The Theory of Strings (Part 2)

The Universe Before the Big Bang

The Theory of Strings

Hence, the observed processes can be suitably described by applying the formalism of quantum field theory following the standard model of elementary particle physics, without any reference to string theory.

The same conclusion applies to the gravitational force if we limit ourselves to sufficiently flat portions of space-time, i.e., space-time regions whose curvature radius is much bigger than Ls. According to the standard cosmological model, however, going backwards in time towards the Big Bang, the curvature radius (i.e., the inverse of the curvature shown in Fig. 1.1) can reach extremely small values. In particular, taking the ratio between Ls and the speed of light c, it is possible to estimate the time at which the curvature radius of our Universe coincides with the string length.

The result is about 10−42 seconds after the Big Bang (a time ten times bigger than the elementary Planck time LP/c). This is the time after which we may trust the cosmological predictions of general relativity. Before such times, the curvature radius of the standard model was indeed smaller than Ls. Hence, the extensions of strings (and of all particles) were greater than the curvature radius and could not be neglected by any means. This implies that, in this regime, the geometry of the Universe should be described by adding those quantum corrections and those extra degrees of freedom predicted by string theory to the equations of general relativity.

The origin and the possible form of those corrections will be discussed at the end of this chapter. The point we wish to stress here is that, as a physical consequence of such corrections, we may expect that the curvature radius cannot become smaller than Ls, i.e., that the curvature scale cannot exceed a maximum value controlled by the reciprocal of the string length parameter, 1/Ls, therefore avoiding a possible singularity. This expectation is linked to an extremely important property of quantum strings, the already mentioned duality symmetry that was introduced and applied in the last chapter, and will be discussed in more detail here.

To begin with, let us consider a point-like object which is con- strained to move along a circle of radius R, following the laws of classical mechanics. We can say that the point somehow feels the dimension of the circle. To complete a round trip, for instance, it takes more time on a large circle than on a small one. The sensitivity to the size of the circle remains valid in the framework of quantum mechanics, despite the fact that the quantization of the moving point-like object forces the velocity (or, more precisely, the momentum of the particle) to take discrete values proportional to the reciprocal radius 1/R.

Consider now a string moving on a circle. As far as classical motion is concerned, the conclusion about the sensitivity to the size of the circle is similar to the previous case, but with an important difference due to the possibility of different types of motion. In particular, a closed string moving on a circle (or in a multidimensional generalization of it) can rotate (like a point), oscillate, and also wrap around the circle (you may think, for in- stance, of a cotton thread wrapped many times around a reel). The energy associated with wrapping, usually called winding energy, is proportional to the radius R and to the number of times the string is wrapped around (see Fig. 4.3). Hence, one could argue that strings feel the size of the circle in various ways.

However, this picture is drastically changed when the string motion is quantized. In fact, the total energy of the string must be computed by adding up the winding energy – which is an integer multiple of the radius – and the kinetic energy due to the rotational velocity – which, once quantized, is an integer multiple of the reciprocal of the radius (as in the case of a point-like object).

An observer identified with the string would be confused, at this point. Since the kinetic energy and the winding energy cannot be distinguished by any possible means, by looking at the energy levels of the string he would not be able to establish whether the string is moving around a circle of radius R or one of radius Ls2/R! (The fundamental length Ls must be introduced in order to guarantee the correct dimensions of length for both radii. See also footnote 1 of Chap. 3.)

These two radii are totally indistinguishable for a string, in all respects. In other words, this means that it is possible to per- form a transformation – called the dual transformation – which interchanges R and L2s/R without modifying any relevant aspect of string physics. This has an important consequence because, if R and L^2s/R are equivalent, the effective minimum value of the radius is not zero but Ls, i.e., the value for which the two radii above coincide (this value is called the fixed point of the duality transformation). Indeed, when R is larger than Ls the effective radius experienced by the string coincides with R. When R is smaller than Ls, on the other hand, the effective radius coincides with L^2s/R, which is still greater than Ls (see Fig. 4.4).

