. A This theory, known as the Standard Model, accurately describes experimental predictions regarding three of the four fundamental forces of nature, and is a gauge theory with the gauge group SU(3) × SU(2) × U(1). These kinds of symmetries, also known as internal symmetries, are distinguished from spacetime symmetries. The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between originally non-interacting fields. ε In the 1970s, Michael Atiyah began studying the mathematics of solutions to the classical Yang–Mills equations. In this case, the G matrices do not "pass through" the derivatives, when G = G(x), The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the Lagrangian. 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. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. with the local gauge symmetries in Yang–Mills theory. ( (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.) b This motivated searching for a strong force gauge theory. F Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. {\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)} ¯ Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. 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. does not vanish.). Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A, with, The general gauge transformations now become not just When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. = ∂ A , while the compensating transformation in Thus, in the abelian case, where 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. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory. Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x. ) The earliest field theory having a gauge symmetry was Maxwell's formulation, in 1864–65, of electrodynamics ("A Dynamical Theory of the Electromagnetic Field") which stated that any vector field whose curl vanishes—and can therefore normally be written as a gradient of a function—could be added to the vector potential without affecting the magnetic field. The formalism of gauge theory carries over to a general setting. {\displaystyle V\mapsto V+C} → D ( μ If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an internal symmetry. are the structure constants of the Lie algebra of the generators of the gauge group. For the mathematical field of gauge theory, see, Use of fiber bundles to describe local symmetries, The Yang–Mills Lagrangian for the gauge field, Learn how and when to remove this template message, A Dynamical Theory of the Electromagnetic Field, Standard Model (mathematical formulation), "Relativistic Field Theories of Elementary Particles", "Conservation of Isotopic Spin and Isotopic Gauge Invariance", "Self-dual connections and the topology of smooth 4-manifolds", "Gauge theory – Past, Present and Future", Mathematical formulation of the Standard Model, Numerical methods for ordinary differential equations, Numerical methods for partial differential equations, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Society for Industrial and Applied Mathematics, Japan Society for Industrial and Applied Mathematics, Société de Mathématiques Appliquées et Industrielles, International Council for Industrial and Applied Mathematics, https://en.wikipedia.org/w/index.php?title=Gauge_theory&oldid=990749065, Short description is different from Wikidata, Articles lacking in-text citations from September 2016, Wikipedia articles needing clarification from February 2014, Articles with unsourced statements from November 2014, Creative Commons Attribution-ShareAlike License. x {\displaystyle {\mathcal {P}}} F − There are many global symmetries (such as SU(2) of isospin symmetry) and local symmetries (like SU(2) of weak interactions) in particle physics. This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. {\displaystyle \wedge } A description of the same thing in different languages is called a Duality. {\displaystyle \Phi } Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or gauge transformation). {\displaystyle A_{\mu }^{a}} A gauge transformation is just a transformation between two such sections. {\displaystyle A} General relativity has a local symmetry of diffeomorphisms (general covariance). {\displaystyle \mu } Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. This can be seen as generating the gravitational force[how?]. Gauge theories may be quantized by specialization of methods which are applicable to any quantum field theory. A stands for the wedge product. When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. Mishandling gauge dependence calculations in boundary conditions is a frequent source of anomalies, and approaches to anomaly avoidance classifies gauge theories[clarification needed]. ) Among the most well known are: A pure gauge is the set of field configurations obtained by a gauge transformation on the null-field configuration, i.e., a gauge-transform of zero. The first methods developed for this involved gauge fixing and then applying canonical quantization. ] ( δ [1], Special relativity only has a global symmetry (Lorentz symmetry or more generally Poincaré symmetry). F where ) V μ x a Any gauge symmetry of the Lagrangian is equivalent to a constraint in the Hamiltonian formalism, i.e. are obtained from potentials Φ where f is any twice differentiable function that depends on position and time. This is evidently not an intrinsic but a frame-dependent quantity. = We know that the all observables are unchanged if we make a global change of the This is seen to preserve the Lagrangian, since the derivative of do not commute with one another. A represents the path-ordered operator. ∂ While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory. {\displaystyle \mathbf {A} } μ Gauge symmetry by definition is a local symmetry. x One nice thing is that if . {\displaystyle T^{a}} {\displaystyle {\mathcal {L}}_{\mathrm {gf} }} where D is the covariant derivative. That is dealt with in the next section by adding yet another term, Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents. The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. x Gauge theories are usually discussed in the language of differential geometry. a non-trivial relation among the coordinates and their canonical momenta. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a connection form (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field in BF theory. There are many global symmetries (such as SU(2) of isospin symmetry) and local symmetries (like SU(2) of weak interactions) in particle physics. One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. 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