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Neumann axioms

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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).

Contents

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• 1 Hilbert space formulation
• 2 Operator algebra formulation
  • 2.1 Example
• 3 See also
• 4 References

Hilbert space formulation[edit]

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[edit]

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[edit]

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[edit]

• Axiomatic quantum field theory