GaugeGroup

In the context of gauge theories in physics, a gauge group is the group of all possible gauge transformations that can be applied to a system without changing its physical predictions. Gauge groups play a fundamental role in the formulation of theories like quantum electrodynamics (QED), quantum chromodynamics (QCD), and the Standard Model of particle physics.

Formally, a gauge group $G$ is a Lie group. In the simplest case of U(1) gauge symmetry in QED, the gauge group is the group of all complex numbers of unit magnitude, which forms a circle in the complex plane and is isomorphic to the group of all rotations in a plane.

In the case of an Abelian U(1) gauge transformation, each transformation can be characterized by a phase factor $e^{i\lambda(x)}$ where $\lambda(x)$ is a real-valued function that can vary from point to point in spacetime. The set of all such phase factors, for all possible $\lambda(x)$, forms the gauge group of the theory.

Mathematically, the U(1) group is described as:

$$U(1) = {e^{i\lambda} | \lambda \in \mathbb{R}}$$

where $i$ is the imaginary unit, and $\lambda$ is a real number.

For non-Abelian gauge groups such as SU(2) and SU(3), which describe the weak and strong nuclear forces respectively, the gauge transformations are represented by special unitary matrices. The set of all such matrices forms the gauge group.

For example, the SU(2) group consists of all 2x2 unitary matrices with determinant 1, which can be written as:

$$SU(2) = \left\{U \in \mathbb{C}^{2\times2} | UU^\dagger = I, \det(U) = 1\right\}$$

where $U^\dagger$ is the Hermitian conjugate of $U$, and $I$ is the identity matrix.

Similarly, the SU(3) group consists of all 3x3 unitary matrices with determinant 1.

In general, a gauge group for a particular gauge theory is a group of continuous transformations parameterized by a set of functions from the spacetime manifold to the group of interest. These functions are chosen in such a way that they leave the action of the theory invariant.