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Superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The word "superspace" was first used by John Archibald Wheeler, John Wheeler in an unrelated sense to describe the Configuration space (physics), configuration space of general relativity; for example, this usage may be seen in his 1973 textbook ''Gravitation (book), Gravitation''. Informal discussion There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym for super Minkowski space. In this case, one takes ordinary Minkowski space, and extend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a Path integral formulation, path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed. Informal discussion Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by :(x,\theta,\bar) where ''x'' is the ( real-number-valued) spacetime coordinate, and \theta\, and \bar are Grassmann-valued spatial "directions". The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Minkowski Space
In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré algebra. Construction Abstract construction Abstractly, super Minkowski space is the space of (right) cosets within the Super Poincaré group of Lorentz group, that is, :\text \cong \frac. This is analogous to the way ordinary Minkowski spacetime can be identified with the (right) cosets within the Poincaré group of the Lorentz group, that is, :\text \cong \frac. The coset space is naturally affine, and the nilpotent, anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group. Direct sum construction For this section, the dimension of the Minkowski space under consideration is d = 4. Super Minkowski space can be concretely realized as the direct sum of Min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude of a -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that v\wedge v=0 implies a skew-sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester James Gates
Sylvester James Gates Jr. (born December 15, 1950), known as S. James Gates Jr. or Jim Gates, is an American theoretical physicist who works on supersymmetry, supergravity, and superstring theory. He is currently the Toll Professor of Physics at the University of Maryland. He also holds the Clark Leadership Chair in Science with the physics department at the University of Maryland College of Computer, Mathematical, and Natural Sciences. He is also affiliated with the University Maryland's School of Public Policy. He previously was the Brown University Theoretical Physics Center Director and the Ford Foundation Professor of Physics. He served on former president Barack Obama's Council of Advisors on Science and Technology. Early life and education Gates, the oldest of four siblings, was born in Tampa, Florida, the son of Sylvester James Gates Sr., a career U.S. Army man, and Charlie Engels Gates. His mother died at age 44 of breast cancer when he was 11. Gates, Sr. raised his chi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Archibald Wheeler
John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr to explain the basic principles of nuclear fission. Together with Gregory Breit, Wheeler developed the concept of the Breit–Wheeler process. He is best known for popularizing the term "black hole" for objects with gravitational collapse already predicted during the early 20th century, for inventing the terms "quantum foam", "neutron moderator", "wormhole" and "it from bit", and for hypothesizing the "one-electron universe". Stephen Hawking called Wheeler the "hero of the black hole story". At 21, Wheeler earned his doctorate at Johns Hopkins University under the supervision of Karl Herzfeld. He studied under Breit and Bohr on a National Research Council (United States), National Research Council fellowship. In 1939 he collaborated with Bohr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice Rogers
Frances Alice Rogers is a British mathematician and mathematical physicist. She is an emeritus professor of mathematics at King's College London. Research Rogers' research concerns mathematical physics and more particularly supermanifolds, generalizations of the manifold concept based on ideas coming from supersymmetry. She is the author of the book ''Supermanifolds: Theory and Applications'' (World Scientific, 2007). Service Rogers has been a member of the British government's Advisory Committee on Mathematics Education,. is the education secretary of the London Mathematical Society (LMS), and represents the LMS on the Joint Mathematical Council of the UK. Education Rogers studied mathematics in New Hall, Cambridge, in the 1960s. Her mother had also studied mathematics at Cambridge in the 1930s and later became a wartime code-breaker at Bletchley Park. Rogers earned her Ph.D. in 1981 from Imperial College London. Recognition In 2016, she was appointed as an Officer of the Or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bryce DeWitt
Bryce Seligman DeWitt (born Carl Bryce Seligman; January 8, 1923 – September 23, 2004) was an American theoretical physicist noted for his work in gravitation and quantum field theory. Personal life He was born Carl Bryce Seligman, but he and his three brothers, including the noted ichthyologist, Hugh Hamilton DeWitt, added "DeWitt" from their mother's side of the family, at the urging of their father, in 1950. Several decades later, when Felix Bloch learned of this name-change, he was so upset that he blocked DeWitt's appointment to Stanford University; consequently, DeWitt and his wife Cecile DeWitt-Morette, a mathematical physicist, accepted faculty positions at the University of Texas at Austin. DeWitt trained in World War II as a naval aviator, but the war ended before he saw combat. He died September 23, 2004, from pancreatic cancer at the age of 81. He is buried in France, and was survived by his wife and four daughters. Academic life He received his bachelor' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Vector Space
In mathematics, a graded vector space is a vector space that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures. Integer gradation Let \mathbb be the set of non-negative integers. An \mathbb-graded vector space, often called simply a graded vector space without the prefix \mathbb, is a vector space together with a decomposition into a direct sum of the form : V = \bigoplus_ V_n where each V_n is a vector space. For a given ''n'' the elements of V_n are then called homogeneous elements of degree ''n''. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
