1. AQFT in brief🔗

Algebraic quantum field theory (AQFT) describes a relativistic quantum system by the observables that can be measured in each region of spacetime, and by the relations between them that relativity and quantum mechanics impose. This chapter introduces it to a reader who knows quantum mechanics and special relativity. It covers nets of local algebras on causal diamonds, the Haag–Kastler axioms, states and the GNS construction, and the points at which this project makes the 1964 axioms precise. Lean appears from the next chapter on.

1.1. Why algebras🔗

As you will recall from a first course in quantum field theory, a field such as the free scalar field \phi is not a function from spacetime to operators. The expression \phi(x) makes sense only inside an integral: the field is an operator-valued distribution, and the objects that act on the Hilbert space are its smearings

\phi(f) = \int f(x)\, \phi(x)\, d^4x,

where f is a smooth test function with compact support. If the support of f lies in a region \mathbf{B} of spacetime, then \phi(f) is something an experimenter confined to \mathbf{B} can, at least in principle, control.

Fields are a convenient way to generate observables, but they are not observables themselves, and they are far from unique. A Dirac field is not self-adjoint and anticommutes, rather than commutes, at spacelike separation; it cannot be measured. A gauge potential changes under gauge transformations that leave every measurable quantity untouched. Even for a scalar field, different fields, such as the free field \phi and \phi + {:}\phi^3{:} with {:}\phi^3{:} the Wick-ordered cube, commute with each other at spacelike separation. Such relatively local fields belong to one Borchers class, and in an interacting theory any two fields of the class that create the same particles give the same scattering matrix. The choice of field is a choice of coordinates on the theory.

What does not depend on that choice is the answer to the question: which observables can be measured in which region of spacetime? Haag's idea is to take this assignment as the primary object. To each region \mathbf{B} one attaches an algebra \mathfrak{U}(\mathbf{B}) of observables measurable in \mathbf{B}, generated, in a theory with a field, by bounded functions of the smeared fields \phi(f) with f real and supported in \mathbf{B}. The physics is then encoded in how these algebras sit inside one another and how they relate across spacetime. The assignment \mathbf{B} \mapsto \mathfrak{U}(\mathbf{B}) is called a net of local algebras.

Two features of the approach matter from the start.

  • The algebras consist of bounded operators, or abstract versions of them. Unbounded operators such as \phi(f) have domains, and their sums and products need not be defined. Bounded functions of them, such as e^{i\phi(f)} or the spectral projections of \phi(f), carry the same physical information without these difficulties.

  • The algebras are abstract: no Hilbert space is fixed in advance. In quantum mechanics with finitely many degrees of freedom, the Stone–von Neumann theorem guarantees that the canonical commutation relations have, up to unitary equivalence, one irreducible representation (with the usual regularity assumptions). A field has infinitely many degrees of freedom, and the theorem fails: the same algebra has many inequivalent representations, describing, for instance, different thermal equilibria or different charge sectors. The algebraic approach keeps them all, and recovers a Hilbert space from a state, by the GNS construction, when it is needed.

Exercise

Let f and g be real test functions, and suppose that the smeared field satisfies the commutation relation [\phi(f), \phi(g)] = i\Delta(f,g)\,\mathbf{1} for a real number \Delta(f,g). Assuming that the formal manipulations are justified, show that the unitaries W(f) = e^{i\phi(f)} satisfy the Weyl relations W(f)\,W(g) = e^{-i\Delta(f,g)/2}\, W(f+g). What does this say about W(f) and W(g) when \Delta(f,g) = 0?

Solution

For operators A and B whose commutator is a multiple of the identity, the Baker–Campbell–Hausdorff formula reduces to e^{A}e^{B} = e^{A+B}e^{[A,B]/2}. With A = i\phi(f) and B = i\phi(g), [A,B] = -[\phi(f),\phi(g)] = -i\Delta(f,g)\mathbf{1}, and linearity of the smearing gives A + B = i\phi(f+g). Hence W(f)W(g) = e^{-i\Delta(f,g)/2}\,W(f+g). Exchanging f and g changes the sign of \Delta, so W(f)W(g) = e^{-i\Delta(f,g)}W(g)W(f). When \Delta(f,g) = 0 the two unitaries commute. For the free field \Delta(f,g) vanishes whenever the supports of f and g are spacelike separated: a first sign of the locality axiom below.

In the library: the net, with its axioms, is the structure Physicslib4.AQFT.HaagKastler.HaagKastlerNet.

