra is surprisingly sim­ ing, re-touting and gluing of loops. These operations will ple; commutators can be expressed in terms of re-routing code the quantum gravity interactions. So, in the Planck of loops and strips, and intersections of loops with strips, regime, the familiar Fevnman diagrams will be replaced ?nd strips with strips. It's a pretty, geometrical picture." by 'topological' diagrams representing these operations. In the loop representation, quantum states arise as Feynman diagrams will arise only in a low energy regime 'functionals' of Wilson loop*. A deep relation exists be­ where, as I indicated above, the concept of gravitons is tween the equivalence classes of loops and the theory of meaningful." knot classes, invariants, and polynomials. There is also What was Profs. Smolin and Rovelli's inspiration to a relation to Chem-Simons theory. come up with the loop representation? Has there been "This comes about because there is a specific 'non­ any collaboration with the Trias and Gambini group, linear duality' between loops and connections: Given which is working on gauge theories? a closed loop and a connection, one obtains a number, "As I recall, they were trying to solve the diffeomor­ the value of the trace of the of the connection phism constraint and realised that the solution is 'obvi­ around the loop. It turns out that this duality implies ous' in the loop representation. At that time, we did not that there is a 1-1 correspondence between certain func­ have the mathematical machinery involving measures on tions on the loop space and measures on the quantum the quantum configuration space and it was hard to see configuration space. Therefore, we can take these loop how to construct solutions to this constraint in the con­ functions as quantum states. One can then formulate the nection representation. Therefore, their realisation was a diffeomorphism constraint rigourously and seek its solu­ real breakthrough. Curiously, none of us knew about the tions. Not surprisingly, in the loop representation, the work by Gambini and his collaborators at that time. One solutions are (certain types of) knot invariants. Thus, ofthe students in our group, Brugmann, came upon their after you have imposed the diffeomorphism constraint, papers while applying the 'loopy ideas' to Yang-Mills the­ you can say that the quantum configuration space is the ory and told us about them. We then contacted Gambini space of knot classes and all knot invariants/polynomials and Trias. Since then, Gambini, Pullin and others have are the potential quantum states. At a heuristic level, made some very significant contributions. We are all ex­ one expects the Chern-Simons theory to define a diffeo­ tremely happy with these synergetic developments." morphism measure on the space of connections You can also bring in supersymmetric matter cou­ and these ideas have in fact been exploited by Gambini plings.

10 "Yea. supersymmetry caa be brought ia. There is the HamUtoaian m—traial. Once we have it and ensure work ia Uus direction by Jacoheon, Matacball, Nkolai that the algebra has no anomalies, then we will be able and others. The work by Nkolai and his colleagues on 3- to face the issue offinding th e inner product on the phys­ itimensinnil has been especially iUumiuat- ical states. However, it is quite possible that to make the ing; it showed that supergravity does allow an iniaite Hamiitonian rrwiti sinl well-defined, we have to bring ia aamber of physical states rather than just, say, a sin­ framed loops and quantum groups and/or supes»/uune- gle solatioa analogoas to the Hartle-Hawfcing wave func­ try. Mathematical machinery does exist to do this. And tion." I would find it quite exr'ting if, e.g., it b the dynam­ ics of quantum gravity that usher quantum groans into Current Work in the Program physics. "At a heunstic level, there has been a lot ot progress Let as talc about some of the current research prob­ on the issue of solutions to the Hamiitonian constraint lems in the field. In the language of connections, yon and Rovdli and Smolin have introduced approximation have to complexify the phase space ofGR, since yon have techniques to find the inner product. These are useful mixed (both complex and real) variables. Now yon have guidelines for rigorous work. However, the problem of to restrict the phase space to real portions, since the inner product on physical states is yet to be faced bead- physical observables mast be real quantities, not com­ on. plex ones. In other words, you must implement conditions in the quantum theory. Have these been dealt "Incidentally, if the program is successful, we will have with satisfactorily? a quantum theory of gravity at a level of rigour that ex­ ceeds the current level in four dimensional quantum field "Several special cases had been treated satisfacto­ theories. This may seem like an overkill. However, since rily over the last few years. These include certain we have so little experience in non-perturbative QFTs Bianchi models and midi superspaces with a single particularly in absence of a background space-time, thb Killing field. Over the past year, we have been able to level of care is needed both to ensure that we have not treat the teaem/ case. In the harmonic oscillator case, ignored some important subtleties and to convince the the Segal-Bargmann transform enables us to pass