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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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    $\begingroup$ A quantum field is a ill-defined mathematical concept. It can be made rigorous, but it is not the way a physicist would like. So a rather good answer could be: a field is an operator-valued distribution that somehow describe the physical content of a relativistic quantum mechanical system. $\endgroup$
    – Phoenix87
    Commented Dec 29, 2014 at 16:32
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    $\begingroup$ @Phoenix87: You just defined a quantum field properly: it is a distribution of operator values over a pre-defined geometry. Mathematicians may not like this approach (it has very serious problems with regards to the usual concepts of calculus that physicists like to apply to this object), but that is what we do and we get the correct results (i.e. the predictions match the experiments). $\endgroup$
    – CuriousOne
    Commented Dec 29, 2014 at 16:36
  • $\begingroup$ @CuriousOne What I have in mind is something more similar to a Segal field, which acts on states of a C*-algebra. To make contact with the operator-valued distributions and test functions one can take some Fourier transforms that send the state to the test function and the Segal field to said distribution. The action of the Segal field on the state (or perhaps the action of the state on the Segal field), which is a well defined evaluation of a linear functional over an element of a C*-algebra, becomes the usual field smeared-out with a test function. $\endgroup$
    – Phoenix87
    Commented Dec 29, 2014 at 16:46
  • $\begingroup$ @Phoenix: To the mathematically uninitiated that sounds a lot like the procedure of defining distributions as linear operators over spaces of test functions in functional analysis? I would very much expect that kind of procedure to be necessary to formalize the problems physicists like to gloss over both in qft as well as in classic field theory. $\endgroup$
    – CuriousOne
    Commented Dec 29, 2014 at 16:57
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    $\begingroup$ @Phoenix87: I certainly hope that we can find a way to get rid of these ugly descriptions. Like a whole generation of physicists I tend to hang on to what stuck to my brain early, but I would like to see a clean, hopefully algebraic and geometric description of QFT that does not suffer the conceptual clutter of the early approaches. $\endgroup$
    – CuriousOne
    Commented Dec 29, 2014 at 17:24

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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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  • $\begingroup$ You say that a quantum field is just a space time manifold with a fock space attached to each point. Yet quantum fields are always represented as just single terms; an example, the Feynman Rules a fermionic field is: ψ(xi). How is such a complex sounding system of fock spaces and Quantum harmonic oscillator be contained in just one single term? $\endgroup$
    – Oliver Gregory
    Commented Jul 23, 2017 at 21:32

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