As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion about it. So let me start with its definition:
The Dirac equation is given by, $$ i \hbar \gamma^\mu \partial_\mu \psi = m c\cdot \psi $$ where the Dirac matrices $\gamma^\mu$ are defined by $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = \eta^{\mu\nu}$ and where $\psi$ is a "solution".
The first deal of confusion already starts with the $\psi$'s. It seems that people freely see them as spinor valued functions or as "operator fields".
But if I understand this correctly, seeing them as operators, is not part of the original picture, but was later added as the so called second quantization. Right?
Now my question is the following: Why do we need this second quantization of the Dirac equation? What experiments can not be described by the original Dirac equation? Maybe there is a list somewhere or such?
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