from skeptics that a solution has really been found." the Sdtrodinger representation where the wave functions 9(f) are complex valued functions of f, to the bolo- How is the problem of time evolution handled? If there morphic representation where they are koiemorjkit*11 are only Hamiitonian constraints, and no Hamiitonian, functions ¥(z) of z. Since we know that « and p are how do you define a time variable, and evolve the quan­ self-adjoint in the Schrodinger representation and since tum system? the Segal-Bargmann transform is unitary, you know au­ "Two approaches have been followed in both of which tomatically that the reality conditions also hold in the the notion of time is an approximate one. The first is boiomorphic representation; you don't have to check it some work I completed in '89 where we saw that, in the explicitly. We have constructed a rigorous analog of tbu> connection representation, one can make a certain trun­ transform for quantum gravity. So, in principle, the prob­ cation of the theory in which the quantum Hamiitonian lem has now been solved. Hov ;er, we still have to de­ constraint can be written as the Schrodinger equation, velop calculational tricks to make the transform useful in where one of the connection components serves as time. practice." Thb shows how the familiar evolution of QFT will arise in the low energy regime. The second and more recent ap­ One of the main problems has been to define an in­ proach is due to Rovelli and Smolin. They couple gravity ner product (vital to make sense of probabilities) in the to a scalar field and use the scalar field as time. So, while Hilbert space. What is the status of this search? we still don't know bow to extract time and evolution in "This problem arises at two levels: kinematical and dy­ the Planck regime - and, like many of my colleagues, I namical. One generally needs an inner product on the the fed thb b not necessarily a meaningful thing to ask - in 'kinematical' states prior to imposition of constraints to suitable approximations, time evolution has been recov- make sure that the constraint operators are well-defined ereo. and that there are no anomalies in the constraint alge­ For posterity, can I have your feelings as to where loop bra. This part is taken care of by the recent mathemat­ variables are heading? ical developments. Furthermore, as I indicated above, "There are three parts to the answer. The first refers the diffeomorphbm constraint has been regularised and to quantum gravity the second to other physical theories solved. There are no anomalies. However, at the rigor­ of connection, especially QCD, and the third to certain ous level, we still do not have a definitive formulation of branches of mathematics. "Within gravity, as I indicated above, for the first time, we have a candidate for the quantum configura­ tion space and the difieomorphbm constraint has been '"A single-rained function f(x) is bolomorphic (or regalar formulated without anomalies, and we know its general or analytic) in a region if it is dnfrrenuaMe at each point of solution. If we can do the same for the Hamiitonian the region. constraint • possibly by bringing in framed loops and

11 quantum groups - one could say that quantum GR ex­ big to be detected by current accelerators, including the ists non-perturbatively. It is like saying, in QCD, that (LHC) being built in Europe. We we have a representation in which the Hamiltonian is are looking at detecting particles as massive as bacteria! rigourously self-adjoint. This would be striking but not Superstrings can come in two varieties. One resembles immediately solve problems of physical interest. The sit­ a line segment with free ends. This type is called the open uation would be the same in QGR. However, knowing superstring. The other type is a loop with the topology that the theory exists, one could then look for reliable of a circle - the closed superstring. Both versions reside approximation methods and know that this is a mean­ in a ten dimensional (9 + 1) spacetime. ingful program. Many such methods are already being Since we only see four (3+1) dimensions in real life, developed. I believe that it is crucial to continue to look it is assumed that the six extra spatial dimensions are for such methods. curled up, in a region as small as the Planck scale. The "Since the mathematical machinery that has been de­ process of curling up is called compactification. There veloped works for any theory of connections, it is tempt­ is no mechanism in the theory to tell how you curl the ing to apply it to QCD. We have already done so in two up, and this arbitrariness is a cause space-time dimensions and obtained several new results. for concern. Theorists have explored Calabi-Yau spaces, The most notable among these are closed expressions for group manifolds, and other spaces as candidates. The the Wilson loop Schwinger functions and a proof of equiv­ idea of compactification is reminiscent of Kaluza-Klein alence of the Hamiltonian and Euclidean path integral theory, which attempts to unify gravity with electromag- methods. One might venture further and try to extend netism using an extended spacetime of 5 or (4 + 1) di­ these methods to three and four space-time dimensions. mensions. Chris says: "This theory was much developed That they will lead to mathematically interesting quan­ in the 1970s to