In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, $<\cal{O}(x)\phi_1(x_1)\phi_2(x_2)..\phi_n(x_n)>$ and this is defined as a path-integral. Typically there are going to be short-distance singularities if any of the two $x_i$ start coinciding. (and that was the topic of my last question)

Though I have done numerous calculations of calculating such correlation functions in various QFTs, it remains unclear to me as to at the fundamental level where does one draw the line between an "operator" and a "field". After quantization aren't all fields actually operators or more precisely Hilbert space operator valued fields on the space-time? This is the confusion that I would like to clarify here.

To start off let me cite some of the definitions that I have seen in this regard.

For the quantities labelled as $\phi$, as used above, one seems to use two terms - "local functionals" and "Wightman fields", namely,

A "local functional" at $x$ is a function of the fields and finitely many derivatives of the fields evaluated at $x$, like $\phi(x)$, $\phi^2(x)$ but NOT $\phi(x) + \phi(2x)$

"Wightman fields" $\phi(x)$ are distributions on the Minkowski space ($V$) with values in the space of operators on the subspace $\cal{D} \subset \cal{H}$ (the Hilbert space of multiparticle states). This means that for any Schwartz function $f$ on $V$, $\phi(f)$ is an honest operator $\phi(f)$ on $\cal{D}$.

It is not clear to me whether "local functionals" and "Wightman fields" are the same things of if there is a natural way to pass between the two things above but I feel that in literature these terms are used interchangeably.

For the quantities $\cal{O}$ I seem to see two statements,

that $\cal{O}(x)$ does

*not*act as an operator (or an operator valued distribution) on any reasonable subspace of the Hilbert space.that since one wants to deal with products of operators at different space-time points, its better to think in terms of "smeared" quantities like $\cal{O}(f) = \int f(x)\cal{O}(x) d^nx $ where $f$ is a compactly supported smooth function on $V$. Then the statement is that,

*"..the product $\cal{O}(f)\cal{O}(f')$ exists in the sense of correlation functions if and only if the "operator" $\cal{O}(x)$ (an operator valued distribution) is actually an honest operator i.e matrix elements of $\cal{O}(f)$ are matrix elements of some operator on $\cal{D}$.."*

Like in the first pair of points, here too there seems to be some interchangeability between the notion of $\cal{O}(x)$ and $\cal{O}(f)$ and they seem quite analogous to the corresponding $\phi(x)$ and $\phi(f)$! Where is the difference?

What exactly is an "operator valued distribution" that $\cal{O}(x)$ usually perhaps is not (first statement of the above pair) but it seem to have to be if products like $\cal{O}(f)\cal{O}(f')$ have to be defined (last statement of the above pair)?

It would be very helpful if someone can disentangle the above (and may be also my linked previous question!) and explain the difference between the notions of a "local functional", "Wightman field" and "operator".