Local Symmetry
In gauge theory, local symmetry refers to a symmetry transformation that can vary at each point in spacetime. Local symmetries play a crucial role in gauge theories as they allow for the consistent formulation of interactions between particles mediated by gauge fields.
To understand local symmetry, it's helpful to contrast it with global symmetry. In a global symmetry, the transformation is the same at all points in spacetime. For example, consider a global phase transformation in quantum mechanics, where a wavefunction is multiplied by a phase factor that is the same everywhere. This global phase transformation does not depend on the position in spacetime.
On the other hand, in a local symmetry, the transformation can vary at each point in spacetime. This means that the transformation parameter can be different at different spacetime locations. In gauge theories, local symmetries are associated with gauge fields that transform under these local symmetry transformations.
The presence of gauge fields introduces a compensation mechanism to maintain the gauge symmetry locally. When a gauge transformation is applied to the gauge fields, an additional term called the gauge transformation term is introduced to the transformation. This gauge transformation term ensures that the total transformation of the gauge fields cancels out the transformation of the matter fields, resulting in an overall gauge invariant theory.
The local symmetry in gauge theories is crucial for understanding the interactions between particles mediated by gauge fields. It allows for the consistent description of the forces associated with the gauge fields. The local symmetry transformations act on both the matter fields and the gauge fields, ensuring that the Lagrangian of the theory remains invariant under these transformations.
A well-known example of a local symmetry is electromagnetic gauge symmetry in quantum electrodynamics (QED). In QED, the local symmetry is associated with the electromagnetic gauge field, the photon. The electromagnetic gauge symmetry transformation involves a phase factor that can vary at each spacetime point. The electromagnetic interaction between charged particles is mediated by the exchange of photons, and the local symmetry ensures the gauge invariance of the theory.
Similarly, other gauge theories, such as the electroweak theory and quantum chromodynamics (QCD), also possess local symmetries associated with their respective gauge fields. The weak force is associated with the weak isospin and hypercharge gauge symmetries, while the strong force is associated with the color gauge symmetry in QCD.
In summary, local symmetry in gauge theory refers to a symmetry transformation that can vary at each point in spacetime. It is associated with gauge fields and plays a fundamental role in gauge theories. Local symmetries ensure the gauge invariance of the theory and provide the framework for understanding the interactions between particles mediated by gauge fields.
