Week 8 — Free-Field Algebras and Weyl Operators

Block B gave us abstract modular theory. Block C turns the abstract machine onto an actual quantum field theory. The free scalar field is the right starting point: it is fully under control, has a Hilbert-space realization we can write down, and its local algebras already display every essential feature we will use in research-level work — including (next week) cyclic-separating vacua and (Week 10) a Bisognano–Wichmann theorem with an explicit modular operator. This week we build the free-field algebra as a Weyl algebra and identify its local subalgebras.

How to use this chapter

• In class: move in the order distribution $\to$ two-point function $\to$ Pauli–Jordan form $\to$ Weyl relations $\to$ quasifree state $\to$ local von Neumann algebra. This keeps bounded observables motivated rather than introduced by fiat.
• For self-study: reproduce the compact-bump symplectic calculation and derive the quasifree Weyl expectation from the covariance inequality. Keep the mostly-plus $i0$ prescription visible throughout.
• Instructor checkpoint: distinguish the global Fock von Neumann algebra from a local algebra, exact compact support from a Gaussian approximation, and a convex mixture of Gaussian states from a quasifree state.

0. Reading

Primary:

• Streater & Wightman, PCT, Spin and Statistics, and All That, ch. 3 (Wightman axioms; field algebra).
• Haag, Local Quantum Physics, ch. II §§5.2–5.4 (Weyl algebras and local nets — the standard reference).

Secondary:

• Bratteli & Robinson Vol. II, §5.2.3 (CCR algebra and quasi-free states; clean operator-algebra treatment).
• Reed & Simon, Methods of Modern Mathematical Physics, Vol. II, §X.7 (the free scalar field rigorously, with operator-domain care).

Optional research reading:

• Witten, “Notes on some entanglement properties of QFT,” arXiv:1803.04993, §§1–2 (modern algebraic introduction).
• Summers, “Yet more ado about nothing: the remarkable relativistic vacuum state,” arXiv:0802.1854 (vacuum properties in AQFT).
• Manuceau, “C*-algèbre de relations de commutation,” Ann. Inst. H. Poincaré (1968) (the Weyl algebra as a C*-algebra, foundational).

1. Why the free scalar?

The free scalar field is the simplest relativistic QFT for which many of our algebraic questions admit explicit answers. We use it as the testbed for Blocks C and D. Three properties make it the right starting point:

1. Fock-space realization. The field acts on a known Hilbert space — Fock space over the one-particle space $L^2({\mathbb{R}}^d,d\mu )$ — and one can write every operator in terms of creation and annihilation operators in modes.

2. Weyl-algebra structure. The bounded local algebras $\mathcal{A}(\mathcal{O})$ are generated by exponentiated smeared fields — the Weyl operators $W(f)=e^{i\varphi (f)}$ — and the multiplicative law between them is encoded in a symplectic form $\sigma$ on test functions. The entire algebraic content of the free scalar is in this one bilinear form.

3. Computable modular theory. For wedge regions, the modular operator is known explicitly via Bisognano–Wichmann (Week 10). For the free massless scalar, Hislop–Longo determine the double-cone modular structure in arbitrary spacetime dimension; later modular-covariance theorems extend the geometric picture to conformal nets under their stated hypotheses. For generic bounded regions, the modular operator is not explicit, although its existence and, with separate hypotheses, its type can still be controlled (Week 12). This makes the free scalar a laboratory in which abstract claims can be checked against concrete models without pretending that every region has a known geometric flow.

The cost is that the free scalar misses interactions and gauge structure. The benefit is that everything in the rest of the course can be tested on it as a controlled limit: the Bisognano–Wichmann boost identification, the type III${}_1$ structure of local algebras (under suitable hypotheses), the crossed product (Block D), the Bell–CHSH violation between Rindler wedges (Week 11), and the Sem II free-field mini-calculations for Witten 2022 and Ahmad–Jefferson.

2. The free scalar field

We work on $(d+1)$-dimensional Minkowski space ${\mathbb{R}}^{1,d}$ with signature $(-,+,\dots ,+)$, metric $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$, and coordinates $x^{\mu }=(x^0,x^1,\dots ,x^d)$. Mass parameter $m\ge 0$.

2.1 Mode expansion

Definition 2.1. The free scalar field of mass $m$ is the operator-valued distribution

$$\varphi (x)=\int \frac{d^{d}k}{(2\pi {)}^d\sqrt{2{\omega }_k}}\left[a_k{e}^{-i{\omega }_k{x}^0+i\vec{k}\cdot \vec{x}}+a_k^{†}{e}^{+i{\omega }_k{x}^0-i\vec{k}\cdot \vec{x}}\right],{\omega }_k=\sqrt{|\vec{k}{|}^2+m^2},$$

acting on Fock space $\mathcal{F}={⨁}_{n=0}^{\infty }{\mathrm{Sym}}^n(L^2({\mathbb{R}}^d))$, with $[a_k,a_{k^{\prime }}^{†}]=(2\pi {)}^d{\delta }^d(\vec{k}-\vec{k}{}^{\prime })$ and a vacuum ${\Omega }_0=|0⟩$ satisfying $a_k|0⟩=0$.

This is a distribution in $x$, not a genuine operator — squaring it gives ultraviolet divergences (because the integrand $1/{\omega }_k$ does not give a normalizable mode at large $|\vec{k}|$). The operators we will actually use are smeared versions.

