Bottom ScienceThe Dirac equation is a relativistic wave equation that describes the behavior of a spin-1/2 particle, such as an electron. The equation is a first-order partial differential equation, which means that it involves the field and its first derivative. The equation is given by:
$\left( i \gamma^\mu \partial_\mu – m \right) \psi(x) = 0$
where $\gamma^\mu$ are the gamma matrices, $\partial_\mu$ is the partial derivative with respect to $x^\mu$, $m$ is the mass of the particle, and $\psi(x)$ is the wave function of the particle.
Complete Derivation
The Dirac equation can be derived by starting with the action for a free spin-1/2 particle, given by:
$S = \int d^4x \left[ \overline{\psi}(x) \left( i \gamma^\mu \partial_\mu – m \right) \psi(x) \right]$
where $\overline{\psi}(x) = \psi^\dagger(x) \gamma^0$ is the Dirac conjugate of $\psi(x)$.
The equation of motion can be obtained by taking the functional derivative of the action with respect to the wave function, and setting it equal to zero:
$\frac{\delta S}{\delta \psi} = 0$
After performing some algebraic manipulation, we get the Dirac equation.
The functional derivative of the action with respect to $\psi$ is given by:
$\frac{\delta S}{\delta \psi} = \int d^4x \left[ \frac{\delta \overline{\psi}(x)}{\delta \psi} \left( i \gamma^\mu \partial_\mu – m \right) \psi(x) + \overline{\psi}(x) \frac{\delta}{\delta \psi} \left( i \gamma^\mu \partial_\mu – m \right) \psi(x) \right]$
Using the properties of the gamma matrices and the partial derivative, we can simplify the above equation to:
$\frac{\delta S}{\delta \psi} = \int d^4x \left[ \left( i \gamma^0 \right) \left( i \gamma^\mu \partial_\mu – m \right) \psi(x) \right]$
Now we can use the property of the dirac conjugate and obtain:
$\frac{\delta S}{\delta \psi} = \int d^4x \left[ \left( i \gamma^0 \right) \left( i \gamma^\mu \partial_\mu – m \right) \psi(x) \right] = 0$
Finally, we can set $\psi$ to the left side of the equation, and we get the Dirac equation:
$\left( i \gamma^\mu \partial_\mu – m \right) \psi(x) = 0$
This equation is the Dirac equation, which describes the behavior of a spin-1/2 particle in a relativistic setting.
It’s also worth noting that the Dirac equation is a 4-component equation, where each component corresponds to a different spin state of the particle.
This means that the wave function, $\psi(x)$, is a 4-component spinor, and the gamma matrices act on these components.
The derivation I have provided is a simplified version, in the sense that it assumes that the spinors are represented in a specific representation of the gamma matrices, and in a specific coordinate system.
