Complex Numbers in Quantum Mechanics: Computational Trick or Fundamental Necessity?

Synopsis

Complex numbers seem to play a fundamental role in the standard Hilbert space formulation of quantum mechanics. However, in other theories of physics, for example electrodynamics, complex numbers are merely useful mathematical tricks. Indeed, the necessity of complex numbers in quantum mechanics has been debated by the pioneers of the theory—a debate which has continued to this day, and motivated new formulations of the theory using only real numbers. After all, a fundamental role for complex numbers in quantum mechanics was “unpleasant” to no one less than Schrödinger himself.

However, less than a year ago, the debate took a new turn as it was shown that a Hilbert space formulation that uses only real numbers cannot reproduce the predictions of standard quantum mechanics in a specific scenario similar to Bell tests. Recent experimental evidence supports the fundamental necessity of complex numbers in the Hilbert space formulation of quantum mechanics.