§ Von neumann: foundations of QM

• I wanted to understand what von neumann actually did when he "made QM rigorous", what was missing, and why we need $C^\star$ algebras for quantum mechanics, or even "rigged hilbert spaces".
• I decided to read Von Neumann: Mathematical foundations of quantum mechanics.
• It seems he provides a rigorous footing for QM, without any dirac deltas. In particular, he proves the Reisez representation theorem, allow for transforming bras to kets and vice versa. On the other hand, it does not allow for dirac delta as bras and kets.
• The document The role of rigged hilbert spaces in QM provides a gentle introduction on how to add in dirac deltas.
• Rigged hilbert spaces (by Gelfand) combine the theory of distributions (by Schwartz), developed to make dirac deltas formal, and the theory of hilbert spaces (by Von Neumann) developed to make quantum mechanics formal.
• To be even more abstract, we can move to $C^\star$ algebras, which allow us to make QFT rigorous.
• So it seems that in total, to be able to write, say, a "rigorous shankar" textbook, one should follow Chapter 2 of Von Neumann, continuing with the next document which lays out how to rig a hilbert space.
• At this point, one has enough mathematical machinery to mathematize all of Shankar.

§ References

• Von neumann: Mathematical foundations of quantum mechanics.
• The role of rigged hilbert spaces in QM