microcausal polynomial observable


Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)


field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory


quantum mechanical system

free field quantization

gauge theories

interacting field quantization


States and observables

Operator algebra

Local QFT

Perturbative QFT

Functional analysis



The microcausal functionals on the space C (X)C^\infty(X) of smooth functions on a globally hyperbolic spacetime (X,e)(X,e) are those which come from compactly supported distributions on some Cartesian product of copies of XX such that the wave front set of the distributions excludes those covectors to a point in X nX^n all whose components are in the closed future cone or all whose components are in the closed past cone

These functionals underly the Wick algebra of free field theories. The condition on the wave front is such that the product of distributions with a Hadamard distribution is well defined, so that the coresponding Moyal star product is well defined, which gives the Wick algebra. At the same time the condition includes local observables and hence in particular the usual (adiabatically switched) point-interaction terms, such as of phi^4 theory.

types of observables in perturbative quantum field theory:

local field linear microcausal polynomial general regular \array{ && \text{local} \\ && & \searrow \\ \text{field} &\longrightarrow& \text{linear} &\longrightarrow& \text{microcausal} &\longrightarrow& \text{polynomial} &\longrightarrow& \text{general} \\ && & \nearrow \\ && \text{regular} }



(polynomial observable)

Let EfbE \overset{fb}{\to} be field bundle which is a vector bundle. An off-shell polynomial observable is a smooth function

A:Γ Σ(E) A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}

on the on-shell space of sections of the field bundle EfbΣE \overset{fb}{\to} \Sigma (space of field histories) which may be expressed as

A(Φ)=α (0)+ Σα a (1)(x)Φ a(x)dvol Σ(x)+ Σ Σα a 1a 2 (2)(x 1,x 2)Φ a 1(x 1)Φ a 2(x 2)dvol Σ(x 1)dvol Σ(x 2)+, A(\Phi) \;=\; \alpha^{(0)} + \int_\Sigma \alpha^{(1)}_a(x) \Phi^a(x) \, dvol_\Sigma(x) + \int_\Sigma \int_\Sigma \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2) \,dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) + \cdots \,,


α (k)Γ Σ k((E *) sym k) \alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)

is a compactly supported distribution of k variables on the kk-fold graded-symmetric external tensor product of vector bundles of the field bundle with itself.


PolyObs(E)Obs(E) PolyObs(E) \hookrightarrow Obs(E)

for the subspace of off-shell polynomial observables onside all off-shell observables.

Let moreover (E,L)(E,\mathbf{L}) be a free Lagrangian field theory whose equations of motion are Green hyperbolic differential equations. Then an on-shell polynomial observable is the restriction of an off-shell polynomial observable along the inclusion of the on-shell space of field histories Γ Σ(E) δ ELL=0Γ Σ(E)\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E). Write

PolyObs(E,L)Obs(E,L) PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})

for the subspace of all on-shell polynomial observables inside all on-shell observables.

By this prop. restriction yields an isomorphism between polynomial on-shell observables and polynomial off-shell observables modulo the image of the differential operator PP:

PolyObs(E,L)restrictionPolyObs(E)/im(P). PolyObs(E,\mathbf{L}) \underoverset{\simeq}{\text{restriction}}{\longleftarrow} PolyObs(E)/im(P) \,.

(microcausal observable)

For Σ\Sigma a spacetime, hence a Lorentzian manifold with time orientation, then a microcausal observable is a polynomial observable (def. 1) such that each coefficient α (k)\alpha^{(k)} has wave front set excluding those points where all kk wave vectors are in the closed future cone or all in the closed past cone.



(non-singular observables are microcausal)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory.

Then a regular observable, hence a polynomial observable (this def.) whose distributional coefficients α a 1a k\alpha_{a_1 \cdots a_k} (eq:RecalledForWickAlgebraExpansionOfPolynomialObservables) are non-singular distributions is a microcausal observable (def. \ref{MicrocausalFunctionals}).

This is simply because the wave front set of non-singular distributions is empty (by definition, via the Paley-Wiener-Schwartz theorem, this prop.).


(compactly averaged point evaluations are microcausal)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory. Assume the field bundle EE is a trivial vector bundle with linear fiber coordinates (ϕ a)(\phi^a).

Let gC c (X)g \in C^\infty_c(X) be a bump function, then for nn \in \mathbb{N} the polynomial observables (this def.) of the form

Γ Σ(E) Φ Xg(x)α˜ a 1a k(x)Φ a 1(x)Φ (a k)dvol Σ(x) \array{ \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C} \\ \Phi &\mapsto& \int_X g(x) \tilde \alpha_{a_1 \cdots a_k}(x) \Phi^{a_1}(x) \cdots \Phi^(a_k)\, dvol_\Sigma(x) }

are microcausal (def. \ref{MicrocausalFunctionals}).

If here we think of ϕ(x) n\phi(x)^n as a point-interaction term (as for instance in phi^4 theory) then gg is to be thought of as an “adiabatically switchedcoupling constant. These are the relevant interaction terms to be quantized via causal perturbation theory.


