Groucho Marxism

In a previous blog post I derived discrete versions of two key equations in quantum mechanics: the canonical commutation relation between position and momentum, and the Schrödinger equation. I did this by considering a particle moving in discrete time and space where time was taken to be imaginary. The eagle-eyed reader would have noticed a problem with the discrete version of the Schrödinger equation though: the imaginary constant i appears on the wrong side! Specifically, the equation I derived was: f_t = -iħf_xx/2m; whereas the ‘true’ Schrödinger equation is: if_t = -ħf_xx/2m. This suggests I didn’t have quite the right formulation. In this blog post I will try again using a slightly different approach, where instead of imaginary time, I will use complex probabilities.

As before, consider a particle moving in discrete time, let x(t) denote the position of the particle at (discrete) time t, and let dt denote the increment between successive time steps. Define the ‘forward’ velocity of the particle by u(t) = [x(t+dt)-x(t)]/dt, and define the commutator between position and velocity by [x,u](t) = u(t)x(t+dt)-x(t)u(t) (note that this has the opposite sign to the definition I used previously). Then it can be easily shown that [x,u] = u²dt. Now suppose that our particle is undergoing a random walk, so that at each time step it moves a short distance dx to the left or right, each with imaginary probability i/2. Then using revised definition [x,u](t) = E[u(t)x(t+dt)-x(t)u(t)] where E is the expectation operator, gives [x,u] = idx²/dt.

Let f(x,t) denote the probability that the particle will be in position x at time t. Then, using the fact that the particle stays in the same place with probability 1-i, we can see immediately that the function f() satisfies the relation: f(x,t+dt)=if(x+dx,t)/2+(1-i)f(x,t)+if(x-dx,t)/2. Analogously to the continuous case, we can define discrete partial derivatives as f_x(x,t) = [f(x+dx,t)-f(x,t)]/dx and f­_t(x,t) = [f(x,t+dt)-f(x,t)]/dt. From our expression for f() above, we have f_t(x,t) = i[f(x+dx,t)/2-f(x,t)+f(x-dx,t)/2]/dt. Let us also define the second partial derivative of f() with respect to x as f_xx(x,t) = [f_x(x,t)-f_x(x-dx,t)]/dx. Expanding this expression gives f_xx(x,t) = [f(x+dx,t)-2f(x,t)+f(x-dx,t)]/dx², and therefore f_t = i(dx²/2dt)f_xx. In other words, the function f() satisfies a discrete version of the diffusion equation, with diffusion constant idx²/2dt.

Let m denote the mass of the particle and set dx = ħ/mc, set dt = ħ/mc², where ħ is the reduced Planck constant, c is the speed of light, and i is the imaginary constant, so that i² = -1. Then idx²/dt = (iħ²/2m²c²)/(ħ/mc²) = iħ/2m = so [x,u] = iħ/m, or equivalently, [x,mu] = iħ, which is a discrete version of the canonical commutation relation between position and momentum. Furthermore, the function f() satisfies a discrete version of the Schrödinger equation: if_t = -(ħ/2m)f_xx, where this time the imaginary constant is on the correct side of the equation. This suggests that considering the motion of a particle as a random walk with complex probabilities is the ‘correct’ formulation. But how are we to interpret such probabilities?

In the standard (continuous) formulation of quantum mechanics, the function f() that satisfies the Schrödinger equation is referred to as the ‘wave function’. The interpretation is that when a measurement is performed, the probability density of the particle being in position x at time t is proportional to |f(x,t)|², where for a complex number c = a+bi, |c|² = cc* = (a+bi)(a-bi) = a²+b². In theory, we can apply exactly the same interpretation in the discrete case, except that now it is the probability rather than the probability density that is proportional to |f(x,t)|². This suggests that in the discrete case, the quantity f(x,t) represents a ‘probability amplitude’ rather than a probability, just as in the continuous case.

There is a problem with this interpretation however. In our derivation of the Schrödinger equation, we used the fact that f(x,t) represent a probability, rather than a probability amplitude, when we wrote down a recursive relation for it. Thus, in our formulation, we must interpret f(x,t) as a probability directly – which again leads to the question of how to interpret it. Perhaps we shouldn’t get too hung up on this though. After all, physicists have been arguing about how to interpret the standard (continuous) formulation of quantum mechanics for 100 years, and they still haven’t come to an agreement on what the ‘correct’ interpretation is, or if it even makes sense to talk about a ‘correct’ interpretation. However there is another formulation which obviates the need to deal with imaginary probabilities.

Let us revert back to our original formulation, where the particle moves left or right with (non-imaginary) probability 1/2, except now let us set dt = -iħ/mc^2­. Then, from the definition of the commutator given above, we have: [x,u] = dx²/dt = (-ħ²/m²c²)/(iħ/mc²) = iħ/m, or [x,mu] = iħ, as before. Also, we have: f_t = (dx²/2dt)f_xx = (iħ/2m)f_xx; multiplying both sides by i yields: if_t = -(iħ/2m)f_xx, which is the ‘true’ Schrödinger equation. Thus, we can derive the canonical commutation relation and Schrödinger equation in a discrete framework using imaginary time by simply flipping the sign of the time increment. This seems like the most promising approach to me. I will leave the question of how to interpret ‘imaginary time’ to a future blog post.