Haag theorem | Physics Library

0.1 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 ℱ_Qc shares the ground state with a free field ℱ₀, implies that 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).

0.2 Haag Theorem

Theorem 0.1. (The Haag theorem in quantum field theory)

Any canonical quantum field, ℱ_Qc that for a fixed value of time t is:

irreducible, and
has a cyclic vector, Ω that is
- ℱ_Qc 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, t,

is itself a free field.

References

[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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