1.2. Regions of spacetime🔗

You will be familiar with Minkowski space: the set \mathbb{R}^4 of points x = (x^0, x^1, x^2, x^3), with time coordinate x^0 and the Minkowski form

\eta(v, w) = -v^0 w^0 + v^1 w^1 + v^2 w^2 + v^3 w^3,

of signature (-,+,+,+), in units with c = 1. A nonzero vector v is timelike if \eta(v,v) < 0, null if \eta(v,v) = 0 and spacelike if \eta(v,v) > 0; a timelike or null vector is causal, and it is future-directed if v^0 > 0. Two points are compared through their difference:

  • p \ll q, read "p chronologically precedes q", if q - p is timelike and future-directed;

  • p \prec q, read "p causally precedes q", if q - p is causal and future-directed, or zero.

The chronological future and chronological past of a point are the open light cones

I^+(p) = \{x : p \ll x\}, \qquad I^-(q) = \{x : x \ll q\},

and the causal future and past J^{\pm} are defined in the same way with \prec in place of \ll. Two points are spacelike separated if neither causally precedes the other, and two sets are completely spacelike if every point of one is spacelike separated from every point of the other.

1.2.1. Causal diamonds🔗

Consider an experiment, carried out by an apparatus with mass, that starts at a spacetime point p and ends at a point q. Every part of the apparatus moves on a timelike curve, so everything the experiment touches lies to the future of p and to the past of q. This is the region

\mathbf{B} = I^+(p) \cap I^-(q),

called the causal diamond (also double cone, or Alexandrov set) of p and q. It is open and bounded, and it is nonempty exactly when p \ll q; when it is empty we still count it as a diamond, the empty diamond. In the frame in which q - p points along the time axis, \mathbf{B} is the domain of dependence of the open ball of radius (q^0 - p^0)/2 centred at the midpoint of p and q: every inextendible causal curve through \mathbf{B} meets the ball, so initial data on the ball fix a hyperbolically propagating field throughout \mathbf{B}.

A causal diamond as the intersection of two light conesA point p near the bottom and a point q near the top. Dashed lines at 45 degrees bound the chronological future of p, opening upwards, and the chronological past of q, opening downwards. Their intersection, shaded, is the diamond B.pqBI⁺(p)I⁻(q)tx

The diamond \mathbf{B} = I^+(p) \cap I^-(q) in one space and one time dimension: the part of the future cone of p that lies in the past cone of q.

Why chronological rather than causal cones? With J^\pm, the region J^+(p) \cap J^-(q) is, when nonempty, compact, so not open, and for q = p it is the single point \{p\}; an operator-valued distribution has no value at a point, so there is no algebra of observables at a point to assign. The chronological cones, and every diamond, are open.

The 1964 axioms of Haag and Kastler (1964)Marginal noteRudolf Haag and Daniel Kastler (1964). “An algebraic approach to quantum field theory”. Journal of Mathematical Physics. 5(7), pp. 848–861. attach algebras instead to all open sets with compact closure. That phrase needs a topology on \mathbb{R}^4, which the axioms do not name. The natural candidate is the one built from the causal structure itself.

The Alexandrov topology is the topology whose open sets are the unions of causal diamonds. For this to define a topology, the diamonds must form a basis: the intersection of two diamonds must be a union of diamonds. In Minkowski space it is, and the Alexandrov topology coincides with the usual Euclidean topology of \mathbb{R}^4, but it is defined by the light cones alone, without reference to a Euclidean metric that has no physical meaning. In a curved spacetime the two topologies can differ; they agree precisely when the spacetime is strongly causal.

So, in this project, Minkowski spacetime is \mathbb{R}^4 with the form \eta, the time orientation given by the vector (1,0,0,0), and the Alexandrov topology, and the regions of the net are its causal diamonds. One fact about them is used throughout: any two diamonds lie in a common, larger diamond (the diamonds are upward directed under inclusion).

Exercise

Show that I^+(p) \cap I^-(q) is nonempty if and only if p \ll q.

Solution

If x lies in the intersection, then p \ll x and x \ll q. The sum of two future-directed timelike vectors is future-directed timelike (both lie in the open forward cone, which is convex), so q - p = (q - x) + (x - p) is future-directed timelike, and p \ll q. Conversely, if p \ll q, the midpoint m = (p+q)/2 satisfies m - p = q - m = (q-p)/2, which is future-directed timelike, so m lies in the intersection.

Exercise

Show that any two diamonds I^+(p_1) \cap I^-(q_1) and I^+(p_2) \cap I^-(q_2) lie in a common diamond.

