Doubt about the physical meaning of Tensor Products in Classical and Quantum Field Theory

Question

In a recent reference $[1]$ the authors evocated a particular Hamiltonian,

$$H = c\chi(\tau)\mu(\tau)\otimes\Phi(x(\tau)), \tag{1}$$

calling it "interaction Hamiltonian". The physics in this situation is described in the following $[1]$:

An Unruh-Dewitt detector, moving along a trajectory $x(\tau)$ Minkowski spacetime $-$$\tau$ is the proper time$-$, where also, $c$ is a coupling constant, $\chi(\tau)$ is a smooth function $-$ compactly supported which satisfies how the interaction "is switched on and off, betwween the detector's monopole moment operator $\mu(\tau)$" and the field pulled back to the detector's worldline $\Phi(x(\tau))$ $-$

Well, my question lies beyond the particularity of that paper. Like, consider,

$$H = A \otimes B, \tag{2}$$

What is the meaning of usage of tensor products like $(1)$ or, in "general case", $(2)$? Is like the field components $A$ interacting with the $B$ components?

$$---\circ---$$

$[1]$ https://arxiv.org/abs/2004.08225

Answer 1

You need to tensor product because you have two systems: the detector and the field that is being measured. In quantum mechanics one always uses a tensor product when you combine two systems. Consider two unentangled spins spins-1/2 whose state vector is the simple product state
$$ |1/2, s_1;1/2,s_2\rangle \equiv |1/2, s_1\rangle\otimes |1/2,s_2\rangle $$ for example. Or perhaps an entangled pair $$ \frac{1}{\sqrt 2}(|1/2, 1/2\rangle\otimes |1/2, -1/2\rangle-|1/2, -1/2\rangle\otimes |1/2, +1/2\rangle). $$ You have surely met the concept of combining systems before?