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Hello everyone. I have a question regarding algebraic QFT. I read that, in order to avoid ill-defined, divergent expressions like the mode expansions for spacetime-dependent field operators φ(x), one starts from the (Wightman?) axioms, using operator-valued 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 annihilation-creation operators.

The question is: why many people use the spacetime-dependence 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 well-defined, or

4b) the φ(x) formalism is actually well-defined, and AQFT just wants to better formalize the theory

? I'm confused, because I rarely see people using the algebraic formalism.

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# Smeared quantum fields in everyday QFT

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