The Universe Before the Big Bang
The Theory of Strings
This chapter is devoted to those readers interested in the physical foundations of pre-Big-Bang cosmology and who wish to learn the basic concepts of the theory of one-dimensional extended objects – usually dubbed string theory. In particular, we will try to outline here the quantum origin of the duality symmetries mentioned in the previous chapter. Further aspects of string theory pertaining to the unified description of all natural forces will be reviewed in Chap. 10.
The standard model of elementary particle physics – not to be confused with the standard cosmological model describing the Universe – is based upon two milestones of 20th century physics, quantum mechanics and special relativity, and upon the heuristic hypothesis that elementary particles, in the classical limit, are point-like objects without any spatial extension. As shown in a book written by Steven Weinberg, these assumptions lead almost uniquely to the description of elementary particle physics in terms of what is called quantum field theory, a theory based upon the principle that any field strength (for instance the electromagnetic field strength) can always be measured with arbitrary precision at any given point in time and space.
The standard model, built upon those grounds, has achieved a success beyond all possible expectations. Many relevant theoretical predictions of this model have been confirmed experimentally, and some of these predictions are very important, especially those tested by the particle accelerators operating at CERN (Geneva, Switzerland) and FERMILAB (Chicago, USA). Unfortunately, how- ever, the standard model leads to a consistent unified description of only three out of the four fundamental forces of nature, i.e., it unifies the electromagnetic, strong nuclear, and weak nuclear forces, but it does not include the gravitational force – which may be negligible at the typical scales of nuclear and subnuclear physics, but which certainly plays an important role at higher densities and energy scales, like those appearing in a cosmological context. In other words, we can say that the standard model effectively com- bines quantum mechanics with special relativity, but is unable to combine quantum mechanics with general relativity.
The physical reason why it is so difficult to reconcile quantum mechanics with general relativity is essentially rooted in Heisenberg’s well-known uncertainty principle. General relativity is in fact a local field theory like the other theories of the standard model, based upon the assumption that the gravitational field can be measured at any given point in time and space. However, according to the above-mentioned uncertainty principle, an arbitrarily large accuracy in the position implies a full uncertainty in the velocity. In other words, if we measure a gravitational field at a given point with great accuracy, we end up with a correspondingly large uncertainty in its energy. On the other hand, such a large energy fluctuation is necessarily associated with a large fluctuation in the gravitational field itself since – as we have already stressed – it is just the energy which plays the role of gravitational “charge”, i.e., source of the gravitational field. Taking into account this intrinsic quantum uncertainty, a gravitational field cannot be measured at a given space-time point with arbitrary accuracy.
For this reason – and only for gravity – an insuperable problem arises when we attempt to combine a local theory like general relativity with quantum mechanics. One may think it would be possible to overcome this problem in various ways, one being the idea that the gravitational force should not be quantized at all (such a drastic alternative would nevertheless require the solution of a number of conceptual and experimental problems). String theory, on the other hand, proposes to overcome this obstacle by abandoning the property of locality, i.e., the requirement that any field should be measurable at any given time and position. Originally, when string theory was first formulated in the early 1970s, inspired by a model developed by the theoretical physicist Gabriele Veneziano, it was aimed at describing strong nuclear interactions. Later in the 1980s the model was extended to include supersymmetry, and to provide a consistent and compelling frame- work for a unified theory of all interactions (see Chap. 10). The number of research scientists that have contributed to this project, and are actively working in this field, is too large to mention all of them here. As for this book, it will suffice to observe that this theory is still based upon quantum mechanics and special relativity, but removes the fundamental hypothesis (present in a quantum field theoretical context) that the building blocks of our physical description should have a point-like nature, assuming instead the existence of elementary fundamental objects with a string-like, i.e., one-dimensional structure.
There are two possible kinds of strings: those characterized by free ends (the so-called open strings) and those closed upon them- selves (the closed strings). For both configurations the elementary building blocks of the theory are characterized by a finite spatial extension so that, using such objects, the possibility of local measurements of any field (electric, magnetic, gravitational, etc.) at a given space-time point is not only practically impossible, but is also forbidden in principle. Within this framework, all the standard- model problems related to the description of the gravitational field at the quantum level are condemned to disappear.
Furthermore, not only is a theory obtained by replacing points with strings compatible with gravity in the quantum regime, but it also automatically predicts that gravity has to be included among the fundamental forces of nature. In fact, strings are not static entities. Besides their center of mass motion (with an associated translational kinetic energy), strings can vibrate and oscillate as elastic bodies. According to quantum mechanics, however, only a set of discrete values is allowed for the energy and the angular momentum assigned to the various oscillatory states (exactly as happens for the energy levels of an atom). These discrete levels of a vibrating string are associated with a spectrum of states of different masses and angular momenta, describing different elementary particles, just in the same way as the different atomic frequencies are associated with the different spectral lines of the various atomic elements. And here we find the “miracle” connecting strings to gravitational interactions.
