Axiomatic Quantum Field Theory

Prerequisites: Canonical Quantum Field Theory, Hilbert Space, Distributions. Full path (25 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Quantum field theory as it is used in practice is a set of calculational recipes. We write down a Lagrangian, expand in a small coupling, meet infinite integrals, and remove the infinities by renormalization. The answers agree with experiment to many digits, but the steps in between are not mathematically well defined, and it is not clear which statements are artefacts of the recipe and which hold for any quantum field theory.

Axiomatic quantum field theory turns the question around. Instead of starting from a Lagrangian, it lists a small number of physical requirements that any relativistic quantum theory of fields should satisfy: states live in a Hilbert space, the theory respects special relativity, energies are bounded from below, there is a vacuum, and measurements at points that cannot exchange signals do not disturb each other. Everything else is then derived from these requirements alone.

A useful comparison is a set of rules for a whole family of board games. The rules do not tell you which game you are playing, but any move they forbid is forbidden in every game of the family. In the same way, results proven from the axioms hold for every theory that satisfies them, whatever its particles and interactions. The best known examples are the CPT theorem and the connection between spin and statistics, which were first proved for particular field theories built from a Lagrangian and later shown to follow from the axioms alone.

The approach also makes clear what is still unknown. Nobody has yet constructed an interacting theory in four spacetime dimensions that is proven to satisfy the axioms.

Concrete

The most widely used set of axioms is due to Wightman. We use the metric signature $(+,-,-,-)$ and state the axioms for a single Hermitian scalar field $\phi$ .

States. There is a separable Hilbert space $\mathcal{H}$ carrying a unitary representation $U(a,\Lambda)$ of the (proper orthochronous) Poincaré group. Writing $U(a,1) = e^{i P^\mu a_\mu}$ , the joint spectrum of the energy-momentum operators $P^\mu$ lies in the closed forward cone $\overline{V}_+ = \{p : p^0 \geq 0,\ p^2 \geq 0\}$ (spectrum condition). There is a unit vector $\Omega$ , unique up to a phase, with $U(a,\Lambda)\Omega = \Omega$ (the vacuum).
Fields. The field is an operator-valued tempered distribution: for every test function $f$ in the Schwartz space $\mathcal{S}(\mathbb{R}^4)$ there is an operator $\phi(f)$ , formally $\phi(f) = \int d^4x\, f(x)\phi(x)$ , defined on a common dense domain $D$ that contains $\Omega$ and is mapped into itself by the $\phi(f)$ and by $U(a,\Lambda)$ . The map $f \mapsto \langle \psi, \phi(f)\chi\rangle$ is a tempered distribution for $\psi,\chi\in D$ , and $\phi(f)^\dagger = \phi(\bar f)$ on $D$ .
Covariance. $U(a,\Lambda)\,\phi(f)\,U(a,\Lambda)^{-1} = \phi(f_{(a,\Lambda)})$ with $f_{(a,\Lambda)}(x) = f(\Lambda^{-1}(x-a))$ .
Locality. If the supports of $f$ and $g$ are spacelike separated, then $[\phi(f),\phi(g)] = 0$ on $D$ (for fermion fields the commutator is replaced by an anticommutator).
Cyclicity. Polynomials in the smeared fields applied to $\Omega$ span a dense subspace of $\mathcal{H}$ .

Why smear the fields? Already for the free field of mass $m$ the object $\phi(x)\Omega$ is not a vector in Hilbert space. With $\phi(x) = \int \frac{d^3p}{(2\pi)^3 2\omega_p}\big(a_p e^{-ip\cdot x} + a_p^\dagger e^{ip\cdot x}\big)$ , $\omega_p = \sqrt{\vec p^{\,2}+m^2}$ and $[a_p, a_q^\dagger] = (2\pi)^3 2\omega_p\,\delta^3(\vec p-\vec q)$ , we find

$\|\phi(f)\Omega\|^2 = \int \frac{d^3p}{(2\pi)^3 2\omega_p}\, |\tilde f(p)|^2 , \qquad \tilde f(p) = \int d^4x\, e^{ip\cdot x} f(x)\Big|_{p^0=\omega_p}.$

For a smooth, rapidly decreasing $f$ this is finite. In the limit $f \to \delta^4(x-y)$ we get $|\tilde f| = 1$ , and the integral diverges quadratically at large $|\vec p|$ . Point-like fields are too singular to be operators.

Wightman functions. All information about the theory sits in the vacuum expectation values

$W_n(x_1,\ldots,x_n) = \langle \Omega, \phi(x_1)\cdots\phi(x_n)\Omega\rangle ,$

which are tempered distributions. For the free field, $W_2(x,y) = \int \frac{d^3p}{(2\pi)^3 2\omega_p}\, e^{-ip\cdot(x-y)}$ . The Wightman reconstruction theorem states that a sequence of distributions $W_n$ with the properties implied by the axioms (covariance, spectrum condition, locality, positivity, cluster property for a unique vacuum) determines the Hilbert space, the vacuum and the field up to unitary equivalence. The spectrum condition implies that $W_n$ are boundary values of analytic functions in a complex domain, and these can be continued to Euclidean points. The Osterwalder-Schrader axioms describe the resulting Euclidean correlation functions, and reflection positivity is the condition that lets one go back to a Wightman theory.

