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I have noticed the differential operator $\partial_0$ in a lot of equations whilst studying quantum field theory. I am used to the notation $\partial_x$ meaning $ \frac{d}{dx} \\\\ $ etc. but just a bit unsure about the zero in the subscript.

What does it explicitly mean?

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  • $\begingroup$ Time derivative? I suppose. The zeroth coordinate ($x^0$) is the "time" coordinate in relativity so... $\endgroup$
    – kali_the_dog
    Commented Oct 18, 2015 at 16:50
  • $\begingroup$ Yeah... because time was found to be a dimension so many years later after Riemann and all the differential geometry, physicists where like "Oh! shi... we forgot this dimension. Hey those index, from where do they start?... from 1?... Oh! good... we can use the 0" $\endgroup$
    – raul
    Commented Oct 18, 2015 at 18:35

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It is common to write $$ \partial_i = \frac{\partial}{\partial x^i}$$ for the derivative with respect to the $i$-th coordinate. Since time is customarily written as the $0$-th coordinate, $\partial_0$ is the time derivative.

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