2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationthe Wightman axioms and Haag's theorem

The axioms of quantum field theory — "the theory exists" said in seven conditions, and infinitely many degrees of freedom force a choice of representation

Physics textbooks calculate on the assumption that an interacting field theory exists (4-03). The mathematical question comes before that. Unless we first decide what must be made for us to say it "exists", we cannot judge whether it has been made. The axioms are the list of conditions for that judgement.

§3 of the problem statement defines a quantum field as 'an operator-valued generalized function on space-time obeying certain axioms', and continues 'Gårding and Wightman gave mathematically precise axioms for quantum field theories on R4 with a Minkowski signature, see [45]'. §4 takes the content of existence to be properties at least as strong as the axioms of [45] (the book of Streater–Wightman) and [35] (the two papers of Osterwalder–Schrader) (0-03) Known. What must be made is the following quadruple.

Example. In the quantum mechanics of one particle (2-05), $L^2(\mathbb R)$, $e^{-itH}$, the ground state and the position operator are these four, and they are at hand from the start. What is new in field theory is that the symmetry is the Poincaré group (translations and Lorentz transformations; 2-04), that fields have no value at a point (section 02), and that with infinitely many degrees of freedom $\mathcal H$ is not determined uniquely (section 07). The axioms on the imaginary-time side are treated in 4-10.

Placing the free field of 2-06 §08 over the whole of space, the vacuum expectation value of the square of the value at a point $\phi(\vec x)$ becomes the sum of the zero-point fluctuations of all modes, and diverges. Averaged with a function $f$, $\phi(f)=\int\phi(\vec x)f(\vec x)\,d^Dx$ ($D$ is the dimension of space) is finite Known.

$\hat f(\vec p)=\int f(\vec x)\,e^{-i\vec p\cdot\vec x}\,d^Dx$ is the Fourier transform: the amplitude of $f$ split into waves of wave number $\vec p$ (1-11). An $f$ of width $\varepsilon$ removes the modes with $|\vec p|\gtrsim1/\varepsilon$, and as $\varepsilon\to0$ (2) diverges. Example. For a normalised Gaussian ($\hat f=e^{-\varepsilon^2p^2/2}$), (2) grows logarithmically for $D=1$, as $e^{\varepsilon^2m^2/2}K_0(\varepsilon^2m^2/2)/(4\pi)$ ($K_0$ is from section 05), and like $1/(8\pi^2\varepsilon^2)$ for $D=3$.

The difference for $D=1$ approaches $(\ln2-\gamma/2)/(2\pi)=0.06439$ ($\gamma$ is Euler's constant) Computed. It is finite for every $\varepsilon\gt0$, and only $\varepsilon=0$ has no meaning. So we define a field as a linear map from test functions $f\in\mathcal S(\mathbb R^4)$ to operators $\phi(f)$. This is an operator-valued distribution. $\mathcal S(\mathbb R^4)$ is the space of functions that are smooth and that, together with their derivatives of every order, decay faster than any power of $|x|$ (1-11). We average over spacetime and not only over space because, for interacting fields, a field at fixed time is thought not necessarily to make sense. In theories that need renormalisation of the field strength (4-04), fixing the time and averaging only over space does not make the quantity corresponding to (2) finite Known.

The axioms are conditions imposed on each part of (1). The numbering differs from book to book, so in this book we call them W1 to W7. Table 2 is for a single real scalar field Known.

We fix the words of the table one line at a time. Dense: every element of $\mathcal H$ can be approached arbitrarily closely by elements of the set. Separable: the linear combinations of countably many vectors are dense. $U(a,\Lambda)$ is the unitary operator assigned to the pair of a spacetime translation $a$ and a Lorentz transformation $\Lambda$ (a linear transformation preserving $x^2$; 2-04 §02), and strongly continuous means that as $(a,\Lambda)$ approaches the identity, $U(a,\Lambda)\psi\to\psi$ for every $\psi$. A tempered distribution is a continuous linear functional on $\mathcal S(\mathbb R^4)$. Two points $x,y$ are spacelike separated if $(x-y)^2\gt0$, that is, if they cannot be connected by a light signal, and the surface $x^2=0$ is the light cone of the origin (2-04 §03).

From (3) follows $H\ge0$, and the mass operator $M=\sqrt{H^2-\vec P^{\,2}}$ makes sense (4-11 §05). The summary in §3 of the problem statement names W1, W2 ('positive energy, 0 ≤ H'), W3 ('unique up to a phase'), W5 and W6 Known. For spin-1/2 fields one uses the double cover of the Poincaré group (a group in which two elements correspond to each Lorentz transformation; the four-dimensional version of $SU(2)\to SO(3)$ in 1-02), and W6 becomes anticommutation between fermions, $\phi(f)\phi(g)+\phi(g)\phi(f)=0$. Example. The free field of mass $m$ satisfies all seven, and its spectrum is the union of $\{0\}$, the hyperboloid $p^0=\sqrt{\vec p^{\,2}+m^2}$ and $p^0\ge\sqrt{\vec p^{\,2}+4m^2}$ (Figure 4 of 2-06 §08).

