I Local Gauge Invariance Explained: Physics & Math Insight

[Silviu]

Hello! Can someone explain to me what exactly a local gauge invariance is? I am reading my first particle physics book and it seems that putting this local gauge invariance to different lagrangians you obtain most of the standard model. The math makes sense to me, I just don't see what is the physical meaning of this local gauge invariance and why would you come up with it? Like, it seems such an easy thing to do, just make the phase of wave function depends on position, but doing this you obtain everything up to Higgs field, and I am not sure I understand, physically, why. Thank you

 

[ohad]

Gauge invariance is about local symmetry of a field.
If you have any field and there is some global symmetry which is preserved by the lagrangian, than requiring this symmetry to be local (i.e. to be position dependent) usually breaks the symmetry, for that you need to introduce the gauge field to make the symmetry possible.
the best example is electrodynamics, you can see that by requiring the local symmetry of the fermion (i.e electron) field you must have the em field which is the gauge field - the photon.
So you got 'for free' the photon and the charge mechanism of the fermions just by requiring the local symmetry.

 

[ChrisVer]

well some fun way to try and imagine things...
With a global gauge symmetry: in each point of spacetime you have the same arc (for the phase degrees) that you are "rotating" your field... Then you can immediately see that you can perform a parallel transport on such a setup without anything (as the flat spacetime).
With a local gauge symmetry, each point gets a different phase that rotates the field... In order to perform a parallel transport, you introduce the gauge fields, as when in GR you made the metric spacetime dependent you introduced the Christoffel Symbols.
That's why the gauge fields can also be seen as connections, for they allow you to define a parallel transport (remember how the partial derivatives change to covariant derivatives).
Why local? because global would be photonless...

 

[my2cts]

Those are all indirect reasons for a fundamental requirement for a field theory.
Global symmetry enforces a corresponding conservation law.
There is no conservation law that requires gauge symmetry.

 

[atyy]

The local gauge principle is just a way of guessing the form of the interaction, also called "minimal coupling". In general relativity, the principle of equivalence" is a minimal coupling type of guess.

https://arxiv.org/abs/1305.0017

 

[vanhees71]

One reason for "local gauge invariance" is the realization of the proper orthochronous Poincare group in terms of massless vector fields. If you want to have a realization with only discrete intrinsic ("spin like") quantum numbers, which is in accordance with observation, you necessarily have to represent the null rotations of the corresponding "little group" trivially, and this leads to the necessity to represent the massless vector fields on a quotient space, i.e., to envoke local (Abelian) gauge invariance. That's how the gauge invariance of electrodyamics comes into view from a group theoretical approach to relativistic QT.

The extension of this insight to non-Abelian local gauge symmetry was an ingenious discovery of Yang and Mills. It's just a natural generalization of the local gauge principle to non-abelian gauge groups. Sometimes in the history of science the mathematically beautiful turns out to be of great utility in physics. That's for sure the case for non-Abelian gauge theories underlying the Standard Model of elementary particles which is more successful than wanted ;-)).

 

[Lapidus]

Hi Silviu,
gauge symmetry is arguably the central tenet of particle physics and qft. But in many older textbooks its physical meaning is only badly explained. Check out Schwartz QFT textbook chapter 8 for a deep yet readable explanation.

One of the best layment yet precise explanation of it can be found in Randall "Warped Passages" in the "Symmetry and Forces" chapter.