Yang Mills Theory - Part - 1

The usual Lagrangian used in our Field theory are,

The Klein-Gordon Lagrangian for scalar field - spin zero particles $$ L = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}\left(\frac{mc}{\hbar}\right)^2\phi^2$$ that gives the Klein Gordon equation, $$ \partial_\mu\partial^\mu\phi + \left(\frac{mc}{\hbar}\right)^2\phi = 0 $$

The Dirac Lagrangian for spinor fields -  spin half particles $$L = i\hbar{c}\tilde\psi\gamma^\mu\partial_\mu\psi - mc^2\tilde\psi\psi$$gives the dirac equations for $\psi$ and $\tilde\psi$ $$i\gamma^\mu\partial_\mu\psi- \frac{mc}{\hbar}\psi = 0$$$$ i\partial_\mu\tilde\psi\gamma^\mu+\frac{mc}{\hbar}\tilde\psi = 0 $$ And finally,

The Proca lagrangian for vector fields - spin one particles $$ L = \frac{-1}{16\pi}F_{\mu\nu}F^{\mu\nu} + \frac{1}{8\pi} \left(\frac{mc}{\hbar}\right)^2 A_\nu{A}^\nu $$ gives the vector field equation $$\partial_\mu{F}^{\mu\nu}+\left(\frac{mc}{\hbar}\right)^2A^\nu = 0 $$

These equations are not the derived ones, rather we take them as an abstract. You may ask how it can suddenly arise from nowhere?

It is more like the reverse engineering. We know the resultant Dirac equation and other field equations for the corresponding particles which is used to construct the appropriate Lagrangian for those fields. Once the Lagrangian is constructed, it is simple to solve the problem in a much formal way.

As far as concerned here, Yang Mills theory deals with the Local gauge invariance.

Let us take the lagrangian for spinor fields $$ L = ic\hbar\tilde\psi\gamma^\mu\partial_\mu\psi - mc^2\tilde\psi\psi $$   
For the transformation, $$ \psi' = e^{i\theta}\psi $$ where $\theta$ some constant, then we can notice that new Lagrangian, $$ L' = ic\hbar\tilde\psi{e}^{-i\theta}\gamma^\mu\partial_\mu\psi{e}^{i\theta} - mc^2 \tilde\psi{e}^{i\theta}e^{-i\theta}\psi $$ which is invariant since, $e^{i\theta}$ being the constant commutes with all other variables. 

But, it is not the same case if we considered functional dependence for $\theta$ by considering local gauge invariance, $$ \theta \,\,\rightarrow\,\,\theta(x)$$ then we have, $$ \partial_\mu\left(e^{i\theta(x)}\psi\right) = e^{i\theta(x)}\partial_\mu\psi + \left[\left(\partial_\mu{e}^{i\theta}\right)\psi = ie^{i\theta(x)}\left(\partial_\mu\theta(x)\right)\psi\right]$$ which gives an extra term$$L ' = L + i^2c\hbar\tilde\psi{e}^{-i\theta(x)}\gamma^\mu\partial_\mu\theta(x){e}^{i\theta(x)}\psi = L - c\hbar\partial_\mu\theta(x)\tilde\psi\gamma^\mu\psi$$ which is rewritten as $$ L' = L + q\tilde\psi\gamma^\mu\psi\left(\partial_\mu\lambda(x)\right)$$ where $$\lambda(x) = -\frac{c\hbar}{q}\theta(x) $$
 
We need to remember that, $\theta$ and its derivatives are considered to be a parameter which is not a matrix, and so the order doesn't matter whether you put it in front or back, wherelse the wavefunction and $\gamma$ matrices don't commute - that makes the order important. 

Now, a new Lagrangian can be constructed to be invariant under a local gauge transformation $$ \psi' = e^{-\frac{iq}{c\hbar}\lambda(x)}\psi$$ as, $$ L_{new} = L_{old} - q\tilde\psi\gamma^\mu\psi\partial_\mu\lambda(x)$$
But, this $\lambda(x)$ is not a constant i.e. not the same everywhere as the previous. It takes different values at different positions. Only after we know the transformation function, we can decide this extra term. Before the transformation, its values are not determined. Thus, we need a function that transforms as,$$ X' = X +\partial_\mu\lambda $$ under the gauge transformation. 

So, far we just used the mathematical arguments involving the transformation rules. Now, we look for this new function X for which we already have a similar tool in our vector fields. $$ A_\mu' = A_\mu + \partial_\mu\lambda(x)$$ 
That is it! 

The New lagrangian is written as, $$ L_{new} = L_{old} - q\tilde\psi\gamma^\mu\psi{A}_\mu $$ 
But, it hasn't completed yet. The last term in our Lagrangian affects the field equations and so introduces its own free Lagrangian i.e.the Proca lagrangian without the mass term which would affect the invariance. 

From which, we arrive at our final lagrangian, $$ L = ic\hbar\tilde\psi\gamma^\mu\partial_mu\psi -  mc^2\tilde\psi\psi - q\tilde\psi\gamma^\mu\psi{A_\mu} - \frac{1}{16\pi}F_{\mu\nu}F^{\mu\nu}$$
 
Things can be written in a compact way with the introduction of covariant derivatives defined as, $$ D_\mu = \partial_\mu + i\frac{q}{c\hbar}A_\mu$$ It is just a notation, not anything more. 



Yang Mills theory applies this same local gauge invariance to spinor fields that involves - SU(2) group.
Let us start with two spinor fields $\psi_1,\psi_2$ and the lagrangian for the combined system is written as, (without interaction term), $$L = ic\hbar\tilde\psi_1\gamma^\mu\partial_\mu\psi_1-m_1c^2\tilde\psi_1\psi_1 +ic\hbar\tilde\psi_2\gamma^\mu\partial_\mu\psi_2 - m_2c^2\tilde\psi_2\psi_2 $$ The combined Lagrangian in Matrix form, $$ L = ic\hbar\tilde\psi\gamma^\mu\partial_\mu\psi - c^2\tilde\psi{M}\psi$$ where M is the diagonal matrix involving mass $m_1, m_2$ If we assume it to be equal, we can get rid of it by simple constant "m".
The matrix notation is $$ \psi = \left(\begin{matrix}\psi_1\\\psi_2\end{matrix}\right)$$
We will see how to implement the local gauge invariance in this SU(2) group in the next post.

Reference book: Introduction to Elementary Particle Physics - Griffiths