# What does $\partial_{\mu}$ mean?

I've stumbled across the following notation a couple times reading physics articles on wikipedia: $$\partial_{\mu}$$ But what does it mean? They don't clarify. Source: https://en.wikipedia.org/wiki/Standard_Model

• And what does that mean? The $x_{\mu}$? May 8, 2017 at 15:47
• It might be helpful to note your level of education/acquaintance with calculus and linear algebra. The depth and content of an answer depends on your current knowledge. In this case, if you're familiar with linear algebra and multivariable calculus, $x_\mu$ denotes the $\mu$-th space(-time) coordinate, and $\partial_\mu$ denotes partial differentiation wrt $x_\mu$. Usually the summation convention also appiles. May 8, 2017 at 16:11

In four dimensional space-time (Minkowski space) a position vector is denoted as $x^\mu$ where the Lorentz index $\mu$ refers to its four components $\mu\in\{0,1,2,3\}$. Lorentz indices are "contracted" as $x\cdot y\equiv x^\mu y_\mu=x^\mu\eta_{\mu\nu}y^\nu$, where $\eta_{\mu\nu}$ is the Minkowski metric. When we want to take a partial derivative on Minkowski space, we usually abbreviate it as

$$\partial_\mu=\frac{\partial}{\partial x^\mu}$$

Note that the free index on the abbreviation $\partial_\mu$ is downstairs. Which means, if it acts on a scalar Lorentz invariant quantity, it should act on an $x^\mu$ with the index upstairs within a contraction, to leave behind a quantity that matches its initial index position. For example:

$$\partial_\mu y\cdot x=\partial_\mu y_\nu x^\nu= y_\nu \partial_\mu x^\nu = y_\nu \delta_\mu^{~~~\nu}=y_\mu$$

where $\delta_\mu^{~~~\nu}$ is a Kronecker delta.

Also keep in mind the Einstein summation convention: Whenever the same index appears twice in an expression, it means that this index is summed over its entire range. E.g. $x_\mu y^\mu\equiv \sum_{\mu=0}^3x_\mu y^\mu$.

$\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$ or $(\frac{\partial}{\partial t},\nabla)$, $x^{\mu}=(t,x,y,z)$ see here