This helps one to understand why Ls represents a sort of effective minimum length within the context of string theory. The above symmetry arguments (first developed by a group of Japanese physicists, Keiji Kikkawa, Masami Yamasaki, Norisuke Sakai, and Ikuo Senda in the 1980s) apply to circles, i.e., to the case of rigid geometries. However, as anticipated in the last chapter, the validity of the duality symmetry can also be extended in the presence of a time-dependent cosmological geometry (as pointed out by Arcady Tseytlin and Gabriele Veneziano), simply by replacing the radius of the circle with what we have called the spatial radius of the Universe (or, more technically, the so-called scale factor of the Robertson–Walker metric). Given the existence of a string-cosmology solution describing a model with spatial radius R (t), duality then implies that there must be a dual cosmological solution with spatial radius equal to the reciprocal 1/R(t). Here, however, we encounter a crucial difference with respect to the case of a circle: the cosmological geometry varies in time, and satisfies a set of equations which (as we shall see below) are different from the ones predicted by general relativity. Those equations contain the dilaton and, when R and 1/R are interchanged, the dilaton must also be properly transformed in order to preserve the validity of the equations themselves. But we refer to Chapter 3 for a detailed discussion of this point and other properties of duality-related cosmological scenarios.

Here we want to comment on another important consequence of the finite size of the strings: the introduction of new energy states, i.e., of elementary new types of energy like the winding energy, whose possible existence is probably one of the most innovative features introduced by string theory into the physics of fundamental interactions.

The possibility for strings to wrap themselves around compact spatial dimensions, besides being at the heart of the duality symmetry, could indeed explain why, in a Universe which (according to unified theories) should have many spatial dimensions – at least nine, as we shall see in Chap. 10 – only three spatial dimensions have enormously expanded with respect to the string length, as is evident from our everyday experience. The mechanism leading to such an explanation has been suggested by work carried out by Robert Brandenberger and Cumrum Vafa at the end of the 1980s, in the so-called string gas cosmological scenario, later extended to the case of higher-dimensional extended objects (the brane gas scenario) by the joint contributions of Stephen Alexander, Robert Brandenberger, Damien Easson, Thorsten Battenfeld, Scott Watson, and others.

This basic idea stems from the fact that immediately after the Big Bang the Universe, in a multidimensional but highly com- pact configuration, should have been filled with a very dense gas of strings produced by the extremely high energies and temperatures.

Not only were such strings moving at relativistic speeds, but they were also wrapped around all spatial dimensions, assumed to be compact. This network of wrapped strings (the so-called winding modes) prevented the Universe from expanding. Indeed, as soon as the expansion switched on (driven by the kinetic energy of the other strings) and the radius started to increase, the energy of the wrapped strings (which is proportional to R) soon increased; such strings then became dominant, balancing and overcoming with their tension the force sustaining the expansion, causing the geometry to contract back to its initial configuration.

The Universe was therefore in a multidimensional equilibrium configuration, with all dimensions equally extended but constrained to a compact size of the order of the string length Ls. So how was it possible for three out of nine spatial dimensions to pass through the net, as it were, and succeed in expanding without any constraint, leading to our currently observed Universe?

In order to answer this question we recall that at very high temperatures there should exist an equal percentage of strings wrapped in both orientations, i.e., wrapped strings and anti-wrapped strings (or winding and anti-winding modes), which annihilate each other by collisions, just as happens for matter and antimatter particles. Therefore, it may be that the wrapped strings gradually tend to disappear by colliding with their opposite counterparts, in such a way that, eventually, the “network” of winding modes breaks up, allowing the Universe to expand. If this is the case, however, why have only three dimensions been expanding?

The answer to this question is quite simple. In order to annihilate, strings must collide. If the space has too many dimensions it is likely that such collisions will never occur, even if the dimensions are compact. Let us think, for instance, of two point-like objects moving around a circle. Unless their velocities are exactly equal and have the same direction, the two objects are doomed to collide, sooner or later. If such point-like objects move instead on the two-dimensional surface of a sphere they may never met, even if their velocities are very different. In contrast, two one-dimensional objects like two strings have a finite probability of colliding even if they move on a sphere. And so on for higher-dimensional extended objects, in spaces with more and more dimensions. Iterating these arguments we arrive at the following general conclusion: given two p-dimensional objects (the so called p-branes that will be discussed in Chap. 10), the maximum number of compact spatial dimensions in which their collision becomes unavoidable is 2p + 1.

