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I'm just tasting a bit of QFT and want to get started. I got stuck right at the start: what is a quantum field, and how should I look at it? This question can be a follow up of the What is a field, really? question.

For me a field can be seen as a function. So saying "vector field" is the shorthand of saying a "function that maps from whatever domain to vectors of some dimensions".

What's the type of the value this function maps to when speaking of quantum fields?

In this another question. They says it's operator valued. As far as I can remember operators can be seen as "ket-bras" so the matrix product of an infinitely long column and row vector. So an infinitely large matrix. Or in other words a function that takes 2 parameters and gives back a scalar. Is that right?

So if I'm right a quantum field is a "scalar field field". So a function that maps space-time coordinates to functions that maps 2 scalars to a scalar.

In this yet another question. It's said it's a scalar valued in path integral formalism. I'm yet to understand that formalism so far. But for now I don't see how a scalar and a function valued field be equivalent...

On the other hand just out of curiosity, how many fields do we need to deal with in QFT? In classical electromagnetics we had two vector fields the electric and magnetic and the Maxwell equations that describes the time evolution. After some googling I can see there are boson fields, fermion fields, higgs field, whatever field... Are all of these quantum fields (so operator valued)? Do we have the field equations that describe all the relationship between these fields (just like the Maxwell-equations do in EM)?

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Calmarius
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1 Answers1

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A quantum field, in general, is not a field of scalar functions (one can construct toy models that are). It can be understood as a field of operators or as a Fock space attached to each space-time point. The good news is that for many purposes we do not have to deal with the full complexity of such an object. Many interactions that are physically relevant can be understood as a convergent perturbation series of individual particle interactions (but one should not be under the impression that this perturbative field theory approach is complete except for a few small "convergence" problems). The other good news is that many properties of quantum fields can be studied using formalisms that look like as if the fields were scalar valued, or, at the very least, one can abstract from a full treatment because of symmetry properties.

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