Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $$S=\int_{t'}^{t''}L(q,\dot{q})$$ is the action evaluated for $$t\in (t',t'').$$ My question is about the insertion of a position operator $$Q(t_1)$$ for the propagator, i.e.

$$\langle q''t''|Q(t_1)|q't'\rangle=\int\mathcal{D}q \ q(t_1)\exp{(iS)}.$$

What does it meane to have a path integral with an integrand $$q(t_1)$$? After all, one way to interpret the path integral measure $$\mathcal{D}q$$ is to imagine an infinite number of slice of time intervals (and therefore slices of paths). In this case, should I understand $$q(t_1)$$ as the position operator at time $$t_1$$, or in the slice-view-point, position operator living in the time interval that contains $$t_1$$?

Or, is there a better way to understand this?

• $q(t_1)$ is the position at time $t_1$. Is that the question? Nov 27 '21 at 15:22
• I understand the path integral as an infinite limit of a product of integrals. My question is, which integration measure among the infinite number of measures? I think it should be understood as integrated over $dq_i$ such that $t_1 \in (t_i, t_{i+1}),$ but is this correct?
– Nugi
Nov 27 '21 at 15:50

The path integral itself is as far as I know not well defined as an "integration over all functions" (i.e. a corresponding measure has not been constructed yet). So let us stay in the "discrete" case where we have finitely many times $$t_i$$ and for each time we have a coordinate $$q_i=q(t_i)$$ and you have an integral over each $$q_i$$. So in the most extreme case with only two times $$t_1, t_2$$ we only have coordinates $$q_1,q_2$$, meaning that our integral would be of the form $$\int dq_1 dq_2 q_1 \exp(iS(q_1,q_2))$$. When this concept is clear you can easily just do the same for more than two different times.