For a point p = 0, we recover the result of our previous ex- ample relative to the one-dimensional circle. Strings, being one- dimensional objects, have p = 1. Hence two strings are very likely to collide and eventually annihilate each other within a space with at most 2 + 1 = 3 dimensions (but not in higher-dimensional spaces!). The winding modes of wrapped strings were thus able to meet and completely annihilate only in a three-dimensional section of our Universe, and this is the reason why only three spatial dimensions have managed to escape from the string net- work, growing large and expanding to form our cosmos. Within the remaining six (or more) dimensions, on the other hand, wrapped strings have not experienced enough collisions, whence the net- work has not broken up, keeping such additional dimensions small and compact, confined to a distance scale of the order of the string length Ls.

At this point of the chapter, we can appropriately summarize the main results of our previous discussion by saying that it is just the finite extension of strings (compared with the point-like nature of classical particles) which provides the key for the new physical effects present in string theory.

Indeed, this finite extension allows the existence of new symmetries (e.g., duality) and new energy states (e.g., winding modes), which in turn suggests new cosmological scenarios and new mechanisms for dynamical determination of the effective dimensionality of our Universe. Furthermore, the presence of a minimal fundamental length Ls introduced by quantization should pro- vide a way of avoiding the cosmological singularity at t = 0, and continuously joining our current standard regime to a primordial (pre-Big-Bang) inflationary regime. It is expected in fact that the quantization process, by providing strings with a finite extension, may also determine a maximum finite value for the curvature, therefore eliminating singularities in the quantum gravity regime – in just the same way as quantum mechanics has solved the singularity and stability problems of atomic orbits determining a minimum atomic radius, forcing the orbits to keep the electrons at a finite distance from the nuclei.

At this stage, however, a careful and expert reader could raise a question, by recalling that within the standard cosmological theory based upon the Einstein equations there are rigorous theorems (proved by George Ellis, Stephen Hawking, and Roger Penrose during the 1960s and the 1970s) stating that – under very general assumptions – it is impossible to avoid the initial singularity. If string cosmology can in fact avoid it, then string theory should yield gravitational equations that differ from those predicted by general relativity. What is the general form of these new equations, and how can they be derived from the theory?

Once again, the answer to these questions is deeply rooted in the symmetries of string theory. With regard to the first question, we recall that the duality symmetry associated with the inversion of the radius requires the presence of the dilaton field, which necessarily introduces a new scalar-type force into the gravitational equations. In the same way, generalized forms of duality are associated with the presence of other fields represented by antisymmetric tensors, which also contribute to the total gravitational force (see Chap. 10). In this context, the Riemannian metric of the curved space-time geometry is only one component of the total force coming into play. Not to mention the presence of fermionic components of the gravitational interaction, associated with the supersymmetry present in superstring models (see Chap. 10), making the resulting model of gravity an effective supergravity theory.

With regard to the second question, the answer calls into play another very important symmetry of string theory, known as conformal invariance (or Weyl invariance, or local scale invariance), which characterizes the motion of a string and its interactions, and which is absent in the case of a point-like object. We shall provide below a short illustration of the origin and properties of the con- formal symmetry, but let us anticipate immediately that, thanks to this symmetry, the quantization of the string motion not only tells us what fundamental fields exist in nature (e.g., gravitational field, electromagnetic field, non-Abelian gauge fields, etc.) but also automatically gives us the equations satisfied by these fields. This is because the consistent quantization of an interacting string imposes rigid constraints on the fields interacting with the string.

This property probably represents the most revolutionary aspect of the theory with respect to conventional models based on the notion of elementary particle. In fact the motion of a point-like test body, even if quantized, does not impose any restriction on the external fields in which the body is embedded and with which it interacts. Such background fields can satisfy arbitrarily prescribed equations of motion, usually chosen on the grounds of phenomenological indications. We can think, for instance, of the Maxwell equations, constructed empirically from the laws of Gauss, Lenz, Faraday, and Ampere. It would be possible, in principle, to formulate sets of equations different from Maxwell’s, but still preserving Lorentz covariance and other symmetry properties (such as the Abelian gauge symmetry) typical of the electromagnetic interactions. Such different equations might well be discarded, in the con- text of quantum field theory, but only for their disagreement with experimental results.

