Disclaimer Some of the writing below is taken directly from [1]. I do so under fair use for pedagogic purposes.
Postulate [First Postulate of Special Relativity]
The laws of physics take the same form in all inertial frames of reference [0].
Definition [Lorentz covariant quantity, Lorentz covariant equation]
- A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. ...[T]hese quantities are built out of scalars, four-vectors, four-tensors, and spinors...
- An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities.... The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame.... This condition is a requirement according to the principle of relativity; i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference. [2]
As shown in [1] and [3], in a frame of reference the homogeneous wave equation can be written in the form
$$
i\,\hbar\,\gamma^\mu\,\frac{\partial\boldsymbol{\psi} }{\partial x^\mu} - m\,c\,\boldsymbol{\psi} = \mathbf{0}.\tag{1}
$$
The $\gamma$ matrices satisfy the Clifford algebra
$$
\gamma^\mu\,\gamma^\nu + \gamma^\nu\,\gamma^\mu = 2\,g^{\mu\nu}\,I,
$$
where $g$ is the metric tensor of special relativity and $I$ is the identity element.
The most general homogeneous Lorentz transformation between two coordinate systems may be written
$$
{x'}^\mu = {a^\mu}_\nu\,x^\nu. \tag{2}
$$
From Equation (2), we write
$$
\frac{\partial}{\partial x^\mu} = \frac{\partial {x'}^\lambda}{\partial x^\mu}\,\frac{\partial}{\partial {x'}^\lambda} =
{a^\lambda}_\mu\,\frac{\partial}{\partial {x'}^\lambda}.\tag{3}
$$
Using Equation (3), we obtain from Equation (1) that
$$
i\,\hbar\,{a^\lambda}_\mu\,\gamma^\mu\,\frac{\partial \boldsymbol{\psi} }{\partial {x'}^\lambda} - m\,c\,\boldsymbol{\psi} = \mathbf{0}.\tag{4}
$$
In analogy with Equation (2), we define
$$
{\gamma'}^\lambda \equiv {a^\lambda}_\mu\,\gamma^\mu. \tag{5}
$$
One can verify with Equation (2) that
the $\gamma'$ matrices satisfy the same Clifford algebra as the $\gamma$ matrices. In other words,
$$
{\gamma'}^\mu\,{\gamma'}^\nu + {\gamma'}^\nu\,{\gamma'}^\mu = 2\,g^{\mu\nu}\,I.
$$
By Pauli's Fundamental Theorem (see [4] for a statement of the theorem and [1] for a proof of the theorem), up to a multiplicative constant there exists only one change-of-basis matrix $S$ such that for each and every $\lambda\in \left\{0,1,2,3\right\}$ we have that
$$
{\gamma'}^\lambda = S^{-1}\,\gamma^\lambda\,S.
\tag{6}
$$
Upon substituting this relation into Equation (4), multiplying on the left by $S$, and changing the dummy index $\lambda$ to $\mu$, we find that
$$
i\,\hbar\,\gamma^\mu\,\frac{\partial S\,\boldsymbol{\psi} }{\partial {x'}^\mu} - m\,c\,S\,\boldsymbol{\psi} = \mathbf{0}.\tag{7}
$$
Defining the Lorentz transformation law for spinors to be
$$
\boldsymbol{\psi}' \equiv S\,\boldsymbol{\psi} ,
$$
we have from Equation (7) that
$$
i\,\hbar\,\gamma^\mu\,\frac{\partial \boldsymbol{\psi}' }{\partial {x'}^\mu} - m\,c\,\boldsymbol{\psi}' = \mathbf{0}.\tag{8}
$$
What we have found is that Equation (1) and Equation (8) have the same form upon changing the dependent variables and independent variables from $\boldsymbol{\psi}=\boldsymbol{\psi}(\mathbf{x})$ to $\boldsymbol{\psi}' = \boldsymbol{\psi}'(\mathbf{x}')$ via
\begin{align}
\boldsymbol{\psi}'(\mathbf{x}') = S\,\boldsymbol{\psi} (\mathbf{x})
\quad\text{and}\quad
{x'}^\mu = {a^\mu}_\nu\,x^\nu .
\end{align}
This finding indicates that the equation of motion given by Dirac's equation conforms to the first postulate of special relativity. Further, all of the quantities in the equation are Lorentz covariant quantities. Hence, the Dirac equation is a Lorentz covariant equation.
As an addendum, it is important to note the following. The physical consequences of the Dirac equation are independent of the frame of reference used to derive them. "Since the trace of a matrix is insensitive to a similarity transformation [see Equation (6)], traces of products of gamma matrices do not depend on the specific representation used. Therefore, physical results determined by the Dirac theory should be expressed in terms of these traces [2]."
Bibliography
[0] https://en.wikipedia.org/wiki/Postulates_of_special_relativity
[1] Bethe and Jackiw, Intermediate Quantum Mechanics, pp. 349-366.
[2] https://en.wikipedia.org/wiki/Lorentz_covariance
[3] Dirac, The Quantum Theory of the Electron
[4] https://math.stackexchange.com/q/5061118
