Dialogue on the Foundations of String Theory


Several years ago there was published in Rome a salutary edict which, in order to obviate the dangerous tendencies of our present age, imposed a seasonable silence upon the Pythagorean opinion that the earth moves. There were those who impudently asserted that this decree had its origin not in judicious inquiry, but in passion none too well informed. Complaints were to be heard that advisers who were totally unskilled at astronomical observations ought not to clip the wings of reflective intellects by means of rash prohibitions. Upon hearing such carping insolence, my zeal could not be contained. Being thoroughly informed about that prudent determination, I decided to appear openly in the theater of the world as a witness of the sober truth.

                                                                                                                Galileo, 1632


Simplicius: I have a very basic questions about string theory. I'm not sure if I can express it clearly, or even if the premise of the question is correct, but here goes...  In general relativity, the effects of gravitation are attributed to curvature of spacetime. Test particles follow geodesics in curved spacetime, and this accounts for how those particles are affected by gravity. The field equations relate the curvature of spacetime to the distribution of mass, energy, and stress.


Now, in string theory we hear about spacetime, and indeed curved spacetime, with curled up dimensions, and so on, but this seems confusing, because if we already have curved spacetime, then we already have gravitation (don't we?), so what are the "strings" doing for us? Are we supposed to understand that, somehow, spacetime consists of strings? Or do strings exist within spacetime? If the latter, is the spacetime allowed to be curved? If so, doesn't this curvature already imply a kind of gravity? On the other hand, if the answer is that spacetime actually consists of strings, then I'm even more baffled, because all the depictions of "strings" that I've ever seen (in popular accounts) seem to show them wiggling around in spacetime. If we are supposed to understand that spacetime consists of (or arises from) strings, then it seems to me the popular accounts are all very incorrect and misleading.


Hopefully you get the gist of my confusion, Salviati. I've heard that one of the great things about string theory is that it “predicts” the existence of gravity, but if strings reside in a "pre-existing" curved spacetime (with its own field equations?), then I have trouble understanding the conceptual foundation of it. Perhaps I can formulate my question more precisely by asking this: In string theory, do the strings reside within spacetime? If so, do we have field equations – analogous to Einstein's equations – governing the metric of that spacetime? And if so, are those field equations separate from the equations governing the behavior of the strings within spacetime? Do the strings experience "Einsteinian gravity" as they move within this spacetime, and is this different than the gravity that is produced (or embodied or entailed) by the strings themselves?



Salviati:  You asked “Are we supposed to understand that, somehow, spacetime consists of strings?" The answer is yes, spacetime somehow consists of strings. Admittedly we start with strings as objects propagating on a pre-existing geometry, but without strings the geometry is fixed, which is to say, there are no additional degrees of freedom that would allow us to change this background geometry. However, when strings are added, we find that if we pump a condensate of strings in a certain vibration mode to the pre-existing spacetime, the physical effect on all other strings and all other objects will be indistinguishable from a deformation of the original geometry. In other words, deformations of spacetimes are equivalent to propagating strings in a certain state. Thus the strings can change the effective metric tensor of the spacetime. (In fact, according to string theory, we could add or subtract strings from spacetime in such a way as to yield an effectively vanishing metric tensor!) Now, the consistency criteria for strings to propagate on a background imply that the background must satisfy Einstein's field equations (with certain corrections which we needn’t worry about here). In summary, one can say that the effective spacetime is created of strings.



Simplicius: You answer is quite helpful and illuminating, Salviati… so naturally it leads me to more questions! First I would like to ask if the original pre-existing “fixed” geometry, in which the strings reside, oscillate, and move about, has a definite number of dimensions and topology. Or doesn’t it matter? For example, does it have some compactified dimensions already, or do those compactifications arise only when strings are added? If adding or subtracting strings can produce a vanishing metric tensor (as you mentioned), then can it also change the number of dimensions and/or the topology?



Salviati: Yes, the pre-existing metric is completely classical and has a very definite number of dimensions, a very definite topology, and a very definite shape and size. In critical (super)string theory the pre-existing metric must be of 10 dimensions.  Some of the dimensions may be infinite, others may be compact (i.e., curled up and closed). All possibilities satisfying Einstein's equations (with suitable corrections) are allowed. For example, circular dimensions may have an arbitrary length.


