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Higher-dimensional Supergravity
Higher-dimensional supergravity is the supersymmetric generalization of general relativity in higher dimensions. Supergravity can be formulated in any number of dimensions up to eleven. This article focuses upon supergravity (SUGRA) in greater than four dimensions. Supermultiplets Fields related by supersymmetry transformations form a supermultiplet; the one that contains a graviton is called the supergravity multiplet. The name of a supergravity theory generally includes the number of dimensions of spacetime that it inhabits, and also the number \mathcal of gravitinos that it has. Sometimes one also includes the choices of supermultiplets in the name of theory. For example, an \mathcal=2, (9 + 1)-dimensional supergravity enjoys 9 spatial dimensions, one time and 2 gravitinos. While the field content of different supergravity theories varies considerably, all supergravity theories contain at least one gravitino and they all contain a single graviton. Thus every super ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra and superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way. Gravitons Like all covariant approaches to quantum gravity, supergravity contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. History Gauge supersymmetry The first theory of local supersymmetry was proposed by Dick Arnowitt and Pra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws, equations, and theories respect Lorentz symmetry: * The kinematical laws of special relativity * Maxwell's field equations in the theory of electromagnetism * The Dirac equation in the theory of the electron * The Standard Model of particle physics The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity. Basic properties The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions. None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In condensed matter physics, some materials can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals. Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term. History The Dirac equation was published in 1928 by Paul Dirac, and was first used to model spin-1/2 particles in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majorana Spinor
In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any (possibly non-relativistic) fermionic particle that is its own anti-particle (and is therefore electrically neutral). There have been proposals that massive neutrinos are described by Majorana particles; there are various extensions to the Standard Model that enable this. The article on Majorana particles presents status for the experimental searches, including details about neutrinos. This article focuses primarily on the mathematical development of the theory, with attention to its discrete and continuous symmetries. The discrete symmetries are charge conjugation, parity transformation and time reversal; the continuous symmetry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or '' holes'', of a topological space. To define the ''n''th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never homeomorphic, but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. Introduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bott Periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, ( real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helicity (particle Physics)
In physics, helicity is the projection of the spin onto the direction of momentum. Mathematically, ''helicity'' is the sign of the projection of the spin vector onto the momentum vector: "left" is negative, "right" is positive. Overview The angular momentum J is the sum of an orbital angular momentum L and a spin S. The relationship between orbital angular momentum L, the position operator r and the linear momentum (orbit part) p is : \mathbf = \mathbf\times\mathbf , so L's component in the direction of p is zero. Thus, helicity is just the projection of the spin onto the direction of linear momentum. The helicity of a particle is positive ("right-handed") if the direction of its spin is the same as the direction of its motion and negative ("left-handed") if opposite. Helicity is conserved. That is, the helicity commutes with the Hamiltonian, and thus, in the absence of external forces, is time-invariant. It is also rotationally invariant, in that a rotation applied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number: electrons and other leptons, quarks, and the fundamental bosons. Subatomic particles such as protons or neutrons, which contain two or more elementary particles, are known as composite particles. Ordinary matter is composed of atoms, themselves once thought to be indivisible elementary particles. The name ''atom'' comes from the Ancient Greek word ''ἄτομος'' ( atomos) which means ''indivisible'' or ''uncuttable''. Despite the theories about atoms that had existed for thousands of years, the factual existence of ato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity (also a vector quantity), then the object's momentum (from Latin '' pellere'' "push, drive") is: \mathbf = m \mathbf. In the International System of Units (SI), the unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is dimensionally equivalent to the newton-second. Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame of reference, it is a ''conserved'' quantity, meaning that if a closed system is not affected by external forces, its total momentum does not change. Momentum is also conserved in special relativity (with a mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Little Group
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Compact Subgroup
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are ''not'' in general unique, but are unique up to conjugation – they are essentially unique. Example An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group. Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. The Dirac spinor is specific to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
