Thanks!Demystifier said:No, in general they are just covariant. For electrodynamics the equations of motion are invariant under gauge transformations, but for Yang-Mills theories they are just covariant. Similarly, the Einstein-Hilbert action is invariant under general coordinate transformations, but the Einstein equation of motion is just covariant.
Gauge transformations are changes made to certain fields in a physical theory that do not alter the observable quantities. They are symmetries of the system, often associated with the redundancy in the description of the physical state. For example, in electromagnetism, gauge transformations involve changes to the vector potential that leave the electric and magnetic fields unchanged.
For equations of motion to be invariant under gauge transformations, they must remain unchanged when the fields in the theory are transformed according to the gauge symmetry. This means that the physical predictions of the theory are not affected by the gauge transformation, ensuring that the redundancy introduced by the gauge symmetry does not lead to different physical outcomes.
Gauge transformations can change the form of the Lagrangian, but the equations of motion derived from the Lagrangian should remain invariant. This is because the Lagrangian is constructed to be gauge-invariant, or it changes in a way that does not affect the resulting equations of motion. The principle of gauge invariance ensures that the physical laws described by the equations of motion are not dependent on the choice of gauge.
One classic example is electromagnetism, described by Maxwell's equations. The gauge transformations in this context involve adding the gradient of a scalar function to the vector potential. Despite this transformation, the electric and magnetic fields, which are the observable quantities, remain unchanged. Consequently, Maxwell's equations, which govern the behavior of these fields, are invariant under such gauge transformations.
Gauge invariance is a cornerstone of modern physics, particularly in the formulation of quantum field theories and the Standard Model of particle physics. It ensures that the physical predictions of a theory do not depend on arbitrary choices made in the mathematical description of the system. This invariance leads to the conservation laws and fundamental interactions observed in nature, providing a deep connection between symmetries and physical laws.