\Psi)$ consisting of a symplectic linear isomorphism from $S_a$ to $S_b$ and a linear morphism $
\delta_a \rightarrow \delta_b$ such that $$\varphi_b \circ
\Psi^{\otimes 2} = \Psi_{g, E_a, E_b}   \circ
\varphi_a.$$ The composition of a morphism $(g,\Psi): a
\rightarrow b$ with a morphism $(g',\Psi
') : b \rightarrow c$ is defined as $$(g', \Psi') {\circ_{{{\mathcal{D}}}}}(g, \Psi) := \bigl( g'g, \zeta ^{1/2} ( g'gE_a,
g' E_b, E_c) \Psi' \circ \Psi \bigr)$$ where the product in the right-hand side is the usual composition of maps.

\[prop:cat\] ${{\mathcal{D}}}$ is a category where each morphism is invertible.

This follows from proposition \[prop:cat\_demi\_forme\]. Indeed, given a morphism $(g, \Psi): a
  \rightarrow b$, one may identify $S_a$ and $S_b$ with $S$ through symplectomorphisms in such a way that $g$ becomes the identity. Then $\Psi$ is a morphism from $(E_a, \delta_a,
  \varphi_a)$ to $( E_b, \delta_b, \varphi_b)$ in the category ${{\mathcal{D}}}(S)$. With these identifications, the composition of morphisms in ${{\mathcal{D}}}$ corresponds to the composition in ${{\mathcal{D}}}(S)$.

In appendix \[sec:linear-quantization\], we will define a functor from the category ${{\mathcal{D}}}$ to the category of Hilbert spaces.

Metaplectic group {#sec:metaplectic-group}
=================

The automorphism group
----------------------

Consider a fixed symplectic vector space $S$ together with a positive polarization $E$. Denote by 