# Correct method for splitting path integral in two

In 3D point particle quantum mechanics we have that the propagator can be represented as a path integral $$$$\langle x|e^{-iHt}|y\rangle =\int_{\gamma(0)=x}^{\gamma(t)=y}\mathcal{D}[\gamma(\tau)]e^{iS[\gamma,\dot{\gamma}]}$$$$ I have two questions regarding how one can split up the propagator into two or more paths

1. What is the correct procedure for splitting the path integral in two? I have derived a formula specific to time independent Hamiltonians to be $$$$\langle x|e^{-iHt}|y\rangle =\frac{1}{t}\int_0^t dt'\int d^3z\int_{\gamma'(t')=z}^{\gamma'(t)=y}\mathcal{D}[\gamma'(\tau')]e^{iS[\gamma',\dot{\gamma}']}\int_{\gamma(0)=x}^{\gamma(t')=z}\mathcal{D}[\gamma(\tau)]e^{iS[\gamma,\dot{\gamma}]}\quad \text{Eq}.(1)$$$$ where now $$t'$$ has the interpretation as the time at which the particle passes through the intermediate point $$z^i$$ and we sum over all such possible times and intermediate points. Is this formula correct? Is there a more general form?

2. On page 33, Eq.97 of Balitsky's paper https://arxiv.org/abs/hep-ph/0101042 he chooses a specific component of the intermediate point to be zero (which I will take to be $$z^1$$ below) and states the result to be $$$$\langle x|e^{-iHt}|y\rangle =\int_0^t dt''\int d^3z\delta(z^1)\int_{\gamma'(t')=z}^{\gamma'(t)=y}\mathcal{D}[\gamma'(\tau')]e^{iS[\gamma',\dot{\gamma}']}\int_{\gamma(0)=x}^{\gamma(t)=z}\mathcal{D}[\gamma(\tau)]\dot{z}^1(t)e^{iS[\gamma,\dot{\gamma}]}$$$$ where one now acquires the velocity factor $$\dot{z}^1(t')$$ at the point $$z^1=0$$. This result appears to be in conflict with Eq.(1) above, leading me to doubt Eq.(1) even further. Is there a difference between breaking up a path integral in two and not specifying the intermediate point (integrating over the intermediate point) versus specifying that the particle was at a particular intermediate point before reaching the final point?

(Note for anyone who consults the equation referenced in Balitsky's paper: He is working in the Schwinger proper time formalism of QFT which leads to a 4+1 point particle quantum mechanics description of the propagator)

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. e.g. arxiv.org/abs/hep-ph/0101042 May 26, 2017 at 9:51
• Relevant - arxiv.org/abs/2110.04969. Fairly recent, presumably still under review
– lux
Jun 5 at 21:07

One can then always trivially multiply by $$1 = \frac{1}{t}\int_0^t dt''$$ to find $$\langle x|e^{-iHt}|y\rangle=\frac{1}{t}\int_0^t dt''\int_{-\infty}^\infty dz\,\int_{\gamma(t_1)=z}^{\gamma(t)=y}\mathcal{D}[\gamma(\tau)]e^{iS[\gamma,\dot{\gamma}]}\int_{\gamma(0)=x}^{\gamma(t_1)=z}\mathcal{D}[\gamma(\tau)]e^{iS[\gamma,\dot{\gamma}]}.$$