For notational convenience, consider the case of the scalar field with k=2k = 2; the general case is directly analogous. Then the local observable coming from ϕ 2\phi^2 (a phi^n interaction-term), has, regarded as a polynomial observable, the delta distribution δ(x 1x 2)\delta(x_1-x_2) as coefficient in degree 2:

A(Φ) =Σg(x)(Φ(x)) 2dvol Σ(x) =Σ×Σg(x 1)δ(x 1x 2)=α (2)Φ(x 1)Φ(x 2)dvol Σ(x 1)dvol Σ(x 2). \begin{aligned} A(\Phi) & = \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x) \\ & = \underset{\Sigma \times \Sigma}{\int} \underset{ = \alpha^{(2)}}{ \underbrace{ g(x_1) \delta(x_1 - x_2) }} \, \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \end{aligned} \,.

Now for (x 1,x 2)Σ×Σ(x_1, x_2) \in \Sigma \times \Sigma and 2nUX×X\mathbb{R}^{2n} \simeq U \subset X \times X a chart around this point, the Fourier transform of distributions of gδ(,)g \cdot \delta(-,-) restricted to this chart is proportional to the Fourier transform g^\hat g of gg evaluated at the sum of the two covectors:

(k 1,k 2) 2ng(x 1)δ(x 1,x 2)e i(k 1x 1+k 2x 2)dvol Σ(x 1)dvol Σ(x 2) g^(k 1+k 2). \begin{aligned} (k_1, k_2) & \mapsto \underset{\mathbb{R}^{2n}}{\int} g(x_1) \delta(x_1, x_2) e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )} \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \propto \hat g(k_1 + k_2) \end{aligned} \,.

Since gg is a plain bump function, its Fourier transform g^\hat g is quickly decaying (according to this inequality) along k 1+k 2k_1 + k_2 (this prop.), as long as k 1+k 20k_1 + k_2 \neq 0. Only on the cone k 1+k 2=0k_1 + k_2 = 0 the Fourier transform is constant, and hence in particular not decaying.

This means that the wave front set consists of the elements of the form (x,(k,k))(x, (k, -k)) with k0k \neq 0. Since kk and k-k are both in the closed future cone or both in the closed past cone precisely if k=0k = 0, this situation is excluded in the wave front set and hence the distribution gδ(,)g \cdot \delta(-,-) is microcausal.

(graphics grabbed from Khavkine-Moretti 14, p. 45)

This shows that microcausality in this case is related to conservation of momentum in the point interaction.

More generally:


(polynomial local observables are microcausal)


Ω poly h,v(E) \Omega_{poly}^{h,v}(E)

for the space of differential forms on the jet bundle of the field bundle EE which locally are polynomials in the field variables.

locC c (Σ)Ω poly 0,0(E)Ω poly d,0(E) \mathcal{F}_{loc} \; \subset \; C^\infty_c(\Sigma) \underset{\Omega_{poly}^{0,0}(E)}{\otimes} \Omega_{poly}^{d,0}(E)

for the subspace of horizontal differential forms of degree dd on the jet bundle (local Lagrangian densities) of those which are compactly supported with respect to Σ\Sigma (local observables) and polynomial with respect to the field variables.

Every L locL \in \mathcal{F}_{loc} induces a functional

Γ Σ(E) \Gamma_\Sigma(E) \longrightarrow \mathbb{R}

by integration of the pullback of LL along the jet prolongation of a given section:

ϕ Σj (ϕ) *L. \phi \mapsto \int_{\Sigma} j^\infty(\phi)^\ast L \,.

These functionals happen to be microcausal, so that there is an inclusion

loc mc \mathcal{F}_{loc} \hookrightarrow \mathcal{F}_{mc}

into the space of microcausal functionals (e.g. Fredenhagen-Rejzner 12, p. 21). In fact this is a dense subspace inclusion (e.g. Fredenhagen-Rejzner 12, p. 23)



Write reg mc\mathcal{F}_{reg} \subset \mathcal{F}_{mc} for that subalgebra of the algebra of microcausal functionals whose coefficients are non-singular distributions.


: reg \langle -\rangle \;\colon\; \mathcal{F}_{reg} \longrightarrow \mathbb{C}

be a state on regular observables which is quasi-free Hadamard. Then this uniquely extends to a state on microcausal functionsal

reg ! mc \array{ \mathcal{F}_{reg} &\overset{\langle -\rangle}{\longrightarrow}& \mathbb{C} \\ \downarrow & \nearrow_{\mathrlap{\exists ! \langle -\rangle}} \\ \mathcal{F}_{mc} }

(Hollands-Ruan 01, remark 1 on p. 12, implied by Brunetti-Fredenhagen 00, Hollands-Wald 01, a special case of Hollands-Ruan 01, theorem III.1 (ii))


Original articles


Revised on January 15, 2018 15:53:43 by Urs Schreiber (