The vacuum expectation values of products of fields are called Wightman functions. By W4 they are tempered distributions.

W1 to W7 translate into properties of $W_n$: Poincaré invariance, the spectral condition, locality (spacelike-separated neighbours may be swapped) and positivity. Positivity says that the length of $\Psi=\sum_n\phi(f_n)\Omega$ ($\phi(f_n)=\int f_n\,\phi(x_1)\cdots\phi(x_n)\,dx$) is not negative, and from $(\phi(x_1)\cdots\phi(x_n))^*=\phi(x_n)\cdots\phi(x_1)$

The reconstruction theorem Known. If $\{W_n\}$ satisfying these exist, a theory (1) with them as its Wightman functions can be made, unique up to unitary equivalence. The abstract of Wightman 1956 writes 'The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.' The construction is of the same type as in 4-10 §05: put (5) as an inner product on finite sequences of test functions $(f_0,f_1,\dots)$, divide out the vectors of length 0 and complete, and the result is $\mathcal H$. $\Omega=(1,0,0,\dots)$, and $\phi(g)$ is the operation that replaces each component by the function with $g$ attached in front ($(g\otimes f_1)(x_0,x_1)=g(x_0)f_1(x_1)$).

Analyticity follows from the spectral condition. Put $\psi=\phi(f)\Omega$ with a spacetime test function $f$, and let $\rho$ be the spectral measure of $H$ for $\psi$ (1-10 §05). Since $\Omega$ does not change under time evolution, the left-hand side is the two-point Wightman function with one of the fields shifted by time $t$ and averaged.

Since $E\ge0$, it extends analytically to $\mathrm{Im}\,t\lt0$, and at $t=-i\tau$ oscillation turns into decay. This is the imaginary-time correlation, and the abstract writes 'The vacuum expectation values are shown to be boundary values of analytic functions.' This direction is the rigorous form of the Euclideanisation of 4-02, and the reverse direction is 4-10 Known. Example. For the free field, $W_n$ is the sum of products of $W_2$ over all pairings, and reconstruction yields the Fock space of 2-06.

W6 is the requirement that fields at two points that cannot be connected by light do not disturb each other. The time order of such two points is reversed for some observers (2-04 §03), so if the order of the operators mattered there would be a contradiction. W6 does not say that the correlation is 0. We compare with the free field in one space dimension.

$\theta$ is the step function, and $J_0(z)=\frac1\pi\int_0^\pi\cos(z\sin\vartheta)\,d\vartheta$ and $K_0(z)=\int_0^\infty e^{-z\cosh u}\,du$ are Bessel functions. The $\theta$ in (8) is W6: outside the light cone the commutator is 0. But the correlation (9) is not 0 even outside, and decreases with distance as $e^{-mr}$. This decay is the spatial face of the mass gap (4-11 §06). Passing to Euclidean space, the spatial direction and the imaginary-time direction are related by rotation, so it decays at the same rate as the imaginary-time decay of (6) (4-11 §02). In four dimensions $W(0,\vec x)=mK_1(mr)/(4\pi^2r)$ Computed.

ComputedThe numbers in this figure were computed on this machine ((8)(9) and the series and integral for $Y_0$ evaluated in JavaScript; agreeing with SciPy to within $10^{-13}$).

On the light cone $\mathrm{Re}\,W$ diverges logarithmically. The divergence at $t=x=0$ is the same as the logarithm for $D=1$ in section 02, a sign that fields have no value at a point.

From the seven conditions alone, properties known from experience come out as theorems. The title of [45], 'PCT, Spin and Statistics and all That', refers to two of them. The PCT theorem says that the combination of spatial reflection, time reversal and the exchange of particles and antiparticles is a symmetry of every theory; the spin–statistics theorem says that, at spacelike-separated points, fields of integer spin commute and fields of half-integer spin anticommute Known. The third is clustering: under W3, when groups of fields are moved apart in a spatial direction, correlations split into products.

With a mass gap the convergence is exponentially fast. §5 of the problem statement writes 'An important consequence of the existence of a mass gap is clustering' (4-11 §06) Known. Example. For the free field, the difference in (10) is a sum of terms $e^{-mr}$ from (9).

Detour: the vacuum reaches everywhere

With finitely many oscillators, the irreducible representation of the commutation relations is unique (Stone–von Neumann; 2-06 §09). With infinitely many it is not. We look at two free fields with different masses. For a single oscillator, the number $n$ of quanta in the ground state of angular frequency $\omega'$, counted with the annihilation operator $a_\omega$ of $\omega$, and the overlap between the ground states are

KnownComputed. For a field on a ring of length $L$ (2-06 (14)), each mode $k=2\pi j/L$ has $\omega_k=\sqrt{k^2+m^2}$ and $\omega'_k=\sqrt{k^2+m'^2}$, and the vacuum is a product over modes, so

$n_k$ is the $n$ of (11) taken for mode $k$, and $N(L)$ is the expected number of quanta of mass $m$ contained in the vacuum of mass $m'$. Since $n_k\approx(m'^2-m^2)^2/(16k^4)$ decreases, for finite $L$ the sum is finite and the overlap is positive. Enlarging $L$, both the number of modes and $N(L)$ are proportional to $L$, and the overlap goes to 0 as $e^{-sL}$.