Solution

Choose p = (t, 0, 0, 0) with t so negative that p_k^0 - t > |\mathbf{p}_k| for k = 1, 2, where \mathbf{p}_k is the spatial part of p_k. Then p_k - p is future-directed timelike, so p \ll p_k, and by transitivity of \ll (the convexity argument of the previous exercise) I^+(p_k) \subseteq I^+(p). Choose q in the same way, far in the future, so that I^-(q_k) \subseteq I^-(q). Then both diamonds lie in I^+(p) \cap I^-(q).

In the library: Physicslib4.MinkowskiSpacetime, with the Alexandrov topology Physicslib4.Spacetime.alexandrovTopology and the diamonds Physicslib4.Spacetime.chronologicalDiamond.

1.3. C*-algebras of observables🔗

Recall that a Hilbert space \mathcal{H} is a complex vector space with an inner product \langle \cdot, \cdot \rangle, linear in the second argument, that is complete in the norm \|\psi\| = \langle \psi, \psi \rangle^{1/2}. A linear operator A on \mathcal{H} is bounded if its operator norm \|A\| = \sup_{\|\psi\| = 1} \|A\psi\| is finite; the bounded operators form an algebra \mathcal{B}(\mathcal{H}). Each bounded A has an adjoint A^*, defined by \langle \psi, A\chi \rangle = \langle A^*\psi, \chi \rangle, and the observables of quantum mechanics are the self-adjoint operators, A = A^*, whose spectra are the possible outcomes of a measurement. (Physicists often write A^\dagger; this chapter writes A^*, as mathematicians do.)

One identity links the algebra \mathcal{B}(\mathcal{H}) to its norm:

\|A^* A\| = \|A\|^2.

(The bound \|A\|^2 \le \|A^*A\| follows from \|A\psi\|^2 = \langle \psi, A^*A\psi \rangle, and \|A^*A\| \le \|A^*\|\,\|A\| = \|A\|^2, since \|A^*\| = \|A\| for bounded operators.) The abstract counterpart of \mathcal{B}(\mathcal{H}) keeps these properties and forgets the Hilbert space.

A C*-algebra is a complex algebra \mathfrak{A} with

  • a norm in which it is complete, with \|ab\| \le \|a\|\,\|b\| (a Banach algebra);

  • an involution a \mapsto a^*: antilinear, with (ab)^* = b^*a^* and a^{**} = a;

  • the C*-identity \|a^*a\| = \|a\|^2 for every a.

It is unital if it has a unit \mathbf{1}, with \mathbf{1}a = a\mathbf{1} = a. All the algebras of this chapter are unital. Examples are \mathcal{B}(\mathcal{H}) itself, any norm-closed subalgebra of it that is closed under adjoints, the matrix algebras M_n(\mathbb{C}), and, as a commutative example, the continuous complex functions on a compact space with the supremum norm and complex conjugation as involution.

A *-homomorphism \pi : \mathfrak{A} \to \mathfrak{B} is a linear map with \pi(ab) = \pi(a)\pi(b) and \pi(a^*) = \pi(a)^*; it is unital if \pi(\mathbf{1}) = \mathbf{1}. A *-homomorphism between C*-algebras is automatically continuous, with \|\pi(a)\| \le \|a\|, and if it is injective it is isometric, \|\pi(a)\| = \|a\|: the norm of a C*-algebra is fixed by its algebraic structure. An injective *-homomorphism is a *-monomorphism, and a bijective one a *-isomorphism. A representation of \mathfrak{A} on a Hilbert space \mathcal{H} is a *-homomorphism \pi : \mathfrak{A} \to \mathcal{B}(\mathcal{H}).

Nothing is lost by the abstraction: by the Gelfand–Naimark theorem, every C*-algebra is *-isomorphic to a norm-closed, adjoint-closed algebra of operators on some Hilbert space. What is gained is that no particular Hilbert space is singled out.

The self-adjoint elements a = a^* of a C*-algebra are its observables. The possible results of measuring a are the points of its spectrum,

\sigma(a) = \{\lambda \in \mathbb{C} : a - \lambda\mathbf{1} \text{ has no inverse in } \mathfrak{A}\},

a nonempty compact subset of \mathbb{R} when a is self-adjoint. The definition refers to the algebra in which inverses are sought. For C*-algebras the choice does not matter: if \mathfrak{A} is a C*-subalgebra of \mathfrak{B} with the same unit, an element of \mathfrak{A} has the same spectrum in both. This spectral permanence fails for general Banach algebras, and it is one reason the axioms ask for C*-algebras: an observable measured in a small region is also an observable of any larger region, and its possible values must not depend on which region we regard it as belonging to.

