# Relativistic path integral

I got totally messed up with Problem 2-6 from Feynman's Quantum Mechanics and Path Integrals. The body of the problem is to find the kernel of a relativistic particle between the points a to b, that is,

$$K(b, a)=\sum_{all\ R}N(R)(i\epsilon)^{R}$$

, where R is the number of reversals of the particle moving in one dimension at the speed of light, i is an imaginary unit, $$\epsilon$$ is an infinitesimal time step, and N(R) represents the number of possible paths with R reversals. I'm being stacked at the first obstruction, which is to find N(R). I first come up with the recurrence relation considering an "n by m" grid

$$N_{+-}^{n,m}(R)=N_{+-}^{n-1, m-1}(R)+N_{-+}^{n-1, m-1}(R-2)+N_{++}^{n-1, m-1}(R-1)+N_{--}^{n-1, m-1}(R-1),$$

$$N_{++}^{n, m}(R)=N_{+-}^{n, m-2}(R-1)+N_{-+}^{n, m-2}(R-1)+N_{++}^{n, m-2}(R)+N_{--}^{n, m-2}(R-2),$$

,where $$N_{++}^{n, m}(R)$$ represents the number of the possible trajectories where the particle is coming out the initial position a at positive(denoted by subscript +/-) velocity and coming in the final position b at positive velocity, with N reversals in the "n by m" grid, and so on. However I finally find that it's really outta my ability to generalize N as a function of n, m, and R. I'm not sure this is the ordinary way to find the solution of the problem. I welcome any kinda help for this problem, thank you.

• I've added the homework-and-exercises tag. In the future, please add this tag to this type of problem.
– user4552
Commented Nov 8, 2019 at 13:41

## 1 Answer

That's the formula, although not really a closed form'' one: \begin{align*} K_{+-}(0,0,x,y)&= \sum_{r=0}^{\lfloor y/2 \rfloor}(-1)^r \binom{(x+y-2)/2}{r}\binom{(y-x-2)/2}{r}\varepsilon^{2r+1}i,\\ K_{++}(0,0,x,y)&= \sum_{r=1}^{\lfloor y/2 \rfloor}(-1)^r \binom{(x+y-2)/2}{r}\binom{(y-x-2)/2}{r-1}\varepsilon^{2r}, \end{align*} for each $$y>|x|$$ with even $$x+y$$, where say $$K_{++}(0,0,x,y)$$ is the sum over all paths from $$(0,0)$$ to $$(x,y)$$ starting and ending with a upwards-right move. See the precise definitions, proofs and references here (both in English and in Russian).

Notice that itself the Feynman problem is getting an asymptotic form for $$y\gg |x|$$, and it is rather hard for rigorous solution; we are currently writing a math paper on that.