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.

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.

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.