include Yang-Mills theory which give, tum field theories is clear. Whether these theories will be for example, the type of non-Abelian gauge theory used as useful in physics is an open issue. In any case, it is ex­ in weak interactions (this involves a spacetime of dimen­ citing that on J and the same methods are being applied sion greater than 5; e.g. 12 for SU(3) internal symmetry). to theories of all basic forces of Nature. The mathematical techniques of these larger-dimensional "On the mathematical side, there have also been some spacetimes (including what is known as 'dimensional re­ interesting new results. A few years ago, it was widely be­ duction' - the means whereby one gets back to the phys­ lieved that spaces of gauge equivalent connections would ical dimension 4) have been much used in discussions of aot admit any non-triv:al diffeomorphism invariant mea­ supergravity and superstring theory. However, it is im­ sures. Several of us have found infinite families of such portant to emphasise that these developments all work measures by now. The relation between knots and these at the level of the classical field equations: the quantum diffeomorphism invariant measures is equally exciting. theory used in quantising them is essentially standard. Our methods also enable one to extend (sufficiently nice) "Surprisingly, perhaps, there has been remarkably lit­ invariants of ordinary knots to those of generalised knots tle work done on trying to directly modify standard QM where loops are allowed to intersect and can have kinks to accommodate the insights of GR. People pay lip ser­ and overlaps. This is of interest to knot and graph the­ vice to this, but not much has been done." orists. Such results arise because we are looking at the String theories have an extraordinarily rich mathemat­ familiar structure from a new angle, with a new perspec­ ical structure. Exploring their , described by tive. Such contributions, I believe, will continue." various algebras, has been a major focus of work in the field. It has emerged that the same mathematical struc­ Superstrings tures appear in many different fields. For example, as it moves through spacetime, the string traces out a world Perhaps the most popularised of all the approaches to sheet or cylinder. Like the world lines (geodesies) of QG is superstrings. point particles, this world surface is constrained to be Superstring theory proposes that the elementary con­ of extremal area. The symmetry properties of these sur­ stituents of matter are one dimensional curves, rather faces are similar to those encountered in two-dimensional than point particles. According to the theory, the , condensed-matter systems. There has been valuable leptons, and gauge bosons of the standard model (SM) cross-fertilisation betveen these fields. arise as excitation states of a truly fundamental entity - Superstring theory is also related to subjects like 2D the superstring. These excitations come from rotational conformal field theories, non-linear sigma models, WZW or vibrational degrees of freedom (rather like the har­ models, Lax pairs, Hamiltonian reduction, the Toda and monic modes in a violin string), or from internal degrees Liouville field theories and their supersymmetric gener­ of freedom like supersymmetry (SUSY) and Lie group alisations, solitons, and other integrable models. It has symmetries. If they exist, superstrings would be the 33 applications in 2D supergravity, Lie , and smallest things in nature. They would be about 10" W-algebras, which are world-sheet symmetry algebras of cm long, and reside in the weird world of Planck scale bosonic string theories. leads to problems physics. Superstring excitations energies are about 1018 5 with 'anomalies' in these algebras in which classical sym­ GeV, or 10" gm. These particle masses are way too metries are violated at the quantum level. However, this

12 problem was resolved by Green and Schwarz in 1984, thus basic excitation of the quantum gravity field would be eliminating a difficulty of long standing. spin-2 radiation, which results from the symmetric nature The vibrational spectrum of open superstrings includes of the metric tensor. How hard would it be to detect the a massless spin-1 gauge associated with a Yang- gravitational analog of the - the , given Mills group - 50(32) or E&x E$. These groups are needed the weakness of the gravitational interaction? Ed replies: for cancellations. For the closed superstring, "Classical gravitational waves (for which there is indirect a massless spin-2 particle arises. It has been identified evidence from timing measurements of the binary pul­ as the graviton, the conjectured carrier of the quantum sar) could be directly detected by laser interferometers gravitational force. - perhaps eventually giving also an important new win­ GR is taken to be the large distance or low energy dow into some exotic phenomena in . It is limit to the superstring. Strings seem like point particles also possible that a possible very long wavelength cosmic at weak energy scales (10~16cm) since it is many orders background of gravitational waves could be detected by of magnitude less than the superstring scale (10-33cm). timing measurements involving space ships and pulsars. A small extended object looks like a point from the per­ These experiments could indeed possibly verify directly spective