2.2 Smeared fields

Definition 2.2. For a real-valued test function $f\in \mathcal{S}({\mathbb{R}}^{1,d})$ (Schwartz, i.e., rapidly decreasing smooth), the smeared field is

$$\varphi (f):=\int d^{d+1}xf(x)\varphi (x).$$

$\varphi (f)$ is a (densely defined) self-adjoint operator on $\mathcal{F}$ for real $f$, with the vacuum in its domain.

Lemma 2.3. [Stated only — refs: Reed–Simon Vol. II §X.7.] The smeared fields $\varphi (f)$ are essentially self-adjoint on the dense subspace of Fock-space vectors with finitely many particles and Schwartz wavefunctions. The closure (still denoted $\varphi (f)$) is self-adjoint.

In computational practice, we work with the dense domain of “nice” vectors and only worry about closures and domain technicalities at the very end. The Weyl-operator construction below sidesteps the domain issue entirely by exponentiating.

2.3 Properties of $\varphi (f)$

The smeared field is:

• linear in $f$: $\varphi (\alpha f+\beta g)=\alpha \varphi (f)+\beta \varphi (g)$ for $\alpha ,\beta \in \mathbb{R}$ (with conjugation rules for complex $f$).
• self-adjoint: $\varphi (f{)}^{\ast }=\varphi (f)$ for real $f$.
• continuous in $f$: $\varphi (f_n)\Omega \to \varphi (f)\Omega$ in norm when $f_n\to f$ in Schwartz topology.
• Klein–Gordon: $\varphi (\square f-m^{2}f)=0$ as an operator identity (the equation of motion lives in the test-function space).

3. Two-point function and symplectic form

3.1 The Wightman function

Definition 3.1. On the complexified test-function space, the Wightman two-point form is

$$W(f,g):=⟨0|\varphi (f{)}^{\ast }\varphi (g)|0⟩=⟨0|\varphi (\overline{f})\varphi (g)|0⟩.$$

For real $f$ and $g$, this is the usual smeared Wightman two-point function. Writing the adjoint explicitly explains why the complexified form is conjugate-linear in its first slot and linear in its second.

By explicit computation in Fock space (use the mode expansion and $⟨0|a_k{a}_{k^{\prime }}^{†}|0⟩=(2\pi {)}^d{\delta }^d(\vec{k}-\vec{k}{}^{\prime })$):

$$W(f,g)=\int \frac{d^{d}k}{(2\pi {)}^{d}2{\omega }_k}\overline{f_{+}(\vec{k})}{g}_{+}(\vec{k}),$$

where, consistently with Appendix B, we use the Minkowski Fourier convention

$$\tilde{f}(p):=\int d^{d+1}xf(x)e^{ip\cdot x},p\cdot x=-p^0{x}^0+\vec{p}\cdot \vec{x},$$

and name the positive-frequency one-particle wavefunction

$$f_{+}(\vec{k}):=\int d^{d+1}xf(x)e^{+i{\omega }_k{x}^0-i\vec{k}\cdot \vec{x}}=\tilde{f}(-{\omega }_k,-\vec{k}).$$

This is a positive sesquilinear form on the complexified test-function space: $W(f,f)=\int d^{d}k|f_{+}(\vec{k}){|}^2/((2\pi {)}^{d}2{\omega }_k)\ge 0$.

3.2 The symplectic form

Definition 3.2 (Convention). Define the symplectic form $\sigma (f,g)$ on real test functions by

$$[\varphi (f),\varphi (g)]=i\sigma (f,g)\cdot 1,$$

where the commutator is a $c$-number on Fock space. Equivalently, $\sigma (f,g)=2\mathrm{Im}W(f,g)$. Concretely,

$$\sigma (f,g)=\int d^{d+1}x{d}^{d+1}yf(x)\Delta (x-y)g(y),$$

where $\Delta (x-y):=-i⟨0|[\varphi (x),\varphi (y)]|0⟩$ is the Pauli–Jordan distribution. It is antisymmetric and supported in the causal cone: it vanishes at spacelike separation. In special cases, such as the massless scalar in $3+1$ dimensions, Huygens’ principle sharpens the support to the null cone itself. The symplectic form is real, bilinear, antisymmetric in $(f,g)$, and vanishes when $\mathrm{supp}(f)$ and $\mathrm{supp}(g)$ are spacelike separated.

This is the Bratteli–Robinson convention. With it, the Weyl relation below takes the form $W(f)W(g)=e^{-i\sigma (f,g)/2}W(f+g)$; with the alternative convention ${\sigma }^{\prime }=\sigma /2$ the relation reads $W(f)W(g)=e^{-i{\sigma }^{\prime }(f,g)}W(f+g)$. Pick one and stay with it; we use the B-R convention throughout.

3.3 Why this symplectic form?

The Pauli-Jordan distribution $\Delta (x-y)$ is the locality-encoding object of the free scalar field. It tells us:

• Spacelike vanishing $\Rightarrow$ commutativity of $\varphi (x)$ and $\varphi (y)$ at spacelike separation;
• Causal-cone support $\Rightarrow$ disturbances propagate causally;
• Antisymmetry $\Rightarrow$ the algebra has a symplectic (not Riemannian) structure.

In the abstract Weyl-algebra construction below, all the geometric information about the free scalar field is encoded in $\sigma$.

