ABSTRACT
The aim of this chapter is to present (in a compact way) the
fundamentals of conventional quantum theory. We select the for-
mulation of QM as an axiomatic theory in the complete accordance
with the Hilbert’s program on axiomatization of physical theories.
We remark that the first step toward the present formulation of
QM was done in the seminal work of Hilbert, Nordheim, and von
Neumann [112]. This work was substantially extended, completed,
and clarified in the book of von Neumann [296]. Another version
of the rigorous mathematical approach to QM was presented in
the book of Dirac [74]. From the mathematical viewpoint, the
main difference between the approaches of von Neumann and
Dirac is that the first one is based on the theory of self-adjoint
operators in complex Hilbert space and the second one is based
on the theory of self-adjoint operators acting (by using the modern
mathematical terminology) in rigged Hilbert spaces. This theory
operates with generalized eigenvectors belonging to the spaces of
distributions. Although the von Neumann’s formalization of QM
became dominant in quantum mathematical physics, the Dirac’s
formalization is widely used in the works on the physical level
of rigorousness. Moreover, the Dirac’s terminology, bra-and ket-
vectors, became the standard for quantum information theory.
However, since quantum information is based on the castrated
version of QM, namely, QM in finite-dimensional Hilbert spaces, the so-called n-qubit spaces, the basic tool of the Dirac’s formalism, usage of generalized eigenvectors, is useless (since in the finite-
dimensional case everything is reduced to manipulation with
matrices). During the last 80 years, there were written hundreds of
books on QM enlightening its foundational aspects. Unfortunately,
the majority of these writings advertise exotic (and some of them
really esoteric) views on QM and, in particular, on interpretation of
the wave function. They are full of such terms as the collapse of the
wave function, quantumnonlocality, manyworlds, thewave function
of universe, the preparation of Schro¨dinger cats, influence of the
future on the past, quantum teleportation. Of course, manipulations
with such terminology create an atmosphere of mystery and a
feeling that something unusual and impossible can happen. And this
atmosphere plays a positive role in attracting young people to QM
and especially to quantum foundations.
