New paper shows how classical physics could explain quantum mechanics
A new paper from MIT proposes a novel approach to quantum mechanics using classical physics. The authors propose that Feynman went overboard when he equated quantum mechanics to a sum over every particle trajectory. You don’t need all the zig-zagging histories of a particle to produce a quantum wavefunction. You just need what they call a multi-valued action where each value of this action gives a different branch or timeline in the quantum history.
Classical physics, going back to the 18th century and the French mathematician Lagrange is based on the concept of the least action. An action is a mathematical expression that in most cases tells you what the balance between kinetic and potential energy is over a given “trajectory”.
If I have a ball and I drop it from a height, then the action is simply going to be the total sum of a quantity called the Lagrangian at each point in the trajectory. The Lagrangian meanwhile is the kinetic energy (given by one half mass times velocity squared) minus the potential energy (mass times the acceleration of gravity times the height). As the ball falls, potential energy decreases while kinetic energy increases. Yet, throughout the fall, the trajectory is such that the action, i.e., sum of the Langrangian over time, is “minimal”.
This minimal action means that no other trajectory produces a smaller action. Since the minimal action occurs when the ball falls straight down, the ball is not allowed to follow another trajectory.
Action is in units of energy times time. Another quantity that has the same units is Planck’s constant. This is why in quantum mechanics when you have an action appear its units are always canceled out by that constant so you get a dimensionless quantity. Planck’s constant is a unit of action.
To understand how we got quantum mechanics from classical and why we may have taken a wrong turn, we need to go back to the beginning. Classical mechanics can be described in three ways, by the least action, by the equivalent Hamilton’s equations, or by the Hamilton-Jacobi equation. The latter equation is an equation where the action itself (some might call it Hamilton’s principal function but I’ll just call it the action) is the variable for which to solve.
Schroedinger used the Hamilton-Jacobi equation to create the famous quantum equation that bears his name by equating the action divided by Planck’s reduced constant (Planck’s constant divided by two pi) with the phase of a quantum wave.
He didn’t invent this concept from whole cloth but inferred it based on the theory of geometric optics which is the theory of the propagation of light wavefronts with small wavelengths.
In geometric optics, the wave is described using complex numbers where the phase is the primary quantity of importance since frequency is neglected. How wave fronts with different phases interact was of primary importance to explain effects like the double-slit experiment which I’ll talk about in a bit.
Using the Hamilton-Jacobi equation to explain geometric optics is all classical physics, but, by equating the action with the phase, he got something new. He got a way of turning mechanics, described by actions that govern things like balls, rockets, and planets, into wavefronts.
All Schroedinger did after that was take the resulting Hamilton-Jacobi equation, which was now in terms of a wavefunction instead of an action, and, taking it as an expression, assume it was minimal rather than exactly zero as classical physics would dictate.
This last one didn’t actually change anything for ordinary particles because the zero was already minimal. It does change things for certain other quantum systems, however.
Schroedinger’s equation turned out to calculate the probabilities of finding particles and other quantities in experiments very well and became the basis of most of the early days of quantum mechanics, displacing the earlier Heisenberg Picture equations.
About 20 years later, a Ph.D. student named Richard Feynman came along and started trying to figure out a way to describe Schroedinger’s waves using classical trajectories. What he came up with was what is now called the path integral which he proved in certain cases was equivalent to Schroedinger’s wavefunction. (There is still no rigorous definition of the path integral in the general case.) This sum-over-histories included every zig-zagging particle path even if it were vanishingly improbable. These trajectories are essentially stochastic, and Many Worlds Interpretation proponents have argued that there is a separate world for every single one.
Like Schroedinger, Feynman used the classical action as the phase. In his case, he calculated the wave for every zig-zag path and summed them all together to get his path integral. His approach also showed that the least action was also the most probable, which fit well with approximating quantum to classical physics in an intuitive way. The path integral also became important to the development of quantum field theory, quantum electrodynamics, and quantum perturbation theory, particularly Feynman diagrams, one of the practical tools for calculating observables in quantum experiments.
All this background is necessary to explain what this new paper is proposing which is that all of Feynman’s zig zag paths and Schroedinger’s deviation from the Hamilton-Jacobi equation of classical geometric optics is unnecessary. All we need, the authors argue, is to take another look at the classical action of geometric optics.
