Quantum Field Theory – How to Show ?/??(x) as a Representation of ?(x)†?

fermionshamiltonian-formalismhilbert-spaceoperatorsquantum-field-theory

I'm following the book of Brian Hatfield, Quantum Field Theory of particles and strings, page 217, eq. 10.89 and the following.
The author is looking for a representation of the operators $\Psi(x)$ and $\Psi(x)^{\dagger}$ that satisfies
$$\begin{align}
\{\Psi(x), \Psi(y)^{\dagger} \} = \delta(x-y).
\end{align}\tag{1}$$
The author then represents $\Psi(x)$ as a multiplication, acting on a functional $\Phi[\psi]$ by
\begin{align}
(\Psi(x) \Phi)[\psi] = \psi(x)\Phi[\psi].
\end{align}
From a previous chapter the author knows that for functionals on $a$-number (grassman) valued functions, $$\{\psi(x), \frac{\delta}{\delta \psi(y)} \} = \delta(x-y),$$ and thus he simply represents $\Psi(x)^{\dagger}$ as $\frac{\delta}{\delta \psi(x)}$.

Now what I want to show is that $\frac{\delta}{\delta \psi(x)}$ actually is the adjoint operator to $\psi(x)$. The book doesn't do this.

Namely, I want to show that if $\Phi_1$ and $\Phi_2$ are two abitrary functionals, then

\begin{align}
\langle \Psi(x) \Phi_1, \Phi_2 \rangle = \langle \Phi_1, \Psi(x)^{\dagger} \Phi_2(x)\rangle
\end{align}
still holds, if I use the mentioned representations. The first thing that I don't know is how to write down a scalar product for functionals – my most naive guess would be
\begin{align}
\langle \Psi(x) \Phi_1, \Phi_2 \rangle = \int \mathcal{D}\psi\ ( \psi(x) \Phi_1[\psi])^{*} \Phi_2[\psi].
\end{align}
And
\begin{align}
\langle \Phi_1, \Psi(x)^{\dagger} \Phi_2 \rangle = \int \mathcal{D}\psi\ \Phi_1[\psi]^*\ \frac{\delta}{\delta \psi(x)} \Phi_2[\psi].
\end{align}
But even if this is true, I don't know how I can show equality between those two. The functional derivative will generate a $\delta(x-y)$ term, that won't appear in the first scalar product.

So my questions would be: For the Schrödinger Wavefunctional theory, is there even the notion of scalar products and adjoint operators – namely, are Schrödinger functionals and the multiplicative $\psi(x)$ and $\frac{\delta}{\delta \psi(x)}$ a representation of the hilbert space and the field operators in the mathematical sense?

If they are, how do you show that $\frac{\delta}{\delta \psi(x)}$, acting on a functional, is indeed the adjoint operator to $\psi(x)$?

Best Answer

Well, the standard 3-step argument goes as follows:

  1. The fundamental equal-time super-Poisson/Dirac bracket is the inverse supermatrix of the supermatrix for the symplectic two-form, cf. e.g. my Phys.SE answer here.

  2. The canonical anticommutation relations (CARs) are the fundamental super-Poisson brackets multiplied with $i\hbar$, cf. the QM correspondence principle.

  3. The Schrödinger representation of the CARs (1) is $$ \hat{\psi}~=~\psi, \qquad \hat{\psi}^{\dagger}~=~\frac{\delta}{\delta \psi} .$$

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