# local gauge symmetry

(In mathematical terms, the theory involves a fiber bundle in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) A {\displaystyle A(x)} ↦ ℏ A A quantity which is gauge-invariant (i.e., invariant under gauge transformations) is the Wilson loop, which is defined over any closed path, γ, as follows: where χ is the character of a complex representation ρ and A gauge symmetry is analogous to how we can describe something within one language through different words (synonyms). Technically, they reduce to the computations of certain correlation functions in the vacuum state. In 1994, Edward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry that enabled the calculation of certain topological invariants (the Seiberg–Witten invariants). The Standard Model of particle physics consists of Yang-Mills Theories. Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity. δ V with the local gauge symmetries in Yang–Mills theory. {\displaystyle V\mapsto V+C} , Invariance of this term under gauge transformations is a particular case of a priori classical (geometrical) symmetry. Often, the term local symmetry is associated[why?] , and so the components of ψ ( = {\displaystyle \mu } After a simple calculation we can see that the gauge field A(x) must transform as follows, The gauge field is an element of the Lie algebra, and can therefore be expanded as. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents. ( The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory. In fact, a result in general gauge theory shows that affine representations (i.e., affine modules) of the gauge transformations can be classified as sections of a jet bundle satisfying certain properties. = ( local gauge transformation), which means that the representation of the symmetry group is a function of the manifold and can thus be taken to act differently on different points of spacetime. μ , to the Lagrangian. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model. This is seen to preserve the Lagrangian, since the derivative of If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory. ( Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. This is because the electric field relates to changes in the potential from one point in space to another, and the constant C would cancel out when subtracting to find the change in potential.  (Ronald Shaw, working under Abdus Salam, independently introduced the same notion in his doctoral thesis.) After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified gauge by replacing the scale factor with a complex quantity and turned the scale transformation into a change of phase, which is a U(1) gauge symmetry. For these local symmetries, one can apply a local transformation (resp. X where D is the covariant derivative. If there is a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a principal homogeneous space of the group of gauge transformations. However, to make this interaction physical and not completely arbitrary, the mediator A(x) needs to propagate in space. The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field. is an element of the vector space spanned by the generators The simplest such group is U(1), which appears in the modern formulation of quantum electrodynamics (QED) via its use of complex numbers. ν ∂ Φ , then {\displaystyle \mathbf {A} } , The complete Lagrangian for the gauge theory is now, As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of perturbation theory by introducing additional fields (the Faddeev–Popov ghosts) and counterterms motivated by anomaly cancellation, in an approach known as BRST quantization. a non-trivial relation among the coordinates and their canonical momenta. {\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)} One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. In terms of vector calculus, the electric field is the gradient of the potential, There are therefore as many gauge fields as there are generators of the Lie algebra. , by. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. , = Gauge theories are usually discussed in the language of differential geometry. 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. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. stands for the wedge product. − As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. Equations have a gauge boson known as internal symmetries in addition to spacetime symmetries, nuclear and energy. 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