Haag's Theorem and Inequivalent Representations

Haag’s theorem does not say that interacting quantum field theory is impossible. It says that, under specific relativistic field, vacuum, spectrum, irreducibility, and equal-time assumptions, a unitary identification of a putatively interacting field with a free field forces too much equality—ultimately forcing the second field to be free as well. The obstruction concerns unitary equivalence of field representations, not abstract isomorphism of separable Hilbert spaces.

Required background. Wightman fields, domains, and axioms supplies the relativistic assumptions; the Wightman reconstruction theorem explains why a full vacuum hierarchy fixes a cyclic representation; and Fock space, vacuum, and particle number supplies the free representation.

Helpful background. Haag’s theorem: physical meaning and scope gives the physical interpretation.

A precise Haag–Hall–Wightman form

Let $\phi_1,\phi_2$ be neutral scalar fields with conjugate momenta $\pi_1,\pi_2$. Suppose at one fixed time $t_0$:

• each equal-time pair gives an irreducible representation of the canonical commutation relations;
• spatial translations and rotations are unitarily implemented;
• each representation has a unique normalized Euclidean-invariant vacuum;
• a unitary $V$ intertwines both fields and both conjugate momenta at $t_0$.

Part I of the Haag–Hall–Wightman theorem then shows that $V$ intertwines the Euclidean groups and maps one vacuum to the other up to phase. If both fields extend to Poincaré-covariant Wightman fields with invariant vacua and nonnegative energy, Part II uses analyticity to show equality of their vacuum expectation values through four fields. In the general case the conclusion is limited to the first four Wightman functions. If $\phi_1$ is a free field, equality of the two-point function plus the Jost–Schroer theorem—or a strengthened argument for all $n$-point functions—forces $\phi_2$ to be free.

This formulation, including the distinction between its two parts, is set out in Earman and Fraser 2006, § 3, author manuscript pp. 8–11. The Wightman analytic proof and generalizations are treated in Streater and Wightman 2016, § 4-5, pp. 161–167.

Several popular summaries are therefore too strong. One equal-time field identity without the momentum, irreducibility, covariance, or vacuum hypotheses is not this theorem. The general theorem does not say that every $n$-point function is equal. And “unitarily inequivalent” does not mean that the underlying infinite-dimensional separable Hilbert spaces are nonisomorphic; it means no unitary intertwines the specified field representations in the required way.

Why equal-time data spread to spacetime

The proof mechanism has two stages. At the chosen time, irreducibility and Euclidean covariance imply that

$$V U_1(\mathbf a,R)V^{-1}=U_2(\mathbf a,R), \qquad V\Omega_1=e^{i\alpha}\Omega_2.$$

The equality of equal-time vacuum functions follows. Poincaré covariance and the spectrum condition make the corresponding Wightman functions boundary values of analytic functions. Lorentz transformations and analytic uniqueness extend equality away from the equal-time slice, first to the low-order spacetime functions. If one two-point function is that of a free scalar, the Jost–Schroer result uses positivity, locality, and the Klein–Gordon equation in norm to conclude that the second field is free.

Thus “infinitely many degrees of freedom” explains why inequivalent representations are possible, but it is not the proof. The relativistic theorem needs its explicit field and analyticity assumptions.

The λφ⁴ interaction-picture test

Assume, for contradiction, that a fixed unitary $V$ identifies the free scalar field and its conjugate momentum with a proposed interacting $\lambda\phi^4$ field at $t_0$, and that both satisfy the hypotheses above. The theorem makes their two-, three-, and four-point vacuum functions equal. The free connected four-point function vanishes:

$$W^{\mathrm T}_{4,\mathrm{free}}=0,$$

because the free hierarchy is Gaussian. A genuinely interacting $\lambda\phi^4$ candidate is expected to have a nonzero connected four-point distribution. The assumptions therefore contradict the intended interaction: if the theorem applies, the candidate has the free correlation structure rather than a nontrivial $\lambda\phi^4$ interaction.

This is a conditional diagnostic, not a construction of four-dimensional $\lambda\phi^4$. The interaction-picture assumptions and their perturbative use are developed on the interaction picture and Dyson series. The theorem tells us that the formal Dyson identification cannot simultaneously be a global unitary equivalence of exact Wightman fields with all the stated hypotheses.

An independent check uses truncated functions. Equality of $W_2$ and $W_4$ makes

$$W_4^{\mathrm T} =W_4-W_2(12)W_2(34)-W_2(13)W_2(24)-W_2(14)W_2(23)$$

equal in the two theories. Since the free value is zero, no nonzero connected four-point interaction can survive.

The finite-cutoff adversary

Put the scalar field in finite volume and retain only finitely many oscillator modes. Stone–von Neumann uniqueness then returns unitary equivalence of regular irreducible canonical representations, and an interacting finite-dimensional Hamiltonian can act on the same Hilbert space as the free one. This is not a counterexample: exact Poincaré covariance, local field structure, and the infinite-volume/continuum degrees of freedom used by Haag-type results are absent. The unitary may fail to converge when the regulators are removed.

Likewise, a time-dependent cutoff intertwiner is not a counterexample to a theorem assuming an exact equal-time intertwiner between two covariant field systems and the accompanying vacuum conditions. One must test the hypotheses at the stage where the theorem is invoked.

Exercises

Why does equality of the first four Wightman functions suffice to rule out a candidate with a nonzero connected four-point function?

Solution

The connected four-point function is an algebraic combination of $W_4$ and products of $W_2$. If both $W_2$ and $W_4$ agree with the free values, the connected combinations agree. Wick’s theorem gives zero for the free connected four-point function, so the candidate’s value must also vanish.

References

• Earman, John, and Doreen Fraser. 2006. “Haag’s Theorem and Its Implications for the Foundations of Quantum Field Theory.” Erkenntnis 64: 305–344. Author manuscript.
• Hall, David, and Arthur S. Wightman. 1957. “A Theorem on Invariant Analytic Functions with Applications to Relativistic Quantum Field Theory.” Matematisk-fysiske Meddelelser, Det Kongelige Danske Videnskabernes Selskab 31 (5): 1–41. Catalog record.
• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.