nLab Møller operator

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 $\vert \psi(t)\rangle_I$ in the interaction picture of quantum mechanics are by definition (this equation) related to the asymptotic free states $\vert\psi\rangle$ by

$$\vert \psi(t)\rangle_I \;=\; \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \vert \psi \rangle$$

and conversely

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

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

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

(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 $\mathcal{S}$ for the given interaction $S_{int}$ with respect to source fields:

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

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

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.)

Related concepts

• quantum master equation

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:

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)