Superstring Theory - A Taste

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantam gravity. Superstring theory is a shorthand for supersymmetric string theory because unlike bosonic string theory, it is the version of string theory that incorporates femions and super symmetry.

The deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures, with quantam mechanics, which describes the other three fundamental forces acting on the atomic scale.

The development of a quantam field theory of a force invariably results in infinite (and therefore useless) probabilities. Physicists have developed mathematical techniques to eliminate these infinities which work for three of the four fundamental forces – electromagnetism, strong nuclear and weak nuclear forces - but not for gravity. The development of a quantam theory of gravity must therefore come about by different means than those used for the other forces.

The basic idea is that the fundamental constituents of reality are strings of the Planck length (about 10^{−33} cm) which vibrate at resonant frequencies. Every string in theory has a unique resonance, or harmonic. Different harmonics determine different fundamental forces. The tension in a string is on the order of the Planck force (10^{44} newtons). The graviton (the proposed messenger particle of the gravitational force), for example, is predicted by the theory to be a string with wave amplitude zero. Another key insight provided by the theory is that no measurable differences can be detected between strings that wrap around dimensions smaller than themselves and those that move along larger dimensions (i.e., effects in a dimension of size R equal those whose size is 1/R). Singularities are avoided because the observed consequences of "Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of a string, at which point it would actually begin expanding.

Our physical space is observed to have only three large dimensions and—taken together with time as the fourth dimension—a physical theory must take this into account. However, nothing prevents a theory from including more than 4 dimensions, per se. In the case of string theory, consistency requires spacetime to have 10, 11 or 26 dimensions. The conflict between observation and theory is resolved by making the unobserved dimensions compactified.

Our minds have difficulty visualizing higher dimensions because we can only move in three spatial dimensions. One way of dealing with this limitation is not to try to visualize higher dimensions at all, but just to think of them as extra numbers in the equations that describe the way the world works. This opens the question of whether these 'extra numbers' can be investigated directly in any experiment (which must show different results in 1, 2, or 2+1 dimensions to a human scientist). This, in turn, raises the question of whether models that rely on such abstract modelling (and potentially impossibly huge experimental apparatus) can be considered scientific. Six-dimensional Calabi-Yau shapes can account for the additional dimensions required by superstring theory. The theory states that every point in space (or whatever we had previously considered a point) is in fact a very small manifold where each extra dimension has a size on the order of the Planck length.

Superstring theory is not the first theory to propose extra spatial dimensions; the Kaluza-Klein theory had done so previously. Modern string theory relies on the mathematics of folds, knots, and topology, which were largely developed after Kaluza and Klein, and has made physical theories relying on extra dimensions much more credible.

Theoretical physicists were troubled by the existence of five separate string theories. This has been solved by the second superstring revolution in the 1990s during which the five string theories were discovered to be different limits of a single underlying theory: M-theory.

The five consistent superstring theories are:

General relativity typically deals with situations involving large mass objects in fairly large regions of spacetime whereas quantum mechanics is generally reserved for scenarios at the atomic scale (small spacetime regions). The two are very rarely used together, and the most common case in which they are combined is in the study of black holes. Having "peak density", or the maximum amount of matter possible in a space, and very small area, the two must be used in synchrony in order to predict conditions in such places; yet, when used together, the equations fall apart, spitting out impossible answers, such as imaginary distances and less than one dimension.

The major problem with their congruence is that, at sub-Planck (an extremely small unit of length) lengths, general relativity predicts a smooth, flowing surface, while quantum mechanics predicts a random, warped surface, neither of which are anywhere near compatible. Superstring theory resolves this issue, replacing the classical idea of point particles with loops. These loops have an average diameter of the Planck length, with extremely small variances, which completely ignores the quantum mechanical predictions of sub-Planck length dimensional warping, there being no matter that is of sub-Planck length.

It is commonly believed that the 5 superstring theories are approximated to a theory in higher dimensions possibly involving membranes. Unfortunately because the action for this involves quartic terms and higher so is not Gaussian the functional integrals are very difficult to solve and so this has confounded the top theoretical physicists. Edward Witten has popularised the concept of a theory in 11 dimensions M-Theory involving membranes interpolating from the known symmetries of superstring theory. It may turn out that there exist membrane models or other non-membrane models in higher dimensions which may become acceptable when new unknown symmetries of nature are found, such as noncommutative geometry for example. It is thought, however, that 16 is probably the maximum since O(16) is a maximal subgroup of E8 the largest exceptional lie group and also is more than large enough to contain the Standard Model. Quartic integrals of the non-functional kind are easier to solve so there is hope for the future. This is the series solution which is always convergent when a is non-zero and negative:

Investigating theories of higher dimensions often involves looking at the 10 dimensional superstring theory and interpreting some of the more obscure results in terms of compactified dimensions. For example D-branes are seen as compactified membranes from 11D M-Theory. Theories of higher dimensions such as 12D F-theory and beyond will produce other effects such as gauge terms higher than U(1). The components of the extra vector fields (A) in the D-brane actions can be thought of as extra coordinates (X) in disguise. However, the known symmetries including supersymmetry currently restrict the spinors to have 32-components which limits the number of dimensions to 11 (or 12 if you include two time dimensions.) Some commentators (e.g. John Baez et al) have speculated that the exceptional lie groups E_{6}, E_{7} and E_{8} having maximum orthogonal subgroups O(10), O(12) and O(16) may be related to theories in 10, 12 and 16 dimensions; 10 dimensions corresponding to string theory and the 12 and 16 dimensional theories being yet undiscovered but would be theories bases on 3-branes and 7-branes respectively. However this is a minority view within the string community. Since E_{7} is some sense F_{4} quaternified and E_{8} is F_{4} octonified, then the 12 and 16 dimensional theories, if they did exist, may involve the noncommutative geometry based on the quaternions and octonions respectively. From the above discussion it can be seen that physicists have many ideas for to extend superstring theory beyond the current 10 dimensional theory but so far none have been successful.

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