Dirac–von Neumann axioms explained
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
is a fixed complex
Hilbert space of
countably infinite dimension.
on
.
of the quantum system is a
unit vector of
, up to scalar multiples; or equivalently, a
ray of the Hilbert space
.
is given by the
inner product
.
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
).
of a state
on an element
is the
expectation value of the observable
if the quantum system is in the state
.
Example
If the C*-algebra is the algebra of all bounded operators on a Hilbert space
, then the bounded observables are just the bounded self-adjoint operators on
. If
is a unit vector of
then
\omega(A)=\langlev,Av\rangle
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.
See also
