Smeared quantum fields in everyday QFT

[rubbergnome]

"Smeared" quantum fields in everyday QFT

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.

 

[A. Neumaier]

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.

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.

 

[rubbergnome]

Thanks a lot. I have many doubts about this, especially since I read papers in which the main objects of the theory where nets of Von Neumann algebras, and there was no reference at things like scattering amplitudes or path integrals to study processes, only observables. I mean, maybe one can derive scattering in AQFT, but I didn't found anything.

 

[A. Neumaier]

Thanks a lot. I have many doubts about this, especially since I read papers in which the main objects of the theory where nets of Von Neumann algebras, and there was no reference at things like scattering amplitudes or path integrals to study processes, only observables. I mean, maybe one can derive scattering in AQFT, but I didn't found anything.

The buzzword here is Haag-Ruelle theory. This is quantum field scattering done rigorously.

The problem is that it doesn't apply directly to QED or QCD as these are gauge theories not covered by the Wightman axioms. And it is not known how to modify the latter to make these theories fit rigorously. (Doing it for QCD without quarks is the essence of one of the Clay millenium problems, whose solution is each worth a million dollars.)

On the other hand, AQFT gives a lot of insight into the mathematical structure of QFTs.
Thus it complements the ''shut up and calculate'' approach that most QFT textbooks follow.