Quantum mechanics formalism

\subsection{Quantum: formalism,  mechanics and field}
% J.-P. Gazeau

Over the last five years J.-P. Gazeau developed his research along five directions. The term ``Quantum'' is a common leitmotiv:
\begin{enumerate}
  \item Covariant integral quantization of various classical systems and applications to models of early cosmology (FRW, Bianchi I, Bianchi 9 and Mixmaster model) with regularisation of models presenting singularities in their classical versions. 

    Covariant integral quantizations  linearly transform functions ("classical observables") on phase spaces (in a wider sense) into operators "quantum observables") on some Hilbert spaces of "quantum states". They  are  based on the resolution of the identity by continuous or discrete families of normalised positive operator valued measures (POVM) which transform in a covariant way  under some symmetry group actions. In the simplest cases these symmetries are described by the   Weyl-Heisenberg group (projective representations of translations in 2d dimensions), or by the affine groups (translations of a subset of variables combined with dilations of the remnant subset of bounded below variables).    Starting from  (quasi-) probability distribution(s) on the  phase spaces  on which the Weyl-Heisenberg or affine groups are acting (classical Hamiltonian models) these quantizations yield their corresponding quantum models and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. In return the tracing of the involved POVM with the operators provides semi-classical portraits of the quantum models which a  regularization of the original classical model.  These quantization methods whose the origin can be traced back to Klauder, Berezin, Toepiltz, are relatively easy to manipulate when compared with geometric or deformation or other quantizations. As a matter of fact they allow to  circumvent the problems   of ordering (canonical quantization), or those due to the presence of singularities in the classical models. Three pedagogical presentations of the method were given in
% Too many citations -- ECM selects last
    % \autocite{Almeida:2018xvj,Gazeau:2018fjd,bergeronBJP19}.
    \autocite{bergeronBJP19}.
% Too many citations -- ECM selects arbitrarily
% See also \autocite{Bergerone20100787} about  the comparison  with phase-space methods \`a la Wigner.

There are multiple developments and applications of the method:   
  \begin{enumerate}
  \item Motions on the half line and on the punctured plane and the fundamental role of the  affine symmetries  in their quantum regularisation
    % Too many citations -- ECM select last
    % \autocite{Gazeau:2017opp,Gazeau:2019nap}
    \autocite{Gazeau:2019nap} 
  \item  Quantum motion on the circle
    % \autocite{Fresneda:2017vnz}
    , on hyperbolic geometry
    %\autocite{delOlmo:2018ecz}
    , on other constrained geometries with variable mass
    % Too many citations -- ECM select last
    % \autocite{Gazeau:2019bpp,Gazeau:2020ltn}
    \autocite{Gazeau:2020ltn}
  \item  Regularised quantum FRW and Mixmaster models for early cosmology and gravitational waves
    % Too many citations -- ECM select last
    % \autocite{Bergeron:2017gte,Bergeron:2017ddo,Bergeron:2018und}
    \autocite{Bergeron:2018und} % and the review \autocite{Bergeron:2019zyl}
\end{enumerate}
  
\item Quantum field theory in de Sitter space-time, and cosmological implications. These works  lie in the continuation of a long-lasting program, initiated more that 25 years ago,  for establishing a rigorous covariant quantum  field theory in de Sitter space-time
  % Too many citations -- ECM select last
  % \autocite{Pejhan:2019qcg,Pejhan:2019ech}
  \autocite{Pejhan:2019ech}

\item A new interpretation of the Cold Dark Matter viewed as a gluonic Bose-Einstein condensate emerging from the transition QGP-Hadrons,   based on \acrshort{qcd} and Anti-de-Sitter symmetry due to the positive curvature provided by the QGP conformal anomaly
  % Too many citations -- ECM select last
  % \autocite{Gazeau:2020jly,Cohen-Tannoudji:2021syj}
  \autocite{Cohen-Tannoudji:2021syj}

  \item Works on generalised coherent states, POVM and Helstrom bound for quantum communication
    % Too many citations -- ECM select last
     \autocite{Gazeau:2021jmp}
  %  \autocite{Kowalski:2018xsw,Gazeau:2018ilc,Curado:21,Gazeau:2021jmp}

\item Quantum formalism and information, and other works
  \autocite{Beneduci:hb21}
  % Too many citations -- ECM select last
  % \autocite{Bagarello_2018,Gazeau:2020ltf,Gazeau:2019amk,GTe21121155,Beneduci:hb21}
\end{enumerate}

\begin{team}
% ECM: commenting out all contributors to the external. Out of scope to list all our collaborators!
\permsci{J.-P.~ Gazeau} % (Emérite, U. Paris),
\permsci{H.~ Bergeron} % (visiteur associ\'e)
%\textbf{E. Czuchry, P. Ma\l kiewicz} (NCNR, Warszawa)
%\textbf{E.M.F. Curado, S. Faci, Ligia M.C.S. Rodriges} (CBPF, Rio de Janeiro)
%\textbf{T. Koide} (UFRJ, Rio de Janeiro)
%\textbf{V. Hussin, J. Moran, K. Zelaya} (CRM, U. Montr\'eal)
%\textbf{R. Beneduci, E. Frion} (U. Calabria)
%\textbf{F. Bagarello} (U. Palermo)
%\textbf{R. Murenzi} (Directeur TWAS, ICTP, Trieste)
\permsci{M.~Takook} % (Invited professor, program Pause)
%\textbf{H. Pejhan, M. Enayati} (Iran)
%\textbf{M. del Olmo} (U. Valladolid)
%\textbf{G. Cohen-Tannoudji} (CEA)
%\textbf{C. Habonimana} (PhD student U. Burundi)
 \end{team}