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i'm having trouble understanding the Proca equation


where $g_{\mu\nu}$ is the metric and $\mu$ is a parameter. Where is the wrong assumption?

$$(g_{\mu\nu}(\partial_{\mu}\partial^{\mu}+\mu^2)-\partial_{\mu}\partial_{\nu})\varphi^{\nu}=0$$ $$((\partial_{\mu}g_{\mu\nu}\partial^{\mu}+g_{\mu\nu}\mu^2)-\partial_{\mu}\partial_{\nu})\varphi^{\nu}=0$$ $$((\partial_{\mu}\partial_{\nu}+g_{\mu\nu}\mu^2)-\partial_{\mu}\partial_{\nu})\varphi^{\nu}=0$$

which implies $$\mu^2=0 $$ not the case. The problem is using $\Box =\partial_{\mu}\partial^{\mu}$ instead of other index, say $\alpha$?. In that case how can I show that

$$\partial_{\nu}\varphi^{\nu}=0 $$

Any hit will be appreciated.

-Edit- Reference Bjorken et.al Relativistic Quantum Fields page 23

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2 Answers

up vote 3 down vote accepted

Hint: Contract with derivative $\partial^{\mu}$ on both sides of your first equation (v1) to deduce that

$$\mu^2 \partial_{\nu}\varphi^{\nu}~=~0.$$

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$$\partial^{\mu}(g_{\mu\nu}(\Box+\mu^2)-\partial_{\mu}\partial_{\nu})\varphi^{\n‌​u}=0$$ $$\partial^{\mu}g_{\mu\nu}\Box\varphi^{\nu}+\partial^{\mu}g_{\mu\nu}\mu^2 \varphi^{\nu}-\partial^{\mu}\partial_{\mu}\partial_{\nu}\varphi^{\nu}=0$$ $$\partial_{\nu}\Box\varphi^{\nu}+\partial_{\nu}\mu^2 \varphi^{\nu}-\Box\partial_{\nu}\varphi^{\nu}=0 $$ Given that derivatives commute: $$\Box\partial_{\nu}\varphi^{\nu}+\mu^2\partial_{\nu} \varphi^{\nu}-\Box\partial_{\nu}\varphi^{\nu}=0 \Rightarrow \mu^2\partial_{\nu} \varphi^{\nu}=0 $$ –  Jorge Nov 22 '12 at 11:24
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I still don't have the reputation to comment everywhere so I leave this as an answer although it's a comment.

Be careful when defining the sum indexes. There are two things that you did wrong in your calculation.

  1. You said correctly that $\square=\partial_\mu \partial^\mu$ but those are indexes to express the sum so wrinting$$ g_{\mu\nu}\square=g_{\mu\nu}\partial_\mu \partial^\mu,$$ is wrong, you should write something like$$ g_{\mu\nu}\square=g_{\mu\nu}\partial_\rho \partial^\rho.$$

  2. The other thing you should be careful with is in the second step you put the metric between the partial derivatives but then you're are taking the partial derivative of the metric and that is not correct.

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About 1: yes, thanks, this is what I tried to mean with my $\alpha$ index About 2: There is no problem because the metric is constant, but thanks for the hint! –  Jorge Nov 22 '12 at 12:06
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