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Topological Yang–Mills Theory
In gauge theory, topological Yang–Mills theory, also known as the theta term or \theta-term is a gauge-invariant term which can be added to the action (physics), action for four-dimensional quantum field theory, field theories, first introduced by Edward Witten. It does not change the classical equations of motion, and its effects are only seen at the quantum level, having important consequences for CPT symmetry. Action Spacetime and field content The most common setting is on four-dimensional, flat spacetime (Minkowski space). As a gauge theory, the theory has a gauge symmetry under the action of a gauge group, a Lie group G, with associated Lie algebra \mathfrak through the usual Lie group-Lie algebra correspondence, correspondence. The field content is the gauge field A_\mu, also known in geometry as the principal connection, connection. It is a 1-form valued in a Lie algebra \mathfrak. Action In this setting the theta term action is S_\theta = \frac\int d^4x \, \tex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
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 under local transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian is invariant under these transformations. 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 th ... [...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\Omeg ... [...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 Multilinear form, 1-form \mathbf over it, we can define a family of Multilinear form, ''p''-forms: In one dimension, the Chern–Simons Multilinear form, 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 \w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras. History and name The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term ''"Killing form"'' first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''"Cartan-Killing form"''. At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \mathrm(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups allow many group-theoretic problems to be reduced to problems in linear algebra. In physics, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the autom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta-angle
In quantum field theory, the theta vacuum is the semi-classical vacuum state of non- abelian Yang–Mills theories specified by the vacuum angle ''θ'' that arises when the state is written as a superposition of an infinite set of topologically distinct vacuum states. The dynamical effects of the vacuum are captured in the Lagrangian formalism through the presence of a ''θ''-term which in quantum chromodynamics leads to the fine tuning problem known as the strong CP problem. It was discovered in 1976 by Curtis Callan, Roger Dashen, and David Gross, and independently by Roman Jackiw and Claudio Rebbi. Yang–Mills vacuum Topological vacua The semi-classical vacuum structure of non-abelian Yang–Mills theories is often investigated in Euclidean spacetime in some fixed gauge such as the temporal gauge A_0 = 0. Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations A_i = i\Omega \nabla_i \Omega^, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients \tbinom nk = \tbinom. The naturalness of the star operator means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Form
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Tensor
In mathematics and theoretical physics, a tensor is antisymmetric or alternating on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ''covariant'' or all ''contravariant''. For example, T_ = -T_ = T_ = -T_ = T_ = -T_ holds when the tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of ''each'' pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order k may be referred to as a differential k-form, and a completely antisymmetric contravariant tensor field may be referred to as a k-vector field. Antisymmetric and symmetric tensors A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. For a general tensor U with comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gluon Field Strength Tensor
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks. The strong interaction is one of the fundamental interactions of nature, and the quantum field theory (QFT) to describe it is called ''quantum chromodynamics'' (QCD). Quarks interact with each other by the strong force due to their color charge, mediated by gluons. Gluons themselves possess color charge and can mutually interact. The gluon field strength tensor is a rank 2 tensor field on the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for necessary definitions). Convention Throughout this article, Latin indices (typically ) take values 1, 2, ..., 8 for the eight gluon color charges, while Greek indices (typically ) take values 0 for timelike components and 1, 2, 3 for spacelike components of four-vectors and four-dimensional spacetime tensors. In all equations, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