Physical picture. The symplectic form is not a quantum object at all — it is the classical phase space structure of the field, in covariant clothing. Recall that the space of real solutions of the Klein–Gordon equation carries the (covariant phase space) symplectic form $\sigma ({\phi }_1,{\phi }_2)={\int }_{\Sigma }({\phi }_1{\partial }_{\mu }{\phi }_2-{\phi }_2{\partial }_{\mu }{\phi }_1)d{\Sigma }^{\mu }$, independent of the Cauchy slice $\Sigma$ by the equations of motion. A test function $f$ determines a solution $\Delta f$ (the Pauli–Jordan propagator applied to $f$), and under this map the smearing form $\sigma (f,g)$ of Definition 3.2 is the classical symplectic form evaluated on the corresponding solutions. So the quantization recipe is exactly canonical: take the classical phase space $(V,\sigma )$, impose $[\varphi (f),\varphi (g)]=i\sigma (f,g)$ — Poisson brackets $\to$ commutators — and exponentiate. Everything relativistic (causal propagation, spacelike commutativity) is inherited from the support properties of the classical propagator. The quantum theory adds exactly one ingredient beyond $(V,\sigma )$: the choice of state (§7), which is where $\hslash$, particles, and temperature enter.

3.4 Example: 2D massless scalar

Infrared qualification. The ordinary massless scalar field in $1+1$ dimensions has a zero-mode obstruction: its Poincaré-invariant vacuum two-point function is not a positive distribution on the full Schwartz space. The commutator distribution below is well defined, but whenever we use the vacuum state we must either (i) restrict to smearings with $\int f=0$, (ii) work with the derivative/current field, or (iii) introduce an infrared regulator. We will use option (i) or (ii) unless a regulator is stated.

In $1+1$ dimensions with $m=0$, the Pauli–Jordan distribution is

$$\Delta (x-y)=-\frac{1}{2}\mathrm{sgn}(x^0-y^0)\Theta \left(-(x-y{)}^2\right),$$

i.e., $-\frac{1}{2}$ inside the future light cone of $y$, $+\frac{1}{2}$ inside the past light cone of $y$, and zero outside. (We use $(x-y{)}^2:=-(x^0-y^0{)}^2+(x^1-y^1{)}^2$, so $-(x-y{)}^2>0$ is the timelike region.)

Verifying spacelike vanishing. For $x,y$ with $(x-y{)}^2>0$ (spacelike separated), $\Theta (-(x-y{)}^2)=0$, so $\Delta (x-y)=0$. ✓

Smearing. For two real compactly supported bumps $f_R,f_L$ in the right and left Rindler wedges, respectively, $\sigma (f_R,f_L)=0$. For $f_R,g_R$ both in the right wedge, $\sigma (f_R,g_R)$ is generally nonzero and is the input to the Weyl-algebra structure below.

3.5 Example: 4D massless scalar

In $3+1$ dimensions with $m=0$,

$$W(x-y)=\underset{ϵ\downarrow 0}{\lim }\frac{1}{4{\pi }^2\left[-(x^0-y^0-iϵ{)}^2+|\vec{x}-\vec{y}{|}^2\right]},\Delta (x-y)=-\frac{1}{2\pi }\mathrm{sgn}(x^0-y^0)\delta ((x-y{)}^2),$$

where the $iϵ$ prescription is written so that at equal times $W(0,\vec{x})=1/(4{\pi }^2|\vec{x}{|}^2)>0$. The sign of $\Delta$ follows from our convention $\Delta =-i⟨[\varphi ,\varphi ]⟩$ and the mostly-plus metric. In four massless dimensions the Pauli–Jordan distribution is supported exactly on the null cone. Smearing against test functions gives a finite distributional pairing.

3.6 Example: 2D massive scalar

In $1+1$ dimensions with $m>0$,

$$\Delta (x-y)=-\frac{1}{2}\mathrm{sgn}(x^0-y^0)J_0\left(m\sqrt{-(x-y{)}^2}\right)\Theta (-(x-y{)}^2),$$

where $J_0$ is the Bessel function. Inside the causal cone, the Pauli–Jordan distribution oscillates with amplitude $\sim 1/\sqrt{m\sqrt{-(x-y{)}^2}}$ at large timelike separation. The important distinction is now explicit: the support fills the causal interior rather than lying only on the null boundary.

4. Weyl operators

The smeared fields $\varphi (f)$ are unbounded. For modular theory we want bounded generators of the local algebras. The standard trick: exponentiate.

4.1 Definition

Definition 4.1. For real $f\in \mathcal{S}({\mathbb{R}}^{1,d})$, the Weyl operator is

$$W(f):=\exp \left(i\varphi (f)\right).$$

$W(f)$ is unitary (since $\varphi (f)$ is self-adjoint), hence bounded with $\Vert W(f)\Vert =1$.

The Weyl operators are bounded, so they generate honest C*-algebras; products of them generate von Neumann algebras after WOT-closure (the closure that produces local-algebra structure).

Physical picture. Exponentiation is more than a domain trick. (1) Operationally, $W(f)$ is a displacement operator: acting on the vacuum it produces a coherent state — the quantum state closest to the classical field configuration $\Delta f$. The Weyl generators are “switch on a classical source $f$, coherently,” which is the most physical family of local operations available in a free theory. (2) Informationally, an unbounded $\varphi (f)$ is not a single measurable quantity but a whole spectral family; the unitary $W(\alpha f)=e^{i\alpha \varphi (f)}$ packages all of it in bounded form, and the parameter family $\alpha \mapsto W(\alpha f)$ recovers $\varphi (f)$ by Stone’s theorem when needed. (3) Structurally, boundedness is what allows the C*/von Neumann machinery of Weeks 1–7 — states, GNS, modular theory — to be applied verbatim: those theorems are theorems about bounded operators. The price of exponentiation is that the linear structure of the field is hidden inside the phase $e^{-i\sigma /2}$ of the Weyl relation; the bargain is that the entire theory becomes a C*-algebra with the phase as its only relation.

