  In this letter I stress the role of causal reversibility (time-symmetry),
together with causality and locality, in the justification of the quantum
formalism. Firstly, in the algebraic quantum formalism, I show that the
assumption of reversibility implies that the observables of a quantum theory
form an abstract real C*-algebra, and can be represented as an algebra of
operators on a real Hilbert space. Secondly, in the quantum logic formalism, I
emphasize which axioms for the lattice of propositions (existence of an
orthocomplementation and the covering property) derive from reversibility. A
new argument based on locality and Soler's theorem is used to derive the
representation as projectors on a regular Hilbert space from the general
quantum logic formalism. In both cases it is recalled that the restriction to
complex algebras and Hilbert spaces comes from the constraints of locality and
separability.
