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Covariant Quantum Mechanics” is a geometric approach to Quantum Mechanics on a curved spacetime equipped with a time fibring and a spacelike riemannian metric. This theory is aimed at implementing the principle of general relativity and the interpretation of gravity as a spacetime connection, in a spacelike riemannian framework (instead of a lorentzian framework), in order to stay close to standard Quantum Mechanics as far as possible.
The classical background of this theory is ruled by a “galileian” linear spacetime connection K , which yields a cosymplectic 2–form Ω of the phase space (replacing the more usual symplectic 2–form). The quantum framework is constituted by the quantum bundle Q over spacetime, the η –hermitian quantum metric h_η and the upper quantum connection Q ↑ .
We derive, in a covariant way, from the upper quantum connection (which lives on the upper quantum bundle, hence involves all possible classical observers) all fundamental objects of quantum dynamics, by following a criterion of “projectability” on spacetime, in order to get rid of observers. Thus, this method turns out to be a way to implement the covariance of the quantum theory. We can exhibit, in a covariant way, a distinguished family spe(J1E,IR) of phase functions called “special phase functions”, equipped with a Lie bracket (replacing the moreusual Poisson bracket), which generates the quantum symmetries and quantum operators.