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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 Minkowski ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Supersymmetry
In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extended supersymmetry is also called \mathcal=2, \mathcal=4 supersymmetry, for example. Extended supersymmetry is very important for analysis of mathematical properties of quantum field theory and superstring theory 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. 'Superstring theory' is a shorthand for supersymmetric string t .... The more extended supersymmetry is, the more it constrains physical observables and parameters. See also * Supersymmetry algebra * Harmonic superspace * Projective superspace Supersymmetry {{Quantum-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Vector Space
In mathematics, a super vector space is a \mathbb Z_2-graded vector space, that is, a vector space over a field \mathbb K with a given decomposition of subspaces of grade 0 and grade 1. The study of super vector spaces and their generalizations is sometimes called super linear algebra. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry. Definitions A super vector space is a \mathbb Z_2-graded vector space with decomposition :V = V_0\oplus V_1,\quad 0, 1 \in \mathbb Z_2 = \mathbb Z/2\mathbb Z. Vectors that are elements of either V_0 or V_1 are said to be ''homogeneous''. The ''parity'' of a nonzero homogeneous element, denoted by , x, , is 0 or 1 according to whether it is in V_0 or V_1, :, x, = \begin0 & x\in V_0\\1 & x\in V_1\end Vectors of parity 0 are called ''even'' and those of parity 1 are called ''odd''. In theoretical physics, the even elements are sometimes called ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 extends ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighbor ... [...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 Minkowski ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-spin Theory
Higher-spin theory or higher-spin gravity is a common name for field theories that contain massless fields of spin greater than two. Usually, the spectrum of such theories contains the graviton as a massless spin-two field, which explains the second name. Massless fields are gauge fields and the theories should be (almost) completely fixed by these higher-spin symmetries. Higher-spin theories are supposed to be consistent quantum theories and, for this reason, to give examples of quantum gravity. Most of the interest in the topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher-spin theories to weakly coupled conformal field theories. It is important to note that only certain parts of these theories are known at present (in particular, standard action principles are not known) and not many examples have been worked out in detail except some specific toy models (such as the higher-spin extension of pure Chern–Simons, Jackiw–Teitelboim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point. The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set (mathematics), set of all real numbers, viewed as a geometry, geometric space (mathematics), space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alpha ... [...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 a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. 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 degree ''n'' are exactly the linear combinations of monomials of degree ''n''. General gradation The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Representation
In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representation of a Lie group, representations of the spin groups, which are Double covering group, double covers of the special orthogonal groups. They are usually studied over the real number, real or complex numbers, but they can be defined over other field (mathematics), fields. Elements of a spin representation are called spinors. They play an important role in the physics, physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor Representation
In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representation of a Lie group, representations of the spin groups, which are Double covering group, double covers of the special orthogonal groups. They are usually studied over the real number, real or complex numbers, but they can be defined over other field (mathematics), fields. Elements of a spin representation are called spinors. They play an important role in the physics, physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
