# This path integral form of Dirac equation (in 3+1 dimension) free space is elegant, but what is it's form in presence of electromagnetic field?

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This arxiv paper generalizes the Feynman checkerboard path integral for Dirac-equation in higher dimensions in a very elegant way.

According to this paper, the amplitude for an electron to travel from one point to another can be obtained in the following way:

Consider $x$, $y$ and $z$ axis as our frame of reference with the electron initially at origin $O$ at time $t=0$. Then, the amplitude for the electron to reach a point P(X,Y,Z,T) will be given by following rule:

1. The electron can only move along $x$, $y$ **'or'** $z$ axis *at a time*. This means that we are considering motion only along an imaginary lattice in space.
2. After every infinitesimally small time $\delta {t}$, the electron is allowed to change its direction of motion if it wants to.
3. Since the electron is limited to move only along the lattice, and still maintain the speed of light in non-discrete way in 3 space, it is allowed to move at a constant speed $\sqrt {3}c$ (or so I guess). Here $c$ is the speed of light.
4. To find the amplitude for the electron to move from $O$ to $P$, we need to consider all possible paths connecting these points. Moving back in time for the electron is not allowed, because it is in free space,  I guess.
5. Each path will be weighted according to the rule given below and the final amplitude will be sum of all these 'weights'.

Now, the rule is: Whenever the electron changes direction in this space-time grid, its amplitude must be multiplied by one of the 3 quaternion numbers. If it changes direction and starts moving towards $x$ axis, multiply the amplitude by $i$. If it starts moving towards the $y$ (or $z$) axis the next time, multiply a $j$ (or $k$, respectively) to the amplitude.

So, my question is: What happens when there is electromagnetic field present? Can you give an analogous expression for the Dirac path integral in electromagnetic field, in the spirit of the above treatment? Do you know of some paper or textbook where a solution exists?

You may skip the text that follows as it is not directly relevant:

In order to evaluate the amplitude of the electron using the above method, we will need to solve the rest of the problem using complicated math. But with this method, at least this 'physical' idea related to the Dirac equation has been reduced to a 'mathematical' one.

But above all, if one can find an elegant path integral for Dirac equation in the presence of electromagnetic field, then this can complete the treatment of Quantum electrodynamics that is given in Feynman and Hibb's book (Quantum mechanics and path integral). They did an amazing job of explaining QED using just path integral and calculus in the book, but they used Schrodinger's equation in the treatment (!) instead of Dirac's, simply because there was no known simple path integral form for Dirac equation in 3+1 dimensions.

So, if one can find a simple path integral form of Dirac equation in the presence of EM field, then it will be of great educational value as an introduction to QED.

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