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? 
 A: I don't know how to extend it from 1+1 dimensions to 3+1 dimensions, but I think I can answer the question asked, which is: why does the trivial approach of just copying the same formula not work?
The gist of it is already alluded to in the question:  in 1-dimension, there is only 1 way to change directions, whereas in 3-dimensions, there are an infinite number of ways in which you can change directions.  Having a continuous infinity of different choices at each time step is very different from having a single binary choice at each time step.
With only 2 choices, you can just multiply by 1 if you remain in the same direction, or i if you change direction by 180 degrees.  But if the direction changes by 23 degrees instead of 180 degrees, what should we multiply by?  Surely something more complicated than just i.  One approach (Hyperdiamonds) is to replace the choices 1 and i with an arbitrary quaternion number, so each number you multiply by has components (1,i,j,k).  This makes this even more complicated than it sounds, since multiplication of quaternions is non-commutative.  (The other approach is hidden behind a paywall which I can't access at the moment; but I'm sure whatever it is, it's also more complicated than the simple 1+1 dimensional formula.)
Another way to see why you might expect the 3+1 dimensional case to be a lot more difficult is to think about what the term "spin 1/2 particle" means in each case.  In 1+1 dimensions, a spin-1/2 particle just means a spinless particle which has one extra binary degree of freedom, that's 0 or 1, + or -.  But in 3+1 dimensions what's meant by a "spin 1/2 particle" is a spinor.  A spinor is not just a binary degree of freedom, it can only be represented by a column of numbers.  For a massless particle with fixed chirality, you need a column of 2 complex numbers to represent it.  But for an arbitrary massive spin-1/2 particle, you need a column of 4 complex numbers to represent it.
