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 Dirac (1930) and von Neumann (1932).

Hilbert space formulation

The space H 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 H, then the bounded observables are just the bounded self-adjoint operators on H. If v is a norm 1 vector of H then defining ω(A) = (v,Av) is a state on the C* algebra, so norm 1 vectors (up to scalar multiplication) give states. 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.

See also

References

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