I have been thinking for a while about the mathematics used to formulate our physical theories, especially the similarities and differences among different mathematical formulations. This was a focus of my 2021 book, Physics, Structure, and Reality, where I discussed these things in the context of classical and spacetime physics.
Recently this has led me toward thinking about mathematical formulations of quantum mechanics, where an interesting question arises concerning the use of complex numbers. (I recently secured a grant from the National Science Foundation for a project investigating this.)
It is frequently said by physicists that complex numbers are essential to formulating quantum mechanics, and that this is different from the situation in classical physics, where complex numbers appear as a useful but ultimately dispensable calculational tool. It is not often said why, or in what way, complex numbers are supposed to be essential to quantum mechanics as opposed to classical physics.
Despite those assertions, however, complex numbers cannot be truly necessary or essential, i.e. genuinely indispensable. Any physical theory can be mathematically formulated in different ways, and it is a straightforward mathematical fact that any expression involving complex numbers can be rewritten in terms of real ones—just think of expressing a complex number in terms of its real and imaginary parts, each of which are themselves real (Complex numbers can be regarded, indeed defined, as ordered pairs of real numbers obeying certain algebraic relations). It should be possible, in principle, to restate any formalism that employs complex numbers in terms of real numbers, the formalism for quantum mechanics being no exception. And, indeed, since the 1960s, there have been formulations of quantum mechanics available that do away with complex numbers (including versions that use only real numbers and ones in terms of quaternions).
Still, complex numbers do seem particularly natural or well-suited to quantum mechanics. Physicists are putting their finger on something characteristic of the theory when they suggest that complex numbers are playing a distinctive and newly central role.
My project aims to figure out why this is. What is it about quantum mechanics and the world it describes that makes a mathematical formulation in terms of complex numbers so well-suited to it? There is no clear or agreed-upon answer to this question, even though just about every famous physicist who has thought about the theory since its inception (including the likes of Schrödinger, Ehrenfest, Pauli, Bohm) has puzzled over the apparent centrality of complex numbers to quantum physics.
I believe that a particularly clear and elucidating answer comes from the nature of spin, a phenomenon with no classical counterpart. The basic idea (which I spell out in a paper in progress) is that there must be enough “room” in the formalism to encode all the possible spin states of a system, and a formalism in terms of complex numbers allows us to do this in a particularly natural and direct way. Complex numbers are not absolutely required, but given the experimental results involving spin, there must be something in the formalism that mirrors the structure of the complex numbers. (Various real versions that have been proposed, for instance, even though they remove complex numbers on the surface, tend to add mathematical structure that mimics the structure of the complex numbers.) Complex numbers themselves allow us to represent the spin facts in a natural, perspicuous, direct way. (For example, using only real numbers requires doubling the number of dimensions of the statespace, while imposing further constraints to limit the allowable states. The complex formulation does not need those extra constraints. It is more direct, lacking the excess mathematics that must be dispensed with by hand in the corresponding real formulation.)
This is not the only explanation for the appearance of complex numbers, which are used to characterize state vectors or wavefunctions without spin degrees of freedom too. It is nonetheless a notably straightforward and illuminating account of their naturalness for certain key phenomena. (All fundamental particles, aside from the Higgs boson, have spin.) For that matter, it is often said that systems with spin are the simplest truly quantum systems. If the simplest paradigmatically quantum mechanical system motivates their use, then this is as good an explanation as any for the entrance of complex numbers into the formalism, one that reveals the distinctive naturalness of complex numbers for quantum mechanics as opposed to classical physics in a physically clear way.
Incidentally, a few years ago, two different groups of physicists claimed to have experimentally verified the true necessity of complex numbers for quantum mechanics (in particular, for representing certain phenomena involving multi-particle systems), a thought that has since permeated the science media. However, given the mathematical facts mentioned above, that cannot be right. And as it happens, those claims rely on certain assumptions, which, when explicitly noted, reveal the conclusion to be quite a bit weaker than having demonstrated the genuine impossibility of doing quantum mechanics without complex numbers. (The conclusion does not extend to certain solutions to the measurement problem or to real formulations that reject the usual tensor-product rule for combined states.) At most, the conclusion we should draw is that one kind of real formulation cannot do the job, not that no real formulation can.
Understanding what it is about a quantum mechanical world that underlies the distinctive role played by complex numbers helps advance our understanding of quantum mechanics and the kind of world it describes, and how it differs from classical physics and the kind of world it describes. This is part of a general project of trying to understand why some part of mathematics works so well for formulating a given theory and characterizing the kind of world it describes, a project that is central to the foundations and interpretation of physics.
There are further philosophical conclusions to be drawn. One is the importance of locating perspicuous formulations of physical theories, formulations that clearly or directly or naturally reflect the nature of physical reality. (In another paper in progress, I discuss the nature of such formulations and why they are an aim of physical theorizing, with historical examples of scientific discoveries that were enabled by a perspicuous formulation.) Another is that the structure of a theory’s space of states is important to understanding the theory and the nature of the world it describes. I have defended this in the past for classical mechanics; quantum mechanics illustrates it further, in a different way.
There are also ramifications for the debate over wavefunction realism (which I plan to explore in another paper). This debate is about whether the mathematical wavefunction appearing in the formulation of quantum mechanics directly represents a real physical field. The literature has focused on wavefunctions without spin, with opponents balking at the huge number of dimensions of the wavefunction and its space. (I myself have defended wavefunction realism in the past.) Once we include spin, however, things become even more complicated in ways that have not been sufficiently acknowledged: the physical field is not just a complex-valued (scalar) field but a spinor-valued field. As Alyssa Ney has argued, the strongest case for wavefunction realism may be that it results in a fundamentally separable and local metaphysics. Separability means that the state of any system is determined by the states of its component parts (on a Humean view in particular, this is ultimately in terms of point-sized parts or values). Wavefunction realism is supposed to be attractive because the wavefunction itself is separable in this view. However, there is a question of whether spinors, or spinor-valued fields, require a structure that is not separable even in the wavefunction’s space. (The concern is that characterizing spinors may involve more holistic facts about their behavior under complete rotations in the space.) This is a new question for wavefunction realism, and I am not yet sure what the answer is.
