It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).


**Is there a way to present solutions of the Dirac equation using path integral?**


Here I assume that there is a distinguished time coordinate orthogonal to spatial coordinates. If necessary one may assume that the metric is Riemannian (rather than Minkowskian).



Path integral presentation of solutions of Dirac equation
There are several relevant papers:


* [Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation](https://academic.oup.com/ptps/article/doi/10.1143/PTPS.92.144/1934972) (1987)
* [Path Integral for
Relativistic Equations of Motion](https://arxiv.org/abs/hep-th/9708121) (1997)
* [Path Integral for the
Dirac Equation](https://arxiv.org/abs/hep-th/9809115) (1998)
* [On the Feynman Path Integral for the Dirac Equation in the General Dimensional Spacetime](https://link.springer.com/article/10.1007/s00220-014-1997-x) (2014)
* [On the construction of the Feynman path integral for the Dirac equation](https://www.semanticscholar.org/paper/On-the-construction-of-the-Feynman-path-integral-Ichinose/8e6e88a8f9004d7e394825ea68fa802166533a4d) (2017)