The background geometry is purely classical. The strings embody all the quantum effects. This is standard practice in particle physics, i.e., we write quantum objects as their vacuum expectation value (i.e., a classical value) plus a quantum fluctuation. In this case, the classical value is ordinary geometry, and the fluctuations arise because of the potential for the existence of strings within that classical background.


You seem to be concerned about the fact that string theory is built upon this fixed pre-existing geometry, Simplicius. But you shouldn’t worry about that, because the detailed attributes of that background turn out to be irrelevant. By starting with different pre-existing geometries, we can arrive at exactly equivalent effective spacetimes, merely by invoking a different arrangement of strings. Let me describe an analogy that may help you understand this. Given an analytic function, we can expand the function into a power series about any chosen point, and this procedure leads to a set of coefficients for the power series. If we then expand the original function into a series about some other point, the coefficients of the resulting series will obviously be different from those of the first series, and yet (assuming the function is analytic over the whole region containing our two chosen points) the two series represent the same function, i.e., they sum to exactly the same values at corresponding points. Likewise for string theory, the choice of a background geometry is analogous to the choice of a point about which to expand a given “function” (which is string theory). In this sense, the choice of a classical background spacetime is rather arbitrary. One can get from any allowed effective geometry to any other by adding strings. This applies not only to the metric, but also to the topology, the numbers of dimensions, and so on.  The classical background geometry invoked by string theorists always has 10 dimensions, but the number of dimensions of the effective spacetime can dynamically change by adding appropriate string condensates.


The background geometries may also have a complicated spatial shape. For example, all Calabi-Yau manifolds that we usually use are very curved but Ricci-flat: they obey vacuum Einstein's equations and can thus be used as backgrounds. There's a whole lot of an interesting structure in Einstein's equations once extra compact dimensions are allowed.



Simplicius: So, as I understand you, once the strings are added, they induce an “effective” metric on top of the pre-existing classical metric. But it sounds as if you are saying two mutually contradictory things. On one hand, you are arguing that the choice of a classical background spacetime is arbitrary, because we can produce any effective spacetime we want (assuming it is in the set of “allowed” spacetimes) simply by a suitable arrangement of strings. We can even change the topology and number of dimensions. But on the other hand, you continue to say that the background spacetime must have 10 dimensions, and some of them must be curled up in certain ways. I presume all the “interesting structure” you mentioned (e.g., Calabi-Yau manifolds) is not totally irrelevant to the resulting theory, so it remains very unclear to me just how arbitrary your background spacetime can be.  Is the original “pre-existing” geometry observable in any sense? Does it have any consequences? If not, why is it necessary to introduce it in the first place? Could we select any arbitrary background geometry and arrive at effectively the same theory?



Salviati: Yes, as said above, any consistent pre-existing metric leads to the same theory: the same function can be Taylor-expanded around any point (corresponding to a geometry) you like. The equivalence between the results with different starting backgrounds is the reason why you really can't universally measure what the background is (one can only measure the full "field", background plus foreground, in labs) and why physics of string theory is said to be background-independent. Because the proof of this fact needs some calculation, we say that this background independence is not manifest and we might speculate that there could also exist a manifestly background-independent language to talk about string theory, a hypothesis that could be right or wrong.



Simplicius: I still feel as if I’m being told two contradictory things about the background spacetime. On one hand, I hear that it is unobservable and ultimately unimportant, like choosing a point about which to expand a function, since the resulting sums will always be the same (assuming convergence), and this extends even to the topology and number of dimensions, because all of these can be compensated by suitable arrangement of strings. But on the other hand, I’m being told that the background spacetime must satisfy some very strict conditions, such as having exactly 10 dimensions, and so on. So I’m perplexed. If the characteristics of the background spacetime ultimately don’t matter, and can be compensated for by a suitable arrangement of strings, then why is it so essential to postulate a background of 10 dimensions? I feel like I’m missing something here. Either the characteristics of the background spacetime matter or they don’t. Which is it?



Salviati: When we say that the background can be "anything", we never mean "really anything". We mean any configuration that satisfies the conditions of string theory - a generalization of Einstein's equations, if you wish. There are quite many solutions - that's why we talk about the large "landscape" of possibilities - but of course this set of solutions is infinitely smaller than the set of possibilities we could think about before we learn the conditions of string theory. For example, for any description of string theory where all degrees of freedom look like geometry, the total number of dimensions is 10 in perturbative superstring theory or 11 in M-theory. These are universal constraints on any full geometry (or the background) required by (super)string theory.


