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Scalar Electrodynamics
In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism. Matter content and Lagrangian Matter content The model consists of a complex scalar field \phi(x) minimally coupled to a gauge field A_\mu(x). This article discusses the theory on flat spacetime \mathbb^ (Minkowski space) so these fields can be treated (naïvely) as functions \phi:\mathbb^\rightarrow \mathbb, and A_\mu:\mathbb^\rightarrow (\mathbb^)^*. The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a principal connection, specifically a principal \text(1) connection. Lagrangian The dynamics is given by the Lagrangian density \ma ... [...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] |
Scalar Field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar (mathematics), mathematical number (dimensionless) or a scalar (physics), scalar physical quantity (with unit of measurement, units). In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. Definition Mathematically, a scalar field on a Region (mathematical analysis), region ''U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Fermion
In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category. Description In particle physics, all fermions in the standard model have distinct antiparticles (''perhaps'' excepting neutrinos) and hence are Dirac fermions. They are named after Paul Dirac, and can be modeled with the Dirac equation. A Dirac fermion is equivalent to two Weyl fermions. The counterpart to a Dirac fermion is a Majorana fermion, a particle that must be its own antiparticle. Dirac quasi-particles In condensed matter physics, low-energy excitations in graphene and topological insulators, among others, are fermionic quasiparticles described by a pseudo-relativistic Dirac equation. See also * Dirac spinor, a wavefunction-like description of a Dirac fermion * Dirac–Kähler fermion, a geometric formulation of Dirac fermions * Majorana fermion, an alternate category of fermion, possibly de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction. In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. History The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W+, W−, and Z0 bosons actually have relatively large masses of around 80 GeV/''c''2. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.This makes spacetime distance an invariant. Becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connection on a principal G-bundle ''P'' over a smooth manifold ''M'' is a particular type of connection which is compatible with the action of the group ''G''. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to ''P'' via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omega(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W+, W−, and Z0 bosons actually have relatively large masses of around 80 GeV/''c''2. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...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, precisely, iso ... [...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] |
