nLab
Møller operator

Context

Algebraic Quantum Field Theory

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

Contents

Idea

In scattering theory the Møller operator intertwines the observables of the free theory with those of the interacting theory.

Details

In quantum mechanics

The quantum states |ψ(t) I\vert \psi(t)\rangle_I in the interaction picture of quantum mechanics are by definition (this equation) related to the asymptotic free states |ψrange\vert\psi\range by

|ψ(t) I=exp(tiH free)exp(tiH free+tiV)|ψ \vert \psi(t)\rangle_I \;=\; \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \vert \psi \rangle

and conversely

|ψ=exp(tiH free+tiV)exp(+tiH free)|ψ(t) I \vert \psi \rangle \;=\; \underbrace{ \exp\left({\tfrac{- t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right) } \vert \psi(t)\rangle_I

the suitable limit for tt \to \mp \infty of the operator under the brace is called the Møller operator

Ω±limtexp(tiH free+tiV)exp(+tiH free) \Omega{\pm} \;\coloneqq\; \underset{t \to \mp \infty}{\lim} \exp\left({\tfrac{- t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right)

(e.g. BEM 01)

In quantum field theory

In perturbative quantum field theory the maps that intertwine the Wick algebra of quantum observables of the free field theory with the interacting field algebra are, on regular polynomial observables. the derivatives of the Bogoliubov formula of the given S-matrix 𝒮\mathcal{S} for the given interaction S intS_{int} with respect to source fields:

()𝒮(S int) 1 H(𝒮(S int) F()) \mathcal{R}(-) \;\coloneqq\; \mathcal{S}(S_{int})^{-1} \star_H (\mathcal{S}(S_{int}) \star_F (-))

(Here H\star_H denotes the star product induced by the Wightman propagator, hence the Wick algebra-product, while F\star_F denotes the star product induced by the Feynman propagator, hence the time-ordered product. The inverse () 1(-)^{-1} is taken with respect to H\star_H.)

This ()\mathcal{R}(-) is referred to as the quantum Møller operator in (Hawkins-Rejzner 16, below def. 5.1). (But notice that in many previous articles in perturbative AQFT, by the same authors and by others, the very same operator is referred to just as the “intertwining operator”, or similar.)

References

Discussion in quantum mechanics:

  • A. Baute, I. Egusquiza, J. Muga, Møller operators and Lippmann-Schwinger equations for step-like potentials (arXiv:quant-ph/0104043)

Discussion in relativistic perturbative quantum field theory in the rigorous formulation of causal perturbation theory/perturbative AQFT:

Created on January 8, 2018 at 15:13:52. See the history of this page for a list of all contributions to it.