PlanetPhysics/Haag Theorem

Introduction

A canonical quantum dynamics (CQD) is determined by the choice of the physical (quantized) `vacuum' state (which is the ground state); thus, the assumption that a field ${\mathcal {F}}_{Qc}$ shares the ground state with a free field ${\mathcal {F}}_{0}$, implies that ${\mathcal {F}}_{Qc}$ is itself free (or admits a Fock representation). This basic assumption is expressed in a mathematically precise form by Haag's theorem in `local quantum physics'. On the other hand, interacting quantum fields generate non-Fock representations of the commutation and anti-commutation relationships (CAR).

Haag Theorem

\begin{theorem} (The Haag theorem in quantum field theory)

Any canonical quantum field, ${\mathcal {F}}_{Qc}$ that for a fixed value of time $t$ is:

1. irreducible, and
2. has a cyclic vector, $\Omega$ that is

 #

• ${\mathcal {F}}_{Qc}$ has a Hamiltonian generator of time translations, and #
• it is unique as a translation-invariant state;

and also,

1. is unitarily equivalent to a free field in the Fock representation at the time instant, $t$,

is itself a free field . \end{theorem}