Quantum theory Axiomatic and algebraic quantum field theory Theorem

Haag's theorem

Builds on Wightman axioms Fock space Unitary representation Hilbert space

Statement

Let $\phi_0$ be the free Hermitian scalar field of mass $m > 0$ on Fock space $H_0$ , with vacuum $\Omega_0$ and conjugate momentum $\pi_0 = \partial_t \phi_0$ . Let $\phi$ be a Hermitian scalar field satisfying the Wightman axioms on a Hilbert space $H$ , with vacuum $\Omega$ and $\pi = \partial_t \phi$ . Assume the following.

At some time $t$ , the fields $\phi(t, \cdot)$ and $\pi(t, \cdot)$ exist as operator-valued distributions on $\R^3$ , and so do $\phi_0(t, \cdot)$ and $\pi_0(t, \cdot)$ .
There is a unitary operator $V \colon H_0 \to H$ with $V \phi_0(t, \mathbf{x}) V^{-1} = \phi(t, \mathbf{x})$ and $V \pi_0(t, \mathbf{x}) V^{-1} = \pi(t, \mathbf{x})$ .
The unitary representations $U_0$ and $U$ of the Euclidean group of $\R^3$ in the two theories satisfy $V U_0(g) V^{-1} = U(g)$ . The vectors $\Omega_0$ and $\Omega$ are, up to a scalar factor, the only vectors invariant under these representations.

Then $V \Omega_0 = c\, \Omega$ for some $c \in \C$ with $\abs{c} = 1$ , the two-point functions of $\phi$ and $\phi_0$ coincide, and $\phi$ is a free field of mass $m$ .

Remarks

The vacuum relation follows from the uniqueness of the invariant vectors. Then the Wightman functions of $\phi$ and $\phi_0$ agree at equal times. Lorentz covariance with analyticity in the forward tube extends the equality of the two-point functions to all points. A Wightman field whose two-point function is that of the free field of mass $m$ is itself free; this is the Jost–Schroer theorem.

The theorem says that the interaction picture of perturbation theory cannot exist in the way it is usually written. An interacting theory cannot live on the Fock space of the free theory with a unitary map that intertwines the fields at a fixed time. Practical calculations escape the theorem by using cutoffs in space or momentum, which break the Euclidean invariance of hypothesis 3, and by removing them only at the end.

Going deeper

For a student

The theorem is a consequence of the infinite number of degrees of freedom of a field. For finitely many degrees of freedom, the Stone–von Neumann theorem says that all regular irreducible representations of the canonical commutation relations are equivalent, but this fails in infinite dimension. Haag's theorem is one reason why rigorous constructions in constructive quantum field theory first introduce a finite volume and an ultraviolet cutoff. Its conclusion does not contradict the success of perturbation theory with renormalization, whose final formulas do not depend on the interaction picture.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Hilbert space concept
Hilbert sum concept
Tensor product of Hilbert spaces concept
Unitary representation concept
Wightman axioms concept
Fock space object
Haag's theorem statement

Tag	`6GF9ZVAJGW`
URI	`m2://6GF9ZVAJGW`
Kind	statement, theorem
Writing	W0 stub
MSC	81T05
Status	This copy of the site has no build data. Run `m2 status 6GF9ZVAJGW`.
About	Quantum field theory
Uses	Wightman axioms, Fock space, Unitary representation, Hilbert space
Same as	Wikidata Q1566372 Wikipedia Haag's theorem
Changed	3 November 2026