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I've been wondering about this forever and I finally decided to ask on the forums. In relativistic index notation (with c= [itex]\hbar[/itex] =1) with the minkowski metric g_{[itex]\mu[/itex]}_{[itex]\nu[/itex]}=diag(1,-1,-1,-1), the 4-vector [itex]x^{\mu}=(t,x,y,z)=(x^0,\vec{x})[/itex], and with the del operator defined as [itex]\partial_{\mu}\equiv \frac{\partial}{\partial x^{\mu}}=(\partial_{t},\nabla)[/itex]. I should have that:

[itex]\partial^{\mu} x_{\mu}=\partial_{\mu} x^{\mu}=1[/itex]

but this is inconsistent with the way I usually think of vector calc because I should have

[itex]\partial_{\mu} x^{\mu}=\partial_{t}t+\nabla\bullet\vec{x}[/itex]

and

[itex]\partial_{t}t=1[/itex]

[itex]\nabla\bullet\vec{x}=3[/itex]

so, with the way I normally think, I should have:

[itex]\partial_{\mu} x^{\mu}=4[/itex]

Where am I going wrong here?

-Adam

P.S. all of this notation is straight out of Peskin and Schroeder's QFT text

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# Relativistic index notation del-operator

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