Haag's theorem

Definition
Haag's theorem is a result in the mathematical foundations of quantum field theory (QFT) stating that the interaction picture—an intermediate representation between the Schrödinger and Heisenberg pictures—cannot be consistently defined for interacting relativistic quantum fields that satisfy the standard axioms of QFT. In essence, the theorem demonstrates that the Hilbert space of a free (non‑interacting) field theory is unitarily inequivalent to that of an interacting field theory, precluding a simple unitary transformation that maps one onto the other.

Historical Background
The theorem is named after the German mathematician and physicist Rudolf Haag, who first proved it in the early 1950s. It emerged from efforts to place quantum field theory on a rigorous mathematical footing, building upon earlier work by Wightman, Lehmann, Symanzik, and others who formalized the axiomatic (or Wightman) approach to QFT. Haag’s original proof appeared in a series of papers culminating in his 1955 article “On quantum field theories” and was later refined and disseminated in textbooks on mathematical physics.

Formal Statement (informal version)
Let $\mathcal{H}_0$ be the Hilbert space of a free scalar field satisfying the Wightman axioms, and let $\mathcal{H}$ be the Hilbert space of a purported interacting scalar field that also satisfies those axioms. If there exists a unitary operator $U$ such that

$$ \phi_{\text{int}}(x) = U , \phi_{0}(x) , U^{-1}, $$

where $\phi_{\text{int}}(x)$ and $\phi_{0}(x)$ are the interacting and free field operators respectively, then the interacting theory must be trivial; i.e., the S‑matrix is the identity and no scattering occurs. Consequently, a non‑trivial interacting QFT cannot be represented in the interaction picture within the same Hilbert space as the free theory.

Implications for Quantum Field Theory

1. Interaction Picture Limitations – The theorem shows that the conventional perturbative construction of the S‑matrix via the interaction picture, widely used in textbook QFT, lacks a rigorous justification for interacting relativistic fields.

2. Unitary Inequivalence – It exemplifies how distinct phases of a quantum field (free vs. interacting) can belong to unitarily inequivalent representations of the canonical commutation relations, a phenomenon also seen in infinite‑dimensional systems such as many‑body physics and statistical mechanics.

3. Alternative Formulations – Because of Haag’s theorem, rigorous approaches to QFT often employ the Heisenberg picture, Euclidean functional integrals, algebraic quantum field theory (AQFT), or constructive field theory methods that sidestep the interaction picture.

4. Renormalization and Effective Theories – In practice, physicists use renormalization and regularization schemes that effectively bypass the theorem, treating the interaction picture as a formal tool whose results are later validated by comparison with experiment.

Mathematical Context

• The theorem relies on the Wightman axioms for relativistic QFT, which include locality, relativistic covariance, spectrum condition (positive energy), and the existence of a unique vacuum state.
• It also utilizes results concerning unitary inequivalence of representations of the canonical commutation relations in infinite‑dimensional Hilbert spaces, a concept originally clarified by von Neumann and later by Haag and others.

Related Results

• Dyson’s argument (1952) concerning the divergence of perturbation series, which also highlights limitations of naïve perturbative expansions.
• The Araki–Woods theorem on the classification of quasifree representations of the canonical commutation relations, providing a broader framework for understanding inequivalent representations.
• Constructive quantum field theory efforts (e.g., the $\phi^4_2$ and $\phi^4_3$ models) succeed in building interacting models that respect the axioms while explicitly avoiding the interaction picture.

Criticism and Misinterpretations

• Some authors have mistakenly inferred that Haag’s theorem invalidates all perturbative calculations. In practice, perturbation theory remains a highly successful heuristic, with the theorem serving primarily as a caution regarding the underlying mathematical assumptions.
• The theorem applies to relativistic, Lorentz‑invariant theories with the full set of Wightman axioms; non‑relativistic quantum many‑body systems or theories on a lattice may not be subject to the same restrictions.

References

1. Haag, R. (1955). “On quantum field theories”. Il Nuovo Cimento 4, 842–861.
2. Streater, R. F., & Wightman, A. S. (1964). PCT, Spin and Statistics, and All That. Benjamin.
3. Bogoliubov, N. N., & Shirkov, D. V. (1959). Introduction to the Theory of Quantized Fields. Wiley.
4. Reed, M., & Simon, B. (1972). Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self‑Adjointness. Academic Press.
5. Strocchi, F. (2005). An Introduction to the Mathematical Structure of Quantum Field Theory. World Scientific.

See Also

• Interaction picture
• Wightman axioms
• Algebraic quantum field theory
• Unitary inequivalence
• Renormalization

This entry summarizes the established understanding of Haag’s theorem as presented in the physics and mathematics literature up to the present date.