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For a massive vector four-field there are only three physical linearly-independent polarizations. For a field excitation at rest, these can be described by \begin{align} \epsilon^1_\mu \equiv \begin{pmatrix} 0 \\ 1 \\ 0 \\0 \end{pmatrix} \, ,\quad \epsilon^2_\mu \equiv \begin{pmatrix} 0 \\ 0 \\ 1\\0 \end{pmatrix} \, , \qquad \epsilon^3_\mu \equiv \begin{pmatrix} 0 \\ 0 \\ 0 \\1 \end{pmatrix} \, . \end{align} We can imagine that a field configuration proportional to $\epsilon^1_\mu$ corresponds to a situation in which a test object gets pushed by the field in the $x$-direction.

For a moving field excitation, e.g. with momentum $\vec p = p \vec{e}_z$, we need to act with the corresponding boost matrix on our polarization vectors and then find \begin{align} \tilde \epsilon^1_\mu \equiv \begin{pmatrix} 0 \\ 1 \\ 0 \\0 \end{pmatrix} \, ,\quad \tilde \epsilon^2_\mu \equiv \begin{pmatrix} 0 \\ 0 \\ 1\\0 \end{pmatrix} \, , \qquad \tilde \epsilon^3_\mu \equiv \begin{pmatrix} p/m \\ 0 \\ 0 \\ E_p/m \end{pmatrix} \, . \end{align} How can we interpret a field configuration that is proportional to $\epsilon^3_\mu$? What does it mean that a test object gets pushed by a field in the temporal direction?

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