What has been proven. From the Wightman axioms one derives the CPT theorem, the spin-statistics theorem, the Reeh-Schlieder theorem and Haag's theorem. Interacting theories satisfying the axioms have been constructed in two and three spacetime dimensions, for example $\phi^4$ theory, by the methods of constructive field theory. No interacting example is known in four dimensions; proving the existence of quantum Yang-Mills theory with a mass gap is one of the Clay Millennium problems.

Abstract

The algebraic formulation of Haag and Kastler moves the emphasis from fields to the algebras of observables attached to regions of spacetime. Fields are then one possible way to generate these algebras, and two fields that generate the same algebras describe the same physics.

A theory on Minkowski space $M$ is a net $\mathcal{O} \mapsto \mathfrak{A}(\mathcal{O})$ that assigns to each bounded open region (one can restrict to double cones) a unital C*-algebra, with the following properties:

Isotony: $\mathcal{O}_1 \subset \mathcal{O}_2 \Rightarrow \mathfrak{A}(\mathcal{O}_1) \subset \mathfrak{A}(\mathcal{O}_2)$ . The quasi-local algebra $\mathfrak{A}$ is the C*-inductive limit of the net.
Locality: if $\mathcal{O}_1$ and $\mathcal{O}_2$ are spacelike separated, then $[\mathfrak{A}(\mathcal{O}_1), \mathfrak{A}(\mathcal{O}_2)] = 0$ .
Covariance: there is a representation $(a,\Lambda) \mapsto \alpha_{(a,\Lambda)}$ of the Poincaré group by automorphisms of $\mathfrak{A}$ with $\alpha_{(a,\Lambda)}\big(\mathfrak{A}(\mathcal{O})\big) = \mathfrak{A}(\Lambda\mathcal{O}+a)$ .

A physical state is a positive normalized linear functional $\omega$ on $\mathfrak{A}$ . The GNS construction turns each state into a representation $\pi_\omega$ on a Hilbert space $\mathcal{H}_\omega$ with a cyclic vector $\Omega_\omega$ such that $\omega(A) = \langle\Omega_\omega, \pi_\omega(A)\Omega_\omega\rangle$ . For a vacuum state one demands Poincaré invariance and the spectrum condition for the implementing unitaries, and one works with the von Neumann algebras $\mathcal{R}(\mathcal{O}) = \pi_\omega(\mathfrak{A}(\mathcal{O}))''$ . The Hilbert space is a derived object here. Inequivalent representations of the same net are the rule, and Haag's theorem is one expression of this.

Several structural results follow in this setting:

Reeh-Schlieder: for a region $\mathcal{O}$ whose causal complement has nonempty interior, the vacuum is cyclic and separating for $\mathcal{R}(\mathcal{O})$ .
Modular theory: since $\Omega$ is cyclic and separating, Tomita-Takesaki theory gives a modular operator $\Delta$ and conjugation $J$ for each such region. For a wedge region and a theory generated by Wightman fields, the Bisognano-Wichmann theorem identifies $\Delta^{it}$ with the one-parameter group of Lorentz boosts that preserve the wedge (with the boost parameter rescaled by $2\pi$ ), and $J$ with a CPT-type reflection combined with a rotation. This is the algebraic origin of the Unruh temperature.
Type of local algebras: under general assumptions the algebras $\mathcal{R}(\mathcal{O})$ are factors of type III $_1$ . They have no pure normal states and no trace, which is why there is no local tensor product factorization of $\mathcal{H}$ into "inside" and "outside" of a region.
Superselection sectors: the theory of Doplicher, Haag and Roberts classifies representations that look like the vacuum outside a bounded region. Their charges carry a composition law and statistics. In four spacetime dimensions the statistics is permutation (Bose or Fermi, possibly parastatistics), while braid statistics occurs in two spacetime dimensions and, for charges localized in spacelike cones, in three. In four dimensions the Doplicher-Roberts theorem reconstructs from the sectors a compact global gauge group and a field net with normal commutation relations.

The Wightman and algebraic frameworks are closely related: the local von Neumann algebras generated by a Wightman field (under mild extra conditions on the field) satisfy the Haag-Kastler axioms. On the Euclidean side, the Osterwalder-Schrader axioms (with a growth condition on the Schwinger functions) are equivalent to the Wightman axioms.

Why is it interesting?

It is well known that people working at the frontiers of physics often don't know what they are talking about. This statement has at least two meanings. First, the experimental evidence available may be incomplete or ambiguous and confusing. Second, the theoretical concepts employed may not have received a precise formulation. In this talk, I will give arguments for the position that it can be useful to refine a theoretical framework to the point at which it is a mathematically precise and consistent scheme, even if, as is sometimes the case, the resulting conceptual frame has physical deficiencies. I will offer several examples to illustrate the point. Of course, it is only occasionally that this sort of enterprise can help in the hunt for the Green Lion, the ultimate Lagrangian of the world.

The usefulness of a general theory of quantized fields by A. S. WIGHTMAN

It was one of the surprises of the 1950s that two of the most prized general results of Lagrangian quantum field theory (also distinguished by being well verified experimentally), the CPT Theorem and the Connection of Spin With Statistics, turned out to be theorems of the general theory of quantized fields with only the above- mentioned hypotheses. The proofs were easy consequences of a characterization of the theories in terms of vacuum expectation values.

The usefulness of a general theory of quantized fields by A. S. WIGHTMAN

Path to this page

Leads to: Haag's theorem.