nLab causal additivity

Idea

The formalization of perturbative quantum field theory via causal perturbation theory states the key properties that the scattering matrix of the theory is supposed to have as axioms (instead of imagining that the scattering matrix is defined by a path integral). The key such axiom is the statement that the S-matrix be causally additive (which gives causal perturbation theory its name). This essentially encodes the idea that effects in the quantum theory propagate causally, hence within the future cone and past cone of the region which causes the effect.

Causal additivity

The S-matrix (as discussed there) is a functional of the form

$$S \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{W}[ [ \tfrac{g}{\hbar} ] ]$$

sending local observables with compact support in spacetime (i.e. adiabatically switched interaction Lagrangian densities $L_{int}$ and source fields $J$) to formal power series in the ratio of a coupling constant $g$ over Planck's constant $\hbar$ with coefficients in the Wick algebra $\mathcal{W}$ of the underlying free field theory.

Causal additivity (Epstein-Glaser 73) is the statement/requirement that

$$\left( supp(J_1) \geq supp(J_2) \right) \;\; \Rightarrow \;\; \left( \underset{L \in \mathcal{F}_{loc}}{\forall} \left( S(L + J_1 + J_2) = S(L + J_1) S(L)^{-1} S(L + J_2) \right) \right) \,.$$

Here $supp(-)$ denotes the spacetime support of the given local observables and $supp(J_1) \geq supp(J_2)$ means that $supp(J_1)$ does not intersect the causal past of $supp(J_2)$. The product on the right is the product in the Wick algebra (hence the normal ordered product).

Causal factorization

As a special case, causal additivity immediately implies causal factorization:

$$\left( supp(L_1) \geq supp(L_2) \right) \;\; \Rightarrow \;\; \left( S(L_1 + L_2) = S(L_1) S(L_2) \right) \,.$$

This in turn directly implies that the perturbative expansion of the S-matrix is given by time-ordered products.

Causal locality of quantum observables

But the reason why the full condition is called causal additivity is that it is equivalent to a simpler-looking condition on the generating function

$$Z_{L}(J) \;\colon\; S(L)^{-1} S(L + J)$$

that is induced by the S-matrix. In terms of these, causal additivity is equivalently (by this lemma) the condition

$$\left( supp(J_1) \geq supp(J_2) \right) \;\; \Rightarrow \;\; \left( \underset{L \in \mathcal{F}_{loc}}{\forall} \left( Z_{L}(J_1 + J_2) = Z_L(J_1) Z_L(J_2) \right) \right) \,.$$

Notice that what these generating functions generate is the perturbative quantum observables

$$\hat A \coloneqq \frac{d}{d \epsilon} T_{\tfrac{g}{\hbar}L_{int}}( \epsilon A ) \vert|_{\epsilon = 0}$$

and causal additivity implies that as long as $L_{int}$ is adiabatically switched only outside the causal closure of $supp(A)$, then the system of quantum observables satisfies causal locality and hence forms a local net of observables (this prop.).

This is how causal perturbation theory gives rise to perturbative AQFT.

References

The concept goes back to

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

Lecture notes include

• Urs Schreiber, geometry of physics – perturbative quantum field theory (around this equation)

See also the references at causal perturbation theory, perturbative AQFT and S-matrix.