algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
The Stone–von Neumann theorem (due to Marshall Stone and John von Neumann) says that there is – up to isomorphism – a unique irreducible unitary representation of the Heisenberg group on finitely many generators (equivalently: of the Weyl relations of the canonical commutation relations).
The analogous statement does not hold for infinitely many generators (as they appear in quantum field theory); this is Haag's theorem.
Explicitly, the canonical commutation relations on two generators (canonical coordinate $q$ and canonical momentum $p$) in the form
may be represented as unbounded operators on the Hilbert space of square integrable functions $L^2(\mathbb{R})$ on the real line by defining them on the dense subspace of smooth functions $\psi \colon \mathbb{R} \to \mathbb{C}$ as
where on the right we have the derivative along the canonical coordinate function on $\mathbb{R}$.
This is often called the Schrödinger representation (after Erwin Schrödinger, e.g. Redei), to be distinguished from “Schrödinger picture” which is a related but different concept.
The original articles are
John von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren , Mathematische Annalen (Springer Berlin / Heidelberg) 104: 570–578,
John von Neumann, Über Einen Satz Von Herrn M. H. Stone (in German), Annals of Mathematics, Second Series 33 (3): 567–573, ISSN 0003-486X
Marc Rieffel, On the uniqueness of the Heisenberg commutation relations (pdf)
Review includes
See also
Last revised on December 28, 2017 at 10:07:10. See the history of this page for a list of all contributions to it.