Are imaginary numbers just a mathematical convenience or do they have a tangible reality?
Does quantum mechanics need imaginary numbers?
The square root of negative one doesn’t correspond to any physical quantity, but that doesn’t mean it has no place in the physical sciences. For example, putting an imaginary number in an exponent changes the behavior of the exponential from rapid growth or decay to a steady sinusoidal oscillation. The result is a useful description of the physics of waves.
In electromagnetism and most other fields of physics, imaginary numbers are merely a mathematical convenience. All the relevant phenomena can still be described using nothing but real numbers. Quantum mechanics is an exception: The observable quantities and probabilities are by necessity all real, but the underlying quantum states and governing equations involve imaginary numbers, and there’s no simple way to remove them. But are they just an artifact of the way the theory was written down, or do they really need to be there?
In their new theoretical work, Miguel Navascués of the Institute for Quantum Optics and Quantum Information in Vienna and colleagues shed some light on that question.1 They find that, subject to some postulates about how a quantum theory must be mathematically structured, no real-valued version of quantum theory can duplicate all the predictions of the familiar complex-valued formulation. Moreover, they designed an experimentally feasible test capable of ruling out real-valued quantum theories. In the time since their proposal was made public in January 2021, two groups carried out the experiment—and both found results in favor of standard complex-valued quantum theory.
Does quantum mechanics need imaginary numbers?
The square root of negative one doesn’t correspond to any physical quantity, but that doesn’t mean it has no place in the physical sciences. For example, putting an imaginary number in an exponent changes the behavior of the exponential from rapid growth or decay to a steady sinusoidal oscillation. The result is a useful description of the physics of waves.
In electromagnetism and most other fields of physics, imaginary numbers are merely a mathematical convenience. All the relevant phenomena can still be described using nothing but real numbers. Quantum mechanics is an exception: The observable quantities and probabilities are by necessity all real, but the underlying quantum states and governing equations involve imaginary numbers, and there’s no simple way to remove them. But are they just an artifact of the way the theory was written down, or do they really need to be there?
In their new theoretical work, Miguel Navascués of the Institute for Quantum Optics and Quantum Information in Vienna and colleagues shed some light on that question.1 They find that, subject to some postulates about how a quantum theory must be mathematically structured, no real-valued version of quantum theory can duplicate all the predictions of the familiar complex-valued formulation. Moreover, they designed an experimentally feasible test capable of ruling out real-valued quantum theories. In the time since their proposal was made public in January 2021, two groups carried out the experiment—and both found results in favor of standard complex-valued quantum theory.
