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 [[../CosmologicalConstant/|field]] ${ℱ}_{Qc}$ shares the ground state with a free field ${ℱ}_0$, implies that ${ℱ}_{Qc}$ is itself free (or admits a Fock [[../CategoricalGroupRepresentation/|representation]]). This basic assumption is expressed in a mathematically precise form by Haag's theorem in `[[../PureState/|local quantum physics]]'. On the other hand, interacting [[../CosmologicalConstant/|quantum fields]] generate non-Fock representations of the commutation and anti-commutation relationships ([[../RepresentationsOfCanonicalAntiCommutationRelationsCAR/|CAR]]).

Haag Theorem

\begin{theorem} (The Haag theorem in [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theory]])

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

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

 #

• ${ℱ}_{Qc}$ has a [[../Hamiltonian2/|Hamiltonian]] [[../Generator/|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}