algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
Euler-Lagrange form, presymplectic current?
quantum mechanical system, quantum probability
state on a star-algebra, expectation value
collapse of the wave function?/conditional expectation value
quasi-free state?,
canonical commutation relations, Weyl relations?
normal ordered product?
interacting field quantization
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.
The S-matrix (as discussed there) is a functional of the form
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
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).
As a special case, causal additivity immediately implies causal factorization:
This in turn directly implies that the perturbative expansion of the S-matrix is given by time-ordered products.
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
that is induced by the S-matrix. In terms of these, causal additivity is equivalently (by this lemma) the condition
Notice that what these generating functions generate is the perturbative quantum observables
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.
See also the references at causal perturbation theory, perturbative AQFT and S-matrix.
Last revised on August 2, 2018 at 03:06:04. See the history of this page for a list of all contributions to it.