In fact, looking at the subset of states describing massless particles, we find that the spectrum of open strings contains a vector field which satisfies all the symmetry properties required to represent an interaction of electromagnetic type. Furthermore, the spectrum of closed strings contains – besides other fields, like the dilaton – a symmetric tensor field which has all the required physical properties of the graviton, representing the “quanta” of the gravitational interaction. On the other hand, closed strings are always (and necessarily) contained in all string models aimed at a unified description of all fundamental interactions (as will be discussed in Chap. 10). It follows that unified models based on strings must necessarily encode a tensor interaction of gravitational type, so that the existence of the gravitational force is guaranteed, at both the classical and quantum level. However, the gravitational field equations predicted by string theory are in principle different from the ones predicted by Einstein (the string equations generalize the Einstein equations in a way that will be illustrated at the end of this chapter).
In a similar fashion – and also because a quantum string vibrates in a multidimensional space (with at least nine spatial dimensions, see Chap. 10) – the quantum spectrum of an oscillating string includes other (possibly massive) states, which are appropriate candidates to describe the quanta of strong and weak nuclear forces. All these particles disappear in the limit where the theory becomes purely classical, so they are associated with intrinsic quantum effects. It is exactly this feature that allows string theory to provide (in principle) a quantum description of all known natural forces, without facing the locality problems arising in the context of field theory.
We can say, therefore, that the most relevant features of string theory are linked to the fact that quantum mechanics itself, when applied to extended objects, becomes somehow helpful, instead of giving problems as happens in a conventional field theory based upon point-like objects. Indeed, it is just quantum mechanics that provides the string with a minimum characteristic size Ls (the analogue of the Bohr radius in the case of atomic physics). Thus, while it would be possible at the classical level to conceive of an arbitrarily small string, eventually allowing local measurements of a field, at the quantum level this turns out to be forbidden – exactly in the same way as stable orbits with the electron too close to the nucleus are forbidden in the quantum mechanics of the atom.
At this point, an interesting question comes to mind: How long should these strings be? Their characteristic quantum length Ls represents a new fundamental constant, which can be expressed in terms of the Planck constant ¯h, the speed of light c, and the string tension T (i.e., the string energy per unit length, a constant parameter appearing in the analytic formulation of the theory). In principle this length (or, equivalently, the string tension) is an arbitrary parameter – actually, it is the only truly arbitrary parameter present in string theory – so that it can be conveniently tuned to any suitable value determined by the kind of forces we aim to de- scribe with the theory. Several years ago, for instance, when strings were used to build a model of strong nuclear interactions, the value of the fundamental string length was assumed to be of the order of the nuclear radius (about 10−13 cm).
Within the context of modern string theory, however, the constant Ls is fixed so that the theory may be able to describe all natural forces in a unified fashion. In this scenario, in fact, there are “additional” spatial dimensions that must be added to our three- dimensional macroscopic space in order to implement a consistent quantization of the string motion. Such extra dimensions are certainly not as large as the three ordinary ones, having escaped direct detection up to now: they are supposed to be wrapped (or more precisely compactified), so as to occupy a finite (and possibly small) volume of space, with a size naturally determined by the string length parameter Ls.
On the other hand, the compactification of the extra dimensions to small scales (i.e., the so-called process of dimensional reduction) is closely related to the process which reduces the higher-dimensional (unified) interactions to the standard form of the interactions that we are currently experiencing. To be consistent with the standard-model interactions, in particular, it turns out that the size of the extra volume of space – and thus the string length – has to be tiny. The expected value for Ls is about 10−32 cm, i.e., one tenth of the Planck length (barring some “membrane” models that will be discussed in Chap. 10).
It should be stressed that the introduction of the new constant Ls does not increase the number of the fundamental constants in nature. Rather, this number drops drastically. String theory has only two fundamental constants, the speed of light c (which is finite, according to special relativity) and the string length Ls (which is necessarily associated with quantization). In such a context, even the Planck constant itself is a derived quantity. A question then arises: What about all other constants of nature, determining for example the gravitational force, the electrostatic force, and even the size of the hydrogen atom?
The answer to this question highlights another peculiar feature of string theory. In contrast to what happens in the standard model of elementary particles, the fundamental constants of nature cease to be arbitrary numbers determined only by experiment. In- stead they are dynamical variables, determined by the expectation values of some fundamental fields – for instance, the already mentioneddilaton – given by the theory. Being expectation values, such constants should be calculable within a given theoretical model, once the current state of the Universe is fixed. However, this procedure, while straightforward in principle, turns out to be difficult to apply in practice to realistic scenarios, owing to computational difficulties.
