Lecture 1.

Wightman axioms.

David Kazhdan

Introduction. In this series of lectures we will present Wightman axioms and show how this set of axioms can be used to derive deep and unexpected results on behavior of quantum field theories. These axioms constitute a consistent mathematical theory which can serve for a mathematician as a gateway to the strange world of quantum field theories. In spite of the failure to describe the structure of gauge theories Wightman axioms are very useful for developing of an intuition on quantum field theories.

1.0 Setup and notations. Let V be a pseudo-Riemannian manifold of signature (1, d − 1). In these lectures we will only consider the case V = ℝ^{1,d−1} with constant quadratic form. We denote by ( , ): V × V → ℝ the corresponding quadratic form and will write v^{2} instead of (v, v) for v ∈ V. We will sometimes use decomposition V = ℝ × ℝ^{d−1}, where ℝ (time) and ℝ^{d−1} (space) are orthogonal subspaces equipped with constant positive (respectively negative) definite metrics. This decomposition is fixed throughout the lectures. We denote: V_{space} = \{v ∈ V |v^{2} < 0\}; and \overline{V}_{space} is the closure of V_{space}. The complement V − \overline{V}_{space} consists of 2 connected components, V_{+} and V_{−} (where V_{±} ⊂ ℝ^{±} × ℝ^{d−1}).

For m ∈ ℝ we put 𝒪_{m}^{±} = \{v ∈ \overline{V}_{±}|v^{2} = m^{2}, v ≠ 0\}. Let dv be the standard measure on V and let μ_{m} = Res _{𝒪_{m}^{+}} \frac{dv}{v^{2}−m^{2}} be the invariant volume form on 𝒪_{m}^{+}.

Let G = Spin(V) be the double (universal for d > 3) cover of connected component of the group SO(1, d − 1). The semidirect product P = G ⋉ V is called the Poincare group. It acts on V by isometries. We denote by τ ∈ G the nontrivial element of the kernel of the map G → SO(1, d − 1).

We will use the quadratic form to identify V^{∗} with V, but we will distinguish between vectors and covectors by reserving the letters v, w, etc. for vectors, and p, q, etc. for covectors.

Note that the cone V_{+} is self-dual (i.e. {p ∈ V |(p, v) > 0 ∀ v ∈ V_{+}} = V_{+}). For a function f on V we will denote by ℱ(f) its Fourier transform; so we have:

ℱ(f)(p) = \frac{1}{(2π)^{d/2}} ∫_{V} e^{−i(p,v)}f(v)dv.

Let U: V → Aut(ℋ) be a unitary representation of the vector group V in a Hilbert space ℋ.

Definition. Representation U is called positive if support of the corresponding spectral measure lies in \overline{V}_{+}.

If U: P → Aut(ℋ) is a unitary representation, then we say that U is positive if U _{V} is positive.

For a G-module H we will write H = H^{even} ⊕ H^{odd}, where H^{even} (respectively H^{odd}) is the space of invariants (respectively antiinvariants) of τ. We will endow H with the structure of superspace given by this decomposition.

Let R = R^{even} ⊕ R^{odd} be a finite dimensional real representation of G. Since P acts transitively on V and stabilizer of 0 is G, we can associate to R a P -equivariant vector bundle ℛ on V.

Let 𝒮^{R} = 𝒮_{even}^{R} ⊕ 𝒮_{odd}^{R} be the space of Schwartz sections of this bundle. (Note that our bundle is trivialized equivariantly with respect to translations, and 𝒮^{R} is identified with the space of smooth rapidly decreasing R-valued functions on V.) We will often abbreviate 𝒮 = 𝒮^{R}.

Finally we need some preparations on signs.

We say that a complex superspace H = H^{even} ⊕ H^{odd} is a superHermitian space if it is endowed with an even ℝ-bilinear pairing ( , ) : H × H → ℂ such that (λx, y) = (x, \overline{λ}y) = λ(x, y) and (x, y) = (−1)^{|x|·|y|}\overline{(y, x)}. Note that since ( , ) is even we have (x, y) = 0 for x ∈ H^{even}, y ∈ H^{odd}.

H is called superHilbert space if (H^{even}, ( , )) and (H^{odd}, −i( , )) are Hilbert spaces. For T ∈ End(H) the (super)adjoint operator T^{∗} is defined by: (T x, y) = (−1)^{|x|·|T |} (x, T^{∗}y). An operator is superHermitian symmetric if T^{∗} = T. We will often omit the word super.