ComputedThe numbers in this figure were computed on this machine ((12) summed over $|k|\le60m$, with the rest filled in by the $k^{-4}$ tail; agrees with Table 4 to 6 digits).

Enlarging $L$, the overlap goes to 0. Precisely, in infinite volume there is no vector in the Fock space of $m$ corresponding to the vacuum of $m'$, and the two give unitarily inequivalent representations of the same commutation relations of the time-zero fields Known. From (12) one can read that the overlap is positive if and only if $\sum_kn_k\lt\infty$. A criterion of the same form is known for general Bogoliubov transformations (transformations that replace $a_k$ by a linear combination of $a_k$ and $a_{-k}^\dagger$; the relation between $a_\omega$ and $a_{\omega'}$ in (11) is an example) (attributed to Shale 1962). The infinite-volume density $\int\frac{d^Dk}{(2\pi)^D}n_k$ ($m'=1.5m$) is finite: $0.012692,\ 0.0049736,\ 0.0030929$ (in units of $m^D$) for space dimension $D=1,2,3$. Even for $D=3$, which corresponds to real spacetime (four-dimensional spacetime), the density is finite, and the breakdown comes from the volume alone. For $D=4$, where space is four-dimensional (spacetime five-dimensional), the density diverges logarithmically, and the overlap is 0 even in a finite box Computed. A figure for the three-dimensional lattice is in 6-11 §07.

Perturbation theory in physics takes the Fock space of the free field as its stage, splits $H=H_0+V$ ($H_0$ free, $V$ the interaction), and uses the picture of passing from the free vacuum to the interacting vacuum (the interaction picture). In section 07, even a mild change that only alters the mass gave a new vacuum that did not lie in the original Fock space.

Haag's theorem says this in general form Known. If the time-zero fields and conjugate momenta are unitarily related to those of the free field, and both theories have a unique vacuum invariant under spatial translations and rotations, then the first four Wightman functions agree with those of the free field, and a theory whose two-point function is the same as the free field's is the free field of the same mass (Haag 1955, Hall–Wightman 1957; the consolidated form is in Streater–Wightman, and a reading in Earman–Fraser 2006). The central lemma is elementary: in the Fock space of mass $m\gt0$, the only vectors invariant under all spatial translations are constant multiples of the vacuum.

The $n$-particle component $\psi_n$ vanishes off the surface where the total momentum is 0, and that surface has volume 0, so as an element of $L^2$ it is 0. Example. The vacuum of $m'$ in (12) is invariant under translations and differs from the free vacuum, so it has no place in the Fock space.

In practice there are two ways of handling this. Physics calculates with a finite box and an ultraviolet cutoff, and takes as answers only the quantities with the cutoff removed by renormalisation (4-04). Constructive field theory builds the Hamiltonian in a space with a cutoff and obtains a new representation in the limit (6-03). Particles are also made from the spectrum. §6 of the problem statement writes 'If m is an isolated eigenvalue of M, then one infers from the Wightman axioms and Haag–Ruelle scattering theory that asymptotic scattering states of an arbitrary number of particles exist' Known.

It is known that the gauge field $A_\mu$ itself does not become a field satisfying W1 to W7 (attributed to Strocchi 1967, 1970) Known. There are three reasons. Observables: $A_\mu$ changes under gauge transformations (3-01). §4 of the problem statement asks that local operators correspond to 'the gauge-invariant local polynomials in the curvature F and its covariant derivatives, such as Tr Fij Fkl (x)', and warns in footnote 1 that, because of renormalisation, there is no natural one-to-one correspondence between classical polynomials and quantum operators Known.

Positivity: treating $A_\mu$ as a field in a Lorentz-covariant gauge needs a space with an indefinite inner product, and W1 is broken. In the Gupta–Bleuler method (for the electromagnetic field), one takes from the large space the subspace of states satisfying a condition, and recovers positivity by dividing out the states of length 0 in it Known. For the non-abelian BRST method (4-08 §04), a construction of the same form (take a subspace and divide; BRST cohomology) is known at the level of perturbation theory (attributed to Kugo–Ojima 1979). At the non-perturbative level this construction is not established, because of Gribov copies (4-08 §05) Physics. The form of observables: Wilson loops (3-02) are quantities on curves and do not fit into W4, which averages fields at points (4-10 §09). §3 of the problem statement also mentions the Haag–Kastler framework, which places an algebra of operators on each region, and writes 'One hopes that the continued mathematical exploration of quantum field theory will lead to refinements of the axiom sets that have been in use up to now'.

Where things stand. §6 of the problem statement writes 'at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions' (an assessment from 2000). In two and three dimensions, interacting scalar fields satisfying the axioms ($\phi^4_2$, $\phi^4_3$ and others) have been constructed (6-03) Known. Three-dimensional Yang–Mills has not been constructed (6-06). The road that makes fields from the Euclidean side is the next chapter.

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