"In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, do not change under local transformations according to certain smooth families of operations (Lie groups)."
Expands the scope of QFT to include gauge fields and symmetry transformations, including the importance of the U(1), SU(2) and SU(3) groups.
Classical gauge theory: This is the classical formulation of gauge theory, which is a mathematical and physical framework for describing the interactions between elementary particles in terms of fields.
Quantum field theory: This is a branch of theoretical physics that studies the behavior of fields and particles in the framework of quantum mechanics.
Group theory: This is the study of groups, which are mathematical structures that describe symmetries of objects and systems.
Lie groups: These are a type of group that have a (continuous) manifold structure, which means they can be described by continuous parameters.
Quantum mechanics: This is the branch of physics that studies the behavior of matter and energy at the microscopic level.
Electromagnetism: This is the branch of physics that studies the properties and interactions of charged particles and electromagnetic fields.
Yang-Mills theory: This is a type of gauge theory that describes the strong interaction between elementary particles based on the exchange of gauge bosons.
Field strength: This is a mathematical quantity that describes the strength and direction of a gauge field.
Covariant derivatives: These are mathematical operators that allow one to take derivatives of fields in a way that preserves the covariance of the gauge theory.
Spacetime: This is the mathematical framework that describes the structure of the universe in terms of the three-dimensional space and one-dimensional time.
Topology: This is the study of the properties of spaces that are preserved under continuous transformations.
Fiber bundles: These are mathematical objects that describe how a space varies over another space, such as a gauge field varying over spacetime.
Quantization: This is the process of taking a classical theory and making it into a quantum theory.
Path integrals: These are a method of computing the probabilities of quantum events by summing over all possible paths in spacetime.
Renormalization: This is a mathematical technique for removing divergences that arise in quantum field theories.
Grand unification: This is the idea that all the fundamental forces of nature can be described by a single unified theory.
Supersymmetry: This is a symmetry that relates particles with integer spin to particles with half-integer spin, and vice versa.
String theory: This is a theoretical framework that attempts to unify all of the forces of nature by describing matter and energy as different vibrations of one-dimensional strings.
Loop quantum gravity: This is a candidate theory of quantum gravity that attempts to reconcile general relativity with quantum mechanics by quantizing the geometry of spacetime.
Quantum chromodynamics: This is the study of the strong force that binds quarks together to form protons, neutrons, and other hadrons.
Abelian gauge theory: In this type of gauge theory, the gauge group is Abelian, which means that the gauge transformations commute with each other.
Non-Abelian gauge theory: In this type of gauge theory, the gauge group is non-Abelian, which means that the gauge transformations do not commute with each other.
Electroweak theory: This theory unifies the electromagnetic and weak forces by introducing the W and Z bosons.
Quantum chromodynamics (QCD): This theory describes the strong nuclear force, which is responsible for holding quarks together inside protons and neutrons.
Grand unified theory (GUT): This theory attempts to unify all the fundamental forces into a single theory.
Supersymmetric gauge theory: This theory introduces supersymmetry, a symmetry that relates particles with different spins.
Topological gauge theory: This theory studies the topological properties of gauge theories, such as knot invariants.
Lattice gauge theory: This theory studies gauge theory on a discrete lattice of points, which allows numerical simulations to be performed.
"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."
"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 the gauge fields are called gauge bosons."
"A global symmetry is just a local symmetry whose group's parameters are fixed in spacetime."
"Gauge theories are important as the successful field theories explaining the dynamics of elementary particles."
"Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson."
"The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons, and eight gluons."
"Gauge theories are also important in explaining gravitation in the theory of general relativity."
"Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor."
"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."
"Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system."
"An alternative theory of gravitation, gauge theory gravity, 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 classical electromagnetism and later in general relativity."
"Today, gauge theories are useful in condensed matter, nuclear, and high energy physics among other subfields." Apologies, but I can only provide a maximum of 10 responses at once.