There also exist non-critical string theories where the dimensionality may be different, but let's omit this class because it would make you even more confused!  With that disclaimer, superstring theory does indeed imply that the total number of dimensions must be 10 or 11 - a prediction that general relativity itself wouldn't be able to make. There is no freedom about it. When we say that the geometry doesn't matter, it certainly doesn't mean that we are allowed to consider, say, 57-dimensional flat background spacetime. String theory is a highly constraining theoretical structure.



Simplicius:  If I understand you, then, string theory only “works” with a highly constrained classical background spacetime. This is somewhat different than the impression I had at first, when you mentioned how strings are capable of creating any allowed geometry from any other, so the choice of background was analogous to arbitrarily choosing a point about which to expand a given function. You seem to be saying now that the choice of background spacetime is highly constrained in some respects, having to do with dimensionality and topology, but arbitrary in others. But I’ve heard that some of the unconstrained features of the background spacetime (the vacuum) are actually consequential, in the sense that the observed laws of physics depend on them. So this, too, leads me to think that the selection of a suitable background spacetime is not as unproblematic as you portray it, and that it does have observable consequences.


Perhaps we should come back to that later. I wanted to ask you about something else that has been bothering me. Assuming the “background” geometry is a solution of Einstein’s equations, then is the “foreground” (i.e., the effective metric induced by the strings) also a solution of Einstein’s equations? If so, are they different solutions? Or are we to understand that these two metrics are one and the same? If so, isn’t there a “chicken and egg” problem here? I mean, we say the effective metric arises from strings propagating in a pre-existing metric, so I don’t see how we can then turn around and say the effective metric serves as the pre-existing metric. But if we don’t identify them with each other, I’m left with two different metrics, both of which apparently are supposed to be solutions of Einstein’s field equations. One of them is a classical spacetime, which presumably satisfies the field equations exactly, just by fiat, whereas the higher-level spacetime is just an effective manifold. Is that right?



Salviati: Yes, the background is a solution of the classical field equations of general relativity (for number-valued fields) while the total effective metric (background plus foreground, as you call it) is also a solution of the full Einstein's equations (imposed upon operators themselves). The latter statement is equivalent to the condition that the foreground strings satisfy their dynamical equations of motion in a given background. On the other hand, the "foreground metric" without the background doesn't satisfy Einstein's equations. (We wouldn’t expect it to, because the field equations are non-linear, so the difference between two solutions is not, in general, a solution.) There is no contradiction in having two different  (and indeed they are nonequivalent) solutions to Einstein's equations. For example, think about empty flat space and a space with a gravitational wave added to it. They are not equivalent. They are related by a physical process that changes one to another - a deformation of metric i.e. addition of strings in the graviton mode.



Simplicius: So you’re saying the background metric satisfies the vacuum field equations of general relativity, which is to say, they are Ricci-flat (the Ricci tensor vanishes, but the full Riemann curvature tensor need not vanish). A couple of things puzzle me about this. First, I can’t figure out what is supposed to be the “source” of the Riemann curvature, in accord with the field equations. I know the field equations support gravitational waves, but I don’t think anyone is suggesting that something like a Calabi-Yau subspace represents a wave solution (are they?)  By comparison, for the plain old vacuum solution of the field equations near a spherically symmetrical mass in ordinary general relativity (for example), the “source” of the curvature is the central mass… but what is the source of the curvature of the background metric in string theory?  Do the strings or membranes act as sources? Surely they have energy, so they ought to act as sources, but then we wouldn’t be dealing with vacuum solutions, would we?



Salviati:  There need not be specific sources in order to have non-vanishing Riemann curvature. Einstein's equations only constrain the Ricci tensor, which has as many components as the metric tensor, so it is the right number of equations for the right number of variables (allowing for four degrees of freedom for the coordinates). The Riemann tensor in the vacuum can be anything as long as the Ricci tensor is zero. In this sense, the curved Calabi-Yau geometry indeed resembles a gravitational wave. The reason why this effect – non-trivial vacuum solutions – is not familiar to us from electromagnetism is that Maxwell's equations are linear and the only vacuum solutions are thus waves. However, Einstein's equations are nonlinear so they allow non-waving non-propagating solutions such as the Calabi-Yau geometry.  Now, if we considered a non-compact (infinite volume) Calabi-Yau, we could say that its Riemann curvature is sourced by matter at infinity, but for compact Calabi-Yau spaces the Riemann curvature is simply not source by anything in the space.


