In this chapter, the physical content of previous chapters is summarized, and the reader is suided towards some key areas of modern physics: the Feynman formulation, the unificaton of quantum theory with special relativity, and dualities in quantum mechanics. This very short chapter is intended to be a bridge aimed at motivating readers to attend first course in quantum field theory.

Towards an overall view

At this stage, the student or the general reader might want to know what picture of the physical world emerges from the material covered so far and from the areas related to it but not covered in our book, so as to be introduced to what is in sight not only for the beginner but also for the scientific community. For this purpose, we summarize very briefly the previous pictures and we arrive at three key concepts: the Lagrangian in quantum mechanics via the Feynman sum over histories; the unification of quantum mechanics and special relativity; and new duality symmetries in quantum mechanics.

From Schrödinger to Feynman

Quantum mechanics provides a probabilistic description of the world on atomic or sub-atomic scale. It tells us that, on such scales, the world can be described by a Hilbert space structure, or suitable generalizations. Even in the relatively simple case of the hydrogen atom, the appropriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbert spaces play a role as well.