nLab
microcausal functional

Context

Functional analysis

Algebraic Quantum Field Theory

Contents

Idea

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 future cone or all whose components are in the 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 is flexible enough to allow the usual (adiabatically switched) point-interaction terms, such as of phi^4 theory, see example 2 below.

Definition

Let (X,e)(X,e) be a spacetime (a pseudo-Riemannian manifold).

Definition

(smooth functionals on smooth functions)

For fC c (X)f \in C_c^\infty(X) a bump function, it induces the function given by integration

C (X) Φ(f) ϕ Xfϕdvol g, \array{ C^\infty(X) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_X f \cdot \phi dvol_g } \,,

where dvol gdvol_g denotes the volume form of the given pseudo-Riemannian metric, and similarly if fC c (X n)f \in C_c^\infty(X^n) is a bump function on the nn-fold Cartesian product manifold, then there is

C (X) Φ(f) ϕ Xfϕ ndvol g, \array{ C^\infty(X) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_X f \cdot \phi^n dvol_g } \,,

Write reg\mathcal{F}_{reg} for the sub-algebra of smooth functions on the smooth space C (X)C^\infty(X) which is generated from the functions Φ(f)\Phi(f) for nn \in \mathbb{N} and fC c (X n)f \in C^\infty_c(X^n).

A further evident generalization of this takes f(X n)f \in \mathcal{E}'(X^n) to be a compactly supported distribution and induces the function

C (X) Φ(f) ϕ f,ϕ n g. \array{ C^\infty(X) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \langle f, \phi^{\otimes^n} \rangle_g } \,.

Write dist\mathcal{F}_{dist} for the subalgebra generated by these functionals.

Definition

(microcausal functionals)

Write mc dist\mathcal{F}_{mc} \subset \mathcal{F}_{dist} for the subalgebra of smooth functionals

C (X) C^\infty(X) \longrightarrow \mathbb{R}

on the smooth space of smooth functions on XX which is generated from those distributions on some Cartesian product X nX^n (as in def. 1) whose wave front set excludes those covectors to a point in X nX^n all whose components are in the future cone or all whose components are in the past cone.

After deformation quantization to the Wick algebra/interacting field algebra, this is the origin of “operator-valued distributions” in perturbative quantum field theory.

Examples

Example

(regular functionals are microcausal)

Every regular functional (def. 1) is a microcausal functional (def. 2), since the wave front set of a distribution that is given by an ordinary function is empty:

reg mc. \mathcal{F}_{reg} \subset \mathcal{F}_{mc} \,.
Example

(adiabtaically switched point interactions are microcausal)

Let gC c (X)g \in C^\infty_c(X) be a bump function, then for nn \in \mathbb{N} the smooth functional

C (X) ϕ Xg(x)(ϕ(x)) ndvol(x) \array{ C^\infty(X) &\overset{}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_X g(x) (\phi(x))^n dvol(x) }

is a microcausal functional (def. 2).

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.

Proof

For notational convenience, consider the case n=2n = 2, the other cases are directly analogous. The distribution in question is the delta distribution

Xg(x)ϕ(x) 2dvol(x)= X×Xg(x 1)ϕ(x 1)ϕ(x 2)dvol(x 1)dvol(x 2)=gδ(,),(ϕpr 1)(ϕpr 2) g. \int_X g(x) \phi(x)^2 dvol(x) \;=\; \int_{X \times X} g(x_1) \phi(x_1) \phi(x_2) dvol(x_1) dvol(x_2) = \langle g \cdot \delta(-,-) , (\phi \circ pr_1)\cdot (\phi \circ pr_2) \rangle_g \,.

Now for (x 1,x 2)X×X(x_1, x_2) \in X \times X and 2nUX×X\mathbb{R}^{2n} \simeq U \subset X \times X a chart around this point, the Fourier transform 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)exp(k 1x 1+k 2x 2)dvol(x 1)dvol(x 2) g^(k 1+k 2). \begin{aligned} (k_1, k_2) & \mapsto \int_{\mathbb{R}^{2n}} g(x_1) \delta(x_1, x_2) \exp( k_1 \cdot x_1 + k_2 \cdot x_2 ) dvol(x_1) dvol(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 (in the sense of wave front sets) with k 1+k 2k_1 + k_2 (this prop.). Thus only on the cone k 1+k 2=0k_1 + k_2 = 0 that function is in fact constant and in particular not decaying.

This means that the wave front set consists of the element of the form (x,(k,k))(x, (k, -k)) with k0k \neq 0. Since kk and k-k are both in the future cone or both in the 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 th point interaction.

More generally:

Example

(polynomial local observables are microcausal)

Write

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)

References

Revised on September 18, 2017 14:16:43 by Urs Schreiber (77.56.177.247)