- #1
Frank Castle
- 579
- 23
The mantra in theoretical physics is that global gauge transformations are physical symmetries of a theory, whereas local gauge transformations are simply redundancies (representing redundant degrees of freedom (dof)) of a theory.
My question is, what distinguishes them (other than being global and local - independent and dependent on spacetime coordinates)? In particular, why is one (global gauge symmetry) considered to be a physical symmetry with an associated Noether current, whereas the other (local gauge symmetry) is not?
For example, the archetypal gauge theory is classical electromagnetism: under a local gauge transformation the 4-potential ##A^{\mu}(x)## transforms as $$A^{\mu}(x)\rightarrow A'^{\mu}(x)=A^{\mu}(x)+\partial^{\mu}\Lambda(x)$$ This, however, leaves the physically measurable field strength ##F^{\mu\nu}(x)## unchanged, i.e. $$F^{\mu\nu}(x)\rightarrow F'^{\mu\nu}(x)=F^{\mu\nu}(x)$$ As I understand it, this is what is meant by a local gauge symmetry being a redundancy in the description of the physical theory, since it simply reflects the fact that the physically unmeasurable quantities used to describe the theory, in this case ##A^{\mu}##, are not uniquely defined. They are introduced as a convenience to describe the theory, and their non-uniqueness is a relic of how we use them to represent physical quantities, and not due to some physical symmetry of the system. (Another example would be in GR - coordinates are introduced to describe physical quantities locally (i.e. project a tensor onto a particular coordinate basis), however, such coordinates are not uniquely defined, one can choose any set of admissable coordinates to describe the same physical quantity).
When it comes to a global symmetry, however, I'm not sure at all how to argue why this is a physical symmetry as opposed to simply a redundancy in the description of the physical theory?!
My question is, what distinguishes them (other than being global and local - independent and dependent on spacetime coordinates)? In particular, why is one (global gauge symmetry) considered to be a physical symmetry with an associated Noether current, whereas the other (local gauge symmetry) is not?
For example, the archetypal gauge theory is classical electromagnetism: under a local gauge transformation the 4-potential ##A^{\mu}(x)## transforms as $$A^{\mu}(x)\rightarrow A'^{\mu}(x)=A^{\mu}(x)+\partial^{\mu}\Lambda(x)$$ This, however, leaves the physically measurable field strength ##F^{\mu\nu}(x)## unchanged, i.e. $$F^{\mu\nu}(x)\rightarrow F'^{\mu\nu}(x)=F^{\mu\nu}(x)$$ As I understand it, this is what is meant by a local gauge symmetry being a redundancy in the description of the physical theory, since it simply reflects the fact that the physically unmeasurable quantities used to describe the theory, in this case ##A^{\mu}##, are not uniquely defined. They are introduced as a convenience to describe the theory, and their non-uniqueness is a relic of how we use them to represent physical quantities, and not due to some physical symmetry of the system. (Another example would be in GR - coordinates are introduced to describe physical quantities locally (i.e. project a tensor onto a particular coordinate basis), however, such coordinates are not uniquely defined, one can choose any set of admissable coordinates to describe the same physical quantity).
When it comes to a global symmetry, however, I'm not sure at all how to argue why this is a physical symmetry as opposed to simply a redundancy in the description of the physical theory?!