Notes on Quantum States, Unitarity and Non-locality

Overview & Synopsis

In this article we will demonstrate that the detection of electrons in an interference experiment with electrons is a transition between two Hilbert spaces caused by interactions with the detecting screen. The first Hilbert space is that of the free electrons before detection and the second is that of the bound electrons in the screen after detection. This transition can be described by unitary quantum-mechanical perturbation theory where the grand Hilbert space required to describe the electrons is the direct sum of these two Hilbert spaces. We can say an electron is detected at the screen only after the interaction between the two has caused an observable effect like an atomic transition. Detection of the electrons does not solely indicate modification of the quantum states taking place in the Hilbert space used to describe the electron before detection. It cannot be used to imply a non-unitary transformation of the quantum states belonging to this Hilbert space only. Similarly, emission of electrons from a material to free space is a transition from the set of bound states to the set of free states. This transition can be described by using time-dependent perturbation theory without referring to any non-unitary transformation. In both the cases, relevant transitions would be given by the time evolution of exact solutions of the interacting theory and rescind the requirement of introducing any non-unitary transformation. This interpretation may be extended to other measurement processes. We will conclude the article with a brief discussion on non-locality in quantum mechanics. This is based on the necessity of complex functions in quantum theory and perception of human brain on imaginary numbers.