  Complex numbers appear in the Hilbert space formulation of quantum mechanics,
but not in the formulation in phase space. Quantum symmetries are described by
complex, unitary or antiunitary operators defining ray representations in
Hilbert space, whereas in phase space they are described by real, true
representations. Equivalence of the formulations requires that the former
representations can be obtained from the latter and vice versa. Examples are
given. Equivalence of the two formulations also requires that complex
superpositions of state vectors can be described in the phase space
formulation, and it is shown that this leads to a nonlinear superposition
principle for orthogonal, pure-state Wigner functions. It is concluded that the
use of complex numbers in quantum mechanics can be regarded as a computational
device to simplify calculations, as in all other applications of mathematics to
physical phenomena.
