form $$\omega_{\mathbb{G}} = \int d^d x\  \delta \Theta(\vec{x}) \wedge
\delta \pi_{\Theta}(\vec{x}), \label{symstruct}$$ provides ${\cal Z}_{\mathbb{G}}$ with the structure of a symplectic supermanifold.

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The Wigner Functional
---------------------

If $\widehat{\rho}$ is the density operator of a quantum state then the functional $\rho_{_W}[\pi_{\Theta},\Theta]$ defined by $$\rho_{_W}[\pi_{\Theta},\Theta] = {\rm tr} \bigg \{
\widehat{\Omega}[\pi_{\Theta},\Theta] \widehat{\rho}\bigg \} =
2^{-\infty} \int {\cal D} {\eta}
\exp \bigg\{ - {2i \over \hbar} \int d^d x \ \pi_{\Theta}(\vec{x})
\cdot \eta(\vec{x}) \bigg\} \langle \Theta +\eta| \widehat{\rho} | \Theta - \eta
\rangle
\label{ttres}$$ is called the Wigner functional corresponding to this state. In particular, the Wigner functional $\rho_{_W}[\pi_{\Theta},\Theta]$ corresponding to the quantum state $\widehat{\rho} = |\Phi\rangle
\langle \Phi^\dag|$ , where $\langle \Phi^\dag| := (|\Phi\rangle)^\dag $ , is given by [@wwmoriginal; @tata; @hillery; @cfz; @imelda] $$\rho_{_W}[\pi_{\Theta},\Theta] =2^{-\infty} \int {\cal D} {\eta}
\exp \bigg\{ - {2i \over \hbar} \int d^d x \ \pi_{\Theta}(\vec{x})
\cdot \eta(\vec{x}) \bigg\} \Phi[\Theta + \eta ] \Phi^\dag[\Theta
- \eta ], \label{tcinco}$$ where $ \Phi[\Theta + \eta ] = \langle \Theta +\eta| \Phi \rangle$ and $\Phi^\dag[\Theta - \eta ] = \langle \Phi^\dag | \Theta - \eta
\rangle$.

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Normal Ordering
---------------

Let $F[\pi_{\Theta},\Theta]$ be a functional on ${\cal
Z}_\mathbb{G}$. We define $$F_{\widehat{\cal N}} := \widehat{\cal N}F,$$ The operator $\widehat{\cal N}$ 