Philosophy of Dimensions: M-Theory. Thought of the Day 85.0


Superstrings provided a perturbatively finite theory of gravity which, after compactification down to 3+1 dimensions, seemed potentially capable of explaining the strong, weak and electromagnetic forces of the Standard Model, including the required chiral representations of quarks and leptons. However, there appeared to be not one but five seemingly different but mathematically consistent superstring theories: the E8 × E8 heterotic string, the SO(32) heterotic string, the SO(32) Type I string, and Types IIA and IIB strings. Each of these theories corresponded to a different way in which fermionic degrees of freedom could be added to the string worldsheet.

Supersymmetry constrains the upper limit on the number of spacetime dimensions to be eleven. Why, then, do superstring theories stop at ten? In fact, before the “first string revolution” of the mid-1980’s, many physicists sought superunification in eleven-dimensional supergravity. Solutions to this most primitive supergravity theory include the elementary supermembrane and its dual partner, the solitonic superfivebrane. These are supersymmetric objects extended over two and five spatial dimensions, respectively. This brings to mind another question: why do superstring theories generalize zero-dimensional point particles only to one-dimensional strings, rather than p-dimensional objects?

During the “second superstring revolution” of the mid-nineties it was found that, in addition to the 1+1-dimensional string solutions, string theory contains soliton-like Dirichlet branes. These Dp-branes have p + 1-dimensional worldvolumes, which are hyperplanes in 9 + 1-dimensional spacetime on which strings are allowed to end. If a closed string collides with a D-brane, it can turn into an open string whose ends move along the D-brane. The end points of such an open string satisfy conventional free boundary conditions along the worldvolume of the D-brane, and fixed (Dirichlet) boundary conditions are obeyed in the 9 − p dimensions transverse to the D-brane.

D-branes make it possible to probe string theories non-perturbatively, i.e., when the interactions are no longer assumed to be weak. This more complete picture makes it evident that the different string theories are actually related via a network of “dualities.” T-dualities relate two different string theories by interchanging winding modes and Kaluza-Klein states, via R → α′/R. For example, Type IIA string theory compactified on a circle of radius R is equivalent to Type IIB string theory compactified on a circle of radius 1/R. We have a similar relation between E8 × E8 and SO(32) heterotic string theories. While T-dualities remain manifest at weak-coupling, S-dualities are less well-established strong/weak-coupling relationships. For example, the SO(32) heterotic string is believed to be S-dual to the SO(32) Type I string, while the Type IIB string is self-S-dual. There is a duality of dualities, in which the T-dual of one theory is the S-dual of another. Compactification on various manifolds often leads to dualities. The heterotic string compactified on a six-dimensional torus T6 is believed to be self-S-dual. Also, the heterotic string on T4 is dual to the type II string on four-dimensional K3. The heterotic string on T6 is dual to the Type II string on a Calabi-Yau manifold. The Type IIA string on a Calabi-Yau manifold is dual to the Type IIB string on the mirror Calabi-Yau manifold.

This led to the discovery that all five string theories are actually different sectors of an eleven-dimensional non-perturbative theory, known as M-theory. When M-theory is compactified on a circle S1 of radius R11, it leads to the Type IIA string, with string coupling constant gs = R3/211. Thus, the illusion that this string theory is ten-dimensional is a remnant of weak-coupling perturbative methods. Similarly, if M-theory is compactified on a line segment S1/Z2, then the E8 × E8 heterotic string is recovered.

