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.

• The observables of a quantum system are defined to be the (possibly unbounded) self-adjoint operators A on H.
• A state φ of the quantum system is a unit vector of H, up to scalar multiples.
• The expectation value of an observable A for a system in a state φ is given by the inner product (φ,Aφ).

Operator algebra formulation

The Dirac–von Neumann axioms can be formulated in terms of a C* algebra as follows.

• The bounded observables of the quantum mechanical system are defined to be the self-adjoint elements of the C* algebra.
• The states of the quantum mechanical system are defined to be the states of the C* algebra (in other words the normalized positive linear functionals ω).
• The value ω(A) of a state ω on an element A is the expectation value of the observable A if the quantum system is in the state ω.

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

• Axiomatic quantum field theory

References

• Dirac, Paul (1930), Principles of Quantum Mechanics
• Strocchi, F. (2008), An introduction to the mathematical structure of quantum mechanics. A short course for mathematicians, Advanced Series in Mathematical Physics 28 (2 ed.), World Scientific Publishing Co., ISBN 9789812835222, MR 2484367
• Takhtajan, Leon A. (2008), Quantum mechanics for mathematicians, Graduate Studies in Mathematics 95, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4630-8, MR 2433906
• von Neumann, John (1932), Mathematical Foundations of Quantum Mechanics, Berlin: Springer, MR 0066944