This document is for persons who have received their graduate degree in theoretical physics and are looking to make their way into the concentration of superstring theory, and what postgraduate mathematics courses are required to do so. Supersting theory is one of the latest forms of theoretical physics and a popular topic with today’s society. However, because of the highly advanced nature of the mathematics involved with Supersting theory, two postgraduate forms of mathematics are required in order to be on the leading edge of work in this field. These are Noncommutative Geometry and K-theory.
Superstring theory is an attempt by humans to model the four fundamental forces of physics as vibrations of tiny supersymmetric strings. Superstring theory seems the most likely to lead to theories of quantum gravity
, an attempt to explain gravity’s relatively weak force when compared to the other forces of physics (“Quantum gravity”, nd). Superstring theory is also "supersymmetric string theory." It is referred to as this because unlike bosonic string theory, the original form of string theory (Bosonic string theory, nd), it is the version of the theory that incorporates fermions, particles that form totally antisymmetric composite quantum states (Fermions, nd), and supersymmetry, which link bosons and fermions (“Supersymmetry”, nd; “Superstring theory”, nd)
As of now, the main goal of theoretical physics is to explain how gravity relates to the other three fundamental forces of natural physics. However with as with every quantum field theory, there are infinite probabilities that result from the calculations. Unlike electromagnetic force, strong nuclear force, and weak nuclear force, physicists have not been able to find a mathematical technique that eliminates these infinities (“Superstring theory”, nd). Therefore, the quantum theory
of gravity must be developed by a different means than those used for the other forces.
Superstring theory dictates that the base of all that is real would be tiny vibrating strings the size of a plank’s length. The proposed messenger particle for gravitational force, a graviton is predicted by the theory to be a string with wave amplitude zero. Another insight the theory provides 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)” (Superstring theory, nd para 3). This is true because according to currant theory, a universe could never become smaller than a string. If a universe were to begin to collapse in on itself it would not destroy itself because once it were the size of a string it would have to begin to expand again (“Superstring theory”, nd).
As humans observe it, physical space has only four large dimensions. String theory takes these four dimensions into account but also goes to say nothing prevents additional dimensions. “In the case of string theory, consistency requires spacetime to have 10, 11 or 26 dimensions”(“Superstring theory”, nd para 4). The reason these higher dimensions can be considered yet remain unseen is that they are compact dimensions, the size of a Plank length and therefore unobservable (“Superstring theory”, nd).
It is difficult to imagine higher dimensions because people only have the ability to move in three spatial dimensions. Moreover, humans only see in two plus one dimensions; having vision in three true dimensions would actually allow for the sight of all sides of an object at the same time. The question raised now is if experiments can be devised to test higher dimension theories where a human scientist can interpret the results in one, two, or two plus one dimensions. This, then, leads to the question of whether models that rely on such an abstract modeling, that is without experimental testing, can be considered 'scientific' rather than philosophy (Groleau, 2003).
Before superstring theory existed, Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University described the six-dimensional geometrical shapes that superstring theory requires to complete its equations. What one of these six-dimensional objects may look like is seen in figure 1. If the spheres in curled-up space are replaced with these Calabi-Yau shapes, the result is the ten dimensions Supersting theory calls for: three spatial, plus the six of the Calabi-Yau shapes, plus one of time (Groleau, 2003). Figure 1- six-dimensional Calabi-Yau shapes from “Imagining Other Dimensions”, PBS.org retrieved
25 August 2004 from http://www.pbs.org/wgbh/nova/
A universe with more than four dimensions is almost unimaginable for humans and there might never be an accurate representation of higher dimensional space a human can accept without actually having to be sucked into that higher dimensional space.
The Five String Theories
Until the mid 1990’s it seemed there were five different String theories. However, the Second Superstring revolution brought about M-theory, which found that the five string theories were all related and part of that M-theory (“Superstring theory”, nd).
The five consistent superstring theories are: type I; type IIA; type IIB; Heterotic E8 X E8, also known as HEt; and Heterotic SO(32), also known as HOt. The type I theory is special in that it is based on unoriented open and closed strings, while the others are based on oriented closed strings. The type II theories have two supersymmetries in the ten-dimensional sense, while the others have only one. And, the IIA theory is special because it is non-chiral or parity conserving, while the rest are chiral or parity violating.
Chiral gauge theories can be inconsistent, this happens “when certain one-loop Feynman diagrams cause a quantum mechanical breakdown of the gauge symmetry”(“Superstring theory”, para 7). When these anomalies cancel, it puts a constraint on possible superstring theories.
