I'm sorry if this is not the right place to ask this question but I've been struggling with this for days now (and I think this is too technical/specific for math stack).

Notation: A conjugation $C$ on a Hilbert space $\mathscr{H}$ is a anti-linear isometry satisfying $C^{2} = I$, the identity map. Let $\mathscr{H}_{C} := \{f \in \mathscr{H}: \hspace{0.1cm} Cf = f\}$. Finally, $\Phi_{s}(\cdot)$ denotes the Segal quantization operator.

On section X.7 on Reed & Simon, vol 2. there is a nice discussion on the rigorous quantization of the free Klein-Gordon field. At some point, the authors introduce the following definition: if $f \in \mathscr{H}_{C}$ then $f \mapsto \varphi(f) := \Phi_{s}(f)$ is called the canonical free field over $(\mathscr{H},C)$ and $f \mapsto \pi(f) := \Phi_{s}(if)$ is called the canonical conjugate momentum.

Then, they especialize to the case where $\mathscr{H} := L^{2}(H_{m},d\Omega_{m})$ where $H_{m}$ is the mass shell $H_{m}$ and $\Omega_{m}$ is associate Lorentz invariant measure. A conjugation $C$ on $\mathscr{S}(\mathbb{R}^{4})$ is defined by: $$(Cf)(p_{0},{\bf{p}}) := \overline{f(p_{0},-{\bf{p}})}$$

In order to define the time-zero field, the authors then introduce the following maps. For each $f \in L^{2}(H_{m},d\Omega_{m})$, let $Ef := \sqrt{2\pi}\hat{f}|_{H_{m}}$ where $\hat{f}$ is the Fourier transform: $$\hat{f}(p) = \frac{1}{(2\pi)^{2}}\int e^{ip\cdot \tilde{x}}f(x) dx $$ with $p\cdot x := p_{0}x_{0}-p_{1}x_{1}-p_{2}x_{2}-p_{3}x_{3}$ is the Lorentz inner product. Then if $f\in \mathscr{S}(\mathbb{R}^{4})$ is real-valued, one defines: $$\varphi_{m}(f) := \varphi(Ef) \quad \mbox{and} \quad \pi_{m}(f) := \pi(\mu Ef) \quad \mu:= \sqrt{m^{2}+|{\bf{p}}|^{2}} $$ and extends to all $\mathscr{S}(\mathbb{R}^{4})$ by linearity.

According to the authors, $\varphi_{m}(\cdot)$ and $\pi_{m}(\cdot)$ are given, in terms of the creation and annihilation operators, by: \begin{eqnarray} \varphi_{m}(f) = \frac{1}{\sqrt{2}}\{ (a^{\dagger}(Ef) + a(CEf)\} \quad \mbox{and} \quad \pi_{m}(f) = \frac{i}{\sqrt{2}}\{a^{\dagger}(\mu Ef) - a(C\mu E f)\} \tag{1}\label{1} \end{eqnarray}

Here are my questions concerning the above. First, if $f \in \mathscr{S}(\mathbb{R}^{4})$, it does not follow that $Ef \in \mathscr{H}_{C}$, so I don't really get the defintions of $\varphi_{m}(\cdot)$ and $\pi_{m}(\cdot)$. One can argue that this is just notation for defining $\varphi_{m}(f) = \Phi_{s}(Ef)$ and $\pi_{m}(f) = \Phi_{s}(i\mu Ef)$, but then how to explain the $C$ factor inside each $a(\cdot)$ in (\ref{1})? It seems to me that these factors are there because we should account for the anti-linearity of $a(\cdot)$ by taking the conjugation $C$, but (as far as I understand) this expression only holds if $Ef \in \mathscr{H}_{C}$, so I'm getting really confused with these definitions. I know these are probably trivial questions, but I'm really stuck there for days and what comes next in the book depends on all these.

I'm sorry if this is not the right place to ask this question but I've been struggling with this for days now (and I think this is too technical/specific for math stack).

Notation: A conjugation $C$ on a Hilbert space $\mathscr{H}$ is a anti-linear isometry satisfying $C^{2} = I$, the identity map. Let $\mathscr{H}_{C} := \{f \in \mathscr{H}: \hspace{0.1cm} Cf = f\}$. Finally, $\Phi_{s}(\cdot)$ denotes the Segal quantization operator.

On section X.7 on Reed & Simon, vol 2. there is a nice discussion on the rigorous quantization of the free Klein-Gordon field. At some point, the authors introduce the following definition: if $f \in \mathscr{H}_{C}$ then $f \mapsto \varphi(f) := \Phi_{s}(f)$ is called the canonical free field over $(\mathscr{H},C)$ and $f \mapsto \pi(f) := \Phi_{s}(if)$ is called the canonical conjugate momentum.

Then, they especialize to the case where $\mathscr{H} := L^{2}(H_{m},d\Omega_{m})$ where $H_{m}$ is the mass shell $H_{m}$ and $\Omega_{m}$ is associate Lorentz invariant measure. A conjugation $C$ on $\mathscr{S}(\mathbb{R}^{4})$ is defined by: $$(Cf)(p_{0},{\bf{p}}) := \overline{f(p_{0},-{\bf{p}})}$$

In order to define the time-zero field, the authors then introduce the following maps. For each $f \in \mathscr{S}(\mathbb{R}^{4})$, let $Ef := \sqrt{2\pi}\hat{f}|_{H_{m}}$ where $\hat{f}$ is the Fourier transform: $$\hat{f}(p) = \frac{1}{(2\pi)^{2}}\int e^{ip\cdot \tilde{x}}f(x) dx $$ with $p\cdot x := p_{0}x_{0}-p_{1}x_{1}-p_{2}x_{2}-p_{3}x_{3}$ is the Lorentz inner product. Then if $f\in \mathscr{S}(\mathbb{R}^{4})$ is real-valued, one defines: $$\varphi_{m}(f) := \varphi(Ef) \quad \mbox{and} \quad \pi_{m}(f) := \pi(\mu Ef) \quad \mu:= \sqrt{m^{2}+|{\bf{p}}|^{2}} $$ and extends to all $\mathscr{S}(\mathbb{R}^{4})$ by linearity.

According to the authors, $\varphi_{m}(\cdot)$ and $\pi_{m}(\cdot)$ are given, in terms of the creation and annihilation operators, by: \begin{eqnarray} \varphi_{m}(f) = \frac{1}{\sqrt{2}}\{ (a^{\dagger}(Ef) + a(CEf)\} \quad \mbox{and} \quad \pi_{m}(f) = \frac{i}{\sqrt{2}}\{a^{\dagger}(\mu Ef) - a(C\mu E f)\} \tag{1}\label{1} \end{eqnarray}

Here are my questions concerning the above. First, if $f \in \mathscr{S}(\mathbb{R}^{4})$, it does not follow that $Ef \in \mathscr{H}_{C}$, so I don't really get the defintions of $\varphi_{m}(\cdot)$ and $\pi_{m}(\cdot)$. One can argue that this is just notation for defining $\varphi_{m}(f) = \Phi_{s}(Ef)$ and $\pi_{m}(f) = \Phi_{s}(i\mu Ef)$, but then how to explain the $C$ factor inside each $a(\cdot)$ in (\ref{1})? It seems to me that these factors are there because we should account for the anti-linearity of $a(\cdot)$ by taking the conjugation $C$, but (as far as I understand) this expression only holds if $Ef \in \mathscr{H}_{C}$, so I'm getting really confused with these definitions. I know these are probably trivial questions, but I'm really stuck there for days and what comes next in the book depends on all these.