Maxwells equations and also the equations of fluid dynamics can be formulated as integral equations. These equations allow so called weak (non-differentiable) solutions, e.g. shock waves in fluid dynamics. That is why the integral equations are often seen as the more fundamental equations, see

Whatever physical model you choose, you have to understand that (the integral equation) is the real equation you care about, and (the partial differential equation) is just a convenient way to write the equation. https://math.stackexchange.com/a/3314745/27609

Absolutely nothing in physics is completely described by a PDE, if you look at a sufficiently small resolution, because space and time are not continuous. ... However almost everything in physics is described at a fundamental level by conservation laws which are most naturally expressed mathematically as integral equations not as differential equations. https://math.stackexchange.com/a/3315144/27609

However I have never seen an integral formulation of the equations of quantum mechanics (e.g. the Schrödinger equation) or an integral equation of Einsteins field equation of general relativity. So is there an integral formulation of QM or GR? And if not, why not? Are weak solutions not possible in QM and GR?

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    $\begingroup$ Propagators! $\endgroup$ – Cosmas Zachos Sep 1 '19 at 23:57
  • $\begingroup$ What kind of integral formulation? Given any PDE describing the behavior of some field or other space-filling entity (like the EM field or a fluid), of course we can always integrate the PDE over some given spatial volume (or area or curve) to obtain an integral equation. In some cases, we can reduce the integral over a spatial region to an integral over the region's boundary. Are you specifically asking when such reductions are possible? $\endgroup$ – Chiral Anomaly Sep 2 '19 at 0:32
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    $\begingroup$ @ChiralAnomaly I actually meant what you are describing. The integral of the PDE over some volume. Now I found out that for QM this formulation is described at en.wikipedia.org/wiki/… . But I haven't seen something like that for GR. $\endgroup$ – asmaier Sep 2 '19 at 19:32

The integral form to the Schrödinger equation



$$\psi(\vec{x})=\psi_0(\vec{x})-\frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\vec{x}-\vec{x}_0|}}{|\vec{x}-\vec{x}_0|}V(\vec{x}_0)\psi(\vec{x}_0)d^3\vec{r}_0$$

where $\psi_0$ is a solution to the free Schrödinger equation $$(\nabla^2+k^2)\psi_0=0$$

The derivation isn't short so I will just give a source. It is done in Griffiths' Quantum Mechanics book, in the scattering chapter (Equation 11.67 in the 2nd Edition). I found a link online which follows most of Griffiths derivation in case you don't have the book.

As for Einstein's equations I'm not a big help. A quick search brought me to this paper, which seems promising, but I can't vouch for it personally.

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  • $\begingroup$ This is not quite what I had in mind, but interesting nevertheless. $\endgroup$ – asmaier Sep 2 '19 at 19:33

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