4.2 The Weyl relation

Theorem 4.2 (Weyl relation). [Proved.] For real $f,g\in \mathcal{S}({\mathbb{R}}^{1,d})$,

$$W(f)W(g)=e^{-i\sigma (f,g)/2}W(f+g).$$

Proof. On the analytic-vector subspace, the Baker–Campbell–Hausdorff identity gives

$$e^X{e}^Y=e^{X+Y+\frac{1}{2}[X,Y]+\cdots },$$

where the higher commutators vanish when $[X,Y]$ is central. Apply with $X=i\varphi (f)$, $Y=i\varphi (g)$. Then $[X,Y]=-[\varphi (f),\varphi (g)]=-i\sigma (f,g)\cdot 1$, so $\frac{1}{2}[X,Y]=-\frac{i}{2}\sigma (f,g)\cdot 1$, and

$$e^{i\varphi (f)}{e}^{i\varphi (g)}=e^{i\varphi (f)+i\varphi (g)-i\sigma (f,g)/2}=e^{-i\sigma (f,g)/2}{e}^{i\varphi (f+g)},$$

using $\varphi (f)+\varphi (g)=\varphi (f+g)$ on the analytic-vector subspace. Extending by closure gives the operator identity on Fock space. $\square$

4.3 Consequences

1. $W(0)=1$, $W(f{)}^{-1}=W(-f)=W(f{)}^{\ast }$.

2. Conjugation by phase:

$$W(f)W(g)W(f{)}^{\ast }=e^{-i\sigma (f,g)}W(g).$$

Weyl operators conjugate one another by phase factors determined by $\sigma$.

3. Spacelike commutativity: if $\sigma (f,g)=0$ (in particular if $f,g$ have spacelike-separated support), then $W(f)W(g)=W(f+g)=W(g)W(f)$, so $W(f)$ and $W(g)$ commute.

4. Continuity in the Fock representation: if $f_n\to f$ in Schwartz topology, then their positive-frequency one-particle wavefunctions converge in norm, and the regular Weyl representation gives $W(f_n)\to W(f)$ strongly (hence weakly). Stone’s theorem controls $\alpha \mapsto W(\alpha f)$ for a fixed $f$; it is not, by itself, the argument for varying $f$.

5. Vacuum expectation (Gaussian): for real $f$,

$${\omega }_0(W(f)):=⟨0|W(f)|0⟩=e^{-W(f,f)/2}.$$

This is the Gaussian-state character formula. Write $\varphi (f)={\varphi }_{-}(f)+{\varphi }_{+}(f)$, where ${\varphi }_{-}$ creates and ${\varphi }_{+}$ annihilates. Since their commutator is a scalar, BCH gives

$$W(f)=e^{i{\varphi }_{-}(f)}{e}^{i{\varphi }_{+}(f)}{e}^{-W(f,f)/2}.$$

The vacuum sandwich then gives the result. The order of the creation and annihilation exponentials matters.

The Weyl operators are the fundamental bounded generators of the free-field algebra. Their algebraic skeleton is captured entirely by the symplectic form $\sigma$.

4.4 The CCR algebra

Abstractly, fixing a real symplectic vector space $(V,\sigma )$, the canonical commutation relation (CCR) algebra $\mathcal{W}(V,\sigma )$ is the unital C*-algebra generated by symbols $W(f)$, $f\in V$, with relations

$$W(f{)}^{\ast }=W(-f),W(f)W(g)=e^{-i\sigma (f,g)/2}W(f+g).$$

For the free scalar field on Minkowski space, $V$ is the space of real-valued Schwartz functions modulo the Klein–Gordon equation (i.e., test functions that integrate to zero against on-shell solutions), and $\sigma$ is the Pauli–Jordan form.

Theorem 4.3 (Slawny). [Stated only — refs: Bratteli–Robinson Vol. II §5.2.3; Manuceau 1968.] Up to isomorphism, there is a unique simple C-algebra $\mathcal{W}(V,\sigma )$ with the Weyl relations, provided $\sigma$ is non-degenerate on $V$.*

This uniqueness is the C*-algebraic statement that every quasi-free representation of the free scalar is fundamentally the same algebra, differing only in the choice of state.

4.5 The vacuum state and its GNS representation

The Fock vacuum state on $\mathcal{W}(V,\sigma )$ is the Gaussian state with covariance given by the symmetric part of the Wightman function:

$${\omega }_0(W(f))=\exp \left(-\frac{1}{2}\mathrm{Re}W(f,f)\right).$$

The GNS representation of this state is the Fock-space representation of §2: the Hilbert space is $\mathcal{F}$, the cyclic vector is $|0⟩$, and the Weyl operators act as $e^{i\varphi (f)}$. This is the Fock vacuum representation. For the full non-degenerate symplectic space this representation is irreducible, so the global von Neumann closure is $\mathcal{B}(\mathcal{F})$. The local closures $\mathcal{A}(\mathcal{O})$ are very different: under the hypotheses discussed in Week 12, they are type III${}_1$ factors.

Different choices of quasi-free state give different (generally inequivalent) GNS representations — this is the failure of Stone–von Neumann in infinite dimensions, and it is why algebra-level statements about the free scalar are more robust than Hilbert-space-level statements.

5. Local algebras

The Weyl algebra has a net structure indexed by spacetime regions.

