Motion Captured: Why Nature is Lazy

June 1, 2015

 The beautiful swirling shape below is known as a Cornu spiral. Here we have obtained it in a strange way. You see it being built up as a number of ‘paths’ are drawn below it. These paths all start and finish at the same points but go via different routes. The slider controls the number of paths to be considered. In a very direct way the Cornu spiral measures the quantum mechanical contribution of each path to the real motion of the particle. The step like function on top is the Cornu spiral again, manifesting less impressively in imaginary time. Paths:

Quantum mechanics was put into, almost, modern form in the 1920s and 30s by Bohr, Heisenberg, Schrodinger, Dirac and others. What you usually learn in university quantum mechanics lectures is the formalism of Schrodinger – differential equations describing probability amplitudes. In more advanced classes you learn the bra-ket notation of Dirac and study spin and mixing and all that other nice stuff. For the average quantum mechanic, interested in tinkering with atoms, this is perfectly fine. Here we are not concerned with practicality and instead look at a wonderful way of formulating quantum mechanics given by Feynman in his PhD thesis. Like almost all pieces of Feynmania you can buy a reprint.

First I have to explain what ‘action’ is. Remember from long ago we talked about the Lagrangian as a method of calculating the motion of particles. We, in a burst of mathematical confidence, used Lagrange’s equations instead of Newton’s to solve for the motion of $N$ coupled pendulums. Lagrange’s equations can be derived from a deeper principle. The Lagrangian is the difference between kinetic energy, $K$, and potential energy, $V$:
$$L = K – V$$
we define the action as
$$S = \int L dt$$
All of classical mechanics can then be derived from the principle of least action: The path taken by the system is the one for which $S$ is minimised.

We think we have some intuition for what energy is. A book perched on a high ledge has a lot of potential energy, if it falls that potential gets converted into kinetic energy and when it lands on your head the energy goes in many directions, causing sound, heat and denting your skull. Energy is a quantity that gets swapped around into different modes as objects interact with each other. What is action? If we look at the definition of $L$, the contribution to the action from an infinitesmal time it is $K – V$. Baez calls this “happeningness” and the principle of least action is that Nature wants the least possible amount to happen. Whatever action is, for some reason nature likes it to be minimised. The paths we get in nature are ones where $K-V$ is as small as possible for as long as possible. (Actually ‘least action’ should be ‘stationary action’. What we mean is that we should somehow differentiate $S$ with respect to the paths and set this derivative equal to zero. As usual this guarantees that the integrand be at a maximum, minimum or other critical point. The path that achieves this critical point is the one nature uses.)

Feynman was the first to give some really good answers about where this principle comes from. How does the particle know which path will give the minimum action and why does it care? The particle could be merrily traveling down some path that seems okay but then does some crazy detour right at the end. In Newtonian mechanics we had acceleration and forces at each instant, steering the particle along the correct trajectory, now we are flying with the guidance system turned off. The ridiculous, and correct, answer is that the particle tries all the paths.

Classical mechanics is ’emergent’ from quantum mechanics. The quantum theory is the fundamental one which tells us how nature behaves. Classically there is nothing else to do other than define the action as above, along with the principle of least action and show that it implies Newton’s laws which have overwhelming experimental support. To derive this classical principle from quantum mechanics we have to start from some quantum postulate. We suppose the following: Each path has a phase, or probability amplitude, $e^{iS/\hbar}$. To get the total probability amplitude we add up all paths. This means ALL the paths, even ones going via Jupiter.
$$\text{Probability Amplitude} = \sum_{\text{all paths}} e^{iS/\hbar}$$

Particles are waves and waves are particles and, just like waves, they can be in and out of phase with each other. The action is a measure of the phase accumulated by a particle as it travels along a given path. An ‘intuitive’ explanation of the principle of least action can be given (though if you have to assume something very non-intuitive like wave particle duality I don’t know if it really qualifies as intuitive). Two paths will tend to have different $S$ and, if we are thinking about macroscopically different paths, this difference divided by $\hbar$ is a huge number. This means the phases are drastically different. The phases for the different paths will be all over the place with no particular bias due to the complicated nature of a general path. This is true except for one region. When the action, and hence the phase, is stationary changing it by a small amount doesn’t change the phase by much. In a small region (compared to $\hbar$) these paths can add up coherently to give a significant contribution to the sum above. This is what we see in the cartoon above for a very small subset of paths. The paths far from a straight line draw the cornu spirals, looping round and round, only those paths relatively close to horizontal add up coherently. Classical mechanics is quantum mechanics using the stationary phase approximation.

We would like to study this on a computer, not just for this article but there is a whole branch of physics, that a large number of people here at SDU have made their life’s work, which is based on this formulation. However, trying to add up a rapidly oscillating function on a computer is very difficult. Computers only have a certain number of decimal places to work with and if changing the 18th decimal can completely rotate the phase angle they will have a hard time arranging the delicate cancellations which are necessary. The loops at the ends of the Cornu spiral need to be controlled.

This is possible using a trick called Wick rotation. Nothing to do with candles but invented by an Italian physicist, Gian-Carlo Wick. You can make anything complicated by studying it enough but for us a Wick rotation is simply a change of variables
$$t \rightarrow -i\tau.$$
The new variable $\tau$ is called ‘imaginary time’. Let’s see how this makes life easier for a computer. The Lagrangian is $L = K – V$. The kinetic energy is given by the usual
$$K = \frac{m}{2} \left( \frac{dx}{dt} \right)^2$$
changing variables sends
$$K \rightarrow -\frac{m}{2} \left( \frac{dx}{d\tau} \right)^2 = -K$$
Ignore the possibility of time dependent potentials for now so that the change of variables doesn’t affect the potential, $V$, thus $L \rightarrow -(K+V)$. In imaginary time the Lagrangian is just the energy of the particle. The action
$$S = \int L dt \rightarrow i \int L d\tau$$
and thus the phases become
$$e^{iS/\hbar} \rightarrow e^{-S/\hbar}.$$

The Wick rotation turns the rapidly oscillating (and lovely) Cornu spiral into a simple (and unlovely) sum of exponentials. Paths far from the minimum hardly contribute anything and so it isn’t necessary to calculate the action arbitrarily accurately. The Wick rotation has turned a dynamic problem into a static problem, quantum mechanics has turned into statistical mechanics! What we have is completely analogous to Boltzmann probability and methods that we used for the Ising model are applicable for studying quantum mechanics. The pulsating quantum heartbeat above is the result of one of these calculations, of quite a complicated system. This is what a quark looks like propagating in imaginary time. I used some pretty fancy code to put a source for quarks in the middle of a lattice and let it propagate through a background of gluon radiation. There isn’t time or space to go into detail here, but there is plenty of local expertise at CP${}^3$.