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I'm in an introductory particle physics class, and in performing manipulations on the Dirac equation, my instructor expands the $\gamma^\mu \partial_\mu$ term as:

$$\gamma^\mu \partial_\mu = \gamma^0 \frac{\partial}{\partial t} + \vec{\gamma}\cdot \vec{\nabla}$$

where $\vec{\gamma} = (\gamma^1,\gamma^2,\gamma^3)$, but to my knowledge,

$$\gamma^\mu \partial_\mu = \gamma^\mu \eta_{\mu \nu} \partial^\nu = \gamma^0\frac{\partial}{\partial t} - \vec{\gamma} \cdot \vec{\nabla}$$

using the convention $\eta_{\mu \nu}=\operatorname{diag}(+,-,-,-)$. Am I missing something here?

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    $\begingroup$ you a forgeting an extra minus sign on the $\partial^\nu = (\frac{\partial}{\partial t}, - \nabla)$ $\endgroup$ – Hydro Guy May 24 '13 at 21:15
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Yes. You are missing the fact that he is using the convention $$ \nabla = (\partial_1, \partial_2, \partial_3) $$ as opposed to $$ \nabla = (\partial^1, \partial^2, \partial^3) $$ The first convention is by far the most common in my experience.

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  • $\begingroup$ I see. Now that I think about it, we didn't actually discuss the difference between $\partial_\mu$ and $\partial^\mu$. Is the only difference a minus sign in the spatial part? $\endgroup$ – Gino May 24 '13 at 21:22
  • $\begingroup$ Yes that's the difference in this case. This is a special case of a more general notational phenomenon used in lots od contexts called "raising and lowering indices" with a metric. $\endgroup$ – joshphysics May 24 '13 at 21:26
  • $\begingroup$ Well, it is not only a convention. Lower indices represent covectors, and the derivative is naturally a covector. $\endgroup$ – Peter Kravchuk May 24 '13 at 21:32
  • $\begingroup$ @PeterKravchuk Well, sure; it's a well-motivated convention that is consistent with other notational conventions used for vectors and covectors, but in this context the distinction is largely artificial anyway because of the tangent-cotangent isomorphism on semi-Riemannian manifolds. Also, besides all of this highbrow talk, at some point you have to choose a definition for the symbol $\nabla$, and you might think one definition is more natural than another, but you certainly could have picked the other one if you really wanted to. $\endgroup$ – joshphysics May 24 '13 at 21:43
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    $\begingroup$ @Gino en.wikipedia.org/wiki/Raising_and_lowering_indices That should answer most of your questions. Also see idv.sinica.edu.tw/ftliang/diff_geom*diff_geometry(II)/3.04/cotangent_tangent.pdf and en.wikipedia.org/wiki/Killing_form (search for the phrase "raise and lower"). Also, consult any book on GR such as the first couple chapters of preposterousuniverse.com/grnotes $\endgroup$ – joshphysics May 24 '13 at 22:03

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