of lower energies. the spin two nature of gravitational waves. By examining string interactions, it is found that the "Unfortunately, there is no in sight that existence of open strings implies closed strings. Closed could directly verify the quantised nature of the gravita­ strings alone can form a consistent system. It follows that tional field." every consistent string theory necessarily includes grav­ The gravity field is different to the other force fields ity, since closed strings have gravitons in their excitation since it plays a dual role. One one hand, it defines the spectrum. arena in which other fields interact. Yet, gravity itself People have tried to generalise strings to higher di­ is a participant in the interaction. Does QFT assume mensional objects, like P-. However it is very that spacetime is flat to all energy scales i.e. has a hard to make quantum theories of extended objects, and Minkowskian spacetime signature even at Planck scales? get them to satisfy physical consistency constraints like "QFT as we know it, without quantising gravity, works and unitarity. In fact, the more extended you on a fixed space-time manifold, which for most obvious go, the harder it is. The successes of scring theory rely purposes we can take to be Minkowski space. There is on conformal symmetry, which is a property solely of 2D then a continuum space-time down to arbitrarily small world sheets. distances." The self-consistency of superstrings is heavily depen­ Gravity is highly non-linear since the gravity field cou­ dent on two experimentally undetected phenomena: su- ples to mass, and gravitons have a mass associated with persymmetry and higher spacetime dimensions. their energy. In other words, the gravitational field grav­ Ed Witten is at the Princeton Institute of Advanced itates. Is this a major conceptual subtlety when we try to Studies, New Jersey. He is a major contributor to su­ quantise gravity? "There are some conceptual issues in perstring theory. He was once & history major at Bran- quantising gravity, but it is not at all clear that they f.re deis, but had some science background. His interest in decisive. What to me is the decisive difficulty is the prob­ superstrings was sparked by a review article by John lem of the infinities that arise when one tries to quantise Schwarz555. He says: "Schwarz's review article made it the usual theory." (It is instructive to contrast this point easier to start learning about developments in superstring of view with Abhay's, stated earlier on.) theory than it had been before - but my concentrating on reading it was, of course, a result of the fact that I Renormalisation, Infinities, and was already very curious about superstrings." Anomalies Quantum field theory began when Planck tried to ex­ plain the emission spectrum of radiation. He QFT works with perturbation expansions around a postulated that energy was emitted discontinuously in Minkowskian background. Terms in these expansions discrete units, or quanta. Later, from a statistical analy­ are represented diagrammatically by Feynman diagrams. sis of the radiation field, Einstein suggested it wasn't just The has an attractive interpretation the emission and absorption of radiation which was quan­ as a representation of an actual physical process, and tised, but the actual electromagnetic field itself - in the (though limited to perturbation theory) this picture has form of . This idea was confirmed experimentally some validity. by the Compton effect. Naturally, there were attempts In QED interactions between photons and electrically to extend the quantum field idea to the classical gravity charged particles are linear. We can see this since cross­ field, described by a metric tensor in Einstein's GR. The ing beams of light don't bend each other. The case is different with gravitons however. Since gravitons have mass, and mass gravitates, two gravitons can bend each others path. Gravitons can also emit and absorb other gravitons, and couple to the other gauge bosons since •"Supeistring Theory, Phys.Rept.89:223,1982. they too have energy.

13 QFT introduces the concept of a particle's self-energy. arbitrary and if one did not have to remove it one would Let's take the electron self-energy as an example. Ac­ lost most of the predictive power In good cases, like cording to QFT there is a difference between a bare elec­ QED, after renormalisation the parameters in the quan­ tron and a physical electron. A bare electron is an un- tum theory are in one-to-one correspondence with the physical entity because it has no radiation field around classical parameters; thus in QED the only parameters it. Such a radiation field is known to exist because of are the charge and mass of the electron (the mass can be Heisenberg's uncertainty principle which allows for the scaled to one by a convenient choice of units)." presence of a sheath of virtual photons. When a bare elec­ The second step is to renormalise the theory by in­ tron interacts with this virtual field, the bare electron is corporating the blanket of virtual particles via sheathing converted to a physical electron. This self-interaction af­ the bare particle. This converts the bare particle into a fects the energy of the bare electron, and hence its mass. physical one. In the process, relations between bare and Quantities like charge are affected as well. In