nLab
local net

Idea

A local net of observables is the assignment associated to a quantum field theory of algebras of local observables to pieces of spacetime.

In the context of AQFT the structure of local nets is used as the very axiomatization of what a quantum field theory is (as opposed to the context of FQFT, where instead the state-propagation is used as the basic axiom).

Definition

In the literature there is a certain variance and flexibility of what precisely the axioms on a local net of observables are, though the core aspects are always the same: it is a copresheaf of (C-star algebra s) on pieces of spacetime such that algebras assigned to causally disconnected regions commute inside the algebra assigned to any joint neighbourhood.

Historically this was first formulated for Minkowski spacetime only, where it is known as the Haag-Kastler axioms. Later it was pointed out (BrunettiFredenhagen) that the axioms easily and usefully generalize to arbitrary spacetimes.

We give the modern general formulation first, and then comment on its restriction to special situations.

Basic general definition

Definition

Write $\mathrm{LorSp}$ for the category whose

• objects are spacetime manifolds; (Lorentzian manifolds equipped with a time-orientation);

• morphism are causal isometric embedding.

Here we say a morphism $f:X\hookrightarrow Y$ is a causal embedding if for every two points $x_1,x_2\in X$ we have that $f(x_1)$ is in the future of $f(x_2)$ in $Y$ only if $x_1$ is in the future of $x_2$ in $X$.

Write $\mathrm{Alg}$ for a suitable category of associative algebras. Usually this is taken to be the category of C-star algebras or that of von Neumann algebras. Write

$${\mathrm{Alg}}_{\mathrm{inc}}\hookrightarrow \mathrm{Alg}$$Alg_{inc} \hookrightarrow Alg

for the subcategory on the monomorphisms.

Definition

A causally local net of observables is a functor

$$𝒜:\mathrm{LorSp}\to {\mathrm{Alg}}_{\mathrm{inc}}\to \mathrm{Alg}$$\mathcal{A} : LorSp \to Alg_{inc} \to Alg

such that whenever $X_1\coprod X_2\hookrightarrow X$ is a causal embedding, def. 1, we have that $𝒜(X_1)\subset 𝒜(X)$ commutes with $𝒜(X_2)\subset 𝒜(X)$.

Remark

The locality axiom encodes the the physical property known as Einstein-causality or micro-causality, which states that physical effects do not propagate faster that the speed of light.

Remark

Many auxiliary operators in quantum field theory do not satisfy causal locality: for instance operators associate to currents in gauge theory. The idea is that those operators that actually do qualify as observables do satisfy the axiom, however, i.e. in particular those that are gauge invariant.

Extra axioms

Strong locality

Commutativity of spacelike separated observables can be argued to capture only part of causal locality.

A natural stronger requirement is that spacelike separated regions of spacetime are literally independent quantum subsystems of any larger region. By the formalization of independent subsystem in quantum mechanics this means the following:

Definition

A local net $𝒜$ satisfies Einstein locality if for every causal embedding $X_1\coprod X_2\to X$ the subsystems

$$𝒜(X_1)\hookrightarrow 𝒜(X)$$\mathcal{A}(X_1) \hookrightarrow \mathcal{A}(X)

and

$$𝒜(X_2)\hookrightarrow 𝒜(X)$$\mathcal{A}(X_2) \hookrightarrow \mathcal{A}(X)

are independent meaning that the algebra $𝒜(X_1)\vee 𝒜(X_2)\in 𝒜(X)$ which they generate is isomorphic to the tensor product $𝒜(X_1)\otimes 𝒜(X_2)$.

This appears as (BrunettiFredenhagen, 5.3.1, axiom 4).

Observation

A local net is Einstein local precisely if it is a monoidal functor

$$𝒜:(\mathrm{LorSp},\coprod )\to (\mathrm{Alg},\otimes ).$$\mathcal{A} : (LorSp, \coprod) \to (Alg, \otimes) \,.

This appears as (BrunettiFredenhagen, 5.3.1, theorem 1).

Remark

Einstein locality implies causal locality, but is stronger.

Other properties implied by Einstein locality are sometimes extracted as separate axioms. For instance the condition that for $X_1\coprod X_2\to X$ a causal embedding, we have

$$𝒜(X_1)\cap 𝒜(X_2)=ℂ.$$\mathcal{A}(X_1) \cap \mathcal{A}(X_2) = \mathbb{C} \,.

Time-slice axiom

Definition

A local net is said to satisfy the time slice axiom if whenever

$$i:X_1\to X_2$$i : X_1 \to X_2

is a causal embedding of globally hyperbolic spacetimes such that $X_1$ contains a Cauchy surface of $X_2$, then

$$𝒜(i):𝒜(X_1)\stackrel{\simeq }{\to }𝒜(X_2)$$\mathcal{A}(i) : \mathcal{A}(X_1) \stackrel{\simeq}{\to} \mathcal{A}(X_2)

is an isomorphism.

Duality

See dual net of von Neumann algebras

Positive energy condition

(…)

Spectrum condition

(…)

Special cases and variants

Minkowski nets / Vacuum representation

• Haag-Kastler vacuum representation

Conformal nets

The notion of local net in the context of conformal field theory is a conformal net.

Examples

Related concepts

• quantum field theory

  • AQFT

    • Haag-Kastler axioms

    • local net of observables

      • net of C-star systems (global gauge group, actions)

      • field net

      • cohomology of local net of observables

      • factorization algebra of observables, chiral algebra, topological chiral homology, blob complex.

  • FQFT

References

For more details see the references at AQFT.

General

The axioms of local nets on general spacetimes were first articulated in

• Romeo Brunetti, Klaus Fredenhagen, Quantum Field Theory on Curved Backgrounds (arXiv:0901.2063)

A comprehensive review, with plenty of background information, is in

• Christian Bär, Klaus Fredenhagen, (eds.) Quantum field theory on curved spacetime , Lecture notes in physics, Springer (2009)

In perturbation theory

The observation that in perturbation theory the Stückelberg-Bogoliubov-Epstein-Glaser local S-matrices yield a local net of observables was first made in

• V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978)

which was however mostly ignored and forgotten. It is taken up again in

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds Commun.Math.Phys.208:623-661 (2000) (arXiv)

(a quick survey is in section 8, details are in section 2).