Now we are ready to formulate:

1.1 Wightman axioms. Data:

1) A finite dimensional real representation ρ: G → Aut R.

2) A positive unitary representation U: P → Aut(ℋ) in a (super)Hilbert space ℋ.

3) A dense P -invariant subspace 𝒟 ⊂ ℋ, and a P -invariant vector Ω ∈ 𝒟, ∥Ω∥ = 1.

4) A linear map ϕ: 𝒮^{R} → End(𝒟), called the field map such that:

α) ϕ is P -equivariant where P acts on End(𝒟) by conjugation.

β) ϕ(f) is superHermitian-symmetric.

γ) For any ψ_{1}, ψ_{2} ∈ 𝒟 the map f → ⟨ψ_{1}, ϕ(f)ψ_{2}⟩ from 𝒮 to ℂ is continuous (𝒮 = 𝒮^{R}

is equipped with the Schwartz topology). δ) The space 𝒟 is generated by vectors ϕ(f_{1})ϕ(f_{2}), …, ϕ(f_{n})Ω, for f_{1}, …, f_{n} ∈ 𝒮. ε) (space-locality) If f_{1}, f_{2} ∈ 𝒮 are such that for any v_{1} ∈ supp(f_{1}), v_{2} ∈ supp(f_{2}) we have (v_{1} − v_{2}) ∈ V_{space} (such f_{1}, f_{2} are called space-like separated), then [ϕ(f_{1}), ϕ(f_{2})] = 0 where [ , ] stands for supercommutator.

Remark. It is often assumed that the space of V -invariants in ℋ is 1-dimensional (uniqueness of vacuum).

Such set of data satisfying the above properties is called QFT in Wightman axioms (or simply Wightman QFT).

1.2 Wightman’s functions. Let us denote: V_{n} = V × V × ⋯ × V (n times); ℛ_{n} = ℛ^{⊠n} (this is a bundle on V_{n}). The space of Schwartz sections of ℛ_{n} will be denoted by 𝒮_{n}.

For n ≥ 1 the Schwartz theorem and the property (γ) imply the existence of unique tempered ℂ-valued distribution 𝒲_{n} ∈ 𝒮′_{n} such that

𝒲_{n}(f_{1} × f_{2} × ⋯ × f_{n}) = ⟨Ω, ϕ(f_{1})ϕ(f_{2}) … ϕ(f_{n})Ω⟩

for f_{1}, …, f_{n} ∈ 𝒮.

These 𝒲_{n} are called Wightman functions, or correlation functions of the theory. (We will use the word “function” for a section of an appropriate vector bundle when it causes no confusion).

1.3 Reconstruction of QFT from Wightman functions. We are going to show that a QFT is uniquely determined by its Wightman functions. So we assume that a representation ρ of G and a collection of distributions 𝒲_{n} ∈ 𝒮_{n} are given, and that 𝒲_{n} are Wightman functions of some Wightman QFT (ρ; ℋ, 𝒟, Ω; ϕ). We want to reconstruct ℋ, 𝒟, Ω and ϕ.

Let us put 𝒮̃ = ⊕_{n} 𝒮_{n}, 𝒲 = Π_{n} 𝒲_{n} ∈ 𝒮̃′ (where we use the standard notation L′ for the

topological dual of a topological vector space L). The space 𝒮̃ carries an algebra structure: f_{1} · f_{2} = f_{1} ⊠ f_{2}. For f ∈ 𝒮_{n} we define f^{∗} ∈ 𝒮_{n} by: f^{∗}(v_{1}, …, v_{n}) = ±f(v_{n}, …, v_{1}) (the sign is determined by the super-rule as usually). Consider a bilinear form { , } on 𝒮̃ given by:

{f_{1}, f_{2}} = 𝒲(f_{1}^{∗} · f_{2}).

Claim. {f_{1}, f_{2}} = ⟨ϕ(f_{1})Ω, ϕ(f_{2})Ω⟩.

Proof. Clear.

We can recover ℋ as the Hilbert space completion of (𝒮̃ ⊗ ℂ, { , }′), where { , }′ is the unique Hermitian scalar product on 𝒮̃ ⊗ ℂ, such that { , }′ 𝒮̃ = { , }. Then

we can take 𝒟 to be the image of the obvious map 𝒮̃⊗ℂ → ℋ and define Ω as the image of 1 ∈ 𝒮̃ under this map. Since P acts on 𝒮̃ preserving { , } this action defines an action of P on ℋ. Finally, we get an algebra map ϕ: 𝒮̃ → End(𝒟).