Strings, branes, and all other objects of course influence the metric, they act as sources, but in the simplest Calabi-Yau geometries, there are no individual strings or branes present. The Calabi-Yau geometry we talked about is a vacuum solution in the sense that the right-hand side of Einstein's equations is zero.



Simplicius:  I understand your point about the non-linearity of the field equations allowing for “sourceless” non-propagating curvature (although the stability of such solutions isn’t self-evident to me). But I’m still trying to understand the significance of the classical spacetime on which you say string theory is based. Let me try to approach it from a slightly different angle. All the popular accounts of string theory talk about the strings vibrating and oscillating, which makes me think there must not only be a pre-existing spacetime in which to carry out those oscillations, but also some kind of “inertia” and some kind of tension or restorative force. (If something has no inertia, it can’t oscillate.) In general relativity, inertia and gravitation are very closely identified, but in string theory, if the basic inertia of the strings (or the parts of strings) is a pre-existing attribute, whereas the effective gravitation is a higher-level consequence of vibrating strings, then is there also a higher-level (“foreground”) version of inertia, and if so, is the mechanism of this meta-inertia distinct from that of the underlying inertia?  Or is this another chicken and egg problem, i.e., the inertia that emerges from the behavior of strings is identified with the inertia that produces the behavior of strings in the first place?



Salviati: The character of inertia in string theory is identical to the same concept in general relativity. The background geometry is what specifies where the centers of strings want to move – along geodesics – as well as how they respond to the internal tension by vibrating: the worldsheets they span in Minkowski spacetime classically extremize the proper area measured by the background geometry. We literally talk about normal fundamental strings and what they would do in a general relativistic spacetime.


What's new here is that if we promote the locations of points along the string to operators and quantize the theory, we find that one of the lowest-lying energy eigenstates of these strings has a physical effect on spacetime that is identical to deforming the spacetime geometry. In other words, if we allow (closed) strings to interact (by simply joining and splitting if they cross), a new string moving in a background geometry equipped with a condensate of other strings will have the same motion – or inertia – as if it propagates in a deformed geometry without strings. So, yes, there is what you called “meta inertia” in the effective spacetime. Even if you cared only about general relativity itself, this fact would probably be highly surprising and important for you. Similar statements apply to all other fields and degrees of freedom: the values of all of them can be changed by an appropriate creation of strings in different states (and superpositions of these states). In point-like field theories, different elementary particles are made out of different material. In string theory, everything is made out of the same stuff.



Simplicious: This is all very interesting, Salviati, but I must admit to being perplexed by what you’ve described. We seem to have two physically very distinct kinds of spacetime curvature (and inertia, and gravitation). We have the classical background spacetime itself, which is curved in accordance with the field equations of general relativity, and then we have the effective metric, which is the net result of the background metric and the foreground metric, the latter being the product of the strings. You stated that neither of these is individually observable, i.e., we can only observe the net effect of the two, and you also indicated (I think) that, starting with a different background, we could get to the same net result simply by postulating a different stringy foreground. Hence the individual natures of the background and foreground my be (in some sense) just an artifact of our choice of the point at which to expand the Taylor series (to use your analogy). Nevertheless…


I’m bothered by the fact that a theory that is supposed to explain the apparent curvature of spacetime in terms of strings, still needs to invoke classical curvature of the “raw” background spacetime, which is the very thing we are trying to explain. For example, you mentioned that “the background geometry is what specifies where the centers of strings want to move (along geodesics)”, so it sounds as though we already have general relativity and a theory of gravity and inertia before we even add the effect of the strings. And this is necessary, because without inertia, the strings have no oscillatory modes, so they can’t get off the ground to produce their meta-inertia.  In order for them to produce their meta inertia (and gravity), they rely crucially on a pre-existing classical inertia (and gravity).


If the classical background spacetime is posited to be a solution of the field equations, prior to any string ingredients being added, then it seems (to me) that spacetime curvature and inertia and the tendency of mass-energy to follow geodesics already exist without strings, so we haven’t really explained anything unless we can totally eliminate that classical background spacetime. Am I reading too much into the background spacetime? Or are we supposed to postulate some sort of meta-string theory to account for the background spacetime dynamics? And then an infinite regress of meta-meta string theories?