Exercise

Let \mathfrak{A} be a nonzero unital C*-algebra. Show that \|a^*\| = \|a\| for every a, that \mathbf{1}^* = \mathbf{1}, and that \|\mathbf{1}\| = 1.

Solution

For a \neq 0, \|a\|^2 = \|a^*a\| \le \|a^*\|\,\|a\|, so \|a\| \le \|a^*\|. Applying this to a^* gives \|a^*\| \le \|a^{**}\| = \|a\|. Next, \mathbf{1}^* a = (a^*\mathbf{1})^* = a and likewise a\mathbf{1}^* = a, so \mathbf{1}^* is a unit, and units are unique. Finally \|\mathbf{1}\| = \|\mathbf{1}^*\mathbf{1}\| = \|\mathbf{1}\|^2, so \|\mathbf{1}\| is 0 or 1, and it is not 0 because \mathbf{1} \neq 0 in a nonzero algebra.

In the library: C*-algebras are Mathlib's class CStarAlgebra (unital, complex); the net assigns one to each diamond in Physicslib4.AQFT.HaagKastler.LocalNet.

1.4. The axioms🔗

A theory, in the Haag–Kastler framework, is a net of C*-algebras on the diamonds of Minkowski spacetime, subject to a short list of axioms, each of which rules something out. They are stated here in the form this project uses.

1.4.1. Axiom 1: local algebras🔗

To every causal diamond \mathbf{B} there is assigned a unital C*-algebra \mathfrak{U}(\mathbf{B}), the algebra of observables measurable in \mathbf{B}. The empty diamond is assigned the one-dimensional algebra \mathbb{C}\mathbf{1} of multiples of the unit.

This axiom makes a theory an assignment of algebras to regions, not a field, and rules out observables attached to single points.

1.4.2. Axiom 2: isotony🔗

An observable that can be measured in a diamond \mathbf{B}_1 can also be measured in any larger diamond \mathbf{B}_2 \supseteq \mathbf{B}_1. Since the algebras are abstract, "is also an element of" must be supplied by a map. Isotony provides, for each such pair, a unital *-monomorphism

i_{\mathbf{B}_1\mathbf{B}_2} : \mathfrak{U}(\mathbf{B}_1) \to \mathfrak{U}(\mathbf{B}_2),

and requires these maps to be coherent:

i_{\mathbf{B}\mathbf{B}} = \mathrm{id}, \qquad i_{\mathbf{B}_2\mathbf{B}_3} \circ i_{\mathbf{B}_1\mathbf{B}_2} = i_{\mathbf{B}_1\mathbf{B}_3} \quad\text{for } \mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}_3.

Because the maps are unital, every local algebra shares the unit of \mathfrak{U}(\emptyset) = \mathbb{C}\mathbf{1}, and \mathbf{1} means the same thing in every region. Because they are injective, they are isometric, and by spectral permanence an observable has the same spectrum in every algebra it is embedded in.

Isotony rules out a theory in which enlarging the region of an experiment changes the observable being measured.

1.4.3. Axiom 3: local commutativity🔗

If \mathbf{B}_1 and \mathbf{B}_2 are completely spacelike, their observables commute. To multiply an element of \mathfrak{U}(\mathbf{B}_1) by one of \mathfrak{U}(\mathbf{B}_2), both must lie in one algebra. Isotony supplies one: the algebra of any diamond \mathbf{B} containing both regions, and in Minkowski spacetime such a diamond exists. The axiom states that for every such \mathbf{B},

i_{\mathbf{B}_1\mathbf{B}}(a_1)\, i_{\mathbf{B}_2\mathbf{B}}(a_2) = i_{\mathbf{B}_2\mathbf{B}}(a_2)\, i_{\mathbf{B}_1\mathbf{B}}(a_1) \qquad\text{for all } a_1 \in \mathfrak{U}(\mathbf{B}_1),\ a_2 \in \mathfrak{U}(\mathbf{B}_2).

Two spacelike-separated diamonds inside a causal diamondA large causal diamond B contains two smaller diamonds B1 and B2 side by side. Dashed light-cone lines at 45 degrees from B1 bound its causal future and past; B2 lies outside them, so B1 and B2 are spacelike separated.BB₁B₂tx

Two completely spacelike diamonds \mathbf{B}_1 and \mathbf{B}_2 inside a diamond \mathbf{B}. The dashed lines bound the causal future and past of \mathbf{B}_1, which \mathbf{B}_2 avoids.

