# The solution of spin-1 massive field equation

I am looking at chapter XI of Luke's QFT notes, where he has stated that the field satisfies the KG equation \begin{equation} (\Box+m^2)A_\nu=0 \end{equation} And then he proceeds to define \begin{equation} A_\nu=\sum_{r=1}^3\int\frac{d^3k}{(2\pi)^{3/2}\sqrt{2w_k}}\left(a_k^r\epsilon_k^re^{-ikx}+a_k^{r ^\dagger}\epsilon^{r*}_k e^{ikx}\right) \end{equation}

For whatever reason, I am havng trouble seeing how this satisfies the above equation. I thought that we were also under the condition that $\partial_\nu A^\nu=0\rightarrow \Box A_\nu=0$ Which means that m=0 as well, but it is massive.

• This seems incorrect: $\partial_\nu A^\nu=0\rightarrow \Box A_\nu=0$. Oct 10, 2016 at 20:15
• @J.Pak well isnt $\Box =\partial_\nu \partial^\nu$ so $\partial_\nu (0)$ =0 Oct 10, 2016 at 22:23
• This is just the gauge condition $\partial_{\mu} A^{\mu} = 0$ , which has been used to obtained the modified Proca equation, i.e. the spin-1 massive field equation mentioned in your question. Please read the answers. Oct 11, 2016 at 12:02
• @yankeefan11 no, because $\Box A_{\nu} = \partial_{\mu} \partial^{\mu} A_{\nu}$. $\nu$ doesn't get contracted, so your logic is incorrect. Oct 11, 2016 at 13:02

To answer your first question: the expression that you have written pretty much satisfies the equations of motion, provided that

$$k ^{\mu} = \left\{ \sqrt{\vec{k}^{\,2} + m^2};\; \vec{k} \right\}.$$

I advice you to carry out the proof.

To answer your second question: no, we only use that condition to gauge-fix a theory with a gauge redundancy, and the theory under consideration is not gauge-invariant, because the mass term is not gauge-invariant.

UPDATE: actually I am mistaken, the Lorentz condition $\partial_{\mu} A^{\mu} = 0$ is used when deriving the equation of motion for the massive field. But nevertheless, the gauge invariance is gone. Thanks to J. Pak for the correction.

• In terms of the proof, I see that $\Box A$ gives a $-k^0^2+\vec{k}^2$ So $-k^0^2+k^2+m^2=0$ or k^2+m^2=k^0^2. Thanks. Oct 10, 2016 at 20:11
• @Solenodon Paradoxus, You are welcome. Oct 10, 2016 at 20:39

About your second question, I should correct the answer of Solenodon Paradoxus. Your spin-1 massive field equation is in fact the Proca equation which has been obtained by gauge condition: $\partial_\nu A^\nu=0$ . This equation is just closely related to the KG equation. Please see the Proca equation for detail.