Fock Space, Propagator 
*

*Is it correct to say that a classical time-varying wave function is a map $\mathbb{R}\times\mathbb{R}^3\xrightarrow{\psi}\mathbb{C}$ (saatisfying certain conditions at infinity)?

*Is it correct to say a quantum field is a map $\mathbb{R}\times\mathbb{R}^3\xrightarrow{\phi}\mathbb{F}^\mathbb{F}$ (satisfying certain conditions at infinity, $\mathbb{F}=\bigoplus_n\mathbb{H}_n$, and $\mathbb{H}_n$ is a Hilbert space)?

*In his book "Quantum Field Theory and the Standard Model," Matthew D. Schwartz on page 30 remarks that "kinetic terms are bilinear, meaning they have exactly two fields." As an example he mentions $\frac{1}{2}\phi\Box\phi$. Is it not standard notation practice in QFT to elide the composition operator $\circ$, so that this example could be written $\frac{1}{2}\phi\,\circ\,\Box\phi$? If $\Box\phi$ is derived from $\phi$, then how is this an example of "exactly two fields"?

*In his Lecture notes, Chapter 3, Feynman Calculus, Professor Etingof explains that calculation of coefficients "$A_i$ reduces to calculation of integrals of the form $\int_VP(x)e^{-B(x,x)/2}dx$ where $P$ is a polynomial and $B$ is a positive definite bilinear form (in fact, $B(v,u)=\partial_v\partial_u(c)$)." Should this not be "nondegenerate" positive definite bilinear form? In his statement of Wick's theorem he writes, "Let $B^{-1}$ denote the inverse form on $V^\ast$." Does this bilinear form relate directly to Schwartz's remark in my question 3.? If $B^{-1}(l_i,l_{\sigma(i)})$ in his Theorem 3.1 is indeed a (Feynman) propagator, how exactly does it relate to Schwartz's $\Pi=-\frac{1}{\Box}$ on page 41?

 A: *

*It is unclear what is meant by "classical time-varying wave function". For example:


*

*The quantum state of a particle without spin in three dimensions is a function $\mathbb{R}^3\to \mathbb{C}$. Its time evolution gives a map $\mathbb{R}\times\mathbb{R}^3\to \mathbb{C}$. But this isn't "classical".

*A classical complex scalar field in $\mathbb{R}\times\mathbb{R}^3$ is also a function $\mathbb{R}\times\mathbb{R}^3\to\mathbb{C}$. But this isn't a "wave function" (at least, in its usual meaning: some kind of quantum state).


*It is true that a quantum field is (roughly) a map from space-time ($\mathbb{R}\times\mathbb{R}^3$) to the endomorphisms of a Hilbert space $F$. However, if the $H_n$ are the Hilbert spaces of $n$-particle states, then it isn't true in general that $F$ is decomposed as their direct sum. That is only true for the free theory.

*The kinetic term is $E(\phi,\phi)$, where $E$ is the bilinear map given by $E(\phi_1,\phi_2)=\phi_1\circ\square\phi_2$. This is the sense in which the kinetic term is bilinear and "has exactly two fields". Perhaps it's clearer to say that the kinetic term is quadratic in $\phi$.

*$B(\phi, \phi)$ is the quadratic part of the action. That includes the kinetic term plus possibly a mass term $\sim m^2 \phi^2 $. For a massless theory, the quadratic part equals the kinetic term. We can rewrite a bilinear form $B$ as a linear transformation $\tilde{B}$ followed by a scalar product:
$$
B(\phi_1, \phi_2)=\left<\phi_1,\tilde{B} \phi_2\right>.
$$
The inverse form is $B^{-1}(\phi_1,\phi_2)=\left<\phi_1,\tilde{B}^{-1} \phi_2\right>$. In the case in which $B(\phi_1,\phi_2)=\int d^4x \phi_1\square\phi_2$, we have $\tilde{B}=\square$ and therefore the propagator is $\tilde{B}^{-1}=\frac{1}{\square}$.
A: 
  
*
  
*Is it correct to say that a classical time-varying wave function is a map $\mathbf{R}\times\mathbf{R}^3\xrightarrow{\psi}\mathbf{C}$ (satisfying certain conditions at infinity)?
  

It could be modeled that way, sure. But this describes a single wave function. If we were interested in the space of wave functions, we would be looking at some subspace of $L^{2}(\mathbb{R})$.


  
*Is it correct to say a quantum field is a map $\mathbf{R}\times\mathbf{R}^3\xrightarrow{\phi}\mathbf{F}^\mathbf{F}$ (satisfying certain conditions at infinity, $\mathbf{F}=\bigoplus_n\mathbf{H}_n$, and $\mathbf{H}_n$ is a Hilbert space)?
  

Hmmm...this is misleading, I think because quantum fields are operator densities, whereas I'm not certain $\mathbf{F}^\mathbf{F}$ captures this aspect to them. (Now, I may be ignorant of various theorems mathematical analysis which supply the necessary & sufficient conditions for an operator density to be represented by a function, so just be aware of that caveat.)
When I say "operator density", I mean -- physically -- we ought not ask the question "What's the value of the field at this point?" But it is meaningful to ask "In this region, what is the average value of the field?"
I keep arguing back and forth whether $\mathbf{F}^\mathbf{F}$ provides the right domain for this. The number operator lives there, but it's also implicitly defined over a region of spacetime. The creation and annihilation operators live there too, and they're operator densities.
There's also subtleties surrounding interacting field theories...


  
*In his book "Quantum Field Theory and the Standard Model," Matthew D. Schwartz on page 30 remarks that "kinetic terms are bilinear, meaning they have exactly two fields." As an example he mentions $\frac{1}{2}\phi\Box\phi$. Is it not standard notation practice in QFT to elide the composition operator $\circ$, so that this example could be written $\frac{1}{2}\phi\,\circ\,\Box\phi$? If $\Box\phi$ is derived from $\phi$, then how is this an example of "exactly two fields"?
  

Well, two things:
First, Schwartz is a little sloppy here. By they "have exactly two fields" he means that there are two factors of the same field in a given kinetic term, or derivatives of the same field. So there are two factors in this kinetic term: the field $\phi$ and the D'Alembertian of the field $\Box\phi$.
Second, remember composing functions $g\circ f$ requires the codomain of $f$ to be the domain of $g$. We cannot write $\phi\circ\Box\phi$ because $\phi\colon\mathcal{M}\to\mathrm{``something"}$ a field is a "function" which takes in a point in spacetime $\mathcal{M}$ and produces "some value"...not a point in spacetime. So composing $\phi\circ\Box\phi$ is a "meaningless" (wrong) statement.
A: Pretty much everything I was trying to understand on this question is handled by A. Zee, "Quantum Theory in a Nutshell." Basically, my confusion was not appreciating that there are multiple approaches to understanding what a Feynman Diagram is. The "canonical" approach versus the "path integral" approach, for me, loses out in terms of my personal "principle of least thought." This is basically that I seek understanding in terms of the least number of steps, each of which is no greater than my intuition can handle.
