Talk:Resolution of the identity

Merging discussion

Well, I suppose the merging is okay, so long as we don't lose too much information from the article. I don't know much about this operator, other than the fact that I can insert it conveniently into certain places when I need to work with Dirac notation. --HappyCamper 16:40, 16 August 2006 (UTC)[reply]

for a Hermitian, or more generally, normal, matrix T, the spectral theorem says T = ∑ x |x><x|. so I = |x><x|. the resolution of the identity refers to the complete set of projection operators {|x><x|}. in physics, when the space is not finite dimensional, the sum is replaced by an integral T = ∫ x d |x><x|, and one speak of a "continuous set of eigenvalues". as you wrote in article, the expression I = ∫ |x><x| is then the resolution of the identity. (while i understand why one might do something like that, i find such casual formal manipulation rather disconcerting.) mathematically, this, the resolution of identity, is the spectral measure obtained via the Borel functional calculus. a section of this name can be added to Borel functional calculus and both points of view can be included. Mct mht 18:25, 16 August 2006 (UTC)[reply]

Okay, I'll let you decide how best to do that. I'm completely unfamiliar with Borel functional calculus. --HappyCamper 18:38, 16 August 2006 (UTC)[reply]