In what follows, I'm following Folland's book and Reed & Simon.
Notation: Points in $\mathbb{R}^{4}$ are denoted by $p =(p_{0},p_{1},p_{2},p_{3})$. Also, I'm using Reed & Simon's notation for the Lorentz inner product: $$p \cdot \tilde{x} = p_{0}x_{0} -p_{1}x_{1} - p_{2}x_{2} - p_{3}x_{3}$$
The mass hyperboloids $X_{m}^{\pm}$ are defined by: $$X_{m}^{+} := \{p \in \mathbb{R}^{4}: \hspace{0.1cm} \mbox{$p\cdot \tilde{p} = m^{2}$ and $p_{0} > 0$}\} \quad \mbox{and} \quad X_{m}^{-} := \{p \in \mathbb{R}^{4}: \hspace{0.1cm} \mbox{$p\cdot \tilde{p} = m^{2}$ and $p_{0} < 0$}\}$$ Also, $\omega_{{\bf{p}}} := \sqrt{m^{2}+|{\bf{p}}|^{2}}$ and $\lambda \equiv \lambda_{\omega_{{\bf{p}}}} = \frac{1}{(2\pi)^{3}}\frac{d{\bf{p}}}{\omega_{{\bf{p}}}}$ is the Lorentz invariant measure on $X_{m}^{+}$.
Quantization of Klein-Gordon
The quantization of Klein-Gordon is done explicitly by Folland. We take the Hilbert space $\mathscr{H} = L^{2}(X_{m}^{+},\lambda)$ and define the Segal quantization operator: $$\mathscr{H} \ni f \mapsto \Phi_{S}(f): F_{0} \to F_{0}$$ where $F_{0}$ is the subspace of the Fock space $\mathcal{F}_{\text{sim}}(\mathscr{H})$ composed by only a finite number of particles and:
$$\Phi_{S}(f) := \frac{1}{\sqrt{2}}(\alpha(f)+\alpha^{\dagger}(f))$$ where I denoted by $\alpha$ and $\alpha^{\dagger}$ the annihilation and creation operators on $F_{0}$, respectively. Now, if $f \in \mathscr{S}(\mathbb{R}^{4})$, let $\mathcal{F}f$ be its Fourier transform and: $$Ef := \mathcal{F}f\bigg{|}_{X_{m}^{+}}$$ The free field is then defined to be $\varphi(f) = \Phi_{S}(Ef)$ if $f \in \mathscr{S}(\mathbb{R}^{4})$ is real-valued and $\varphi(f) = \varphi(\operatorname{Re}f) + i \varphi(\operatorname{Im}f)$ otherwise. To obtain the usual expressions of the free field, we take $J: L^{2}(X_{m}^{+},\lambda) \to L^{2}(\mathbb{R}^{3})$ to be a unitary transformation and extend it to a unitary operator $\Gamma(J): \mathcal{F}_{\text{sim}}(\mathscr{H}) \to \mathcal{F}_{\text{sim}}(L^{2}(\mathbb{R}^{3})$ in a natural way. Then, we identify $\alpha$ and $\alpha^{\dagger}$ to new operators on $L^{2}(\mathbb{R}^{3})$ by: $$a(f) := \Gamma(J)\alpha(J^{-1}f)\Gamma(J)^{-1} \quad \mbox{and} \quad a^{\dagger}(f) := \Gamma(J)\alpha^{\dagger}(J^{-1}f)\Gamma(J)^{-1}$$ and analogously for the free field: $$\phi(f) := \Gamma(J)\varphi(f)\Gamma(J)^{-1}.$$ With these tools, every expression found in physics books can be derived.
The Dirac Field
Folland says that the construction for the Dirac field follows in an analogous fashion. Only this time, the creation and annihilation operators are supposed to be defined on a tensor product $\mathcal{F}_{1}\otimes \mathcal{F}_{2}$ of two copies of the Fock space associated to the state space of a single Dirac particle.
Question: Is it enough to simply replace $\mathscr{H} = L^{2}(X_{m}^{+},\lambda; \mathbb{C}^{4})$ (now the functions take values on $\mathbb{C}^{4}$) and adapt the above defintions fo $\mathcal{F}_{1}\otimes \mathcal{F}_{2}$? I'm not sure if this is the right way to formalize it, specially because the answer of my previous question mentions that now $\mathscr{H}$ should be a Direct sum of orthogonal spaces. Maybe this is just another approach, equivalent to consider the tensor product $\mathcal{F}_{1}\otimes \mathcal{F}_{2}$? I'd be glad if someone could give me some details.
