What are $\partial_t$ and $\partial^\mu$?

Question

I'm reading the Wikipedia page for the Dirac equation:

$\rho=\phi^*\phi\,$

......

$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$

with the conservation of probability current and density following from the Schrödinger equation:

$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0.$

The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace by probability density by the symmetrically formed expression

$\rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).$

which now becomes the 4th component of a space-time vector, and the entire 4-current density has the relativistically covariant expression

$J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)$

1. What exactly are $\partial_t$ and $\partial^\mu$?

2. Are they tensors?

3. If they are, how are they defined?

Answer 1

$\partial_t\equiv\frac\partial{\partial t}$ and $\partial^\mu\equiv g^{\mu\nu}\frac\partial{\partial x^\nu}=\left(\sum_{\nu=0}^3g^{\mu\nu}\frac\partial{\partial x^\nu}\right)_{\mu=0}^3$ are differential operators. $\partial^\mu$ is formally contravariant (upper index) and obeys the corresponding transformation laws. $\partial_t$ has a lower index and is (up to a constant factor) a component of the formally covariant operator $\partial_\mu$ via $\partial_0=\frac1c\partial_t$, which, in general, is not equal to $\partial^0$, the zeroth component of $\partial^\mu$.

The differential operator $\partial^\mu$ is known as gradient, which derives vector fields from potential functions. The gradient is not a natural operation on arbitrary manifolds and only available if there's a metric. Its dual $\partial_\mu\equiv\frac\partial{\partial x^\mu}$ on the other hand is a natural operation corresponding to the differential $\mathrm d$, taking potentials to 1-forms (covectorfields).

As a side note, $\partial_t$ can also be understood as a local vector field, as one of the intrinsic definitions of vectors on manifolds is via their directional derivatives. In mathematical literature, it is common to write the basis of the tangent space as $\{\frac\partial{\partial x^\mu}\}$ and its dual space as $\{\mathrm dx^\mu\}$.

Answer 2

Depending on notation used, $\partial^\mu$ could be a tensor, referring to the four-vector $(\partial^0,\partial^1,\partial^2,\partial^3)$. However, in the context of the above equation, you can treat both $\partial_t$ and $\partial^\mu$ as scalars referring to the partial differentiation by $x^t$ and $x_\mu$ correspondingly.

The exact definition of $\partial_t$ is not uniform in literature, sometimes it is set to $\partial_0$, otherwise to $\frac{1}{c}\partial_0$. However, with $c = 1$, this rarely matters.

Note furthermore that time is either assigned the zeroth or fourth component of space-time.