The Importance of Gauge Theories in Particle Physics

[ghery]

Hi:
What's a gauge theory?, Is it just some kind of theory invariant with respect to some transformation? (like electrodynamics where the potentials are not sigle valued) and what is the importance of gauge theories in particle physics?

Thanks

 

[humanino]

From the perspective of quantum mechanics, the gauge principle can be understood as the inobervability of the absolute phases of wavefunctions, so all phases can be shifted by a constant, and this can be done locally at every point in spacetime. The corresponding change in the derivative of the wavefunctions creates interactions with vector bosons (in the standard model).

 

[malawi_glenn]

We can not determine the phase in experiments, so each observer may choose his own phase = gauge.

And in math, global phase change:
\psi \rightarrow \psi ' = \psi*e^{i\theta}
where \theta is the phase.

local change:
\psi \rightarrow \psi ' = \psi*e^{i\theta (x)}
where x is a space-time coordinate (4 indicies)

If a formula is invariant under such local gauge transformation, you'll call it gauge invariant.

And as humanino pointed out, since you'll have derivatives in the Lagrangian for equation of motion, and the fact that derivatives usally don't commute with the functions which the operate on, you must impose that the derivative under such gauge transformation transforms as:

Derivative -> Derivative_prime = Derivative + Field

Where the field describes the interaction with the particle with so called Gauge bosons (which are vectors).

So that is what you must to to get the Lagrangian gauge invariant, find out how the derivative should transform.