K-THEORY AND STING THEORY
Though Supersting theory is a highly advanced form of theoretical physics, it is not the first theory to propose extra spatial dimensions. String theory relies on the “mathematics of folds, knots, and topology, which was largely developed after Kaluza and Klein, and has made physical theories relying on extra dimensions much more credible” (Witten, 1998 p. 9).
In 1919, Theodor Kaluza theorized that the existence of a fourth spatial dimension would allow the linking of theories of general relativity and electromagnetism. Oskar Klein later refined this idea by proposing that space consisted of both extended and curled-up dimensions (Groleau, 2003). The extended dimensions are the three spatial dimensions humans exist in, and the curled-up dimension found deep within those extended dimensions. Based on later experiments, Kaluza and Klein's curled-up dimension could not unite general relativity and electromagnetic theory, but now string theorists find the idea useful and necessary.
The mathematics used in superstring theory requires at least ten dimensions in order for the equations used to work out. Equations need to make use of additional dimensions in order to connect general relativity and quantum mechanics, explain the nature of particles, or unify forces. String theorists believe the extra dimensions are in the curled-up space Kaluza and Klein first described (Superstring theory, nd).
In order to extend the curled-up space to include these added dimensions, the Kaluza-Klein circles are replaced with spheres. Then there are two dimensions if only the spheres' surfaces are considered and three if the space with in the sphere is taken into account, creating a total of six (Groleau, 2003)
NONCOMMUTATIVE GEOMETRY IN STING THEORY
There is the possible relation of noncommutative geometry to string theory. It has been mentioned, “since noncommutative geometry is pointless a field theory on it will be divergence-free” (Madore, 1999 p. 16). In particular, provided of course that the geometry in which monopole configurations are constructed can be approximated by a noncommutative geometry, monopole configurations will have finite energy since the point on which they are localized has been replaced by a volume of fuzz. This is one characteristic noncommutative geometry shares with string theory. Because a throat to an adjacent D-brane replaces the point in space where certain monopoles are located, solutions to these have a finite energy (Madore, 1999).
In noncommutative geometry the string is replaced by a certain finite number of elementary volumes of ‘fuzz’, each of which can contain one quantum mode. Because of the nontrivial commutation relations the ‘line’ _qµ = qµ′−qµ joining two points qµ′ and qµ is quantized and can be characterized by a certain number of creation operators aj [sic] each of which creates a longitudinal displacement. They would correspond to the rigid longitudinal vibrational modes of the string. Since it requires no energy to separate two points the string tension would be zero. This has not much in common with traditional string theory. (Madore, 1999 p. 16)
Noncommutative Kaluza-Klein theory has much in common with the M-theory of D-branes, but is lacking is a supersymmetric extension. There have been speculations that string theory might give rise naturally to space-time uncertainty relations and to a noncommutative theory of gravity. There have also been attempts to relate a noncommutative structure of space-time to the quantization of the open string in the presence of a non-vanishing B-field (Witten, 1998).
In order to become involved in advanced theoretical physics, such as Superstring theory, it is suggested that K-theory and noncommutative geometry are studied and understood. Superstring theory is on the leading edge of explaining the universe and perhaps in time will unify the four fundamental forces of physics. Though man may never be able to comprehend the extra dimensions required for this, Supersting theory is man’s best hope at understanding all that has happened and will happen in the universe at the present time.
Bosonic string theory. Retrieved 25 August 2004 from http://en.wikipedia.org/wiki/
Fermions. Retrieved 25 August 2004 from http://en.wikipedia.org/wiki/Fermions
Groleau, R (2003, July). Imagining Other Dimensions. http://www.pbs.org/wgbh/nova/
Kaluza-Klein theory. Retrieved 25 August 2004 from http://en.wikipedia.org/wiki/Kaluza Klein_theory
Madore, J (1999, July 25). Noncommutative Geometry for Pedestrians. Munich, Germany: Author
Quantum gravity. Retrieved 25 August 2004 from http://en.wikipedia.org/wiki/Quantum_gravity
Superstring theory. Retrieved 23 August 2004 from http://en.wikipedia.org/wiki/
Supersymmetry. Retrieved 25 August 2004 from http://en.wikipedia.org/wiki/Supersymmetry
Witten, E (1998, October). D-branes and K-theory. Princeton, NJ: School of Natural Sciences, Institute for Advanced Study