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Haag theorem (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)



"Haag theorem" is owned by bci1.
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See Also: QFT, AQFT, QED, quantum field theories (QFT)

Other names:  Haag's theorem
Also defines:  cyclic vector, $\mathcal{F}_{Qc}$, \Omega, CQD, Fock representation of quantum free fields, canonical quantum dynamics
Keywords:  CAR, canonical quantum dynamics, $\mathcal{F}_{Qc}$, commutation and anti-commutation relationships, Fock representation, vacuum/ground state, canonical quantum dynamics, free fields, Hamiltonian generator, time translations, Fock representation of free fields, local quantum physics

Cross-references: generator, Hamiltonian, quantum field theory, CAR, quantum fields, local quantum physics, representation, field

This is version 16 of Haag theorem, born on 2009-01-29, modified 2009-01-31.
Object id is 447, canonical name is HaagTheorem.
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Classification:
Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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