View source for Haag theorem

Haag's theorem ([[#References|[a1]]], see also [[#References|[a4]]]), in the context of canonical [[Quantum field theory|quantum field theory]], states in its generalized form [[#References|[a2]]] that a canonical quantum field which for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046040/h0460401.png" /> 1) is irreducible; 2) has a cyclic vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046040/h0460402.png" /> that is a) annihilated by the Hamiltonian (i.e., the generator of time translations) and b) unique as a translation-invariant state; and 3) is unitarily equivalent to a free field in Fock [Fok] representation at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046040/h0460403.png" />, is itself a free field.

Haag's theorem reflects the fact that canonical quantum dynamics is determined by the choice of the ground state [[#References|[a3]]] or  "vacuum" . Since by the assumptions the field shares the ground state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046040/h0460404.png" /> with a free one, it is free itself; interacting fields generate non-Fock representations of the CCR (cf. [[Commutation and anti-commutation relationships, representation of|Commutation and anti-commutation relationships, representation of]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Haag,   "On quantum field theories"  ''Danske Mat.-Fys. Medd.'' , '''29''' :  12  (1955)  pp. 17–112</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Emch,   "Algebraic methods in statistical mechanics and quantum field theory" , Wiley  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Streit,   "Energy forms: Schroedinger theory, processes. New stochastic methods in physics"  ''Physics reports'' , '''77''' :  3  (1980)  pp. 363–375</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.F. Streater,   A.S. Wightman,   "PCT, spin and statistics, and all that" , Benjamin  (1964)</TD></TR></table>