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Gauge Anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e. of a gauge theory. All gauge anomalies must cancel out. Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel degrees of freedom with a negative norm which are unphysical (such as a photon polarized in the time direction). Indeed, cancellation occurs in the Standard Model. The term gauge anomaly is usually used for vector gauge anomalies. Another type of gauge anomaly is the gravitational anomaly, because coordinate reparametrization (called a diffeomorphism) is the gauge symmetry of gravitation. Calculation of the anomaly Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams. Vector gauge anomalies In vector gauge anoma ... [...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] |
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called chiral symmetry. Chirality and helicity The helicity of a particle is positive (“right-handed”) if the direction of its spin is the same as the direction of its motion. It is negative (“left-handed”) if the directions of spin and motion are opposite. So a standard clock, with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards. Mathematically, ''helicity'' is the sign of the projection of the spin vector onto the momentum vector: “left” is negative, “right” is positive. The chira ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Schwarz Mechanism
The Green–Schwarz mechanism (sometimes called the Green–Schwarz anomaly cancellation mechanism) is the main discovery that started the first superstring revolution in superstring theory. Discovery In 1984, Michael Green and John H. Schwarz realized that the anomaly in type I string theory with the gauge group SO(32) cancels because of an extra "classical" contribution from a 2-form field. They realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496 and then demonstrated this to be so. In the original calculation, gauge anomalies, mixed anomalies, and gravitational anomalies were expected to arise from a hexagon Feynman diagram. For the special choice of the gauge group SO(32) or E8 x E8, however, the anomaly factorizes and may be cancelled by a tree diagram. In string theory, this indeed occurs. The tree diagram describes the exchange of a virtual quantum of the B-fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anomaly Matching Condition
In quantum field theory, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly for the flavor symmetry must not depend on what scale is chosen for the calculation if it is done by using the degrees of freedom of the theory at some energy scale. It is also known as the 't Hooft condition and the 't Hooft UV-IR anomaly matching condition.In the context of quantum field theory, “UV” actually means the high-energy limit of a theory, and “IR” means the low-energy limit, by analogy to the upper and lower peripheries of visible light, but not actually meaning either light or those particular energies. 't Hooft anomalies There are two closely related but different types of obstructions to formulating a quantum field theory that are both called anomalies: chiral, or ''Adler–Bell–Jackiw'' anomalies, and 't Hooft anomalies. If we say that the symmetry of the theory has a t Hooft anomaly'', it means that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Gauge Theory
In quantum field theory, a chiral gauge theory is a quantum field theory with charged chiral (i.e. Weyl) fermions. For instance, the Standard Model is a chiral gauge theory. For topological reasons, chiral charged fermions cannot be given a mass without breaking the gauge symmetry, which will lead to inconsistencies unlike a global symmetry. It is notoriously difficult to construct a chiral gauge theory from a theory which does not already contain chiral fields at the fundamental level. A consistent chiral gauge theory must have no gauge anomaly (or global anomaly). Almost by necessity, regulators will have to break the gauge symmetry. This is responsible for gauge anomalies in the first place. Fermion doubling on a lattice Lattice regularizations suffer from fermion doublings leading to a loss of chirality. See also * Chiral anomaly In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Simons Form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose. Definition Given a manifold and a Lie algebra valued 1-form \mathbf over it, we can define a family of ''p''-forms: In one dimension, the Chern–Simons 1-form is given by :\operatorname \mathbf In three dimensions, the Chern–Simons 3-form is given by :\operatorname \left \mathbf \wedge \mathbf-\frac \mathbf \wedge \mathbf \wedge \mathbf \right= \operatorname \left d\mathbf \wedge \mathbf + \frac \mathbf \wedge \mathbf \wedge \mathbf\right In five dimensions, the Chern–Simons 5-form is given by : \begin & \operatorname \left \mathbf\wedge\mathbf \wedge \mathbf-\frac \mathbf \wedge\mathbf\wedge\mathbf\wedge\mathbf +\frac \mathbf \wedge \mathbf \wedge \mathbf \wedge \mathbf \wedge\mathbf \right\\ pt= & \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Product
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists r > 0, such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket Of Vector Fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth manifold ''M'' a third vector field denoted . Conceptually, the Lie bracket is the derivative of ''Y'' along the flow generated by ''X'', and is sometimes denoted ''\mathcal_X Y'' ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by ''X''. The Lie bracket is an R- bilinear operation and turns the set of all smooth vector fields on the manifold ''M'' into an (infinite-dimensional) Lie algebra. The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems., nonholonomic systems; , feedback linearization. Definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (differential Topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Introduction In its most elementary form, the theorem addresses the problem of finding a maximal set of inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Fermion
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
