In the most general possible terms, the propagator is the Green's function of what operator?

Question

In non-relativistic quantum mechanics (NRQM), the propagator is the Green's function of $D_\text{NRQM} = i\partial_t - H$, where $H$ is the Hamiltonian of the non-relativistic system in question. In QFT (taking the simplest possible case of a Klein-Gordon field), the propagator is the Green's function of $D_\text{KG} = \partial^2 + m^2$. Naively, both theories involve a wave equation of the form $D\psi=0$, where $D$ is some differential operator, and the propagator is the Green's function of $D$. My main question is how this wave equation fits into the standard Schrödinger picture of QM, and how QFT can be understood as a special case of this.

I want to be able to say something like, "Given a quantum theory with Hilbert space $\mathcal H$ and Hamiltonian $H$, the propagator is the Green's function of the ___ operator, which acts on the Hilbert space ____ ," and then, "For example, for a scalar field in QFT, we have $\mathcal H = \mathcal H_\text{KG} = $ ___ and $H = H_\text{KG} = $ ___, from which it follows [derivation] that $D_\text{KG} = $ ___, which acts on the space ____."

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Note that my question has three aspects, which I think are related enough (or at least, my confusion is broad enough to encompass all of them at once) to count as a single question belonging in a single post.

• Terminological aspect: What is the name of the operator $i\partial_t-H$?
• Quantum mechanical aspect: On what Hilbert space does $i\partial_t-H$ act, and how is it formally related to the Hilbert space $\mathcal H$ of the system?
• Quantum field theory aspect: How can a QFT be viewed in the above "standard QM, Schrödinger picture" framework, in which $\partial^2 + m^2$ is a special case of $i\partial_t - H$?

Answer 1

Terminology
This may sound disappointing, but $i\partial_t - H$ is often called Schrödinger operator.

This is a rather common (even if unofficial) naming convention for operators to be called in the same way as the related equation - e.g., Laplace operator, Sturm-Liouville operator, etc.

"Quantum mechanical aspect"
The question about Hilbert space is not per se of quantum quantum mechanical nature - similar problems are ubiquitous whenever one uses differential equations. More precisely, the eigenvalue problems usually arise when dealing with a boundary value problems, while they are of less importance when solving initial value problems (such as Newton's equations.)

Schrödinger equation ($\mathcal{L}=i\partial_t - H(x)$) and diffusion equation ($\mathcal{L}=\partial_t - D\partial_x^2$ or more generally $\mathcal{L}=\partial_t - D(x)$ for Fokker-Planck equation) are partial differential equations. Separation of variables typically reduces them to a boundary value problem for the spatial part (dealt using Hilbert space formalisms) and initial value problem for the time component.

QFT aspect
$\partial_t^2-c^2\partial_x^2+m^2$ is again not particularly tied to quantum mechanics. Indeed, without $m^2$ term this is just the well-known wave equation, ubiquitous in electrodynamics, acoustics and elasticity theory. Separation of variables results in Helmholtz equation for the coordinate part, i.e., with operator $\partial_x^2\pm k^2$. Full "Klein-Gordon" form arises, e.g., when dealing with waveguides, where one separates variables for only the transverse part of the field, but keep the wave part for the longitudinal component.

The mathematical theory for these equations is well-known and pre-dates quantum mechanics. (Pierre-Simon Laplace 1749-1827, Siméon Denis Poisson 1781-1840, Hermann von Helmholz 1821-1894, Erwin Schrödinger 1887-1961, Quantim mechanics - b. mid-1920s.)