Mathematical Representation in Physics: the Curious Case of the Complex Numbers
We use mathematical objects and structures to formulate our physical theories and to represent physical reality. This gives rise to a host of philosophical questions surrounding the use of mathematics in physics, brought to the fore by a particular case. Physicists often remark that whereas in classical physics complex numbers are used as a dispensable calculational tool, in quantum mechanics they become essential to formulating the theory. Less often do physicists offer an explanation as to why this is, and among those who do, there is no clear or agreed-upon answer, despite the long and illustrious history of physicists wondering about this. Instead this is regarded as an outstanding puzzle.
In this talk, I discuss a few interrelated issues brought out by this curious situation. I argue that complex numbers are not truly necessary for quantum mechanics, against conventional wisdom, though they do yield a particularly perspicuous formulation, for reasons I will mention. Yet the main focus of my talk is not complex numbers or quantum mechanics per se. Rather, I use this case as a springboard for explicating and illuminating aspects of mathematical representation in physics in general. Getting clear on these aspects, I argue, is crucial to the foundational project of figuring out what our best physical theories, which are formulated in abstract mathematical terms, are saying about the nature of physical reality.