Rahul Sharma (Editor)

Dirac–von Neumann axioms

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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

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.

    References

    Dirac–von Neumann axioms Wikipedia
    (Text) CC BY-SA