PlanetPhysics/Haag Theorem

Introduction

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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).

Haag Theorem

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\begin{theorem} (The Haag theorem in quantum field theory)

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

  1. irreducible, and
  2. has a cyclic vector, that is
 #
  • has a Hamiltonian generator of time translations, and #
  • it is unique as a translation-invariant state;

and also,

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

is itself a free field . \end{theorem}

All Sources

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[1] [2] [3] [4]

References

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  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: Schr\"odinger 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)