5.1 Definition

Definition 5.1. For an open region $\mathcal{O}\subset {\mathbb{R}}^{1,d}$, the local Weyl algebra is

$$\mathcal{W}(\mathcal{O}):={\mathrm{C}}^{\ast }\left(W(f):\mathrm{supp}(f)\subset \mathcal{O}\right)\subset \mathcal{W}(V,\sigma ),$$

i.e., the C*-subalgebra generated by Weyl operators with test-function support in $\mathcal{O}$. The local von Neumann algebra (in the Fock-space representation) is the WOT-closure:

$$\mathcal{A}(\mathcal{O}):=\mathcal{W}(\mathcal{O}{)}^{\prime \prime }.$$

5.2 Properties

The local-net data $\{\mathcal{O}\mapsto \mathcal{A}(\mathcal{O})\}$ satisfies (most of) the Haag–Kastler axioms (Week 9):

• Isotony: ${\mathcal{O}}_1\subset {\mathcal{O}}_2\Rightarrow \mathcal{A}({\mathcal{O}}_1)\subset \mathcal{A}({\mathcal{O}}_2)$. [Proved here, immediate from the definition.]

• Locality: ${\mathcal{O}}_1,{\mathcal{O}}_2$ spacelike $\Rightarrow$ $[\mathcal{A}({\mathcal{O}}_1),\mathcal{A}({\mathcal{O}}_2)]=0$. [Proved here.] This follows from the Weyl relation and the spacelike vanishing of $\sigma$: $W(f)W(g)=W(g)W(f)$ when $\sigma (f,g)=0$, and commuting generators give commuting algebras. The argument extends from generators to the full algebra by linearity, multiplication, and WOT-continuity.

• Covariance: the Poincaré group acts on the test-function space ($f\mapsto f\circ {\Lambda }^{-1}$ for $\Lambda \in {\mathcal{P}}_{+}^{\uparrow }$) and hence on the Weyl algebra by automorphisms, mapping $\mathcal{A}(\mathcal{O})$ to $\mathcal{A}(\Lambda \mathcal{O})$. [Sketched — uses the unitary implementer of $\Lambda$ on Fock space, which exists because the vacuum is Poincaré-invariant and the Weyl algebra is generated by smeared fields.]

• Vacuum invariance: $U(\Lambda )|0⟩=|0⟩$ for $\Lambda \in {\mathcal{P}}_{+}^{\uparrow }$, by construction of the Fock vacuum.

• Spectral condition: the joint spectrum of the translation generators lies in the closed forward cone. Each one-particle momentum $({\omega }_k,\vec{k})$ is future-directed on shell, and finite sums of such momenta remain in the forward cone; Fock-space closure gives the full statement. Positivity of $P^0$ alone would be weaker.

These five properties are the Haag–Kastler axioms for a local net of algebras; they are introduced systematically next week.

5.3 Locality made explicit

For Weyl operators $W(f),W(g)$ with $\mathrm{supp}(f)\subset {\mathcal{O}}_1$, $\mathrm{supp}(g)\subset {\mathcal{O}}_2$, and ${\mathcal{O}}_1,{\mathcal{O}}_2$ spacelike separated:

$$\sigma (f,g)=\int \int f(x)\Delta (x-y)g(y)d^{d+1}x{d}^{d+1}y=0$$

(because $\Delta (x-y)=0$ for spacelike $x-y$, and every pair $x\in \mathrm{supp}(f),y\in \mathrm{supp}(g)$ is spacelike). Hence

$$W(f)W(g)=e^{-i\cdot 0/2}W(f+g)=W(f+g)=W(g)W(f).$$

For arbitrary supports the Weyl relation gives $W(f)W(g)=e^{-i\sigma (f,g)}W(g)W(f)$. Spacelike separation is the case $\sigma (f,g)=0$, where the phase is trivial and the operators commute.

6. Worked example: 2D massless scalar in two wedges

Take $d=1$ (so $(d+1)=2$), $m=0$. The right and left Rindler wedges are

$$W_R=\{x:x^1>|x^0|\},W_L=\{x:x^1<-|x^0|\}.$$

These are spacelike separated, and the same one-parameter boost subgroup

$$x^0\mapsto \cosh (\eta )x^0+\sinh (\eta )x^1,x^1\mapsto \sinh (\eta )x^0+\cosh (\eta )x^1$$

preserves each wedge separately for every $\eta \in \mathbb{R}$. The opposite orientation that later appears for the commutant is a modular-time statement, not a different Lorentz transformation needed to preserve $W_L$.

6.1 Test functions

An exact localization argument needs compact support: a Gaussian is never supported inside a wedge. Let

$$b(u)=\{\begin{array}{ll}C\exp \left[-\frac{1}{1-|u{|}^2}\right],& |u|<1,\\ 0,& |u|\ge 1,\end{array}$$

with $C$ chosen so that ${\int }_{{\mathbb{R}}^2}b(u)d^{2}u=1$. For a center $x_c=(t_c,s_c)$ and width $\epsilon >0$, define

$$b_{x_c,\epsilon }(x)={\epsilon }^{-2}b\left(\frac{x-x_c}{\epsilon }\right).$$

If the closed Euclidean ball $\overline{B_{\epsilon }(x_c)}$ lies in $W_R$, this is a real $C_c^{\infty }(W_R)$ test function of integral $1$. Choose $f_R=b_{x_f,{\epsilon }_f}$ and $g_R=b_{x_g,{\epsilon }_g}$.

