Dirac–von Neumann axioms
In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann in 1932. Hilbert space formulation. The space formula_1 is a fixed complex Hilbert space of countable infinite dimension. Operator algebra formulation. The Dirac–von Neumann axioms can be formulated in terms of a C*-algebra as follows. Example. If the C*-algebra is the algebra of all bounded operators on a Hilbert space formula_1, then the bounded observables are just the bounded self -adjoint operators on formula_1. If formula_17 is a unit vector of formula_1 then formula_19 is a state on the C*-algebra, meaning the unit vectors (up to scalar multiplication) give the states of the system. This is similar to Dirac's formulation of quantum mechanics, though Dirac also allowed unbounded operators, and did not distinguish clearly between self-adjoint and Hermitian operators.