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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?

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    $\begingroup$ $q(t_1)$ is the position at time $t_1$. Is that the question? $\endgroup$
    – Qmechanic
    Nov 27 '21 at 15:22
  • $\begingroup$ 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? $\endgroup$
    – Nugi
    Nov 27 '21 at 15:50
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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.

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  • $\begingroup$ Do you have any personal reference regarding the measure for the path integral? $\endgroup$
    – Nugi
    Nov 30 '21 at 6:58
  • $\begingroup$ Are you looking for something like a book? You could look in the chapters called "Path Integral in Quantum Mechanics" in Mark Srednickis book "Quantum Field Theory" and Peskin and Schröders "An introduction to Quantum Field Theory" (in both books the chapter has the same name I think). You can also probably find a good description in some lecture notes online. $\endgroup$
    – unsure
    Nov 30 '21 at 14:18

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