Just as a given string theory has a corresponding supergravity in its low-energy limit, eleven-dimensional supergravity is the low-energy limit of M-theory. Since we do not yet know what the full M-theory actually is, many different names have been attributed to the “M,” including Magical, Mystery, Matrix, and Membrane! Whenever we refer to “M-theory,” we mean the theory which subsumes all five string theories and whose low-energy limit is eleven-dimensional supergravity. We now have an adequate framework with which to understand a wealth of non-perturbative phenomena. For example, electric-magnetic duality in D = 4 is a consequence of string-string duality in D = 6, which in turn is the result of membrane-fivebrane duality in D = 11. Furthermore, the exact electric-magnetic duality has been extended to an effective duality of non-conformal N = 2 Seiberg-Witten theory, which can be derived from M-theory. In fact, it seems that all supersymmetric quantum field theories with any gauge group could have a geometrical interpretation through M-theory, as worldvolume fields propagating on a common intersection of stacks of p-branes wrapped around various cycles of compactified manifolds.

In addition, while perturbative string theory has vacuum degeneracy problems due to the billions of Calabi-Yau vacua, the non-perturbative effects of M-theory lead to smooth transitions from one Calabi-Yau manifold to another. Now the question to ask is not why do we live in one topology but rather why do we live in a particular corner of the unique topology. M-theory might offer a dynamical explanation of this. While supersymmetry ensures that the high-energy values of the Standard Model coupling constants meet at a common value, which is consistent with the idea of grand unification, the gravitational coupling constant just misses this meeting point. In fact, M-theory may resolve long-standing cosmological and quantum gravitational problems. For example, M-theory accounts for a microscopic description of black holes by supplying the necessary non-perturbative components, namely p-branes. This solves the problem of counting black hole entropy by internal degrees of freedom.

Local Gauge Transformations in Locally Gauge Invariant Relativistic Field Theory


The question arises of whether local space-time symmetries – arbitrary co-ordinate transformations that leave the explicit form of the equations of motion unaffected – also have an active interpretation. As in the case of local gauge symmetry, it has been argued in the literature that the introduction of a force is required to ‘restore’ local symmetry.  In the case of arbitrary co-ordinate transformations, the force invoked is gravity. Once again, we believe that the arguments (though seductive) are wrong, and that it is important to see why. Kosso’s discussion of arbitrary coordinate transformations is analogous to his argument with respect to local gauge transformations. He writes:

Observing this symmetry requires comparing experimental outcomes between two reference frames that are in variable relative motion, frames that are relatively accelerating or rotating….One can, in principle, observe that this sort of transformation has occurred. … just look out of the window and you can see if you are speeding up or turning with respect to some object that defines a coordinate system in the reference frame of the ground…Now do the experiments to see if the invariance is true. Do the same experiments in the original reference frame that is stationary on the ground, and again in the accelerating reference frame of the train, and see if the physics is the same. One can run the same experiments, with mechanical forces or with light and electromagnetic forces, and observe the results, so the invariance should be observable…But when the experiments are done, the invariance is not directly observed. Spurious forces appear in the accelerating system, objects move spontaneously, light bends, and so on. … The physics is different.

In other words, if we place ourselves at rest first in an inertial reference frame, and then in a non-inertial reference frame, our observations will be distinguishable. For example, in the non-inertial reference frame objects that are seemingly force-free will appear to accelerate, and so we will have to introduce extra, ‘spurious’, forces to account for this accelerated motion. The transformation described by Kosso is clearly not a symmetry transformation. Despite that, his claim appears to be that if we move to General Relativity, this transformation becomes a symmetry transformation. In order to assess this claim, let’s begin by considering Kosso’s experiment from the point of view of classical physics.

Suppose that we describe these observations using Newtonian physics and Maxwell’s equations. We would not be surprised that our descriptions differ depending on the choice of coordinate system: arbitrary coordinate transformations are not symmetries of the Newtonian and Maxwell equations of motion as usually expressed. Nevertheless, we are free to re-write Newtonian and Maxwellian physics in generally covariant form. But notice: the arbitrary coordinate transformations now apply not just to the Newtonian particles and the Maxwellian electromagnetic fields, but also to the metric, and this is necessary for general covariance.