fact, all physical quantities are introduced. These relations con­ physical quantities are affected by the self-interaction. tain renormalisation constants. Thirdly we revert from When the Feynman integrals are evaluated for these the regularised and renormalised theory to the QFT by self-energy terms, the integrals diverge to infinity. A sim­ taking the limit when the cut-offs go to infinity. This ilar phenomena happen when we calculate the photon restores the original theory. The divergences reappear self-energy in the form of vacuum polarisation diagrams. again, but only in the renormalisation constants relat­ These divergences are known as ultra-violet and infra-red ing physical and bare quantities. Like the bare quan­ divergences respectively. tities, the constants aren't observable. The original in­ The first order terms (tree level) are finite, but the tegrals, written in terms of physical quantities, become higher order terms (the radiative corrections) involving well-defined and finite to all energy scales! So in a sense closed loops diverge. Physically, the difficulties arise as a we have shifted all the infinity problems aside. According result of the concept. Heisenberg's princi­ to Ed: " is natural and necessary even ple says that energy conservation can be violated locally if the effects are finite. So there is no issue of not having such that the relation AE • At > ft is obeyed. Since an to renormalise. That would be so even if we were deal­ electron is taken to be a point, emission and reabsorption ing with a fundamental theory valid to arbitrarily short of a virtual photon need take no time at all. So the virtual distances. photon can have an infinite amount of energy. This con­ "In practice we are always (except maybe in string tributes an infinite mass to the physical electron, since theory) dealing with theories only valid down to very it must haul around the virtual photon sheath. Other short distances, so there really is a cutoff- we just do not properties of the electron affected similarly. know what it is. Renormalization then extracts whatever Despite these shortcomings, QFT is one of the most can be learned without knowing what the cutoff is." successful theories we have. On this Ed says: "Just be­ It would seem that you have to apply this procedure to cause we write down a quantum field theory Lagrangian all the terms in the S-matrix expansion, for all the phys­ containing some fields does not mean that we know what ical quantities you want to compute. Now the expansion the particles are going to be. QCD is an example where is an infinite series of terms, which seems to require an the particles (mesons and baryons) do not even appear infinite number of regularisation and renormalisation op­ in the Lagrangian (which contains quarks and gluons). erations. Amazingly QED has a gauge invariance which Even if the particles are in natural one-to-one correspon­ allows you to remove all the divergences from the mea­ dence with the fields - as in QED - the masses of par­ surable quantities to all orders of perturbation expansion, ticles cannot be just read off from the constants in the via just one type of renormalisation. Lagrangian. It is necessary to compute all kinds of quan­ Why can't we compute physical quantities directly, in­ tum corrections. The fact that the nature and masses of stead of going via this roundabout? Feynman himself the particles is - in large part - computed, not postulated, thought renormalisation was 'sweeping the problem un- is a manifestation of the fact that the QFT description d>?r the carpet'. He thought that we should be able to of nature is more thorough than its predecessors." formulate a theory such that we get zero To handle these divergences, QFT invokes the princi­ energy from the very start. What physics could we be ple that energy cannot be measured, only differences in missing here? One possibility, already mentioned by Ab- energy can be. A scheme called renormalisation was de­ hay, is that the infinities ultimately arise from neglecting veloped by Feynman, Tomonaga, and Schwinger back in effects in Planck scale physics which would have a subtle 1947 to take care of this, for QED. effect on low energy physics. To handle divergent graphs, we firstly dimensionally When we try to fit gravity into QFT, the algorithm is regularise the offending integrals by imposing cut-offs. to linearise the GR field by choosing a co-ordinate system This modifies the theory into another one. The cut-offs in which the metric tensor can be expressed as g^ = set some energy scale limit to the theory's predictions. i)iiv + hf,v where rj^ is the Minkowski metric and /»M„ Ed comments: "The cutoff theory has physically unac­ is a small perturbation such that |/i^„| < 1. This small ceptable properties, so it is necessary to remove the cutoff perturbation - a , is subject to quantisation and 'renormalise.' This is just as well, since the cutoff is on a Minkowski background which provides the causal

