Using Dirac’s formalism of Quantum Mechanics, if a complex Hilbert space $\mathscr{H}$ is given, the elements of $\mathscr{H}$ are denoted by $|x\rangle$ and elements of its dual $\mathscr{H}’$ are denoted by $\langle x|$. Sometimes, if a countable basis $\{ |n\rangle : n \in \Bbb N\}$ is given, a formula like the following holds
$$ \mathbf{1} = \sum_{n \in \Bbb N} |n\rangle\langle n|.$$
This is quite an obvious formula if $\mathscr{H}$ is a separable space (i.e. with a countable basis). Often, however, if there is an uncountable basis $\{ |x\rangle : x \in \Bbb R\}$, a similar formula is used:
$$ \mathbf{1} = \int |x\rangle\mathrm{d}x\langle x | $$
which is understandable in the Dirac’s notation, but I don’t understand it formally.
In the theory of Hilbert spaces,
- What are the projection operators $|x\rangle\langle x|$ and what is their meaning?
- In which sense the integral holds?
Thank you for any help with this.
Best Answer
The continuous case is based around the Fourier transform, as opposed to the Fourier series. For example, consider the case of the Fourier transform on $\mathbb{R}$: $$ f = \frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(s)e^{isx}ds = \int_{-\infty}^{\infty}\langle f,e_s\rangle e_sds,\;\;\; e_s(x)=\frac{1}{\sqrt{2\pi}}e^{isx}. $$ Notice the similarity to the formalism of the Fourier series $$ f = \sum_{s=-\infty}^{\infty}\langle f,e_s\rangle e_s,\;\;\; e_s(x)=\frac{1}{\sqrt{2\pi}}e^{isx}. $$ Dirac's treatment is over-simplified in the continuous case, and the notation is not so nice when there is a mix of continuous and discrete spectrum. But it's the thought that counts when it comes to Dirac's formalism, and the thought is an elegant extension of the Fourier series to a Fourier integral and/or a mixed Fourier integral and Fourier series expansion.
General expansions are not fully handled by a "discrete" and an "absolutely continuous" expansion, but Physicists rarely need to consider a measure that is "singular" with respect to Lebesgue measure, but which is not discrete. The problems Dirac considers do not require it.