Gauge theory

In <a href="/facts/Physics/a7aH2y08">physics</a>, a gauge theory is a type of <a href="/facts/Field_theory_(physics)/FJdAKeRA">field theory</a> in which the <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a>, and hence the dynamics of the system itself, does not change under <a href="/facts/Local_symmetry/gJmgsooI">local transformations</a> according to certain smooth families of operations (<a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>). Formally, the Lagrangian is <a href="/facts/Invariant_(physics)/UfHcehYs">invariant</a> under these transformations. 
The term "gauge" refers to any specific mathematical formalism to regulate redundant <a href="/facts/Degrees_of_freedom_(physics_and_chemistry)/Rrjl0lnq">degrees of freedom</a> in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> or the gauge group of the theory. Associated with any Lie group is the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of <a href="/facts/Group_generator/B9q5lnFY">group generators</a>. For each group generator there necessarily arises a corresponding <a href="/facts/Field_(physics)/FJdAKeRA">field</a> (usually a <a href="/facts/Vector_field/ccFNuPPp">vector field</a>) 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 <a href="/facts/Canonical_quantization/AKHX2JQn">quantized</a>, the <a href="/facts/Quantum/4x5QTxGB">quanta</a> of the gauge fields are called <a href="/facts/Gauge_boson/TEEpAwxJ">gauge bosons</a>. If the symmetry group is non-commutative, then the gauge theory is referred to as <a href="/facts/Non-Abelian_group/KQYHrwJl">non-abelian</a> gauge theory, the usual example being the <a href="/facts/Yang%E2%80%93Mills_theory/m3g0C7hB">Yang–Mills theory</a>.
Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> in the <a href="/facts/Spacetime/uRIN1x4b">spacetime</a> in which the physical processes occur, they are said to have a <a href="/facts/Global_symmetry/gJmgsooI">global symmetry</a>. <a href="/facts/Local_symmetry/gJmgsooI">Local symmetry</a>, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).
Gauge theories are important as the successful field theories explaining the dynamics of <a href="/facts/Elementary_particles/CTFDM97U">elementary particles</a>. <a href="/facts/Quantum_electrodynamics/zPteAr6G">Quantum electrodynamics</a> is an <a href="/facts/Abelian_group/FfWwmx4R">abelian</a> gauge theory with the symmetry group <a href="/facts/Unitary_group/lEvudIhU">U(1)</a> and has one gauge field, the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a>, with the <a href="/facts/Photon/mFNhAEzA">photon</a> being the gauge boson. The <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a> is a non-abelian gauge theory with the symmetry group U(1) × <a href="/facts/Special_unitary_group/Qgy2orcu">SU(2)</a> × <a href="/facts/Special_unitary_group/Qgy2orcu">SU(3)</a> and has a total of twelve gauge bosons: the <a href="/facts/Photon/mFNhAEzA">photon</a>, three <a href="/facts/Weak_boson/gAncY5XV">weak bosons</a> and eight <a href="/facts/Gluons/t6YH8I21">gluons</a>.
Gauge theories are also important in explaining <a href="/facts/Gravitation/bqAN7DdQ">gravitation</a> in the theory of <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>. Its case is somewhat unusual in that the gauge field is a <a href="/facts/Tensor/pNjFo5tV">tensor</a>, the <a href="/facts/Lanczos_tensor/oR2eKBa1">Lanczos tensor</a>. Theories of <a href="/facts/Quantum_gravity/t7whBMNE">quantum gravity</a>, beginning with <a href="/facts/Gauge_gravitation_theory/wwRu94cD">gauge gravitation theory</a>, also postulate the existence of a gauge boson known as the <a href="/facts/Graviton/MVJUBjek">graviton</a>. Gauge symmetries can be viewed as analogues of the <a href="/facts/Principle_of_general_covariance/KzTG2t9g">principle of general covariance</a> of general relativity in which the coordinate system can be chosen freely under arbitrary <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, <a href="/facts/Gauge_theory_gravity/76H1kBKV">gauge theory gravity</a>, replaces the principle of general covariance with a true gauge principle with new gauge fields.
Historically, these ideas were first stated in the context of <a href="/facts/Classical_electromagnetism/vK0YI6dj">classical electromagnetism</a> and later in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>. However, the modern importance of gauge symmetries appeared first in the <a href="/facts/Relativistic_quantum_mechanics/DhkDNnHf">relativistic quantum mechanics</a> of <a href="/facts/Electron/L0KBo5cW">electrons</a> – <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a>, elaborated on below. Today, gauge theories are useful in <a href="/facts/Condensed_matter_physics/EBVI5nmL">condensed matter</a>, <a href="/facts/Nuclear_physics/fzQbt1aF">nuclear</a> and <a href="/facts/High_energy_physics/YgIuyj2t">high energy physics</a> among other subfields.

Gauge theory open-in-new

Gauge theory