Groucho Marxism

Consider a particle moving in discrete time in one dimension. Let X(t) denote the position of the particle at (discrete) time t; let dt denote the increment between time steps; let U(t) = [X(t+dt)-X(t)]/dt denote the discrete velocity of the particle at time t; and let [X,U](t) = E[X(t+dt)U(t)-X(t)U(t)] denote the discrete commutator between position and velocity at time t, where E is the expectation operator. Then it is straightforward to show that [X,U] = U²dt. Now suppose that at each time step the particle moves a distance dx or -dx each with probability 1/2. Then it is clear that [X,U] = dx²dt. In what follows we will set dx = ħ/mc, the particle’s ‘Compton wavelength’, where m is the mass of the particle; and dt = -iħ/mc², the particle’s ‘Compton time’ multiplied by a factor of -i, where i is the imaginary constant.

The particle’s Compton wavelength, ħ/mc, is the wavelength of a photon whose energy is the same as the rest energy of that particle; and the particle’s Compton time, ħ/mc², is the time for the particle to travel the Compton wavelength when moving at the velocity of light. Given these definitions it is straightforward to show that [X,U] = iħ/m, or [X,P] = iħ where P = mU, which is a discrete version of the canonical commutation relation between position and momentum. Now let us denote by f(x,t) the probability that the particle will be in position x at time t, and define the partial derivatives of f() by f_t(x,t) = [f(x,t+dt)-f(x,t)]/dt, f_x(x,t) = [f(x+dx,t)-f(x,t)]/dx, and f_xx(x,t) = [f_x(x,t)-f_x(x-dx,t)]/dx. Then it is straightforward to show that if_t = -ħf_xx/2m, which is a discrete version of the Schrödinger equation.

The Compton wavelength and Compton time are considered to be fundamental quantum mechanical properties of a particle. Our choice of these units was therefore not arbitrary. What about the fact that we multiplied the Compton time by -i? This was not arbitrary either and relates to a concept in physics known as ‘imaginary time’, whereby time is represented using imaginary numbers. Mathematically, imaginary time is real time which has undergone a so-called Wick rotation so that its coordinates are multiplied by the imaginary unit i. The fact that we used -i and not +i in our definition need not concern us as +i and -i are algebraically indistinguishable, because both defined as roots of the same polynomial equation, x²+1 = 0.

A standard mathematical result known as the Cauchy-Schwartz inequality says that for any complex-valued random variables Y and Z, E(|Y|²)E(|Z|²) ≥|E(YZ*)|², where Z* denotes the complex conjugate of Z. If we set Y = X(t)-E[X(t)] and Z = P(t)-E[P(t)] we get S[X(t)]²S[P(t)]² ≥|E[X(t)P(t)*]|², where S[X(t)] and S[P(t)] are the standard deviations of X(t) and P(t) respectively. Taking the positive square root of both sides gives S[X(t)]S[P(t)]≥|E[X(t)P(t)*]|, and a similar argument gives S[X(t+1)]S[P(t)]≥|E[X(t+1)P(t)*]|. Another standard mathematical result known as the triangle inequality says that for any complex numbers y and z, |y|+|z| ≥  |y+z|. If we set y = E[X(t+1)P(t)*] and z = -E[X(t)P(t)*] we get |E[X(t+1)P(t)*]|+|E[X(t)P(t)*]| ≥ |E[X(t+1)P(t)*]-E[X(t)P(t)*]|.

Combining the above inequalities and using the definition of the commutator [X,P*] (see above) gives S[X(t)]S[P(t)]+S[X(t+1)]S[P(t)] ≥ |[X,P*]|. Let us now set s[X](t) = S[X(t)]+S[X(t+1)] and s[P](t) = 2S[P(t)], so that s[X] and s[P] represent the sum of the standard deviations of X and P taken over the time points t and t+1. Then we have 2s[X]s[P] ≥ |[X,P*]|. By definition, P = -mc²[X(t+dt)-X(t)]/iħ, and as 1/i = -i, we have P = imc²[X(t+dt)-X(t)]/ħ, and therefore P* = -P. It follows from this and the definition of the commutator above that [X,P*] = -[X,P], and therefore |[X,P*]| = |[X,P]|. Combining this with the inequality above, we get 2s[X]s[P] ≥ |[X,P]|. This is a discrete version of the well-known Heisenberg uncertainty principle, named after the German physicist Werner Heisenberg.

Thus, we have shown that in a discrete framework it is possible to derive an analogues of the three most well-known equations of quantum mechanics: the canonical commutation relation between position and momentum; the Schrödinger equation; and the  Heisenberg uncertainty principle. Moreover, we have done this without having to use the heavy-duty Hilbert space machinery that must be employed in the continuous case. However, we have only done this for the special case of a 1-dimensional particle undergoing a simple random walk. Whether our approach can be generalized to other cases is an interesting question that I will leave for a future blog post.