Before we formulate the next Claim let us fix some notations. Let Δ: V → V_{n} be the diagonal embedding, and \overline{V}_{n} = V_{n}/Δ be the quotient space. Since ℛ_{n} is P -equivariant it defines a bundle \overline{ℛ}_{n} on \overline{V}_{n}. Let \overline{𝒮}_{n} be the space of Schwartz sections of \overline{ℛ}_{n}. For f ∈ 𝒮_{n} let \overline{f} ∈ \overline{𝒮}_{n} be its averaging over Δ(V), and let pr^{∗} be the dual map on distributions.

We identify \overline{V}_{n}→^{∼}V_{n−1} by:

(v_{1}, v_{2}, …, v_{n}) → (v_{2} − v_{1}, …, v_{n} − v_{n−1}).

Claim. 𝒲_{n} satisfy the following properties:

1) 𝒲_{n} is P -invariant for all n.

In particular, 𝒲_{n} is Δ(V)-invariant; hence 𝒲_{n} = pr^{∗}(W_{n}) for unique W_{n} ∈ \overline{𝒮}′_{n}.

2) supp ℱ(W_{n}) ⊂ (\overline{V}_{+})^{n−1}.

3) 𝒲_{n}(f^{∗}) = \overline{𝒲_{n}(f)}.

4) 𝒲(v_{1}, …, v_{i}, v_{i+1}, …, v_{n}) = ±𝒲(v_{1}, …, v_{i+1}, v_{i}, …, v_{n}) if v_{i}−v_{i+1}∈V_{space}. (Here the sign comes from the ℤ/2 grading on the space of test function 𝒮_{n} = 𝒮_{n}^{+} ⊕ 𝒮_{n}^{−}.)

5) 𝒲(f^{∗} · f) ≥ 0.

Proof. 1) and 3)–4) are clear. Let us prove 2).

We have to check that W_{n}(f) = 0 if for some i supp(ℱ(f)) lies in the domain {(v_{1}, .., v_{n}) ∈ V_{n}|v_{i} ∈ V − \overline{V}_{+}\}. For such f we can find g ∈ 𝒮_{n+1} with \overline{g} = f satisfying: supp ℱ(g) ⊂ \{(v_{1}, …, v_{n+1}) ∈ V_{n+1}|v_{1} + ⋯ + v_{i+1} ∈ V − \overline{V}_{+}\}. So we have to show that for such g: 𝒲_{n}(g) = 0. We can assume g = g_{1} ⊗ g_{2} for g_{1} ∈ 𝒮_{i+1}, g_{2} ∈ 𝒮_{n−i}, and supp ℱ(g_{1}) ⊂ \{(v_{1}, …, v_{i+1}) ∈ V_{i+1}|v_{1} + ⋯ + v_{i+1} ⊂ V − \overline{V}_{+}\}. We have 𝒲_{n}(g) = ⟨ϕ(g_{1})^{∗}Ω, ϕ(g_{2})Ω⟩. But the spectral support of ϕ(g_{1}^{∗})Ω is contained in the set {v_{1} + ⋯ + v_{i+1}|(v_{1}, …, v_{i+1}) ∈ supp ℱ(g_{1})\}. On the other hand, it is contained in \overline{V}_{+} by the positivity condition. Hence, it is empty, and ϕ(g_{1}^{∗})Ω = 0 ⇒ 𝒲_{n}(g) = 0.

Remark. By a standard theorem from analysis property 2) is equivalent to the following: 2′) W_{n} is a boundary value of an analytic function on V_{n−1} + i(V_{+})^{n−1}. It is not hard to see the following

Theorem. The above constructions provide a bijection between the following sets of data:

a) ρ; ℋ ⊃ 𝒟 ∋ Ω; ϕ satisfying Wightman’s axioms;

b) representation ρ of G and a collection of distributions {𝒲_{n}} satisfying properties 1) – 5) above.

Remark. Sometimes one considers theories satisfying weaker axioms. Namely, let 𝒮_{n}^{0} ⊂ 𝒮_{n} be the subspace of sections which vanish on diagonals together with all partial derivatives. A QFT satisfying weak Wightman’s Axioms is (a representation ρ together with) a collection of 𝒲_{n} ∈ (𝒮_{n}^{0})′ satisfying 1), 2′), 3), 4) above.

