Quote by rubbergnome
Hello everyone. I have a question regarding algebraic QFT. I read that, in order to avoid illdefined, divergent expressions like the mode expansions for spacetimedependent field operators φ(x), one starts from the (Wightman?) axioms, using operatorvalued distribution on compact support functions, φ(f), instead. Formally this is achieved by integrating the product f(x)φ(x) which results in a smearing that encodes the uncertainty in spacetime position. This is, I think, to avoid having arbitrairly high frequency modes in the mode expansion in terms of annihilationcreation operators.
The question is: why many people use the spacetimedependence formalism anyway? Is that because:
1) it's operationally simpler
2) experiments give extremely accurate results anyway
3) renormalization takes care of every divergence
4a) phycisists don't bother that much with quantum fields being welldefined, or
4b) the φ(x) formalism is actually welldefined, and AQFT just wants to better formalize the theory
? I'm confused, because I rarely see people using the algebraic formalism.

All of 1) through 4a). The smeared version is needed only when you want to impose some rigor on what is done, as the field operators are distributions only, so its value at a spacetime point is typically not defined. But the extra baggage in the formulas is of no significant help in actual computations, so most people avoid it.