I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator).
Let $\{ e_i \}$ be an orthonormal frame and $x^i$ its associated global inertial coordinates. The Dirac operator is then $$D = \sum_{i = 0}^3 \gamma(e_i) \frac{\partial}{\partial x^i}. $$ To satisfy $D^2 = \Delta$, we need that $$\gamma(e_i) \gamma(e_j) + \gamma(e_j) \gamma(e_i) = 2 \eta(e_i,e_j) \mathbb{I}$$ i.e. a matrix representation of $Cl(1,3)$. Since a complex representation of this algebra is the same as a complex representation of the complex Clifford algebra $\mathbb{C}l(4)$, which itself is isomorphic to $\mathrm{End}(\mathbb{C}^4)$, we are led to the gamma matrices and in fact told by the structure theorem of complex clifford algebras how to get them from the Pauli matrices (The isomorphism singles out a basis so somehow this will need to be reflecting in going to another reference frame). Simultaneously, we establish a Dirac spinor field $\Psi(x^i)$ as a map from Minkowski space to $\mathbb{C}^4$.
Now suppose another inertial frame wants to write down the Dirac operator. They would write $$D = \sum_{j = 0}^3 \gamma(\tilde{e_j}) \frac{\partial}{\partial \tilde{x}^j}.$$ Now by the linearity of the Clifford map $\gamma: T_p M \to Cl(T_p M)$, we have $$\gamma(\tilde{e_{j}}) = \Lambda_{j}^i \gamma(e_i).$$ Since the spinor representation is just the identity, going to the level of representation (i.e. actual matrices), it must be that $$\tilde{\gamma^j} = \Lambda_{j}^i \gamma^i.$$ Now what I don't understand in the notes I'm following is that there must exist some spacetime dependent matrix $S(x^i)$ such that $$S \gamma^i S^{-1} = \tilde{\gamma^i}$$ and thus for relativistic invariance we must have $$\tilde{\Psi}(\tilde{x^i}) = S(x^i) \Psi(x^i)$$ and that we already know that the $S$ matrix must pointwise belong to $\mathrm{Spin}(1,3)$. Can someone help me finish my mathematical physics motivation for the Dirac spinor representation?