The most peculiar (and most relevant, for the purpose of this book) example of “promotion” of fundamental constants to dynamical quantities is provided by the dilaton, a new field which is not contained in the standard model, but which is unavoidably present in all string theory models. This field determines the coupling strength of all the fundamental forces, as outlined already during the 1980s by the theoretical physicist Edward Witten – one of the world’s leading experts on the theory of superstrings (i.e., string models whose formulation includes both boson and fermion variables, which get interchanged under the so-called supersymmetry transformations). The non-trivial link between the dilaton field and the effective coupling strengths is a typical property of string theory, important enough to deserve at least a short illustrative discussion.
Consider a string, embedded in an external (higer-dimensional) space. Like a particle, its propagation from one spatial position to another describes a continuous trajectory in the external space-time manifold spanned by the whole set of space and time coordinates.
However, the trajectory of a point particle is represented by a one-dimensional curve – the so-called world-line of the particle – parametrized by a single time-like coordinate, while the trajectory of the string is represented by a two-dimensional surface – the world-sheet – parametrized by one time-like coordinate describing the evolution in time of the string, and one space-like coordinate describing the spatial positions of the different points of the string at a given instant of time (see Fig. 4.1).
Consider now a closed string, represented by a circle, whose propagation in the space-time manifold describes a cylindrical world-sheet surface. Due to quantum effects (or because of external interactions), the string may split into two strings, which subsequently recombine to form the initial string once again. This process may also occur for a particle in a quantum field theoretical context, and in that case it is represented by a picture – called a one-loop Feynman graph – describing the splitting of the particle world-line. In the string case, the splitting of the cylindrical world- sheet surface will produce a surface with the topology of a torus (namely, a sphere with a hole through it), as illustrated in Fig. 4.2 (top). The same description applies to quantum processes of higher order, as shown in Fig. 4.2 (bottom) for the two-loop case.
As time goes on the particle moves from one point to another, and the continuous sequence of its (point-like) spatial positions at different times describes a one-dimensional trajectory, the particle world-line (left). The string is also moving in space but, at any given time, it is characterized by a one-dimensional (finite) spatial extension. The continuous sequence of the spatial positions of all points of the string describes a two-dimensional trajectory, the string world-sheet (right)
A string process with n loops will be described, in general, by a two-dimensional world-sheet surface with n “handles”, also called (more technically) a surface of genus n. Quantum interactions among strings can then be approximated by a series of world- sheet configurations of higher and higher genus. The genus, being a topologically invariant property of the surface, can be expressed in terms of the intrinsic curvature of the world-sheet surface (more precisely, as an integral of the two-dimensional scalar curvature). The dilaton, by definition, is directly coupled to such a curvature.
In particular, a given dilaton φ appears as a multiplicative factor of the curvature and thus, if it is constant, also of the genus n. The quantum description of string interactions, on the other hand, is based on what is known as the partition function, which is proportional to the exponential of the dilaton–curvature coupling (in our case, to the exponential of the factor φn). An expansion of the partition function in a series of higher-genus world sheets (i.e., a sum of terms with n = 0, n = 1, n = 2, and so on) thus becomes an expansion in powers of the exponential of the dilaton, exp φ. But, by definition, the loop approximation is an expansion in powers of the string coupling constant g^s 2. This leads us to identify (at least in the approximate, perturbative regime) the exponential of the dilaton with the string coupling parameter. The above result is valid even if φ is not a constant, in which case exp φ still plays the role of a local effective coupling. In general, in fact, the dilaton is a field that can take different values in different regions of space and at different times. The same will be true for g^s2 and all the other coupling constants, i.e., for the “charges” that determine the strengths of the various forces, and which are obtained from g^s2 in the context of string models of unified interactions.
Concerning this point, there is a nice analogy with the transition from Newton’s theory of gravity to Einstein’s. In the context of general relativity the rigid (Euclidean) geometry of Newtonian gravity becomes “soft”, i.e., variable in space and time; the strength of the gravitational interaction remains “rigid”, however. In a similar way the transition to string theory removes this residual rigidity because, thanks to the dilaton, even the gravitational strength (as well as the strength of the other forces) becomes “malleable” and space-time dependent. Note that it is just the variability of the dilaton (and the couplings) at early times that provides us with the new cosmological scenarios introduced in the last chapter.
Let us now go back to the physical effects associated with the finite size of strings. Given their tiny extension, it is evident that fundamental strings cannot be distinguished from point-like objects in any process where the typical length scale is much greater than Ls. On the other hand, current experiments involving particle accelerators are unable to resolve distances much smaller than about 10^−15 cm. This means that they are only sensitive to length scales much greater than Ls, if we assume for the string length the standard value of about 10^−32 cm suggested by unification models.
People also ask
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