# Interpreting plane waves solutions of the Dirac equation

The Dirac equation $$(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$$ has plane wave solutions of the form: $$\psi_{+,k}(t,x) = e^{-iE(k)t+ik\cdot x}u(k) \quad \mbox{and} \quad \psi_{-,k}(t,x) = e^{iE(k)t -ik\cdot x}v(-k)$$ where $$k, x \in \mathbb{R}^{3}$$.

Question: The above expressions are functions of $$t,x$$ for each fixed $$k$$. How do I interpret these solutions? Does it mean that the Dirac equation have solutions with constant well-defined momentum? Each such pair of solutions represent a fermion with constant momentum $$k$$?

## 2 Answers

The Dirac equation is linear, meaning that sums of solutions with arbitrary weights are also solutions.

You can take multiple plane waves with different $$k$$ and add them together.

Or you can take a distribution in $$k$$-space and integrate your plane wave over $$k$$-space weighted by that distribution.

I'd just like to say that, in physics and field theory, whenever you have a solution to a linear theory which is proportional to $$e^{-i k_\mu x^\mu}$$ (with maybe some constant n dimensional vector out front) where $$k^2 = m^2$$, you always interpret this quantum mechanically as a particle of four momentum $$k_\mu$$. This is justified when you go to quantum field theory, as you can build an oscillator in the mode of this solution with integer occupation numbers.