  We show that quantum theory (QT) is a substructure of classical probabilistic
physics. The central quantity of the classical theory is Hamilton's function,
which determines canonical equations, a corresponding flow, and a Liouville
equation for a probability density. We extend this theory in two respects: (1)
The same structure is defined for arbitrary observables. Thus we have all of
the above entities generated not only by Hamilton's function but by every
observable. (2) We introduce for each observable a phase space function
representing the classical action. This is a redundant quantity in a classical
context but indispensable for the transition to QT. The basic equations of the
resulting theory take a "quantum-like" form, which allows for a simple
derivation of QT by means of a projection to configuration space reported
previously [Quantum Stud.:Math. Found. (2018) 5:219-227]. We obtain the most
important relations of QT, namely the form of operators, Schr\"odinger's
equation, eigenvalue equations, commutation relations, expectation values, and
Born's rule. Implications for the interpretation of QT are discussed, as well
as an alternative projection method allowing for a derivation of spin.