Salviati: We don't have any general relativity in string theory before strings are added. Before strings are allowed to be condensed, the spacetime is non-dynamical (haven't we discussed this already at length? I feel that we start to repeat ourselves) and its shape is unconstrained. Both Einstein's equations and the links between different shapes of spacetime arise only once we add strings. You say "we haven't explained anything". I don't know what you mean by "anything". What has been derived from string theory is dynamical geometry following Einstein's equations, i.e. gravity, because strings are equivalent to dynamics (changes) of spacetime geometry and the consistency of their equations of motion requires that the spacetime geometry must satisfy Einstein's equations. As explained above, saying that all of spacetime geometry is equivalent to strings is a bit of a stretch because the geometry when it's zero is singular and ill-defined.


I think there is something irrational about the way you are thinking about this whole issue. We don't need to get rid of spacetime altogether to derive general relativity. Quite the contrary, every good theory of physics should explain that spacetime should exist, and only shapes that look similar to solutions of general relativity are allowed. If you want to derive (or assume) that there is no spacetime from the theory, you clearly want just the opposite of the goal with which we stated at the beginning, because our goal was to derive the fact that spacetime in accord with general relativity does exist, not that it doesn't exist. In some other approaches, spacetime as we know it (or some of its dimensions) are not manifest at the beginning but they emerge. But the main victory here is not the fact that spacetime is not manifest (at the beginning) but rather the fact that spacetime does appear at the end.


You might want to satisfy some religious dream where everything is really "created out of nothing", in the same way as God's Creation. But from the scientific viewpoint, this is not the goal.



Simplicius:  Perhaps some of my confusion is due to the terminology. At one point you said the background geometry can be “time-dependent”, but later you said it is “non-dynamical”. If the background spacetime is a time-dependent solution of Einstein’s field equations (as you said), and the mass-energy of the strings moves inertially along geodesics in this spacetime (as you said), then I would call this a dynamical spacetime. (Surely the mass-energy of a string can’t be treated as purely passive in the background spacetime, i.e., it must also act as a source.) Hence we have all the ingredients of classical general relativity. But I see that you’re using the terms differently, because you say the time-dependent background spacetime is not dynamical. Can you clarify the distinction between “time-dependent” and “dynamical”?


This may just be semantics, but I think it actually gets to the heart of my uneasiness with the whole subject. We begin from the standpoint of an existing classical theory of gravity, called general relativity, in which spacetime is a primitive element. In other words, a basic premise of general relativity is the existence of a continuous spacetime manifold endowed with a metric that satisfies Einstein’s field equations. Objects follow geodesics in this manifold, and also serve as sources of curvature (bearing in mind that gravity itself gravitates, so we may have “sourceless” curvature too). Now, there are no “gravitons” or anything like that in classical general relativity, so it is (seemingly) incompatible with quantum field theory. As a result, we have this unsatisfactory dualism in physics, with most forces being explicable in the context of quantum field theory as phenomena within spacetime, but with gravitation being the result of curvature of spacetime itself… making it appear to be a qualitatively different kind of thing. We would like to understand all of these forces in a single unified context – if possible – one that is consistent with the principles of quantum mechanics.


Now, my impression (and I may be very wrong about this) is that string theorists say “The effects of gravity are actually due to the behavior of strings, which can be described in a way that is consistent with quantum theory, so we will no longer need to account for gravity by appealing to a classical geometrical curvature of the spacetime manifold (as general relativity does). Instead, we will explain gravity as a manifestation of effective curvature in an effective manifold induced by some condensate of strings.”


This sounds impressive to me, because it sounds like I’m about to learn how the appearance of a curved spacetime manifold (with all the consequences of general relativity) arises from a more primitive set of premises, so that we no longer need to simply postulate (as general relativity does) the existence of a raw classical spacetime manifold endowed with a metric that somehow satisfies Einstein’s field equations, and in which objects follow geodesic paths and act as sources of curvature. So I say “Wonderful! Tell me about it!”  But then I’m told that the very first step in explaining the appearance of the curved spacetime of general relativity is to begin by simply postulating a classical curved spacetime satisfying the field equations of general relativity, in which objects inertially follow geodesic paths.


