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Haag theorem

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Haag's theorem ([a1], see also [a4]), in the context of canonical quantum field theory, states in its generalized form [a2] that a canonical quantum field which for fixed 1) is irreducible; 2) has a cyclic vector 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 , is itself a free field.

Haag's theorem reflects the fact that canonical quantum dynamics is determined by the choice of the ground state [a3] or "vacuum" . Since by the assumptions the field shares the ground state 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).

References

[a1] R. Haag, "On quantum field theories" Danske Mat.-Fys. Medd. , 29 : 12 (1955) pp. 17–112
[a2] G. Emch, "Algebraic methods in statistical mechanics and quantum field theory" , Wiley (1972)
[a3] L. Streit, "Energy forms: Schroedinger theory, processes. New stochastic methods in physics" Physics reports , 77 : 3 (1980) pp. 363–375
[a4] R.F. Streater, A.S. Wightman, "PCT, spin and statistics, and all that" , Benjamin (1964)
How to Cite This Entry:
Haag theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Haag_theorem&oldid=15077