Groucho Marxism

In a previous blog post I suggested it may be the case that many ideas in physics that seem strange when formulated in a continuous framework based on infinite sets make more sense in a discrete framework based on finite sets. In this blog post I will attempt to flesh out this claim in a bit more detail. To fix ideas, consider a particle moving in discrete time and let x(t) denote the position of the particle at (discrete) time t. Let dt denote the increment between successive time steps. There are two ways to define velocity at time t: a ‘forward’ velocity u(t) = (x(t+dt)-x(t))/dt, and a ‘backward’ velocity u(t) = (x(t+dt)-x(t))/dt. We can already see therefore that it doesn’t make any sense to talk of ‘the’ velocity at time t. In a discrete context there is no notion of instantaneous velocity.

Nevertheless, for the sake of argument let us go with the first definition (the forward velocity – this choice makes no difference to the argument that follows). Suppose that we want to multiply together the position and velocity of the particle. If we measure position first we will get x(t)u(t), whereas if we measure velocity first we will get u(t)x(t+dt). Let us define the difference, or commutator, between the two as [x,u](t) = u(t)x(t+dt) – x(t)u(t). Then it is easy to show that [x,u](t) = -u(t)²/dt. 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 probability p≤1/2. Using the slightly modified definition of the commutator [x,u](t) = Ev(t)x(t+dt)-Ex(t)v(t), where E is the expectation operator, gives the expression [x,u] = -2pdx²/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 1-2p, we can see immediately that the function f satisfies the relation f(x,t+dt) = pf(x-dx,t)+(1-2p)f(x,t)+pf(x+dx,t). Analogously to the continuous case, we can define the discrete partial derivatives of f 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. Similarly we can define the discrete second partial derivatives of f as f_xx(x,t) = (f_x(x+dx,t)-f_x(x,t))/dx, and so on. Then again, using a little algebra, it is easy to show that f_t=pdx²f_xx. In other words, the function f satisfies a discrete version of the well-known diffusion equation, with diffusion constant pdx².

Now for the interesting bit. Let m denote the mass of the particle and set dx = ħ/mc, dt = iħ/mc², and p = 1/2, where ħ is the reduced Planck constant, c is the speed of light, and i is the imaginary constant, so that i² = -1. Then pdx²/dt = -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: f_t = (ħ/2m)f_xx. We have just derived discrete versions of two of the most important relations in quantum mechanics using nothing but high-school algebra. Moreover, in deriving these results we may have uncovered some insights into why they hold, as we will now see.

If m is the mass of the particle then ħ/mc is referred to as the particle’s Compton wavelength, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle. Similarly, ħ/mc² is referred to as the particle’s Compton time, which is simply the time of a step length of the Compton wavelength taken at the velocity of light. These are considered to be fundamental quantum mechanical properties of a particle. Now suppose that m is the Planck mass, the largest possible mass of a particle, so that m = √(ħc/G), where G is is Newton’s gravitational constant. Then ħ/mc is the Planck length, the smallest unit of length, and ħ/mc² is the Planck time, the smallest unit of time. These amounts of mass, length, and time are constructed from the values of fundamental physical constants.

Thus, our choice of units in the derivation above was far from arbitrary. What about the fact that we multiplied the Compton time by the imaginary constant? This was not arbitrary either. In fact, the concept of ‘imaginary time’ is well established in physics, and was popularized by Stephen Hawking in his book The Universe in a Nutshell. It should be noted that the imaginary constant is no more ‘imaginary’ than any other number; this unfortunate misnomer is only retained as it his become so well established in the mathematics world. In fact, I would argue that the imaginary constant i is more real than most ‘real’ numbers, as it can be constructed from the axioms of mathematics in a finite number of steps, whereas most ‘real’ numbers cannot.

We have seen therefore that in a discrete framework, we can derive fundamental relations of quantum mechanics in a natural way without having to resort to the usual heavy-duty Hilbert space machinery. Furthermore, these derivations used parameters based on fundamental physical constants with a clear physical interpretation. Admittedly, we have only derived these relations in a special case, and more work would need to be done to see if these results could be generalized. There is no reason to think they couldn’t be though. In my view, a discrete framework is much more natural for quantum mechanics than a continuous framework. After all, quantum mechanics is fundamentally concerned with quanta – the smallest possible, and therefore indivisible, units of nature.