Now, I don’t dispute that it’s possible to posit a framework in which an effective metric, distinct from the underlying metric, emerges at some level, nor even that it’s possible to make the effective field compatible with quantum theory. And I understand that this might turn out to be mathematically possible only if the underlying structure satisfies certain rigorous and seemingly non-trivial conditions, so in this sense one might argue that such a theory has explained gravity by requiring that the total combination of elements yields a consistent theory of gravity that appears, at some level, to be a quantum theory. However, if the effective metric is still built upon an underlying classical spacetime background, it still seems to me that we haven’t really accomplished our intent, which was to eliminate the dualism in physics. After all, if we are satisfied with a partly classical and partly quantum physics, then we already have that with general relativity and the standard model. This is what I meant when I said earlier that “we haven’t really accomplished anything” unless we can totally dispense with the classical background spacetime. If we are willing to accept a classical background spacetime with curvature affecting the motions of objects, then we may as well just accept general relativity and be done with it.


Earlier you seemed to be headed in the direction of saying that the classical background spacetime is totally arbitrary, and therefore insignificant in the sense that it has no effect only anything, so I thought you were going to say that this is why the dualism has essentially been eliminated. But in subsequent responses you emphasized that the background spacetime definitely needs to satisfy a number of specific requirements, e.g., the number of dimensions, and you also seem to agree that the range of possible vacuua leads to a range of physically distinct possible universes, so this makes me think the background spacetime can’t really be discounted as physically meaningless.


The other approach that I thought you might take (to eliminate the dualism) is to say that the background spacetime could, itself, just be an effective manifold of some meta-string theory, which is based on a meta-background that is just an effective manifold of a meta-meta-string theory, and so on, ad infinitum. This may be kind of silly (turtles all the way down), but at least I could understand how someone might claim to have eliminated the dualism on that basis. But since you’re not claiming this, it seems that we’re stuck with the dualism that we started with. So I guess my basic problem is I don’t understand what string theory is trying to accomplish. From what you’ve said, Salviati, it sounds like string theory, even if it were to succeed in its own terms, would still leave us with the partly classical and partly quantum dualism with which we started – unless we can somehow get rid of that classical background spacetime.  Am I totally wrong about this?



Salviati:  The goal of science is to explain how things work and what equations the reality satisfies. It is the scientific sense, not the religious sense, in which string theory predicts gravity. One can derive equations from string theory that were not derivable before strings were included. The goal is not to see God gluing spacetime out of nothing. Such a way to see the origin of spacetime could turn out to be an alternative way to look at the situation in the future. Or not. At any rate, it is not needed for the fact that spacetime following general relativity may be derived from string theory.



Simplicius:  You’ve made two distinct claims for string theory there Salviati. First, you claimed that one can derive equations from string theory that were not derivable before strings were included. Second, you claim that the existence of spacetime conforming to general relativity may be derived from string theory. These are quite different claims. We haven’t really discussed the first claim, so I don’t know what (if any) new physical predictions may have been derived from string theory, nor whether they have been empirically confirmed. Perhaps we can have that discussion on another day.


As to your second claim, I understand the sense in which you believe string theory provides a derivation of the spacetime of general relativity. Based on what you’ve said, I gather that a self-consistent quantum field theory of strings exists mathematically only within certain restricted context, and one of the requirements is that the classical background spacetime must have a certain number of dimensions and must satisfy Einstein’s field equations. Given this pre-condition, you say the resulting quantum theory of strings resembles (or can be made to resemble) the universe of our experience. Perhaps we could make an analogy to a system of linear equations, for which a self-consistent solution exists only for a suitable eigenvalue. In other words, the system imposes constraints on the allowable eigenvalue, and once a suitable value has been selected, the corresponding eigenvector is determined. Similarly, string theory imposes constraints on the background, and once a suitable background has been chosen, the corresponding solution (eigenvector) is determined. In this sense, string theory plays the role of a linear operator, from which theorists hope to infer the required eigen-spacetime, allowing them to determine the corresponding eigen-theory of physics. One of the difficulties presently facing string theory is due to the apparent fact that the possible “eigen-spacetimes” (vacuua) are very numerous, and there is no known way of assigning amplitudes (probabilities) to them.


