Why 10 dimensions?
From Wikipedia, the free encyclopedia
Although the human mind comprehends the universe with three spatial dimensions, some theories in physics, including string theory, include the idea that there are additional spatial dimensions. Such theories suggest that there may be a specific number of spatial dimensions such as 10. The question, "Why 10 dimensions?" arises from these theories.
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[edit] Why 10, 11, or 26 physical dimensions in string theory?
This is one of the questions discussed by Michio Kaku in his book Hyperspace, which attempts to translate the mathematics of hyperspace theory into readily understandable language. This article is devoted to the same goal, leaving the details of the mathematics to the hyperspace theory article.
Kaku traces the number of dimensions to Srinivasa Ramanujan's modular functions, but this article will start with some fundamentals and work its way into the mathematics.
[edit] Foundation
[edit] Spatial dimensions
Macroscopic physical objects such as people are free to move in three different directions, so we conclude that there are three spatial dimensions. Theoretical physicists have speculated that there might be additional spatial dimensions that escape our notice because they are too small. Some theoretical objects like strings are small enough that they would be able to vibrate in the compact dimensions while moving through the familiar three extended dimensions.
[edit] String theory
String theory is a proposed physical theory. There are several versions or types of string theory. Attempts are being made to discover which version of the theory (if any) is in agreement with observations of the physical universe (M-Theory is a progressive step toward reaching this goal). All string theories include the idea of a hyperspace of more than three spatial dimensions. The "extra" spatial dimensions are theoretically "compact" or "collapsed" dimensions. This means that they are not as extended in space as the three familiar spatial dimensions. The collapsed dimensions are too small to observe directly.
It is not clear how many collapsed dimensions are required for a string theory that is in best agreement with observations of the physical universe, but mathematical constraints currently favor string theories with 10, 11, or 26 dimensions observed and so attention has turned to theoretical attempts to describe the diversity of subatomic particles in an elegant physical theory. Why are there so many different particles? Why do they have the physical properties that they are observed to have? Have things always been this way or have the properties of subatomic particles changed since the formation of our universe? Do they continue to change?
Some theoretical physicists are exploring the idea that the diversity of subatomic particles can be accounted for in terms of symmetry breaking. Maybe under the high energy conditions of the early universe all particles were initially indistinguishable, a condition called supersymmetry. As the universe cooled, some spatial dimensions compacted and particles distributed themselves among the available stable energy states provided by three extended spatial dimensions and six or more compact dimensions. This line of reasoning suggests that it might be possible to explain the diversity of subatomic particles and fundamental forces in terms of a theory of how an original hyperspace "broke" into two "parts"; our extended 4 dimensional space-time and an "invisible" group of several additional compact spatial dimensions. String theory is a popular hyperspace theory in part because it easily accommodates gravity in terms of a spin-2 graviton.
[edit] Strings
Within string theory, a string is a one-dimensional object. These hypothetical one-dimensional strings are very small, on the order of the planck length (about 1.6 × 10-35 meters ). String theory explores the implications of strings that are either open (they have free ends) or closed (they form loops and have no free ends). Within string theory, the stable vibrational states of strings are taken to correspond to physical particles like the graviton. When a string moves through spacetime it sweeps out a 2-dimensional surface called a worldsheet. One of the features of string theory that has appeal to physicists is that when particle interactions are thought of in terms of interacting worldsheets it is possible to overcome some of the mathematical problems confronted when considering particles as points.
String theory grew out of attempts to find a simple and elegant way to account for the diversity of particles and forces observed in our universe. The starting point was to assume that there might be a way to account for that diversity in terms of a single fundamental physical entity (string) that can exist in many "vibrational" states. The various allowed vibrational states of string could theoretically account for all the observed particles and forces. Unfortunately, there are many potential string theories and no simple way of finding the one that accounts for the way things are in our universe.
One way to make progress is to assume that our universe arose through a process involving an initial hyperspace with supersymmetry that, upon cooling, underwent a unique process of symmetry breaking. The symmetry breaking process resulted in conventional 4 dimensional extended space-time AND some combination of additional compact dimensions. Mathematics might be able to tell us how many additional compact dimensions exist.
[edit] Conformal anomaly
String theory over a worldsheet of fixed genus is a conformal field theory. The conformal symmetry is really a remnant of the worldsheet diffeomorphism after gauge-fixing. The diffeomorphism symmetry gives rise to a stress-energy tensor constraint, constraining it to zero. From the Virasoro algebra, such a constraint can only be implemented if the conformal anomaly is zero. The BRST ghosts contribute -24 to the conformal anomaly. Each spatial dimension contributes 1 and each time dimension contributes -1. With 25 spatial dimensions and 1 time dimension, the conformal anomaly is -24+25-1=0.
[edit] Modular functions
Modular functions are a subclass of the more general modular forms. An example of a modular function is the Dedekind eta function, given by the infinite product
Like other modular forms, this function is defined over the domain of complex numbers z = x + iy where x and y are real and y > 0.
For complex numbers i is the square root of −1. In the function, e is Euler's number (2.71828....) and π is pi (3.14159265359....).
[edit] The connection between modular functions and string theory
Closed strings can be viewed as a set of loops arrayed on the manifold of spacetime. The study of how simple loops behave under deformations is known as homology, and is intensely studied in K-theory. Imagine, for example, a Green's function defined on a manifold: it can be thought of as defining a measure-preserving flow on the manifold. As the loops flow along the manifold, they trace out cylinders which sometimes merge and join, and sometimes split apart. Places where two cylinders join are known as pairs of pants. Riemann surfaces of negative curvature can be formed by stitching together pairs of pants. Thus, the natural setting for a string theory of closed loops is a Riemann surface.
Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann surface theory is relevant to describing the behavior of strings as they move through space-time. When strings move they maintain a kind of symmetry called "conformal invariance".
Conformal invariance (also called "scale invariance") is related to the fact that points on the surface of a string's world sheet need not be evaluated in a particular order. As long as all points on the surface are taken into account in any consistent way, the physics should not change. Equations of how strings must behave when moving involve the Ramanujan function.
[edit] Ramanujan modular functions
The 1968 Veneziano model Euler beta function describes the strong nuclear force.
When a string moves in space-time by splitting and recombining (see worldsheet diagram at right), a large number of mathematical identities must be satisfied. These are the identities of Ramanujan's modular function.
The KSV loop diagrams of interacting strings can be described using modular functions.
The Ramanujan function (an elliptic modular function, satisfying the need for "conformal symmetry") has 24 "modes" that correspond to the physical vibrations of a bosonic string.
When the Ramanujan function is generalized, 24 is replaced by 8 (8 + 2 = 10) for fermion strings.
[edit] See also
[edit] External links
- Why hyperspace? (PDF)
- Examination of Fernando Loup's Macroscopic shortcut article (PDF)
- Scientific American: Extra dimensional theories are claimed to work in 10 or 11 dimensions. Why these numbers and not, say, 42?
- The theory of longitudinal and transversal rotational momentum A mathematical theory that unites all physical forces and eliminates the wave/particle duality.
