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In our personal human experiences, we seem to exist in a 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. Binomial name Homo sapiens Linnaeus, 1758 Subspecies Homo sapiens idaltu (extinct) Homo sapiens sapiens Human beings define themselves in biological, social, and spiritual terms. ...
The deepest visible-light image of the cosmos. ...
Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ...
String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...
Why 10, 11, or 26 physical dimensions in string theory?
This is one of the questions discussed by Michio Kaku in his book Hyperspace. That book is an attempt to translate the mathematics of hyperspace theory into ordinary language that can be understood by a wide audience. This article is devoted to the same goal, leaving the details of the mathematics to the hyperspace theory article. Michio Kaku (加來 紀雄) is a Japanese American theoretical physicist known for his popular accounts of string field theory. ...
For an account of the concept of hyperspace used in science fiction, largely unrelated to the topic of this article, see hyperspace (science fiction). ...
Hyperspace theories are concerned with theoretical systems that have more than the familiar three spatial dimensions. ...
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. The goal here is to use ordinary language when possible and be careful to clearly define jargonistic terms that come to us from the mathematics and physics of hyperspace. Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...
A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...
Jargon redirects here. ...
Foundation: let's be clear what we are talking about All in one place: short definitions
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. Macroscopic means measurable and observable by the naked eye; describes existence as we perceive it. ...
Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small (Planck length). ...
Strings (as a sound (voice) in electronic musical instruments and synthesizers) is an imitation of classical string ensembles sound. ...
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. 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. Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, and predicting physical phenomena through a physical theory. There are three types of theories in physics: mainstream theories, proposed theories and fringe theories. ...
String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...
For an account of the concept of hyperspace used in science fiction, largely unrelated to the topic of this article, see hyperspace (science fiction). ...
In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small (Planck length). ...
Hyperspace What explanatory power comes from including "extra" compact dimensions in a physical theory? Since the time of the first written speculations about the possible existence of an atom, a goal of physics has been to understand the fundamental physical components of the universe. Unfortunately, many subatomic particles (each subject to some combination of the four fundamental forces) have been 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 particle changed since the formation of our universe? Do they continue to change? Some theoretical physicists are exploring the idea that the diversity of subatomic particle 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. In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small (Planck length). ...
Properties For alternative meanings see atom (disambiguation). ...
Helium atom (not to scale) Showing two protons (red), two neutrons (green) and a probability cloud (gray) of two electrons (yellow). ...
A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ...
Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some Lie group goes into a vacuum state that is not symmetric. ...
According to the Big Bang theory, the universe originated in an extremely dense and hot state (bottom). ...
In particle physics, supersymmetry is a hypothetical symmetry that relates bosons and fermions. ...
In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ...
Gravitation is the tendency of masses to move toward each other. ...
In physics, spin is an intrinsic angular momentum associated with microscopic particles. ...
In physics, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. ...
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. The Planck length is the natural unit of length, denoted by . ...
Recap before getting to the mathematics 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. What can mathematics tell us about how many additional compact dimensions might exist? Buckminsterfullerene is a spherically symmetrical carbon-containing molecule. ...
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 A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...
The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...
In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ...
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. In mathematics, the domain of a function is the set of all input values to the function. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Remember that for complex numbers i is the square root of −1. In the function, e is Euler's number (2.71828....) and π is pi (3.14159....). In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ...
In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ...
The mathematical constant e (occasionally called Eulers number after the Swiss mathematician Leonhard Euler, or Napiers constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. ...
The mathematical constant π represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ...
What is the connection between modular functions and string theory? 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" In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ...
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. In physics, scale invariance is the feature of physical objects of laws that do not change if the space is magnified, i. ...
A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ...
Ramanujan modular functions 1968 "Veneziano model" Euler beta function describes the strong nuclear force. Worldsheet diagram. ...
A separate article treats the beta-function (written with a hyphen) of physics. ...
The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ...
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. Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ...
A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ...
Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...
A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...
The KSV loop diagrams of interacting strings can be described using modular functions.
The "Ramanujan function" (an elliptic modular function? satisfies the need for "conformal symmetry") has 24 "modes" that correspond to the physical vibrations of a bosonic string. Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ...
When the Ramanujan function is generalized, 24 is replaced by 8 (8 + 2 = 10) for fermion strings. Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ...
See also Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ...
M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ...
M-theory is a cutting-edge theory of physics that deals with the extension of superstring theory. ...
External links - Why hyperspace? (http://doc.cern.ch//archive/electronic/other/ext/ext-2004-121.pdf)
- Examination of Fernando Loup's Macroscopic shortcut article (http://doc.cern.ch/archive/electronic/cern/preprints/open/open-2004-037.pdf)
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