6.2 Pauli–Jordan integration

Using the 2D massless formula from §3.4,

$$\sigma (f_R,g_R)=\int d^{2}x{d}^{2}y{f}_R(x)\left[-\frac{1}{2}\mathrm{sgn}(x^0-y^0)\Theta (-(x-y{)}^2)\right]g_R(y).$$

This commutator calculation is infrared safe. The individual normalized bumps have nonzero integral, so vacuum expectation values of their Weyl operators require the infrared-regulated scalar. In the zero-mode-free vacuum algebra one instead uses zero-integral linear combinations of bumps.

There is a simple exact case. Choose the supports so that every $x\in \mathrm{supp}{f}_R$ lies chronologically to the future of every $y\in \mathrm{supp}{g}_R$. Then $\mathrm{sgn}(x^0-y^0)=+1$ and $\Theta (-(x-y{)}^2)=1$ throughout the integration region. Hence

$$\sigma (f_R,g_R)=-\frac{1}{2}\left(\int f_R\right)\left(\int g_R\right)=-\frac{1}{2}.$$

Reversing the chronological order gives $+1/2$; spacelike-separated supports give $0$. For generic overlapping causal relations no universal closed formula exists, but the displayed distributional integral is well posed and can be evaluated numerically.

6.3 Weyl algebra in the wedge

The von Neumann algebra

$$\mathcal{A}(W_R)=\{W(f):\mathrm{supp}(f)\subset W_R{\}}^{\prime \prime }$$

contains $W(f_R),W(g_R)$ and all their finite products. The Weyl relation gives

$$W(f_R)W(g_R)=e^{-i\sigma (f_R,g_R)/2}W(f_R+g_R),$$

which is again in $\mathcal{A}(W_R)$ because $f_R+g_R$ has support in $W_R$.

6.4 Locality across the wedges

For $f_R$ supported in $W_R$ and $g_L$ supported in $W_L$, every pair $x\in \mathrm{supp}(f_R),y\in \mathrm{supp}(g_L)$ has $x^1>0$ and $y^1<0$, with $(x-y{)}^2>0$ (spacelike). So $\Delta (x-y)=0$ on this region, and $\sigma (f_R,g_L)=0$. Hence $W(f_R)W(g_L)=W(g_L)W(f_R)$, and $[\mathcal{A}(W_R),\mathcal{A}(W_L)]=0$.

6.5 Why this calculation matters

This calculation is the workhorse for everything that follows in Block C:

• Week 9 (Reeh–Schlieder): the vacuum is cyclic-separating for $\mathcal{A}(W_R)$, with the dense subspace built from polynomials in $\varphi (f_R)$.
• Week 10 (Bisognano–Wichmann): the modular flow on $\mathcal{A}(W_R)$ acts on Weyl operators by $W(f_R)\to W(f_R\circ {\Lambda }^{\mathrm{boost}}(-2\pi t))$ — a geometric boost on test functions.
• Week 11 (Bell–CHSH): the contractions $\cos (\alpha \varphi (f_R))$ and $\cos (\alpha \varphi (g_L))$ are explicit bounded local observables. Whether a chosen family violates CHSH—and how closely it approaches the Tsirelson bound—is a separate optimization problem. Structural maximal-correlation theorems concern the supremum over all admissible local contractions.

The free 2D massless scalar is the testbed where all of these are explicitly computable.

7. Quasi-free states beyond the vacuum

The choice of state on $\mathcal{W}(V,\sigma )$ fixes the GNS representation; the vacuum is by no means the only physically relevant choice.

7.1 Definition

Definition 7.1. A centered state $\omega$ on $\mathcal{W}(V,\sigma )$ is quasi-free (or Gaussian) if

$$\omega (W(f))=\exp \left(-\frac{1}{2}\mu (f,f)\right),$$

where $\mu$ is a real symmetric positive bilinear form satisfying the CCR uncertainty inequality

$$|\sigma (f,g){|}^2\le 4\mu (f,f)\mu (g,g).$$

Equivalently, the complex two-point form

$$S(f,g):=\mu (f,g)+\frac{i}{2}\sigma (f,g)$$

is positive on the complexified test-function space. For the Fock vacuum, $\mu (f,g)=\mathrm{Re}W(f,g)$ for real $f,g$. The factor of $i$ is essential: the antisymmetric part of a complex two-point function encodes the commutator.

The Fock vacuum corresponds to

$$S(f,g)=\mathrm{Re}W(f,g)+\frac{i}{2}\sigma (f,g),$$

which is the standard complex Wightman two-point form in these conventions.

Physical picture. The algebra $(V,\sigma )$ fixes the commutator but not a notion of particle. A pure quasi-free state determines a compatible complex structure, hence a positive/negative-frequency split and creation and annihilation operators. A mixed quasi-free state, such as a thermal state, is described by a covariance with additional occupation numbers and need not define a new vacuum split in the same way. Bogoliubov transformations compare pure splits; their implementability decides whether the corresponding Fock representations are unitarily equivalent. The safe algebraic moral is: the field algebra and its causal commutator are state-independent, while particles and occupation numbers depend on the chosen state and dynamics.

7.2 Examples of other quasi-free states

• Thermal states. For a free scalar in a fixed inertial frame, the KMS state at inverse temperature $\beta$ is quasi-free, with real covariance

$${\mu }_{\beta }(f,f)=\int \frac{d^{d}k}{(2\pi {)}^{d}2{\omega }_k}\coth (\beta {\omega }_k/2)|f_{+}(\vec{k}){|}^2.$$

At $\beta \to \infty$, $\coth \to 1$ and we recover the vacuum.

