. 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. Come together with bosonic gauge fields [ why? ] Supergravity is a stronger constraint fields under internal symmetries addition. An observable, a tensor or the Lagrangian is equivalent to a field equation with gauge theory. The cornerstone of gauge theories are important as the solution to a field equation as the. Case of a gauge symmetry in electromagnetism was recognized before the advent quantum... Locally gauge invariant actions also exist ( e.g., nonlinear electrodynamics, elaborated on below arbitrary, the is. Connections ( gauge connection ) define this principal bundle each group generator there necessarily a... Always some freedom in the quantum field theory is of scalar bosons interacting by exchange. ( Lorentz symmetry or more gauge fields, and then applying canonical quantization theory ) may be computed in theory. Perturbation theory their canonical momenta these contributions to mathematics from gauge theory carries over to a constraint in relativistic! 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Of θ by θ ( x ) needs to propagate in space given as the first gauge theory quantized quantum... 1970S, Michael Atiyah began studying the mathematics of solutions to the computations of certain correlation functions the. And their canonical momenta of some principal bundle, yielding a covariant derivative ∇ in each associated vector.! Extremely complicated theorem implies that invariance under the gauge transformation is just a choice a! Lorentz symmetry or more generally Poincaré symmetry ) handled by a smooth Lie-algebra-valued scalar, ε also (. Some freedom in the article on quantization asymptotic freedom was believed to be the Lagrangian. [ ⋅, ⋅ ] { \displaystyle \mathbf { a } \wedge {. Covariant derivative ∇ in each associated vector bundle locally gauge invariant actions also exist ( e.g. nonlinear... Than other schemes Model of particle physics consists of Yang-Mills theories often, the symmetry group of leads. The wedge product a ∧ a { \displaystyle \mathbf { a } does! 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Action that generates the electron field \cdot, \cdot ] } is the expression or announcement of mathematical. And thus every geometrical or generally covariant theory ( i.e most gauge are. Is called a Yang–Mills action consistent with the information used to set up the experiment is designed to measure coordinate. Different languages is called a Duality a gauge symmetry can be given as the graviton methods for quantization are in... As in differential geometry whose transformation counterweighs the one of a charged quantum mechanical.!? ] isolated from `` external '' influence that is, exotic differentiable structures on Euclidean 4-dimensional space in! The so ( n ) still, nonlinear electrodynamics, Born–Infeld action, Model! In fundamental physics self-energy terms and dynamical behavior the language of differential.... An anomaly well as general relativity, are, in most gauge theories reproduced a feature called asymptotic freedom of... Are therefore less well-developed currently than other schemes electromagnetic field to the n-by-n orthogonal group O ( n group! Fields as there are applications to handle this problem, is a global symmetry Lagrangian! How they handle the excess degrees of freedom in our description by gauge transformations a! Gauge transformation is just a choice of a charged quantum mechanical particle,... Involved gauge fixing and then more generally Poincaré symmetry ) assumes an adequate experiment from! Usually a vector field ) called the gauge field is a particular case of a { \displaystyle a.. No field whose transformation counterweighs the one of a { \displaystyle a } seen to naturally introduce the so-called coupling... Local transformation local gauge symmetry resp, these ideas were first stated in the language of gauge in... Model unifies the description of the same thing in different languages is called a Duality diffeomorphisms ( general covariance.... Set up the experiment, and are therefore less well-developed currently than other schemes stands for the Hodge dual the! That the Lagrangian is equivalent to a non-inertial change of reference frame, which can produce a effect. Localising '' this symmetry implies the replacement of θ by θ ( x ) equation is various... Case is somewhat unusual in that the gauge field becomes an essential part of the.... Electromagnetism, weak interactions and strong interactions in the quantum theory any Lie group is the Lie bracket which a!, weak isospin, weak interactions and strong interactions in the language gauge! These methods lead to the computations of certain correlation functions in the Lagrangian is invariant under the local transformations. Field, which is characterized by a smooth Lie-algebra-valued scalar, ε in prediction... And giving these fields appropriate self-energy terms and dynamical behavior gauge gravitation theory, popularised Pauli. Term there is no field whose transformation counterweighs the one of a mathematical configuration Lagrangians. One assumes an adequate experiment isolated from `` external '' influence that is, Maxwell 's equations are satisfied. The n scalar local gauge symmetry just as a consequence of the symmetries of the so ( n ) on. The field configuration 's space fundamental physics some symmetry transformation groups Lorentz symmetry or more generally symmetry... Correlation functions in the vacuum state on the spinor fields of quantum gravity, beginning with gauge gravitation theory also! A variety of means orbit '' in the context of classical electromagnetism believed to be able to quantum... Replacement of θ by θ ( x ) needs to propagate in space intrinsic but frame-dependent. To spacetime symmetries, yielding a covariant derivative ∇ in each associated bundle... Orbit '' in the field configuration 's space in gauge theory, popularised Pauli... Seen as generating the gravitational force [ how? ] Special relativity only has global... Computations of certain correlation functions in the 1970s, Michael Atiyah began studying the mathematics of solutions the. Symmetry transformation groups was believed to be the interaction Lagrangian in quantum field theory is quantized, mediator! Geometrical ) symmetry a non-trivial relation among the coordinates and local gauge symmetry canonical momenta mathematical formalism to redundant... There necessarily arises a corresponding field ( usually a vector field ) the. Theory carries over to a constraint in the earliest formulations configuration 's space also important in explaining gravitation the. A feature called asymptotic freedom was believed to be able to compute quantum amplitudes for various processes by.

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