This is Einstein causality. Commuting observables can be measured together, and measuring one does not change the statistics of the other (see the exercise below), so the axiom rules out signalling faster than light. It does not rule out correlations: the vacuum of a quantum field is entangled across spacelike separation.

1.4.4. Axiom 4: the quasilocal algebra🔗

The local algebras fit together into a single algebra. Take all the \mathfrak{U}(\mathbf{B}) and identify each a \in \mathfrak{U}(\mathbf{B}_1) with its images i_{\mathbf{B}_1\mathbf{B}_2}(a) in the larger algebras. The result is the inductive limit of the local algebras (in category-theory language, the directed colimit). Elements from \mathfrak{U}(\mathbf{B}_1) and \mathfrak{U}(\mathbf{B}_2) are added or multiplied in any diamond containing both, and the coherence laws of isotony make the answer independent of the choice; the norm is inherited, since the embeddings are isometric. This normed *-algebra completes to a C*-algebra \mathfrak{U}, the quasilocal algebra. It comes with unital *-monomorphisms \iota_{\mathbf{B}} : \mathfrak{U}(\mathbf{B}) \to \mathfrak{U}, compatible with isotony: \iota_{\mathbf{B}_2} \circ i_{\mathbf{B}_1\mathbf{B}_2} = \iota_{\mathbf{B}_1}.

An element of \mathfrak{U} need not belong to any single diamond, but it is approximated arbitrarily well, in norm, by elements that do: hence quasilocal. Local commutativity can equally be stated in \mathfrak{U}: \iota_{\mathbf{B}_1}(a_1) and \iota_{\mathbf{B}_2}(a_2) commute.

The 1964 axiom adds that \mathfrak{U} contains "all observables of interest". This is a claim about physics, not mathematics: it says that every quantity that can actually be measured is represented by a self-adjoint element of \mathfrak{U}. For a set S of bounded operators, the commutant S' is the set of bounded operators that commute with every element of S. In a representation \pi, the bicommutant \pi(\mathfrak{U})'' = (\pi(\mathfrak{U})')' is, by von Neumann's bicommutant theorem, the closure of \pi(\mathfrak{U}) in the weak operator topology, and it is in general larger than \pi(\mathfrak{U}). But by the von Neumann density theorem, for every T \in \pi(\mathfrak{U})'', every finite set of vectors \psi_1, \dots, \psi_n and every \varepsilon > 0 there is an a \in \mathfrak{U} with \|\pi(a)\psi_k - T\psi_k\| < \varepsilon for all k, so no finite set of measurements distinguishes them. What the principle rules out is a measurable quantity that cannot be approximated by local observables.

1.4.5. Axiom 5: Lorentz covariance🔗

As you will recall, the Poincaré group consists of the maps x \mapsto \Lambda x + a with a \in \mathbb{R}^4 and \Lambda a Lorentz transformation, \eta(\Lambda v, \Lambda w) = \eta(v, w). (Haag and Kastler call it the inhomogeneous Lorentz group, hence the name of the axiom.) It has four connected components. The component of the identity, the proper orthochronous group, has \det\Lambda = 1 and \Lambda^0{}_0 \geq 1; the other three are reached by space and time reflections, which are not symmetries of nature (the weak interaction violates parity), so the axiom uses only the identity component. Its elements L preserve \ll, and therefore map diamonds to diamonds: L(I^+(p) \cap I^-(q)) = I^+(Lp) \cap I^-(Lq).

The axiom provides, for each such L and each diamond \mathbf{B}, a unital *-isomorphism

\alpha_L : \mathfrak{U}(\mathbf{B}) \to \mathfrak{U}(L\mathbf{B}),

with \alpha_e = \mathrm{id} for the identity e of the group, \alpha_{L'L} = \alpha_{L'} \circ \alpha_L, and compatibility with isotony, \alpha_L \circ i_{\mathbf{B}_1\mathbf{B}_2} = i_{L\mathbf{B}_1\,L\mathbf{B}_2} \circ \alpha_L. The maps then extend, by continuity, to automorphisms of \mathfrak{U}. The laws of physics are thus the same at every place and in every inertial frame; the axiom rules out a preferred origin or a preferred rest frame.

1.4.6. An aside: primitivity🔗

The 1964 list has a sixth axiom: \mathfrak{U} is primitive, that is, it has a representation that is both faithful (injective) and irreducible (no closed subspace other than 0 and the whole space is invariant). Its physical motivation is thin, and Haag's later account in Local Quantum Physics omits it. This project omits it too. For orientation: a commutative C*-algebra is primitive only if it is \mathbb{C}\mathbf{1}, so the axiom excludes classical nets other than the trivial one.