14 structure. It is treated as the 'graviton' particle. There are currently three d — 10 superstring theo­ "Formally there is no problem in constructing the per­ ries (Type I, Type II, heterotic) and although they have turbation expansion of the S matrix along the above lines. no adjustable dimensionless parameters, there are many The only real difficulty is that one meets ultra-violet mathematically consistent solutions which seem to sat­ divergences that cannot be renormalised away (without isfy the superstring equations of motion. Choosing phys­ introducing new parameters absent in the classical the­ ically representative solutions phenomenologically seems ory)." no different to inserting SM parameters by hand. The The problem is that in gravity the problem of finding physical solutions can only be solved if has dimensions. The renormalisation algorithm works we know how compactification works. Doing this we can for QED because there the coupling is a dimensionless find the ground (vacuum) state of the theory. From there, number, the fine structure constant. Because of this, the we can work out the low energy excitation spectrum, find Feynman graphs of different loop order can be shuffled the ratios of the excitation masses, and determine cou­ around, cancelling the infinities which arise from evalu­ plings constants from the topology of the compactified ating the integrals. In QG we cannot do this. So QG space. must be finite term by term or diverge. It is found that Ed is optimistic about this arbitrariness: "The attrac­ pure QG diverges at the 2-loop level. tive near-uniqueness that string theory appears to pos­ On the associated problems of anomalies in QG he sess in principle is largely lost in practice because one comments: "The existence of a quantum theory is always finds a plethora of possible classical solutions. Hopefully, delicate. Once one writes down a classical Lagrangian, in time we will learn something new - perhaps connected one does not automatically get a quantum theory. In­ with the question of why the van­ finities and anomalies are two of the main ways to fail. ishes - and the nature of the issue will change." Anomalies that affect gauge invariance are even worse On the problem of formulating a non-perturbative the­ than lack of renormalisability; they mean that the per­ ory of superstrings Ed says: "My work with Nathan turbation expansion of the quantum theory just does not Seiberg (and related work he did by himself) led to some make sense." Essentially the time rate of change of some new results about non-perturbative behaviour of some quantity is zero i: the classical theory and upon quanti­ quantum field theories, but not yet to new results on sation, this rate Decomes non-zero, breaking some sym­ non-perturbative behaviour of string theory. metry, and hence a . String theory appears to admit a group of discrete field transformations, called 5-dualities, as exact non- Recent Work in Superstrings perturbativf: quantum symmetries. How could these symmetries help in formulating a non-perturbative QG How does extending the idea of particles to super- theory of strings? strings change things? An analogy would be helpful here. "I don't know yet, but I think it is a very exciting clue. Let us take the decay of the free neutron, which is me­ J-dualities are symmetries that have no real analog in our diated by the weak force. We know that the old Fermi previous experience in physics (they do persist in some theory of weak decay is not renormalisable. However, field theories - including some of those I studied with the electroweak theory is. The electroweak theory re­ Seiberg - which are related to low energy limits of string places the point interaction of the old Fermi theory by theories)." inserting a massive W boson , which mediates Particles trace out world lines, but strings trace out the decay. The theory is now renormalisable. Virtual su- world sheet.i or cylinders, which can be treated as Rie- permassive superstring excitation states affect the renor­ mann surfaces. When the Feynman diagrams are worked malisability of superstring interactions by 'softening' or out for the closed string, it is found they finite order by spreading out the interactions. order in the genus expansion. Ed says: "Certainly, in going to string theory, every­ Ed says: "In string theory everything is finite - there thing becomes softened, and eventually finite. How best is no need for renormalisation. to understand it is not yet clear, but somehow, the ul­ "Feynman diagrams are replaced by Riemann surfaces; traviolet region is missing in string theory." these have much additional symmetry and structure, and Although the superstrings are extended objects, their this is related to some of the known wonders of the sub­ interactions still take place at single spacetime points. ject - there are surely also unknown wonders (5-duality However this point depends on the Lorentz frame of the is a hint of them), but since they are unknown I cannot observer (the pants diagram phenomena) unlike the point tell you about them." particle case. There, the spacetime interaction point is On the matter of progress on the physical principles identical in all Lorentz observer frames. Thus there less behind superstrings which should generalise Einstein's freedom in constructing string theories compared to the , he replies: "The new physical prin­ point particle case. This is why we have only three super- ciples haven't yet been found. It is hard to say how much string theories instead of the infinite number of point field progress there has been, since some of the things that theories. (We have to choose the correct point-particle have been learned may well turn out to be important QFT via .) clues."

