The mystery of square root of minus one in quantum mechanics, and its demystification

TL;DR

This paper explores the role of the imaginary unit i in quantum mechanics, proposing a perspective that potentially eliminates the conceptual mystery of complex numbers in the theory.

Contribution

It introduces a new interpretation of the imaginary unit i, demonstrating its possible elimination in key quantum relationships and clarifying its role in quantum formulations.

Findings

Elimination of i in the fundamental commutation relation pq - qp = h/(2πi)

Reinterpretation of the role of complex exponentials in quantum theory

Clarification of the conceptual basis of complex numbers in quantum mechanics

Abstract

To most physicists, quantum mechanics must embrace the imaginary number i = square root of minus one is at least a common belief if not a mystery. We use the famous example pq -qp = h/(2 pi i) to demonstrate the possible elimination of i when constructing this noncommutative relationship. We then discuss the role of i in the formulation of Schroedinger's wave equation. Common to the original development of these two quantum theories was the use of complex exponential to represent the fundamental variables (i.e., p, q, and the wave function). Understanding this complex function from the right perspective, as we suggest in this essay, removes the mysteries surrounding the complex nature of quantum mechanics.