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lugita15
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Dirac's bra-ket formalism implicitly assumed that there was a Hilbert space of ket vectors representing quantum states, that there were self-adjoint linear operators defined everywhere on that space representing observables, and that the eigenvectors of any such operator formed an orthogonal basis of the ket space. But this runs into a few problems: first of all, some operators like position and momentum are unbounded, so they cannot be defined everywhere on the ket space due to the Hellinger-Toeplitz theorem. Second, the criterion that you have an orthogonal basis seems to require the ket space to be seperable, but seperable Hilbert spaces can only have countably many dimensions, which is incompatible with the idea that position eigenstates form an uncountable basis.
The RHS construction tries to alleviate these concerns as follows (this is mostly gleamed from reading old PF threads on the topic). We start with the standard Hilbert space [itex]H=L^{2}(ℝ^{3})[/itex], and we pick out a so-called nuclear subspace [itex]\Phi[/itex] which is dense in [itex]H[/itex] and on which any continuous function [itex]f(P,Q,H)[/itex] (defined via Taylor series) of the position, momentum, and Hamiltonian operators is defined. We then basically use [itex]\Phi[/itex] as the space of test functions for a theory of Schwartz distributions - the space [itex]\Phi^{\times}[/itex] of continuous anti-linear functionals will be our ket space, and the space [itex]\Phi'[/itex] of continuous linear functionals will be our bra space.
My question is, does RHS sucessfully implement all of Dirac's formalism unscathed, so that we can completely forget about the original Hilbert space once we've set up the rigged Hilbert space? I've heard that the Gelfand-Maurin nuclear spectral theorem proves that every self-adjoint linear operator acting on [itex]H[/itex] has an eigenbasis in [itex]\Phi^{\times}[/itex], but can the same be said of all self-adjoint linear operators acting on [itex]\Phi^{\times}[/itex]? Is it possible to define a distribution-valued inner product on [itex]\Phi^{\times}[/itex] (or do we run into the age-old problem of multiplication of distributions?), and does this make [itex]\Phi^{\times}[/itex] an uncountable-dimensional non-seperable Hilbert space? Can [itex]\Phi'[/itex] also be considered as the dual space of [itex]\Phi^{\times}[/itex], and is there a one-to-one correspondence between these two spaces? Finally, can things like delta-function potentials be rigorously justified in RHS framework?
Sorry for asking so many questions, but I only recently discovered Rigged Hilbert Spaces, and I'm absolutely fascinated by them. Can anyone recommend a good book on the subject? (I've already read the couple pages Ballentine has to say on the subject, and I wish there was a whole quantum mechanics text using this approach.)
Any help would be greatly appreciated.
Thank You in Advance.
The RHS construction tries to alleviate these concerns as follows (this is mostly gleamed from reading old PF threads on the topic). We start with the standard Hilbert space [itex]H=L^{2}(ℝ^{3})[/itex], and we pick out a so-called nuclear subspace [itex]\Phi[/itex] which is dense in [itex]H[/itex] and on which any continuous function [itex]f(P,Q,H)[/itex] (defined via Taylor series) of the position, momentum, and Hamiltonian operators is defined. We then basically use [itex]\Phi[/itex] as the space of test functions for a theory of Schwartz distributions - the space [itex]\Phi^{\times}[/itex] of continuous anti-linear functionals will be our ket space, and the space [itex]\Phi'[/itex] of continuous linear functionals will be our bra space.
My question is, does RHS sucessfully implement all of Dirac's formalism unscathed, so that we can completely forget about the original Hilbert space once we've set up the rigged Hilbert space? I've heard that the Gelfand-Maurin nuclear spectral theorem proves that every self-adjoint linear operator acting on [itex]H[/itex] has an eigenbasis in [itex]\Phi^{\times}[/itex], but can the same be said of all self-adjoint linear operators acting on [itex]\Phi^{\times}[/itex]? Is it possible to define a distribution-valued inner product on [itex]\Phi^{\times}[/itex] (or do we run into the age-old problem of multiplication of distributions?), and does this make [itex]\Phi^{\times}[/itex] an uncountable-dimensional non-seperable Hilbert space? Can [itex]\Phi'[/itex] also be considered as the dual space of [itex]\Phi^{\times}[/itex], and is there a one-to-one correspondence between these two spaces? Finally, can things like delta-function potentials be rigorously justified in RHS framework?
Sorry for asking so many questions, but I only recently discovered Rigged Hilbert Spaces, and I'm absolutely fascinated by them. Can anyone recommend a good book on the subject? (I've already read the couple pages Ballentine has to say on the subject, and I wish there was a whole quantum mechanics text using this approach.)
Any help would be greatly appreciated.
Thank You in Advance.