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Stone-von Neumann theorem

Contents

Context

AQFT and operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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 qq and canonical momentum pp) in the form

[q,p]=i [q,p] = i \hbar

may be represented as unbounded operators on the Hilbert space of square integrable functions L 2()L^2(\mathbb{R}) on the real line by defining them on the dense subspace of smooth functions ψ:\psi \colon \mathbb{R} \to \mathbb{C} as

(qψ)(x)xψ(x)AAAA(pψ)(x)ixψ(x), (q \psi)(x) \coloneqq x \psi(x) \phantom{AAAA} (p \psi)(x) \coloneqq -i \hbar \frac{\partial}{\partial x} \psi(x) \,,

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.

References

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

  • Miklós Rédei, Von Neumann’s proof of Uniqueness of Schrödinger representation of Heisenberg’s commutation relation (pdf)

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.