Note that an essential property 5) can not be formulated if 𝒲_{n} are defined on 𝒮_{n}^{0} only because f^{∗} · f does not lie in 𝒮^{0}.

1.4 Spin-statistics Theorem. This is a statement that our convention on the parity of the basic superspace R is the only right one. More precisely, we have:

Theorem. Consider the Wightman axioms with the sign in the axiom of space-locality changed. Any theory satisfying this modified set of axioms is trivial. (The field map ϕ = 0.)

Proof. Let us take a 2-dimensional subspace U ⊂ V of signature (1,1). Let σ′ be the element of SO(d − 1, 1) which acts by −1 on U and by 1 on the orthogonal complement; let σ be an element in the complexification of Spin(d − 1, 1) which projects to σ′. Then σ^{2} = τ is the spin element. Also note that the eigenspaces of σ in the space of R ⊗ ℂ are defined over ℝ. If τ acts trivially in R then this is obvious because σ is real; otherwise τ acts by −1 and we use the identity σ̄ = τσ to see that the ±i eigenspace of τ is invariant under the conjugation.

In the next lecture (§2.2) we will show that W_{n} extends to an analytic G_{ℂ}-equivariant function on a certain open domain in the complexification \overline{V}_{n} ⊗ ℂ. In particular W_{2} is analytic in the domain 𝒯̃ := {v | v^{2} ∉ ℝ^{≥0}}. This analytic function is also denoted W_{2}.

Let now r ∈ R be a σ eigenvector; let v be a point in U ⊗ ℂ ∩ 𝒯̃. We claim that ⟨r ⊗ r, W_{2}(v)⟩ = 0. Indeed we have ⟨r ⊗ r, W_{2}(v)⟩ = ⟨σ(r) ⊗ σ(r), W_{2}(−v)⟩ = spin(R) · ⟨r ⊗ r, W_{2}(−v)⟩ where spin(R) is the τ eigenvalue on R; here the first equality follows from σ-equivariance, and the second is obvious from σ^{2} = τ.

On the other hand by our assumption ⟨r_{1} ⊗ r_{2}, W_{2}(v)⟩ = −spin(R) · ⟨r_{2} ⊗ r_{1}, W_{2}(−v)⟩ for v ∈ V^{space}. Since V^{space} ⊂ 𝒯̃, we get ⟨r ⊗ r, W_{2}(v)⟩ = 0 for v ∈ V^{space}, and hence for all v ∈ 𝒯̃.

For any real valued Schwartz function f on V we have ⟨ϕ(f · r)Ω, ϕ(f · r)Ω⟩ = ∫ ⟨f · r × f · r, 𝒲_{2}⟩ = ∫ ⟨ℱ(f) · r × ℱ(f) · r, ℱ(𝒲_{2})⟩ = ∫ ℱ(f) · \overline{ℱ(f)} ⟨r ⊗ r, ℱ(W_{2})⟩. This shows that the distribution ⟨r ⊗ r, ℱ(W_{2})⟩ is nonnegative.

Now let us take v ∈ V^{+}. Then we get

0 = ⟨r ⊗ r, W_{2}(iv)⟩ = ∫ exp(−v, p) ⟨r ⊗ r, ℱ(W_{2})⟩ .

Since exp(−v, p) is positive this implies that ⟨r ⊗ r, ℱ(W_{2})⟩ = 0, hence (ϕ(f · r)Ω)^{2} = 0, so the theory is trivial.

We finish by some elementary properties of Wightman functions, and their relation to properties of the field map.

1.5 Mass spectrum of a theory. Let us denote Δ_{+}(p, m^{2}) = ℱ(δ_{𝒪_{m}^{+}} )(p).

Claim. For any scalar field theory ϕ: 𝒮(V) → End(𝒟) there exists a positive measure dρ on [0, ∞) such that:

W_{2}(p) = ∫ Δ_{+}(p, m)dρ(m) .

Proof. We have to show that ℱ(W_{2}) = ∫ dρ(m)δ_{𝒪_{m}^{+}} for a positive measure dρ. Since 𝒪_{m}^{+} is an orbit of the group G it is obvious that any G-invariant distribution supported in ⊔𝒪_{m}^{+} = \overline{V_{+}} can be written down in such form for some distribution ρ. In order to see that ρ is positive we have to show that ℱ(W_{2}) is positive.

