Formalism Theorems
These cards summarize the structural results behind self-adjoint observables, unitary generators, canonical representations, ray symmetries, and projection probabilities. They are lookup projections of full treatments in the rigorous volume.
Released cards and owners
Section titled “Released cards and owners”| Card | Key hypothesis to check | Full treatment |
|---|---|---|
| Spectral Theorem | self-adjointness and the spectral-integrability domain | Unbounded Spectral Theorem |
| Stone’s Theorem | strong continuity of a one-parameter unitary group | Stone’s Theorem |
| Stone–von Neumann | finite degrees, regularity, irreducibility, fixed nonzero central character | Stone–von Neumann Theorem |
| Wigner’s Theorem | bijective preservation of all ray transition probabilities | Wigner’s Theorem |
| Gleason’s Theorem | dimension at least three and countable orthogonal additivity | Gleason’s Theorem |
Shared cautions
Section titled “Shared cautions”- Infinite-dimensional domains and continuity topologies are part of a theorem statement, not optional annotations.
- A finite-dimensional special case can hide unbounded-generator assumptions.
- A uniqueness theorem is only as broad as its representation class.
- A structural probability theorem does not by itself derive state update, dynamics, or interpretation.
- The cards intentionally omit proof sketches when compression would obscure a nontrivial imported lemma.
References
Section titled “References”- G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
- B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
- N. P. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Springer, 2017.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.