However, even setting aside the apparent lack of uniqueness (which negates the very aspect of string theory that was originally considered to be its most compelling feature), the construction of a effective field theory approximating general relativity in the context of a pre-existing spacetime that already obeys classical general relativity does not seem to address the fundamental issue, which is the problematic dualism in the combination of classical and quantum physics. If we are seeking to quantize all the forces of nature, can we really claim to have succeeded by making use of a primitive un-quantized theory of gravity as the background?



Sagredo:  Alas, at this point the dialogue came to an end, leaving our poor Simplicius still thirsting for knowledge. Incidentally, as Salviati was departing I mentioned to him that I had been quite interested in his discussion of how the string condensates in a background spacetime could alter not only the metric but also the topology and dimensionality of the effective spacetime, and that only this effective spacetime was observable. It reminded me of a debate about ten years ago on the subject of general relativity and Poincare’s conventionalism. Recall that Poincare talked about observable experience as being the net sum of two individually unobservable components, namely, the posited background geometry G, and the posited physical laws P. He argued that the true nature of geometry is ultimately unknowable, or rather conventional, because we never observe G (or P) individually, we only observe the sum G+P. For example, we could maintain that the Earth’s surface is flat, provided we were willing to posit physical laws that entail distortions of measuring rods, lines of sight, etc., in order to yield the appearance of living on a spherical surface. In general (Poincare argued), we can postulate any G we like, provided we are willing to augment it with whatever (possibly outlandish) physical laws P are necessary to give the same net G+P. Of course, Poincare advised choosing G so that P is as simple as possible, but he insisted that this is just a matter of taste and convenience, not a logical necessity.


When this was mentioned ten years ago, many physicists, including those in the string theory community, strenuously objected to the idea, saying things like “That’s clearly invalid, because no physics could ever alter the dimensionality or topology of the geometry”. It’s interesting that this very schema now apparently forms an integral part of string theory.



Salviati: Well, Sagredo, physics is something different than a particular axiomatic system in geometry. In geometry as a branch of mathematics, transitions that change topology or dimension are forbidden because one would have to go through configurations that look singular - where some submanifolds shrink to zero size - and these are not treated as manifolds etc. and mathematicians have to omit them from the list of allowed choices. However, these points happen to be completely non-singular in string theory. It is the dynamics of string theory that physically - I mean really experimentally - allows you to go from spacetime whose compact dimensions have one topology to another spacetime with a different topology.


String theory also allows you to change the "effective" number of dimensions. For example, the secret 11th dimension of type IIA string theory is not visible at weak coupling but grows larger and visible at stronger coupling. There are many more non-trivial examples of this kind. In some sense, the "total" number of dimensions is always 10 or 11 but the number of dimensions that behave as we are used to in geometry (that are large enough) can be different. But general mathematicians have clearly nothing to do with these facts. These are not facts about some rudimentary geometry that mathematicians know once they finish the college. These are non-trivial facts about the rules that govern string theory – rules that we call "physics" even though they are again formulated very mathematically – rules that only the best mathematicians learn once they become at least postdocs and spend a year with string theorists.



Sagredo:  Yes, I see, although Poincare was not actually talking about mathematical transformations, he was specifically talking about physical laws and processes that would induce an effective metric, so that the postulated background geometry was unobservable, since we only observe the net effect of the background plus the physical processes. Your “string condensate” inducing an effective metric is precisely the kind of thing Poincare was talking about when he explained how both the effective dimensionality and the effective topology of space could be altered by suitable physical phenomena. Granted, Poincare talked in terms of a “switchboard” instead of “string condensates”, but his generic mechanism was actually more general than that of string theory, and included things like “string condensates” as a tiny subset. In general, he was talking about the same kind of things as you’ve described for string theory, and this is exactly the concept that at least one well-known string theorist strenuously argued was inconceivable ten years ago. I’m sure Poincare would be gratified to learn that string theorists have (re)discovered how to conceive of such things!


By the way, Salviati, another interesting question that I was hoping would come up in your dialogue with Simplicius concerns how the lightcone structure of special relativity behaves in a small cylindrical dimension, such as those postulated in string theory. Are we to understand that null (lightlike) intervals circle around these dimensions, and therefore intersect with themselves at infinitely many points? If so, does the theory exploit this in some way?  [But it was too late. Salviati had departed.]


Return to MathPages Main Menu