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}