• The Rindler vacuum. As we’ll see in Week 10, the “vacuum” defined by accelerated observers (annihilated by Rindler-positive-frequency modes) is a different quasi-free state from the Minkowski vacuum. The Bogoliubov transformation between them is the Unruh effect.

• Bisognano–Wichmann (advance preview). Under the theorem’s Wightman hypotheses, the Minkowski vacuum restricted to the wedge algebra $\mathcal{A}(W_R)$ is exactly KMS for the boost automorphism at $T_U=1/(2\pi )$ in Rindler units. This is the algebraic content of the Unruh effect; the free-mode calculation of Week 10 is a consistency check, not the proof of the KMS theorem.

7.3 Mixtures need not be quasi-free

A convex combination of two quasi-free states is certainly a state, but it is generically not quasi-free: a mixture of two Gaussian characteristic functions is not one Gaussian characteristic function. Passing to a larger GNS representation does not change that fact about the state. Quasi-free states remain the main exactly solvable family, while their convex mixtures provide simple non-Gaussian states on the same Weyl algebra.

8. What to take away

• Stated only: the free scalar field is an operator-valued distribution on Fock space; smeared fields $\varphi (f)$ are densely defined self-adjoint operators (Reed–Simon Vol. II §X.7).
• Proved: the bounded Weyl operators $W(f)=e^{i\varphi (f)}$ satisfy the Weyl relation $W(f)W(g)=e^{-i\sigma (f,g)/2}W(f+g)$, with $\sigma$ the Pauli–Jordan symplectic form (B-R convention).
• Proved: local algebras at spacelike separation commute. Locality at the algebraic level follows from the spacelike vanishing of $\sigma$.
• Computed here: the 2D massless Pauli–Jordan distribution and an exact symplectic pairing for compactly supported wedge test functions, with the infrared qualification stated explicitly.
• Physical interpretation: the Weyl algebra carries the state-independent CCR and causal structure; the choice of state (vacuum, thermal, Rindler vacuum, etc.) determines its correlation functions and GNS representation.

9. Looking ahead

Week 9 introduces the Reeh–Schlieder theorem: under the standard Wightman hypotheses, the vacuum is cyclic for every nonempty open region and, by locality, separating when the causal complement contains a nonempty open region. This is what licenses applying Tomita–Takesaki to such local algebras: the vacuum supplies their distinguished modular data. Reeh–Schlieder is the bridge between abstract modular theory and QFT.

10. Problem set

Core problems.

1. The Weyl relation in a wedge. Use the compactly supported bumps of §6.1 and choose their supports so that every point of $\mathrm{supp}f$ is chronologically later than every point of $\mathrm{supp}g$. Normalize both integrals to $1$. (a) Compute exactly that $\sigma (f,g)=-1/2$. (b) Insert this value into the abstract Weyl relation. (c) In an infrared-regulated massless scalar, verify the relation on the dense set of coherent vectors. Explain why the vacuum check cannot be performed on the unregulated $1+1$-dimensional scalar for these nonzero-integral smearings.

2. Spacelike commutativity. Let ${\mathcal{O}}_1,{\mathcal{O}}_2$ be open regions with ${\mathcal{O}}_1\subset {\mathcal{O}}_2^{\prime }$ (i.e., ${\mathcal{O}}_1$ is spacelike to ${\mathcal{O}}_2$). Show that $\mathcal{A}({\mathcal{O}}_1)$ and $\mathcal{A}({\mathcal{O}}_2)$ commute. (Hint: show the generators commute via the Weyl relation, then extend to the WOT-closures.)

3. Continuity in the test function. Let $f_n\to f$ in $\mathcal{S}({\mathbb{R}}^{1,d})$. Show that the positive-frequency one-particle wavefunctions converge in norm and hence $\varphi (f_n)|0⟩\to \varphi (f)|0⟩$. Using the explicit action of Weyl operators on exponential vectors—a dense set in Fock space—prove that $W(f_n)\to W(f)$ strongly. Explain why convergence only on the vacuum vector would not be sufficient.

4. The vacuum is Gaussian. Verify ${\omega }_0(W(f))=e^{-W(f,f)/2}$ for real $f$, using the Fock-space realization. (Hint: compute $⟨0|e^{i\varphi (f)}|0⟩$ by expanding $\varphi (f)={\varphi }_{+}+{\varphi }_{-}$ into annihilation and creation parts. Then ${\varphi }_{+}(f)|0⟩=0$ and $⟨0|{\varphi }_{-}(f)=0$. Use the BCH identity $e^{X+Y}=e^X{e}^Y{e}^{-[X,Y]/2}$ for $X=i{\varphi }_{-}(f),Y=i{\varphi }_{+}(f)$.)

5. Symplectic form for compact bumps. For two compact bumps centered at $(t_1,s_1)$ and $(t_2,s_2)$ in the infrared-regulated two-dimensional massless scalar: (a) write $\sigma (f_1,f_2)$ as the four-dimensional distributional integral over $x$ and $y$; (b) prove that it vanishes when the supports are spacelike separated; (c) compute it when one support is entirely chronologically later than the other; (d) for a generic placement, set up a numerical quadrature and explain why there is no universal notion of “maximal” without fixing normalization and an admissible class of test functions.

6. Pauli–Jordan in 4D massless. With metric $(-,+,+,+)$ and $\Delta =-i⟨0|[\varphi (x),\varphi (y)]|0⟩$, verify

$$\Delta (x-y)=-\frac{1}{2\pi }\mathrm{sgn}(x^0-y^0)\delta ((x-y{)}^2)$$

from the Fock-space mode expansion. Check the sign against the $i0$ prescription in §3.5 rather than importing a formula written in the opposite metric or commutator convention.

