Commutative (Abelian) Gauge Transformations

These are associated with U(1) gauge symmetry, which is the gauge symmetry of quantum electrodynamics (QED).

In an Abelian gauge transformation, the gauge fields and wavefunctions transform as follows:

$$\psi(x) \rightarrow \psi'(x) = e^{i\lambda(x)}\psi(x)$$

A_{\mu}(x) \rightarrow A'_{\mu}(x) = A_{\mu}(x) + \partial_{\mu} \lambda(x)

Here, $\psi(x)$ is the wavefunction of the system, $A_\mu(x)$ is the gauge field (the photon field in the case of QED), and $\lambda(x)$ is the gauge parameter, which can vary from point to point in spacetime.

Non-commutative (Non-Abelian) Gauge Transformations

These are associated with non-commutative gauge groups like SU(2) and SU(3), which describe the weak and strong nuclear forces, respectively.

In a non-commutative gauge transformation, the gauge fields and wavefunctions transform as follows:

$$\psi(x) \rightarrow \psi'(x) = U(x)\psi(x)$$

A_\mu(x) \rightarrow A'_\mu(x) = U(x)A_\mu(x)U^{-1}(x) - \frac{i}{g}(\partial_\mu U(x))U^{-1}(x)

Here, $\psi(x)$ is again the wavefunction of the system, $A_\mu(x)$ is the gauge field, $U(x)$ is a unitary operator in the group space that depends on the gauge parameters, and $g$ is the coupling constant of the gauge theory.

$$\psi(x) \rightarrow \psi'(x) = \psi(y(x))$$

Here, $g_{\mu\nu}(x)$ is the metric tensor that describes the geometry of spacetime, $\psi(x)$ is a matter field, and $y(x)$ is the coordinate transformation.

General Coordinate Transformations

These are associated with general relativity, where the gauge group is the group of all diffeomorphisms (smooth invertible mappings) of spacetime onto itself.

In a general coordinate transformation, the metric tensor and matter fields transform as follows:

g_{\mu\nu}(x) \rightarrow g'_{\mu\nu}(x) = \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\sigma}{\partial x'^\nu} g_{\rho\sigma}(y(x))

$$\psi(x) \rightarrow \psi'(x) = \psi(y(x))$$

Here, $g_{\mu\nu}(x)$ is the metric tensor that describes the geometry of spacetime, $\psi(x)$ is a matter field, and $y(x)$ is the coordinate transformation.