Haag theorem

Introduction

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 $ \mathcal{F}_{Qc}$ shares the ground state with a free field $ \mathcal{F}_{0}$ , implies that $ \mathcal{F}_{Qc}$ 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).

Haag Theorem

Theorem 0.1 (The Haag theorem in quantum field theory)  

Any canonical quantum field, $ \mathcal{F}_{Qc}$ that for a fixed value of time $ t$ is:

  1. irreducible, and
  2. has a cyclic vector, $ \Omega$ that is
    • $ \mathcal{F}_{Qc}$ has a Hamiltonian generator of time translations, and
    • it is unique as a translation-invariant state;

    and also,

  3. is unitarily equivalent to a free field in the Fock representation at the time instant, $ t$ ,

is itself a free field.

Bibliography

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)



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.