In the context of string theory, on the other hand, such alternative equations must be discarded a priori, as they would be inconsistent with the quantization of a charged string interacting with an external electromagnetic field. Indeed quantum string theory requires the electromagnetic field to satisfy a set of differential equations which, to lowest order, miraculously reduce precisely to the Maxwell equations (the same is true for the gravitational field, for non-Abelian Yang–Mills fields, and so on).

The above property of string theory is grounded in the geometrical properties of the two-dimensional surface spanned by the string evolving in the external space-time manifold – the world- sheet surface already mentioned, as illustrated in Fig. 4.1. Such a surface is curved, in general, and is thus characterized by a Riemannian metric associated with its intrinsic geometry. How- ever, the area of this surface is an invariant, and does not change if the world-sheet metric is deformed by an arbitrary multiplicative factor which is local, i.e., variable in space and time.

This invariance is called conformal invariance, and represents a symmetry of the classical string motion. Thanks to this symmetry it is always possible to introduce a reference frame in which the world-sheet metric reduces to the flat Minkowski metric. More- over, it is always possible to eliminate the “longitudinal” oscillations of the string, leaving only the degrees of freedom describing oscillations transverse to the string. Conformal invariance thus plays a crucial role in the process of determining the correct set of physical variables to be quantized, in order to obtain the correct quantum spectrum of physical string states.

When we have a test string interacting with any one of the fields present in its spectrum (for instance the dilaton field, or the gravitational field, or the electromagnetic field if the string is charged), we must then require, for consistency, that the conformal invariance (determining the string spectrum) be preserved by the given interaction, not only at the classical but also at the quantum level. In other words, the quantization of a string including its background interactions must avoid the presence of conformal anomalies, i.e., quantum violations of conformal invariance which is already associated with the world-sheet geometry at the classical level.

This observation leads us to the crucial point of our discussion: the only background-field configurations admissible in a string theory context are those satisfying the conditions of conformal invariance. Such conditions are represented by a set of differential equations corresponding, in every respect, to the equations of motion of the field we are considering. The field equations predicted by string theory – for any field, and in particular for the gravitational field – can thus be obtained directly by imposing conformal invariance on the quantum string interactions.

Unfortunately, however, such equations are hard to derive in closed and exact form for any given model of interacting strings. In practice, we have to follow a perturbative method: the quantized interaction of the string world-sheet with the background fields is approximated by a series of higher-order corrections as in standard quantum field theory, but with the difference that the fields are defined on a two-dimensional space-time, the string world-sheet.3 The absence of conformal anomalies is then imposed at any order of this approximation, determining the corresponding differential conditions. As a result, the exact equations predicted by string theory for the background fields are approximated by an infinite series of differential equations, containing higher and higher derivative terms as we consider approximations of higher and higher order.

To a first approximation (i.e., to lowest order) we then recover the second-order differential equations already well known for classical fields (i.e., the Maxwell, Einstein, and Yang–Mills equations, and also the Dirac equations for the fermion fields). To higher or- der, there are quantum corrections to these equations in the form of higher derivatives of the fields, appearing as an expansion in powers of the string length parameter Ls (also conventionally called the α� expansion, in terms of an equivalent parameter α� defined by L2s = 2πα�). Such corrections are a typical effect of the theory due to the finite extension of strings. Indeed, they disappear in the point-particle limit Ls → 0, while they become important in the strong field limit in which the length scale of a given process (for instance, the space-time curvature scale in the case of gravity) becomes comparable with the string length Ls.

In conclusion, the new symmetries present in string theory tell us that the Einstein equations – and hence the gravitational equations to be used for the formulation of our cosmological models – are to be modified in two ways. Not only by the addition of new fields (like the dilaton), but also by the addition of quantum corrections due to strong fields (expansion in powers of Ls2) and/or strong couplings (topological expansion in powers of gs2). In the context of pre-Big-Bang cosmology, both these corrections may play an important role in the transition to the phase of standard decelerated evolution, as we shall discuss in Chap. 8.

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The Theory of Strings (Part 1)