Exercise

Let \omega be a state, a positive normalised expectation functional (defined in the next section), and let P \in \mathfrak{U}(\mathbf{B}_1) be a projection, P = P^* = P^2. A measurement of P whose result is not recorded changes the state to \omega'(b) = \omega(PbP) + \omega((\mathbf{1}-P)\,b\,(\mathbf{1}-P)). Show that \omega'(b) = \omega(b) for every b that commutes with P, in particular for every b in a completely spacelike region. What does this say about signalling?

Solution

If bP = Pb, then PbP = bP^2 = bP and (\mathbf{1}-P)b(\mathbf{1}-P) = b(\mathbf{1}-P)^2 = b(\mathbf{1}-P), because \mathbf{1}-P is also a projection. The sum is bP + b - bP = b, so \omega'(b) = \omega(b). By local commutativity this holds for every observable b of a region completely spacelike to \mathbf{B}_1: an observer there sees the same expectation values whether or not the measurement in \mathbf{B}_1 took place, so the measurement cannot be used to send a signal.

In the library: Physicslib4.AQFT.HaagKastler.LocalNet, Physicslib4.AQFT.HaagKastler.Isotony, Physicslib4.AQFT.HaagKastler.LocalCommutativity, Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra and Physicslib4.AQFT.HaagKastler.LorentzCovariance.

1.5. States and the GNS construction🔗

To make predictions, the axioms need a state: an assignment of expectation values to observables.

A state on a unital C*-algebra \mathfrak{A} is a linear functional \omega : \mathfrak{A} \to \mathbb{C} that is

  • positive: \omega(a^*a) \geq 0 for every a;

  • normalised: \|\omega\| = 1, which for a positive functional is the same as \omega(\mathbf{1}) = 1.

The number \omega(a) is the expectation value of the observable a. The familiar states of quantum mechanics are examples: for a representation \pi on \mathcal{H}, a unit vector \psi gives the vector state \omega(a) = \langle \psi, \pi(a)\psi \rangle, and a density matrix \rho gives \omega(a) = \operatorname{tr}(\rho\,\pi(a)). The states form a convex set; its extreme points, the states that are not mixtures of two others, are the pure states. A state is faithful if \omega(a^*a) > 0 whenever a \neq 0.

1.5.1. The GNS construction🔗

A state determines a Hilbert space representation, uniquely up to unitary equivalence, by a theorem of Gelfand, Naimark and Segal.

Let \omega be a state on a unital C*-algebra \mathfrak{A}. There are a Hilbert space \mathcal{H}_\omega, a representation \pi_\omega of \mathfrak{A} on it and a unit vector \Omega_\omega such that

\omega(a) = \langle \Omega_\omega, \pi_\omega(a)\,\Omega_\omega \rangle \quad\text{for all } a \in \mathfrak{A},

and \Omega_\omega is cyclic: the vectors \pi_\omega(a)\Omega_\omega are dense in \mathcal{H}_\omega. Any two such triples are related by a unitary map. If \omega is faithful, so is \pi_\omega.

The proof builds the Hilbert space out of the algebra, in four steps.

  1. A semi-inner product. Define \langle a, b \rangle = \omega(a^*b) on \mathfrak{A}. It is linear in b, conjugate-linear in a, and \langle a, a \rangle \geq 0 by positivity, so it satisfies the Cauchy–Schwarz inequality |\omega(a^*b)|^2 \le \omega(a^*a)\,\omega(b^*b). It is not an inner product, because \omega(a^*a) may vanish for a \neq 0.

  2. The null space. Let \mathcal{N} = \{n : \omega(n^*n) = 0\}. By Cauchy–Schwarz, \mathcal{N} is a subspace and a left ideal: an \in \mathcal{N} for all a and all n \in \mathcal{N}. On the quotient \mathfrak{A}/\mathcal{N}, the form \langle [a], [b] \rangle = \omega(a^*b) is well defined and is an inner product.

  3. Completion and representation. \mathcal{H}_\omega is the completion of \mathfrak{A}/\mathcal{N}. The algebra acts by left multiplication, \pi_\omega(a)[b] = [ab], which is well defined because \mathcal{N} is a left ideal, and bounded, \|[ab]\| \le \|a\|\,\|[b]\|, because b^*a^*ab \le \|a\|^2\, b^*b. It extends to the completion, and \pi_\omega(a^*) = \pi_\omega(a)^* follows from \omega((a^*b)^*c) = \omega(b^*(ac)).

