The Dirac Equation

2024-07-28  ยท Prakash Gautam

Klein Gordan Equation

We have seen in this post that Klein-Gordan equation describes a spinless particle with mass \(m\). The Klein-Gordan equation formally is

\[ \begin{aligned}\left( \square + m^{2} \right) \psi = 0\end{aligned} \]

Unlike Schrödinger's equation, this equation doesn't treat time differently from space derivatives. It can be shown that this in indeed Lorentz invariant. However, the problem with Klein-Gordan equation is that it violates the conservation of probability, which is a serious problem. We need a first order differential equation which would indeed conserve the probability.

Dirac Equation

This dynamic equation also doesn't take into account the spin of particles, electron for example. Paul Dirac attempted to obtain a first order equation which would conserve probability and also take into consideration the spin of the particle. The first line of attack is to start from Schrödinger's equation and add an extra paramter in the wae function which would correspond to different spins. Taking a wave function solution

\[ \begin{aligned}\psi(x,a)\end{aligned} \]

where the different values of \(a\) would correspond to different spin of the particle.

we could try to factorize the equation as

\[ \begin{aligned}\left( \delta p^\mu + \chi m \right) \left( p_\mu \alpha - \beta m \right) = \left(\square - m^{2} \right)\end{aligned} \]

Like we did in quantum harmonic oscillator. We could multiply through this expression and see what should we have for the coefficients \(\alpha\) and \(\beta\). Working out the product on the left side we get

\[ \begin{aligned}\delta p_{\mu} p^{\mu} \alpha + \chi m p_\mu \alpha - \delta \beta m p^\mu -\chi \beta m^{2} = \square - m^{2}\end{aligned} \]

Since there are no cross terms in the RHS of the above expression we necessarily eed to have

\[ \begin{aligned}\alpha \beta - \beta \alpha = 0\end{aligned} \]

Such situation is referred to as being commuting and is usually written as

\[ \begin{aligned}\left[ A, B \right] = AB - BA\end{aligned} \]

Thus in this notation \(\left[ \alpha, \beta \right] = 0\)

\[ \begin{aligned}\left( i \gamma^\mu \partial_\mu - m \right) \psi = 0\end{aligned} \]

As in tensor notation the expressions with a raised and a repeated lower index contract. Similarly the expression like \(\gamma^\mu \partial_\mu\) contract and Feynman suggested a notation to use for these kinds of expression as

\[ \begin{aligned}\gamma^\mu \partial_\mu \equiv \partial \llap{/}\end{aligned} \]

whith the use of this notation we can write the dirac equation in the form.

\[ \begin{aligned}\left( i \partial \llap{/} - m \right) \psi = 0\end{aligned} \]

Which is our glorius Dirac equation, which describes the particle dynamics of particle with spin.