Difference between revisions of "Haag theorem"
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| − | 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 | + | {{TEX|auto}} |
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| + | 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 $ t $ | ||
| + | 1) is irreducible; 2) has a cyclic vector $ \Omega $ | ||
| + | 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 $ t $, | ||
| + | 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 $ \Omega $ | ||
| + | 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==== | ====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> | <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> | ||
Latest revision as of 19:42, 5 June 2020
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 $ t $
1) is irreducible; 2) has a cyclic vector $ \Omega $
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 $ t $,
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 $ \Omega $ 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://encyclopediaofmath.org/index.php?title=Haag_theorem&oldid=15077
Haag theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haag_theorem&oldid=15077