These lecture notes present a concise and introductory yet as far as possible coherent view
of the main formalizations of quantum mechanics and of quantum field theories their
interrelations and their theoretical foundations. The standard formulation of quantum mechanics
(involving the Hilbert space of pure states self-adjoint operators as physical observables
and the probabilistic interpretation given by the Born rule) on one hand and the path integral
and functional integral representations of probabilities amplitudes on the other are the
standard tools used in most applications of quantum theory in physics and chemistry. Yet other
mathematical representations of quantum mechanics sometimes allow better comprehension and
justification of quantum theory. This text focuses on two of such representations: the
algebraic formulation of quantum mechanics and the quantum logic approach. Last but not least
some emphasis will also be put on understanding the relation between quantum physics and
special relativity through their common roots - causality locality and reversibility as well
as on the relation between quantum theory information theory correlations and measurements
and quantum gravity. Quantum mechanics is probably the most successful physical theory ever
proposed and despite huge experimental and technical progresses in over almost a century it
has never been seriously challenged by experiments. In addition quantum information science
has become an important and very active field in recent decades further enriching the many
facets of quantum physics. Yet there is a strong revival of the discussions about the
principles of quantum mechanics and its seemingly paradoxical aspects: sometimes the theory is
portrayed as the unchallenged and dominant paradigm of modern physical sciences and
technologies while sometimes it is considered a still mysterious and poorly understood theory
waiting for a revolution. This volume addressing graduate students and seasoned researchers
alike aims to contribute to the reconciliation of these two facets of quantum mechanics.
