# Why is the trivial analogous expression for Feynman's checkerboard approach to Dirac's equation in 3+1 dimensions (as described below) not correct?

Feynman's checkerboard approach to Dirac's equation in 1+1 space says that a half spin particle can be assumed to be traveling at speed of light and switching directions only after discrete intervals of time. So, the amplitude will be given by the expression: $$K=\sum \limits_{n=1}A(n)(iE)^{n},$$ where $E$ is infinitely small time interval, $n$ is the number of time the particle changes direction and $A(n)$ is a function equal to the number of individual paths possible for given $n$.

I do not understand why this expression cannot be extended to 3+1 dimensions in following way: let there be a spin half particle moving at speed of light, changing direction of motion only after discrete interval of time. Then the Kernel will be given by $$K=\sum \limits_{n=1}A(n)(iE)^{n},$$ where $n$ is the number of times the particle changes direction.

Here, the analogy will only suffer from the fact that in 1+1 dimensions, there are only two direction to choose from and change, while in 3 space, there are infinite ways in which the particle can change its direction of motion.

Why is this analogy wrong? Can you explain a basic physical reason why this analogy cannot be extended to 3+1 dimensions in such a simple way? There must be some physical reason prohibiting this.

Is it possible to express path integral form of Dirac equation in 3+1 dimensions in a simple, mathematical way?

• The wikipedia article you link to cites two different ways to extend Feynman's 1+1 dimension approach to work in 3+1 dimensions. One is to take a 3+1 dimensional lattice of points, and to sum over different paths traversing the lattice. Another way is to embed the 1+1 lattice into two continuous spatial dimensions. It's not clear to me from your question what you are asking. Are you asking how one or both of those approaches works? Or is your question about why there isn't a third option? – Andrew Jan 4 '17 at 5:25
• @andrew Actually I followed the papers cited by wikipedia and tried to read the solutions, but could not understand them because they introduced many physical quantities like spinor which i currently do not understand. So meanwhile as I understand these physical quantities, I was wondering why a simple mathematical extention of the model cannot be made in which we allow the electron to move as it may, and use analogous equation for kernel. I am not asking how the two approaches given in wikipedia are equivalent. I am asking why the approach I have described in the question does not works – Prem kumar Jan 4 '17 at 5:36
• Also, you said, " another way is to embed the 1+1 lattice into two continous spatial dimension." This sounds like the solution I have suggested in the question. Could you please explain this method as an answer. The problem actually is that I tried, but could not, understand the extention of feynman's model in 3+1 dimension as suggested in wikipedia. – Prem kumar Jan 4 '17 at 5:46
• Yes, I agree, it does sound similar to what you have in your answer. I don't have access to the papers right now as I am traveling, but I will try to take a look when I do. It wouldn't surprise me to learn that your basic intuition is right and leads to something along the lines of one of the two approaches on wikipedia (probably the second one, as you point out). – Andrew Jan 4 '17 at 5:49