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Batalin–Vilkovisky Formalism
In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart. Batalin–Vilkovisky algebras In mathematics, a Batalin–V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faddeev–Popov Ghost
In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov. A more general meaning of the word "ghost" in theoretical physics is discussed in Ghost (physics). Overcounting in Feynman path integrals The necessity for Faddeev–Popov ghosts follows from the requirement that quantum field theories yield unambiguous, non-singular solutions. This is not possible in the path integral formulation when a gauge symmetry is present since there is no procedure for selecting among physically equivalent solutions related by gauge transformation. The path integrals overcount field configurations corresponding to the same physical state; the measure of the path integrals contains a factor which does not allow obtaining various results directly from the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theories
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the '' symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge boso ... [...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 a superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way. Gravitons Like any field theory of gravity, a supergravity theory 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 Pran Nath in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Formalism
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (p,q) and Hamiltonian H Let (M, \mathcal L) be a mechanical system with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity describing how a physical system has dynamics (physics), changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a constant velocity (uniform linear motion), the action is the momentum of the particle times the distance it moves, integral (mathematics), added up along its path; equivalently, action is twice the particle's kinetic energy times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined. More formally, action is a functional (mathematics), mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensional analysis, dimensions of energy × time or momentu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (physics)
In physics, a field is a physical quantity, represented by a scalar (mathematics), scalar, vector (mathematics and physics), vector, or tensor, that has a value for each Point (geometry), point in Spacetime, space and time. For example, on a weather map, the surface temperature is described by assigning a real number, number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an vector (mathematics and physics), arrow to each point on a map that describes the wind velocity, speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of Mathematical descriptions of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BRST Formalism
BRST may refer to: * BRST Films, a Serbian video production company * BRST algorithm, an optimization algorithm suitable for finding the global optimum of black box functions * BRST quantization in Yang-Mills theories, a way to quantize a gauge-symmetric field theory * Brasília Summer Time Time in Brazil is calculated using standard time, and the country (including its offshore islands) is divided into four standard time zones: UTC−02:00, UTC−03:00, UTC−04:00 and UTC−05:00. Time zones Fernando de Noronha time (UTC−02 ... * Burst.com, from its stock ticker, now Democrasoft {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Mills Theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics. History and theoretical description In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
