According to multiple online sources, antimatter was discovered through the Dirac equation because there were multiple solutions; a positive energy solution, to be expected and a negative energy solution. What does this mean. Also I read something about this giving rise to a symmetry called CPT symmetry, however that is not the main focus of this question.
I would just like an explanation of what negative energy is and why it results in antimatter.
Answer
In brief :
From relativity, the Dirac equation gives the following relation : \begin{equation}\tag{1} E^2 = p^2 c^2 + m_0^2 \, c^4, \end{equation} which is a second order algebraic equation, with two roots : \begin{equation}\tag{2} E = \pm \, \sqrt{p^2 c^2 + m_0^2 \, c^4}. \end{equation} Now, quantum mechanics requires that all solutions be considered, since any superposition of solutions is another solution to the linear Dirac equation. Thus, you have negative solutions to consider. You can't just throw them away just because they don't have "physical" sense to you. Now, the Dirac equation admits an operation (complex conjugate and a matrix multiplication) which can convert a negative solution to a positive solution traveling in the opposite direction, and reversed spin and electric charge. This operation gives another solution of the Dirac equation : \begin{equation}\tag{3} \psi_{\text{c}}(t, x) = \gamma^2 \, \psi^{\ast}(t, x). \end{equation} This suggest that the negative solution may be interpreted as an "anti-particle", i.e. one with a reversed spin and electric charge.
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