Starred problems.

7*. The Weyl algebra is simple (theorem-guided). Read the standard Slawny/CCR simplicity proof and reconstruct its averaging step for a finite linear combination of Weyl operators. Use

$$W(f)W(g)W(f{)}^{\ast }=e^{-i\sigma (f,g)}W(g)$$

and non-degeneracy of $\sigma$ to isolate the identity coefficient. Explain how approximation then shows that every nonzero closed two-sided ideal contains $1$ and therefore equals $\mathcal{W}(V,\sigma )$. The conjugation formula alone is not a complete proof; identify where the averaging and norm estimate enter.

8*. Boosted Weyl operators in the wedge. In the 2D massless free scalar, let ${\Lambda }_{\eta }$ be a boost by rapidity $\eta$ in the $(x^0,x^1)$ plane. Show that under ${\Lambda }_{\eta }$, $W(f)\mapsto W(f\circ {\Lambda }_{\eta }^{-1})$, and that this maps $\mathcal{A}(W_R)$ into itself for every $\eta \in \mathbb{R}$.

9*. Type-I regularization—and what it does not prove. Put the scalar in a finite spatial box and retain a finite set of oscillator modes. Show that the represented global algebra is $\mathcal{B}({\mathcal{F}}_{\mathrm{fin}})$, a type I${}_{\infty }$ factor, and that the vacuum is represented by the rank-one density operator $|0⟩⟨0|$. Explain:

• why a momentum cutoff alone still leaves infinitely many continuum modes unless a box/discretization is also imposed;
• why a bosonic Fock space over finitely many modes is infinite-dimensional;
• why sharp momentum cutoffs destroy strict spacetime locality; and
• why the local type III${}_1$ result in the continuum is a separate theorem, not a consequence of saying “the cutoff goes to infinity.”

10*. Massive vs. massless 2D. Repeat the 2D Pauli–Jordan computation for $m>0$. Show that the support of $\Delta (x-y)$ is still inside the light cone (the field commutator is supported timelike-or-null even for massive fields), but the coefficients involve Bessel functions $J_0(m\sqrt{-(x-y{)}^2})$.

11*. Minkowski and Rindler representations: criterion and scope. Use either a massive two-dimensional scalar or the massless derivative field, so that the infrared sector is controlled. Formulate the Minkowski and Fulling–Rindler quasi-free states on the wedge Weyl algebra. State the Shale–Stinespring/Hilbert–Schmidt criterion for unitary implementability of their Bogoliubov transformation, and consult a standard reference to determine its failure for the full wedge. Distinguish this global inequivalence from local quasi-equivalence on suitably compact subregions. The phrase “Stone–von Neumann fails” is motivation, not a proof.

Self-study answer checkpoints

These checkpoints cover the core problems. Starred, project, and explicitly research-level problems remain source-led; they should be completed with the references and hypotheses named in the problem.

1. Chronological ordering and unit normalization give $\sigma (f,g)=-1/2$, hence $$W(f)W(g)=e^{i/4}W(f+g).$$ The coherent-vector calculation must reproduce this phase. The nonzero integrals of $f$ and $g$ are precisely why the unregulated $1+1$-dimensional scalar vacuum is unavailable for this check.
2. Spacelike separation gives $\sigma (f,g)=0$ for every pair of generators. Thus the Weyl generators commute, and taking double commutants yields $$\mathcal{A}({\mathcal{O}}_1)\subseteq \mathcal{A}({\mathcal{O}}_2{)}^{\prime },[\mathcal{A}({\mathcal{O}}_1),\mathcal{A}({\mathcal{O}}_2)]=0.$$
3. Schwartz convergence implies $\Vert (f_n-f{)}_{+}{\Vert }_{1p}\to 0$ in the one-particle norm. The explicit coherent-vector formula then gives $W(f_n)\Psi \to W(f)\Psi$ on a dense set; since all Weyl operators have norm one, density extends this to strong convergence on all of Fock space. Convergence on $\Omega$ alone would test only one vector and is therefore insufficient.
4. Normal ordering with $X=i{\varphi }_{-}(f)$ and $Y=i{\varphi }_{+}(f)$ leaves the scalar BCH factor, while the remaining vacuum expectation is one. The result is $$⟨0|W(f)|0⟩=e^{-W(f,f)/2}.$$
5. The required pairing is $$\sigma (f_1,f_2)=\int d^{2}x{d}^{2}y{f}_1(x)\Delta (x-y)f_2(y).$$ It is $0$ for spacelike-separated supports and $-\frac{1}{2}(\int f_1)(\int f_2)$ when the first support is entirely to the chronological future of the second; reversing the order reverses the sign. A generic numerical calculation must resolve the causal-domain boundaries, and no maximum is defined before fixing normalization and the admissible smearing class.
6. The angular integral in the mode expansion reduces the distribution to $$\Delta (t,r)=-\frac{1}{4\pi r}[\delta (t-r)-\delta (t+r)]=-\frac{1}{2\pi }\mathrm{sgn}(t)\delta (r^2-t^2).$$ Since $x^2=r^2-t^2$, this is the stated mostly-plus formula. The equal-time check is ${\partial }_t\Delta (0,\vec{x})=-{\delta }^{(3)}(\vec{x})$ in the present convention.

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Notes prepared for the algebraic-QFT course, Semester I Block C. Last revised 2026-09-29.