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Jay-qu · 07-06-2009

Ok, I will give it a shot. (sorry for the ugly TeX )

The gauge principle can be is used to usher in three of the four fundamental forces (so far). Back in the early 20th century Herman Weyl discovered that General Relativity was invariant under a change in length scale, he noted this as an invariance wrt change of gauge (size). Eventually the term became generalised to cover all types of invariance under changes.

The gauge theories of the forces happen to be the manifestation of internal symmetries of our particles. These internal symmetries can be represented by Lie groups. The simplest case is the invariance of the wavefunction with respect to the Lie group U(1). Physically this amounts to the fact that the phase of the wavefunction is irrelevant. We shall see this necessitates the introduction of the electromagnetic field. Beginning with the Schrodinger equation for a particle in free space
$i\partial _t \Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi$
It is trivial to show that this equation will remain invariant under a global U(1) change of phase - $\Psi -> \Psi e^{i \theta}$ , where $\theta$ is a constant. This is because a U(1) transformation is just rotating the phase of the wavefunction and the phase does not effect the observed probability density $|\Psi|^2$ . But why stop at just global phase changes? What if we want the wavefunction to be invariant under an arbitrary transformation of the phase of the form
$\Psi -> \Psi e^{ie\theta(\vec{x},t)}$
When transforming the wavefunction by this arbitrary phase change the Schrodinger equation is changed. Let us now consider the Schrodinger equation in the presence of an electromagnetic field (in rationalised Gaussian units and $\hbar=c=1$
$-i\partial _t \Psi = [\frac{1}{2m}(\nabla + ie\vec{A})^2 - e\phi]\Psi,$

where the vector and scalar potentials ( $\vec{A}$ and $\phi$ ) are related to the electromagnetic fields via the following
$\vec{B}=\nabla \times \vec{A}$
$\vec{E}= -\nabla \phi-\partial_t\vec{A}$
Taking the Schrodinger equation and performing an arbitrary phase transformation yields a mess of terms that can be neatly factorised into (try this, it may take a page of algebra though

):
$-e^{ie\theta(\vec{x},t)}\partial _t \Psi = e^{ie\theta(\vec{x},t)}[\frac{1}{2m}(\nabla + ie(\vec{A}+\nabla \theta))^2-e(\phi-\partial _t \theta)]\Psi$
Cancelling the $e^{ie\theta(\vec{x},t)}$ terms we find that we can essentially summarise the changes as
$\vec{A} -> \vec{A}' = \vec{A}+\nabla \theta$
$\phi -> \phi ' = \phi-\partial_t \theta$
Taking these new potentials and substituting them into the equations that relate the potential to the E and B fields (above) will result in the electromagnetic fields being unchanged; and hence the physical situation is invariant under arbitrary phase (U(1)) transformations. The transformation of the form above is formally known as a gauge transformation. Here we have the freedom to choose a range of potentials that can be formed from any arbitrary choice of the function theta - and it will still describe the same physical situation. This is called gauge freedom.

In maintaining the gauge invariance of the Schrodinger equation we had to introduce the interaction terms for the EM field. This is known as the principle of minimal gauge invariance. You can then think of the interaction terms as been part of the derivative. In the relativistic formalism of Lagrangian dynamics the derivatives become gauge covariant derivatives
$D_{\mu} = \partial_{\mu} -ieA_{\mu}$
Here the gauge field $A_{\mu} = (\phi,\vec{A})$ .
In a similar way gauge theories of the strong and weak nuclear forces are able to be developed by using corresponding symmetries. These other internal symmetries are described by the lie groups SU(2) and SU(3) for the weak and strong force respectively.

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