  4. The cyclic vector. \Omega_\omega = [\mathbf{1}]. Then \pi_\omega(a)\Omega_\omega = [a], so these vectors are all of \mathfrak{A}/\mathcal{N}, which is dense, and \langle \Omega_\omega, \pi_\omega(a)\Omega_\omega \rangle = \omega(\mathbf{1}^*a) = \omega(a).

Here the unit is used: without it there is no vector [\mathbf{1}]. Uniqueness comes from mapping \pi_1(a)\Omega_1 \mapsto \pi_2(a)\Omega_2, which preserves inner products because both equal \omega(a^*b). Faithfulness: if \pi_\omega(a) = 0, then [a] = 0, so \omega(a^*a) = 0 and, for faithful \omega, a = 0.

The GNS construction connects the abstract algebras to the Hilbert spaces of ordinary quantum theory: applied to a state on \mathfrak{U}, it gives the Hilbert space of the theory in that state, with \pi_\omega(a) the operator of the observable a. The state is pure exactly when \pi_\omega is irreducible.

1.5.2. The vacuum🔗

A vacuum state is a state \omega on \mathfrak{U} that is invariant and has positive energy.

It is invariant if \omega \circ \alpha_L = \omega for every L in the identity component of the Poincaré group. Then U(L)\,\pi_\omega(a)\Omega_\omega = \pi_\omega(\alpha_L a)\Omega_\omega defines unitaries on \mathcal{H}_\omega that represent the group, leave \Omega_\omega fixed and implement the symmetry: U(L)\,\pi_\omega(a)\,U(L)^{-1} = \pi_\omega(\alpha_L a) (see the second exercise below).

Recall Stone's theorem: a strongly continuous one-parameter group of unitaries has the form U(t) = e^{itH} with a self-adjoint, in general unbounded, generator H. For the time translations, H is the Hamiltonian. Positive energy, or the spectrum condition, requires that for every future-directed timelike vector a the translations t \mapsto U(ta) form such a group with nonnegative generator; equivalently, the joint spectrum of the energy-momentum operators lies in the closed forward light cone. This makes the vacuum a state of lowest energy.

1.5.3. Superselection sectors🔗

Different states can give inequivalent representations of the same algebra \mathfrak{U}. Two irreducible representations \pi_1 and \pi_2 are either unitarily equivalent or disjoint: no nonzero bounded operator T satisfies T\pi_1(a) = \pi_2(a)T for all a. The equivalence classes of the irreducible representations of physical interest are the superselection sectors of the theory, labelled by conserved charges such as electric charge. No observable in \mathfrak{U} connects two sectors, so a superposition of state vectors from different sectors cannot be told apart from the corresponding mixture: the relative phase is not measurable.

Exercise

Let \mathfrak{A} = M_2(\mathbb{C}) and \omega(a) = a_{11}, the state given by the unit vector e_1. Find \mathcal{N}, \mathcal{H}_\omega, \pi_\omega and \Omega_\omega. What changes for the faithful state \omega(a) = \tfrac{1}{2}\operatorname{tr} a?

Solution

\omega(n^*n) = (n^*n)_{11} = \|n e_1\|^2, so \mathcal{N} is the set of matrices whose first column is zero. The map [a] \mapsto a e_1 identifies \mathfrak{A}/\mathcal{N} with \mathbb{C}^2, preserving inner products since \omega(a^*b) = \langle a e_1, b e_1 \rangle. Under it, \pi_\omega(a) is the matrix a acting on \mathbb{C}^2 and \Omega_\omega = e_1: the GNS construction returns the defining representation. For \omega = \tfrac12 \operatorname{tr}, \mathcal{N} = 0, and \mathcal{H}_\omega is M_2(\mathbb{C}) itself, four-dimensional, with \langle a, b \rangle = \tfrac12 \operatorname{tr}(a^*b). The algebra acts by left multiplication and \Omega_\omega = \mathbf{1}. The representation is faithful but reducible: the two columns of a matrix are invariant subspaces, each a copy of \mathbb{C}^2. This matches the facts that the first state is pure and the second is a mixture.

Exercise

Show that the operators U(L) defined above for an invariant state are well defined and preserve inner products.

