Resolution of The Identity
Let T be a self-adjoint operator. If E is a Borel subset of R, and 1E is the indicator function of E, then 1E(T) is a self-adjoint projection on H. Then mapping
is a projection-valued measure called the resolution of the identity for the self adjoint operator T. The measure of R with respect to Ω is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral I = ∫ 1 dΩ. Sometimes the term "resolution of the identity" is also used to describe this representation of the identity operator as a spectral integral.
In the case of a discrete measure (in particular, when H is finite dimensional), I = ∫ 1 dΩ can be written as
in the Dirac notation, where each |i> is a normalized eigenvector of T. The set { |i〉 } is an orthonormal basis of H.
In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as
and speak of a "continuous basis", or "continuum of basis states", { |i〉 }. Mathematically, unless rigorous justifications are given, this expression is purely formal.
Read more about this topic: Borel Functional Calculus
Famous quotes containing the words resolution and/or identity:
“We often see malefactors, when they are led to execution, put on resolution and a contempt of death which, in truth, is nothing else but fearing to look it in the faceso that this pretended bravery may very truly be said to do the same good office to their mind that the blindfold does to their eyes.”
—François, Duc De La Rochefoucauld (16131680)
“Growing has no connection with audience. / Audience has no
connection with identity. / Identity has no
connection with a universe. / A universe has no
connection with human nature.”
—Gertrude Stein (18741946)