To what extent should the treatment of wave functions in physics be aligned with mathematics?

The wave function ψ is something that always appears when learning quantum mechanics. In textbooks from the last century, this ψ was treated as a highly mysterious object, and issues such as the measurement problem were discussed seriously. At that time, the only definition of ψ was that it was a complex function satisfying the Schrödinger equation, and that the square of its absolute value provided the probability density distribution of a particle's position.

However, in modern times, it has been properly defined operationally through the method of quantum state tomography.

By measuring the probability distributions of several physical quantities, state vectors and wave functions have become concepts that can be determined experimentally.

However, in the last century, many people felt that in order to pursue the physical meaning of the mysterious ψ, a solid mathematical foundation had to be constructed for it. Consequently, a value system was fostered over a long period that considered it superior to treat wave functions using rigorous mathematical functional analysis. Even today, there are still quite a few instructors who are dragged along by that value system, and there are cases where functional analysis is taught in quantum mechanics lectures and textbooks.

My opinion is that teaching physics students from the perspective of mathematical functional analysis is almost meaningless in the 21st century. From the standpoint of empirical science, mathematical prerequisites such as the completeness required in Hilbert spaces, for example, have not been necessary at all based on experiments to date. Experiments in domains where completeness can be checked are not currently being conducted, and in physics experiments, which are always accompanied by experimental error no matter how small, verification of completeness will be impossible in the future as well. Furthermore, there is the fact that theoretical models without completeness can sufficiently explain experimental results obtained so far. Please refer to the article below regarding these points.

In my book below, which is a textbook based on the positivism of theoretical physics, state vectors and wave functions are generated by applying all possible quantum operations to a system in a specific state prepared by reference measurements.

Basic physical operations are mathematically written as unitary operators, which are called unitary operations. Thanks to quantum computer theory, it is now known that, at least for finite systems even at the macro level, any unitary operation can be approximated with arbitrary precision by a combination of a finite number of types of quantum logic gates, i.e., a "quantum circuit." By performing that operation on a quantum system in a specific state, any quantum state can be generated with arbitrary precision.

It is expected that high-precision macro quantum computers will be realized in the future. In such an era, it is thought that the quantum computer itself will be used for the experimental generation of state vectors. The set of state vectors generated in this case is densely distributed within the Hilbert space, but due to the actual discreteness and finiteness of quantum circuits, it will never satisfy completeness. It is like the distribution of rational numbers within the real number space, similar to how any real number is approximated by rational numbers with arbitrary error. Computers, whether classical or quantum, are fundamentally "digital." However, in quantum mechanics as an empirical science, such state vectors are sufficient.

Believing arbitrarily in "completeness," which has not been directly required by experimental results to date and can never be verified by experiment in the future either, is, from the standpoint of positivism, the same as a superstition without grounds.

In the first place, the fact that physics is a discipline that explores the principles of things means that we explore precisely because the definition and essence of "things" are not yet understood. Unlike mathematics, where one can construct logic by strictly defining the subject in advance, such as "satisfying completeness," in physics, which is an empirical science, one has no choice but to face the one and only, complex, and mysterious reality of nature.

The domain and range of the wave function ψ considered in physics are also based on the common understanding that "ψ: Nature → Nature." For that domain and range, we set what is reasonable in the regions known from experimental observations so far for the time being, and we will investigate the details from here on out. That is the very attitude of physics as an empirical science.