Solution

For a, b \in \mathfrak{U},

\langle \pi_\omega(\alpha_L a)\Omega_\omega, \pi_\omega(\alpha_L b)\Omega_\omega \rangle = \omega(\alpha_L(a^*b)) = \omega(a^*b) = \langle \pi_\omega(a)\Omega_\omega, \pi_\omega(b)\Omega_\omega \rangle,

using that \alpha_L is a *-homomorphism and that \omega is invariant. So the map preserves inner products on the dense set of vectors \pi_\omega(a)\Omega_\omega. In particular, if \pi_\omega(a)\Omega_\omega = 0, its image has norm zero, so the map is well defined. It extends by continuity to an isometry of \mathcal{H}_\omega, and it is onto because U(L^{-1}) inverts it.

In the library: Physicslib4.GNS.State, Physicslib4.GNS.gns_construction, and for the vacuum Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState.

1.6. What this project sharpens🔗

The axioms of Haag and Kastler (1964)Marginal noteRudolf Haag and Daniel Kastler (1964). “An algebraic approach to quantum field theory”. Journal of Mathematical Physics. 5(7), pp. 848–861. leave open some details that a proof needs. The project's blueprint fixes them, and the formal statements are checked against the blueprint. The forms above differ from the 1964 text in these points.

  • Topology and regions. Minkowski spacetime carries the Alexandrov topology, and the regions are the causal diamonds, including the empty one, rather than all open sets with compact closure in an unnamed topology.

  • The unit. The 1964 isotony axiom allows the two algebras of each nested pair to have a common unit or neither to have one. Here all have one, fixed by \mathfrak{U}(\emptyset) = \mathbb{C}\mathbf{1} and unital embeddings; the GNS construction needs it.

  • Isotony as data. "\mathfrak{U}(\mathbf{B}_1) \subset \mathfrak{U}(\mathbf{B}_2)" becomes a chosen family of *-monomorphisms with identity and composition laws, without which the union of the local algebras has no well-defined product.

  • Local commutativity in a containing diamond, using only the net and isotony; in Minkowski spacetime this is equivalent to commutation in \mathfrak{U}.

  • The quasilocal algebra is constructed from the local algebras and isotony, not assumed. The claim that it contains all observables of interest is kept apart as a physical principle.

  • The symmetry group. Covariance uses the identity component of the Poincaré group, and the transported algebras must respect isotony.

  • Primitivity is dropped.

The sharpened axioms carry over to a curved spacetime, a four-dimensional Lorentzian manifold used as a fixed classical background, provided its Alexandrov topology is Hausdorff (equivalently, the spacetime is strongly causal; the Alexandrov and manifold topologies then agree). The local algebras and isotony are unchanged, and the Poincaré group is replaced by the isometries connected to the identity.

One thing fails: two diamonds need not lie in a common diamond. In a Schwarzschild black hole, a diamond just below the singularity and a diamond outside the horizon at sufficiently late times have none: the top vertex of a common diamond would have to lie in the future of both, and inside the hole the future of the first ends on the singularity before it meets the future of the second. Their algebras then lie in no common algebra, and there is no quasilocal algebra. Local commutativity is stated, as above, inside every diamond that contains both regions, and says nothing when there is none: no experiment could compare the two. "All observables of interest" becomes a statement about the local algebras.

This differs from the locally covariant framework of Brunetti, Fredenhagen, and Verch (2003)Marginal noteRomeo Brunetti, Klaus Fredenhagen, and Rainer Verch (2003). “The generally covariant locality principle: a new paradigm for local quantum field theory”. Communications in Mathematical Physics. 237(1–2), pp. 31–68., in which a theory is a functor on a category of globally hyperbolic spacetimes; here it is a net on one fixed spacetime. Later chapters develop both the Minkowski and the curved-spacetime axioms in Lean.

Exercise

Let M be the strip -1 < x^0 < 1 of two-dimensional Minkowski space, with the causal structure of Minkowski space. Show that two small diamonds centred at (0, 0) and (0, 10) lie in no common diamond of M.

Solution

A diamond of M is I^+(p) \cap I^-(q) with p, q \in M, so q^0 - p^0 < 2. A point x of it satisfies |x^1 - p^1| < x^0 - p^0 and |q^1 - x^1| < q^0 - x^0. Adding the two inequalities, |2x^1 - p^1 - q^1| < q^0 - p^0 < 2, so all points of the diamond have x^1 within 1 of (p^1 + q^1)/2. The two given diamonds contain points whose x^1 coordinates differ by almost 10, so no diamond of M contains both.

In the library: the curved-spacetime net is Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet, and the physical principle of the quasilocal algebra is Physicslib4.AQFT.HaagKastler.ObservableCorrespondence.

1.7. Further reading🔗