Haag theorem
A canonical quantum dynamics (CQD) is determined by the choice of the physical (quantized) `vacuum' state (which is the ground state); thus, the assumption that a field
shares the ground state with a free field
, implies that
is itself free (or admits a Fock representation). This basic assumption is expressed in a mathematically precise form by Haag's theorem in `local quantum physics'.
On the other hand, interacting quantum fields generate non-Fock representations of the commutation and anti-commutation relationships (CAR).
Theorem 0.1 (The Haag theorem in
quantum field theory)
Any canonical quantum field,
that for a fixed
value of time
is:
- irreducible, and
- has a cyclic vector,
that is
-
has a Hamiltonian
generator
of time translations, and
- it is unique as a translation-invariant state;
and also,
- is unitarily equivalent to a free field in the Fock representation at the time instant,
,
is itself a free field.
- 1
-
R. Haag, ``On quantum field theories.'', Danske Mat.-Fys. Medd. , 29 : 12 (1955) pp. 17-112 .
- 2
-
[a2] G. Emch, ``Algebraic methods in statistical mechanics and quantum field theory.'' , Wiley (1972)
- 3
-
L. Streit, ``Energy forms: Schrödinger theory, processes. New stochastic methods in physics.'' Physics reports , 77 : 3 (1980) pp. 363-375.
- 4
-
R.F. Streater, and A.S. Wightman, ``PCT, spin and statistics, and all that''. , Benjamin (1964)
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