Representations of Quantum Mechanics

Considering general aspects of motion, the content of Lie symmetries is derived as a central structure for mechanical theories. This algebraic apparatus is associated with the Schwinger variational principle, which is related to the Feynman formulation, as is well known. In this realm of symmetries, different representations of non-relativistic quantum mechanics is addressed. The first, and most natural in this context, is the Liouville-von Neumann formulation in terms of density matrix. Looking for solutions of the Liouville-von Neumann equation, the Schrödinger-equation representation is derived as a theorem. In this realm we emphasize the meaning of the Heisenberg algebra for deriving the generators of the Galilei group. This procedure, based on symmetries, only, brings another perspective to consider an old but not fully solved problem: the Heisenberg relation for the time observer and the generator of time translation (the Hamiltonian). From the density matrix representation, the phase space Wigner representation of quantum mechanics is presented, followed by the symplectic representation. The most general presentation of quantum mechanics is based on the Fock space. This is discussed considering symmetries. As a consequence, for any type of representation, quantum mechanics is basically a quantum field theory. This interpretation is achieved by consistency and leads naturally to solve initial problems of quantum mechanics, such as the wave-particle notion and the principle of complementarity.

Keywords:
Quantum Mechanics Structure; Representations; Symmetry