Quantum mechanics has always relied on a very specific kind of mathematics: complex numbers arranged in a Hilbert space. Textbooks simply assume this structure and move on. But why should nature use this exact mathematical language? Why not real numbers, or quaternions, or something else entirely? This has remained one of the quiet mysteries at the heart of modern physics—not because the equations don’t work (they work extraordinarily well), but because no one has fully explained why this structure, and no other, underlies the quantum world.
In our new work, we start from something far more primitive than wavefunctions or operators: the idea that the universe is made of configurations that can be told apart, and that physical predictions must respect the symmetries of these configurations. When we demand only two things—(1) that different possibilities can interfere, and (2) that physics must be invariant under the internal symmetries of whatever number system we use—we find something striking: complex numbers are not optional; they are forced. Real numbers are too limited to describe smooth interference, and quaternions have too much internal symmetry, which clashes with the symmetry between distinguishable states. Complex numbers sit precisely in the “Goldilocks zone.” Once they are selected, the entire structure of quantum mechanics—Hilbert space, inner products, unitary evolution, even the Hamiltonian—emerges naturally.
If this reconstruction is correct, it transforms our understanding of quantum theory. The mathematics of quantum mechanics stops looking like a mysterious human choice and instead becomes the unique way to describe a world where distinguishable possibilities can combine and interfere. It also gives new insight into why quantum computing works—the power of quantum algorithms follows directly from this uniquely determined geometric structure. In short, the complex Hilbert space isn’t a convenient assumption: it is the only consistent language for the physics of interference, making quantum theory feel not arbitrary, but inevitable.