Editing Haag's theorem (section)

== Formal description ==
In its modern form, the Haag theorem has two parts:<ref name=":0" /><ref>{{Cite book |last=Emch |first=Gerard&nbsp;G. |title=Algebraic Methods in Statistical and Quantum Field Theory |publisher=Wiley Interscience |year=1972 |location=New York |pages=247–253}}</ref>

# If a quantum field is [[Free field|free]] and [[Euclidean group|Euclidean-invariant]] in the spatial dimensions, then that field's vacuum does not polarize.
# If two [[Poincare invariance|Poincaré-invariant]] quantum fields share the same vacuum, then their first four [[Wightman function]]s coincide.  Moreover, if one such field is free, then the other must also be a free field of the same [[Mass (physics)|mass]].

This state of affairs is in stark contrast to ordinary non-relativistic [[quantum mechanics]], where there is always a [[Self-adjoint operator|unitary equivalence]] between the free and interacting representations. That fact is used in constructing the [[interaction picture]], where operators are evolved using a free field representation, while states evolve using the interacting field representation. Within the formalism of [[quantum field theory]] (QFT) such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the quantum field theorist is confronted with the so-called ''choice problem'': One must choose the ‘right’ representation among an [[uncountable|uncountably-infinite]] set of representations which are not equivalent.