15 Superstrings are supposed to reside in 10 spacetime di­ Moreover, if supersymmetry is observed, it won't be just mensions, but we are really only speaking approximately. that experimentalists will say 'Yes, there is supersymme­ The phase space of the superstring consists of all the pos­ try.' They will observe a plethora of superpartner masses sible orientations and states the string can be in. What and couplings. There is a reasonable hope that in all that are the problems of formulating superstrings directly in data there will be clues to how string theory should be four spacetime dimensions from the start? Aren't super- done - or tests of theoretical ideas that may have emerged strings supposed to generate spacetime, instead of just in the meantime." residing in it, since superstrings have gravitons in their I asked Ed what the inspiration for his ideas was. Did excitation spectrum? he manipulate pagj-long equations in his head? Did he "It is true that in our present understanding 10 space- deal with concepts, and then phrase everything in maths time dimensions is the classical solution of string theory later on? Where Einstein, Galileo, and Newton were dif­ with the maximum symmetry. Exactly how strings can ferent from everyone else was they could ask really simple be so smart as to generate space-time is still a mystery, questions which had shattering consequences for existing but they do. physical theories. Was this Ed's modus operandi too? "You can formulate strings 'directly in four dimensions To this he replied: "I am afraid I don't work on the from the start' but then you do not have the degree of cosmic scale you are suggesting. Most of the time I am uniqueness that one has in 10 dimensions. That is why bogged down doing nothing, and whatever successes I the theory is often said to be 10 dimensional." have had come by focussing on a very little bit of the There is no mechanism in superstring theory which total picture." tells how us the extra dimensions should compactify. Feynman said about superstrings, "I don't like it that Outlook they're not calculating anything. I don't like it that they don't check their ideas. I don't like it that for anything It is clear that unless someone has a world-shattering that disagrees with experiment, they cook up an expla­ idea, we are a long way off from a full theory of QG. I nation - a fix-up to say 'Well, it still might be true.' For asked Chris about possible lines of future investigation. example the [superstring] theory requires 10 dimensions. "I think that the superstring programme and the Well, maybe there's a way of wrapping up six of the di­ Ashtekar programme are currently the only two really mensions. Yes, that's possible mathematically but why viable approaches to trying to construct a full quantum not seven? When they write their equation, the equation theory of gravity, and I therefore support strongly work should decide how many of these things get wrapped up, on both fronts! not the desire to agree with experiment..." What is your "Myself, I have long been bothered by the use of con­ response to this remark? tinuum ideas in both quantum theory and classical gen­ "Some of the predictions are indeed checked. For in­ eral relativity, whereas both programmes do take this for stance, strings predict gravity while pre-string physics granted. So I would like to be able to get away from makes quantum gravity impossible. This prediction has this, but the problem is to find a theoretical framework been tested - and the result is one of the main reasons that does not seen to contrived and artificial. One of for the interest in string theory." the many reasons why the superstring and Ashtekar pro­ grammes are so important is that they do not have this Are you familiar with the Ashtekar canonical gravity contrived nature: within their own terms each has quite program? If you are, do you think that the program is compelling reasons for taking it seriously. founded on valid physical assumptions? "Some of the work in that program is interesting to "Similarly, I (like Roger Penrose) believe in my heart me. I'd peisonally be surprised, however, if it turns that quantum theory itself needs to be changed radically out that conventional general relativity exists as a non- to reflect our changing conception of space and time. perturbative quantum field theory." But, again, it is difficult to do this in a way that is be­ lievable to one other person than the author of the paper Do you think the LHC will r"-ed any light on low energy (Salam's definition of what it means to say that a theory superstring phenomenology? The detection of supersym- is worth taking seriously!)" metry partners could be crucial. To explain the absence of SUSY partners at low energies we have introduced the Ed Witten gives this advice to those who want a try at idea of the spontaneous breaking of supersymmetry, in superstrings: "It is hard to give general advice to young analog to the for electroweak symme­ researchers. But people who want to learn strings should try breaking. One facet of the SM which is unconfirmed definitely make sure that along the way they get a thor­ is this mysterious Higgs field which pervades all space- ough knowledge of what is known in particle physics and time, which makes the electroweak theory work by giving how it can be (more or less) derived from strings." masses to particles, and which we have no shred of evi­ As to where superstrings are heading he has this to dence for. say: "Lots of really nice things have been discovered, and that will continue. Sooner or later, probably far in "Apart from gravity, supersymmetry is one of the gen­ the next century, the underlying unifying ideas will be eral predictions of string theory. Therefore, confirming it discovered and then the physical landscape will be very at the LHC would give a big boost to the whole effort.

