Rudolf Haag showed in 1955 that the interaction picture cannot be rigorously defined in quantum field theory, a result now commonly cited as Haag's Theorem. This is in stark contrast to the successes of perturbative quantum electrodynamics. In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Quantum electrodynamics (QED) is a quantum field theory of electromagnetism. ...

Citing the formulation used by Arageorgis:

• If two pure ground states are not equal, then they generate unitarily inequivalent irreducible representations.
• If two local quantum fields are unitarily equivalent at any given time, then both fields are free if one of them is free.

For unitary equivalence of bounded operators in Hilbert space, see spectral theorem. ... In mathematics, the term irreducible is used in several ways. ... Classically, a free field is a field described by linear partial differential equations which has a unique solution given initial data. ...

References

Arageorgis, A.: 1995, Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime, Ph.D. Thesis, Univ. of Pittsburgh.