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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?

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  • $\begingroup$ 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? $\endgroup$
    – Andrew
    Commented Jan 4, 2017 at 5:25
  • $\begingroup$ @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 $\endgroup$
    – Prem
    Commented Jan 4, 2017 at 5:36
  • $\begingroup$ 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. $\endgroup$
    – Prem
    Commented Jan 4, 2017 at 5:46
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    $\begingroup$ 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). $\endgroup$
    – Andrew
    Commented Jan 4, 2017 at 5:49

1 Answer 1

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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.

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  • $\begingroup$ thank you. Your answer pretty much answers my question, given that i understand the whole of it. Could you explain why spin or spinor affects the motion of a particle in vaccum? For example, a standard layman definition of spin is that it can be assumed as a classical rotating charged sphere, although such definition is of course not accurate. So, why does this physical quantity affects the motion even in absence of any field. Or, does the electron creates and interacts with EM field even if it is moving alone? Why exactly should spin affect motion? $\endgroup$
    – Prem
    Commented Jan 5, 2017 at 8:20
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    $\begingroup$ I don't think of spin as something that affects the motion of a particle through the vacuum. The motion of a particle, whether it has spin or not, is determined by the Klein-Gordon equation which relates its energy, momentum, and mass. The Dirac equation is the Klein-Gordon equation plus constraints on how the spin of the particle changes as it moves. I think Feynman's insight was that, at least in 1-dimension, the spin dynamics seem to be derivable from the motion as long as you make this strange assumption about what factor to multiply by every time it changes direction. $\endgroup$
    – reductionista
    Commented Jan 7, 2017 at 22:49

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