For any f ∈ 𝒮 we have 𝒲_{2}(f ⊠ f) = ⟨ϕ(f)Ω, ϕ(f)Ω⟩ ≥ 0. Hence for g = ℱ(f) we have ℱ(𝒲_{2}) (g ⊠ g) ≥ 0.

We know that ℱ(𝒲_{2}) is concentrated on the anti-diagonal and

ℱ(𝒲_{2})(p, −p) = ℱ(W_{2})(p)

Now note that f ∈ 𝒮 iff g = ℱ(f) ∈ 𝒮^{ℂ} and g(−p) = \overline{g(p)}. Thus for any such g

ℱ(W_{2})(g · \overline{g}) = ℱ(𝒲_{2})(g ⊠ g) ≥ 0

which means that ℱ(W_{2}) is positive.

Definition. The mass spectrum of a QFT is support of the measure dρ. The mass of the theory is the infinum of its mass spectrum with 0 removed.

1.6 Asymptotics of Wightman functions. The next statement is a baby example of relation between analytical behavior of Wightman functions and properties of the field map. Estimates of this kind will be used essentially in the lecture on scattering theory (see lecture 4, §4.3, Proposition 1 and Claim 1).

Claim.

a) Uniqueness of vacuum in a QFT is equivalent to the equality:

lim_{λ→∞} 𝒲_{n}(v_{1}, …, v_{i}, v_{i+1} + λa, …, v_{n} + λa) = 𝒲_{i}(v_{1}, …, v_{i})𝒲_{n−i}(v_{i+1}, …, v_{n})

being true for all i, n and a such that a^{2} < 0.

b) Assume that mass of the theory is m_{0} > 0. Then W_{2}(v) ∼ e^{−m_{0}\sqrt{−v^{2}}} as v^{2} → −∞.

Proof. a) For any f_{1}, …, f_{n} ∈ 𝒮 we have

𝒲_{n} (f_{1}, …, f_{i}, T_{λa}(f_{i+1}), …, T_{λa}(f_{n})) = ⟨ϕ(f_{i}) … ϕ(f_{1})Ω, ϕ(T_{λa}(f_{i+1})) … ϕ(T_{λa}(f_{n}))Ω⟩

= ⟨ϕ(f_{i}) … ϕ(f_{1})Ω , U(T_{λa})(ϕ(f_{i+1}) … ϕ(f_{n})Ω)⟩

where T_{λa} stands for translation by λa. From the positivity condition it follows that the weak limit lim_{λ→∞} U(T_{λa}) exists as a bilinear form (i.e. all matrix coefficients converge), and

equals to the projection on ℋ^{V}.

Hence

lim_{λ→∞} ⟨ϕ(f_{i}) … ϕ(f_{1})Ω, U(T_{λa}) (ϕ(f_{i+1}) … ϕ(f_{n})Ω)⟩ = ⟨pr(ϕ(f_{i}) … ϕ(f_{1})Ω), pr(ϕ(f_{i+1}) … ϕ(f_{n})Ω)⟩

where pr is the projection on ℋ^{V}. The statement follows.

b) We have

W_{2}(v) = ∫_{V} ℱ(W_{2})(p)e^{i(v,p)}dp = ∫_{0}^{∞}dρ(m) ∫_{𝒪_{m}^{+}} e^{(iv,p)}dμ_{m}(p)

where μ_{m} is the invariant measure on 𝒪_{m}^{+}. For v ∈ ℝ^{d−1} − {0} it can be rewritten as:

const + ∫_{m_{0}}^{∞}dρ(m) ∫_{ℝ^{d−1}} e^{i(v,q)}dq/\sqrt{m^{2} − q^{2}}

(Remember that q^{2} = (q, q) is negative for q ∈ ℝ^{d−1} − {0}).

The integrand is analytic in the region Re(m_{0}^{2} −q^{2}) > 0; it also decreases as (p⃗)^{2} → −∞ for fixed p_{0}, where p = (p⃗, p_{0}) is decomposition into space and time components. Hence we can shift the contour of integration from ℝ^{d−1} to Γ ⊂ ℂ^{d−1} where Γ is ℝ^{d−1} + im_{0}v/\sqrt{−v^{2}} squeezed slightly towards ℝ^{d−1} in a small neighborhood of the singular point im_{0}v/\sqrt{−v^{2}} to avoid the singularity. Then the estimate becomes obvious.

We will usually assume that uniqueness of vacuum holds for the theory under consideration.