16 different from what we can imagine today. If we are lucky 94-9-4, n.d. (reed Oct 1994) 13pp. e-Print Archive: gr- and work hard, maybe we will live to see the day!" [email protected] 9411055 Acknowledgements Overview and Outlook, Abhay Ashtekar (Penn State U.), in Canonical Gravity from Classical to Quantum, This article was based on interviews with Chris Isham, edited by J.Ehlers and H. Friedrich, Springer-Verlag Ed Witten and Abhay Ashtekar. (1994). Quantum Gravity: A Perspec­ Further Reading tive, Abhay Ashtekar (Penn State U.), in The Proceed­ The Problem of Time ings of the International Conference on Mathematical Physics Towards the 21st Century, Negev, Beer Sheva, Prima Facie Questions in Quantum Gravity, C.J. Israel, 14-19 Mar 1993; edited by R. N. Sen and Ger&ten Isham, Canonical Relativity: Classical and Quantum, (Beer-Sheva University Press, 1994). eds J. Ehlers and H. Friedrich, Springer-Verlag, Berlin Maihem 'ical Problems of Nonperturbative Quantum (1994). General Relativity, Abhay Ashtekar (Syracuse U.), SU- Time, Gravity, and Quantum Mechanics, William GP-92-11-2, Dec 1992. 87pp. Invited talk at Lectures G. Unruh (British Columbia U.), e-Print Archive: gr- given at Les Houches Summer School on Gravitation and [email protected] - 9312027. Quantisations, Session 57, Les Houches, France, 5 Jul - 1 Aug 1992. e-Print Archive: [email protected] 9302024 The Physical Foundations of General Relativity, Den­ nis Sciama, Anchor Books, Doubieday, 1969. Conceptual Problems of Quantum Gravity. Proceed­ ings, Osgood Hill Conference, North Andover, USA, Conceptual and Geometrical Problems in Quantum MAY 15-18, 1988, A. Ashtekar, (ed.) (Syracuse U.), Gravity, C.J. Isham in "Recent Developments of Quan­ J. Stachel, (ed.) (Boston U), 1991. Boston, USA: tum Fields", ppl23-225, eds H. Mitter and H. Gaustener, Birkhaeuser (1991) 602 p. (Einstein studies, 2). Springer-Verlag (1992). Lectures on Nonperturbative Canonical Gravity, A. Canonical Quantum Gravity and the Problem of Time, Ashtekar (Syracuse U.), 1991. Singapore, Singapore: C.J. Isham In "Integrable Systems, Quantum Groups, World Scientific (1991) 334 p. (Advanced series in as­ and Quantum Field Theories", ppl57-288, eds L. A. trophysics and cosmology, 6). Ibort and M. A. Rodriguez, Kluwer Academic Publishers, London (1993). New Variables for Classical and Quantum Gravity, A. Asbtekar (Syracuse U. and UC, Santa Barbara), 1986. Canonical quantum gravity and the question of time, Phys. Rev. Lett. 57 ( 1986) 2244-2247. C.J. Isham in "Relativity: Classical and Quantum", eds J. Ehlers and H. Friedrich, Springer-Verlag, Berlin Superstrings (1994). e-Print Archive: [email protected] 9310031. Superstrings: A ?, P.C.W. Davies The Ashtekar Program and J. Brown, Cambridge, Cambridge University Press, 1988. Loops of Space, Marsha Bartusiak, Discover Magazine, April 1993. : a scientific odyssey through parallel uni­ verses, time warps, and the 10th dimension, Michio Gravity Quantised?, John Horgan, Scientific American, Kaku, New York, Oxford University Press, 1994. September 1992. Introduction to superstrings, , New York: Bibliography of Publications related to Classical and Berlin: Springer-Verlag, 1988. Quantum Gravity in terms of the Ashtekar Variable?, B. Briigmann (Syracuse U), SU-GP-BIB, Mar 1993. 14pp. The God particle: if the universe is the answer, vihat is pre-Print Archive: [email protected] - 9303015. the question?, Leon Lederman with Dick Teresi, Boston, Houghton Mifflin, 1993. Canonical Quantum Gravity, Karel V. Kuchar (Utah U.), UU-REL-92-12-10, n.d. (reed Apr 1993) 40pp. e- The in the : a discussion of the mysteries of Print Archive: [email protected] - 9304012. quantum physics, P.C.W. Davies and J.R. Brown, Cam­ bridge [Cambridgeshire]: New York: Cambridge Univer­ Lectures on Quantum Mechanics, Paul A.M. Dirac, sity Press, 1986. New York, Belfer Graduate School of Science, Yeshiva University, 1964. Superstrings and the Search for the Theory of Every­ thing, F. David Peat, Chicago, Contemporary Books, Recent Mathematical Developments in Quantum Gen­ 1988. eral Relativity, Abhay Ashtekar (Penn State U.), CGPG- Quantum field theory, Franz Mandl and Graham Shaw,

17 Chichester, Wiley, 1990. Superstrings, John H. Schwarz, Physics Today, Novem­ ber 1987. Superstrings, Michael B. Green, Scientific American, September 1986. Superstring Theory. Vol. 1: Introduction, M.B. Green (Queen Mary - Westfield Coll.), J.H. Schwarz (Cal Tech), E. Witten (Princeton U), Cambridge, Uk: Univ. Pr., 1987. Superstring Theory. Vol. 2: Loop Amplitudes, Anoma­ lies and Phenomenology, M.B. Green (Queen Mary - Westfield Coll.), J.H. Schwara (Cal Tech), E. Witten (Princeton U.), Cambridge, Uk: Univ. Pr., 1987.