Kosso’s example is given in terms of passive transformations – transformations of the coordinate systems in which we re-coordinatise the fields. In the Kosso experiment, however, we re-coordinatise the matter fields without re-coordinatising the metric field. This is not achieved by a mere coordinate transformation in generally covariant classical theory: a passive arbitrary coordinate transformation induces a re-coordinatisation of not only the matter fields but also the metric. The two states described by Kosso are not related by an arbitrary coordinate transformation in generally covariant classical theory. Further, such a coordinate transformation applied to only the matter and electromagnetic fields is not a symmetry of the equations of Newtonian and Maxwellian physics, regardless of whether those equations are written in generally covariant form.

Suppose that we use General Relativity to describe the above observations. Kosso suggests that in General Relativity the observations made in an inertial reference frame will indeed be related by a symmetry transformation to those made in a non-inertial reference frame. He writes:

The invariance can be restored by revising the physics, by adding a specific dynamical principle. This is why the local symmetry is a dynamical symmetry. We can add to the physics a claim about a specific force that restores the invariance. It is a force that exactly compensates for the local transform. In the case of the general theory of relativity the dynamical principle is the principle of equivalence, and the force is gravity. … With gravity included in the physics and with the windows of the train shuttered, there is no way to tell if the transformation, the acceleration, has taken place. That is, there is now no difference in the outcome of experiments between the transformed and untransformed systems. The force pulling objects to the back of the train could just as well be gravity. Thus the physics, all things including gravity considered, is invariant from one locally transformed frame to the next. The symmetry is restored.

This analysis mixes together the equivalence principle with the meaning of invariance under arbitrary coordinate transformations in a way which seems to us to be confused, with the consequence that the account of local symmetry in General Relativity is mistaken.

Einstein’s field equations are covariant under arbitrary smooth coordinate transformations. However, as with generally covariant Newtonian physics, these symmetry transformations are transformations of the matter fields (such as particles and electromagnetic radiation) combined with transformations of the metric. Kosso’s example, as we have already emphasised, re-coordinatises the matter fields without re-coordinatising the metric field. So, the two states described by Kosso are not related by an arbitrary coordinate transformation even in General Relativity. We can put the point vividly by locating ourselves at the origin of the coordinate system: I will always be able to tell whether the train, myself, and its other contents are all freely falling together, or whether there is a relative acceleration of the other contents relative to the train and me (in which case the other contents would appear to be flung around). This is completely independent of what coordinate system I use – my conclusion is the same regardless of whether I use a coordinate system at rest with respect to the train or one that is accelerating arbitrarily. (This coordinate independence is, of course, the symmetry that Kosso sought in the opening quotation above, but his analysis is mistaken.)

What, then, of the equivalence principle? The Kosso transformation leads to a physically and observationally distinct scenario, and the principle of equivalence is not relevant to the difference between those scenarios. What the principle of equivalence tells us is that the effect in the second scenario, where the contents of the train appear to accelerate to the back of the train, may be due to acceleration of the train in the absence of a gravitational field, or due to the presence of a gravitational field in which the contents of the train are in free fall but the train is not. Mere coordinate transformations cannot be used to bring real physical forces in and out of existence.

It is perhaps worthwhile briefly indicating the analogy between this case and the gauge case. Active arbitrary coordinate transformations in General Relativity involve transformations of both the matter fields and the metric, and they are symmetry transformations having no observable consequences. Coordinate transformations applied to the matter fields alone are no more symmetry transformations in General Relativity than they are in Newtonian physics (whether written in generally covariant form or not). Such transformations do have observational consequences. Analogously, local gauge transformations in locally gauge invariant relativistic field theory are transformations of both the particle fields and the gauge fields, and they are symmetry transformations having no observable consequences. Local phase transformations alone (i.e. local gauge transformations of the matter fields alone) are no more symmetries of this theory than they are of the globally phase invariant theory of free particles. Neither an arbitrary coordinate transformation in General Relativity, nor a local gauge transformation in locally gauge invariant relativistic field theory, can bring forces in and out of existence: no generation of gravitational effects, and no changes to the interference pattern.