Editing Haag's theorem (section)

== Introduction ==
{{Main|Algebraic quantum field theory}}
Traditionally, describing a quantum field theory requires describing a set of operators satisfying the [[Canonical commutation relation|canonical (anti)commutation relations]], and a [[Hilbert space]] on which those operators act.  Equivalently, one should give a [[Representation of an algebra|representation]] of the [[free algebra]] on those operators, modulo the canonical commutation relations (the [[CCR and CAR algebras|CCR/CAR algebra]]); in the latter perspective, the underlying algebra of operators is the same, but different field theories correspond to different (i.e., [[Self-adjoint operator|unitarily inequivalent]]) representations.

Philosophically, the action of the CCR algebra should be [[Irreducible representation|irreducible]], for otherwise the theory can be written as the combined effects of two separate fields.  That principle implies the existence of a [[Cyclic vector|cyclic]] [[vacuum state]].  Importantly, a vacuum uniquely determines the algebra representation, because it is cyclic.

Two different specifications of the vacuum are common: the [[Ground state|minimum-energy]] [[Eigenvalues and eigenvectors|eigenvector]] of the field [[Hamiltonian (quantum mechanics)|Hamiltonian]], or the state annihilated by the [[number operator]] {{Math|''a''<sup>†</sup>''a''}}.  When these specifications describe different vectors, the vacuum is said to [[Vacuum polarization|polarize]], after the physical interpretation in the case of [[quantum electrodynamics]].{{Sfn|Fraser|2006}}

Haag's result explains that the same quantum field theory must treat the vacuum very differently when interacting vs. free.{{Sfn|Fraser|2006}}<ref name=":0">{{Cite book |last1=Bogoliubov |first1=N.&nbsp;N. |title=Introduction to Axiomatic Quantum Field Theory |last2=Logunov |first2=A.&nbsp;A. |last3=Todorov |first3=I.&nbsp;T. |publisher=W.&nbsp;A. Benjamin |year=1975 |editor-last=Fuling |editor-first=Stephen&nbsp;A. |series=Mathematical Physics Monographs 18 |location=Reading, MA |pages=548–562 |translator-last=Fuling |translator-first=Stephen&nbsp;A. |translator-last2=Popova |translator-first2=Ludmila&nbsp;G.}}</ref>