Josh Mitteldorf Sheds Light on the Mindboggling Complexity of Many-Body Interactions

April 8, 2023

This article was written by Josh Mitteldorf and originally published on April 1, 2023. It is reprinted here with his permission. Mitteldorf earned his BS in physics at Harvard University and his Ph.D. in theoretical astrophysics at the University of Pennsylvania.

When Isaac Newton discovered the equations that govern motion of the planets through the heavens, he was able to solve them with pencil and paper much faster than the planets themselves were moving. Thus he was able to make useful predictions. Solving the equation — even when it involves oodles of numerical computations by hand and pad after pad of yellow paper, it’s still a whole lot easier and faster than actually doing the experiment and making the measurement.

From the Twentieth Century, we have better theories than Newton had. The two most fundamental theories of physics are Quantum Mechanics and General Relativity. We are tantalized to think they must be better than Newton. because for some simple cases, we can solve the equations and we get better agreement with experiment than we get using Newton’s equations. We think they are fantastically accurate. Maybe they are the Ultimate Reality.

But here’s the cosmic joke. Both Schrodinger’s Equation of Quantum Mechanics and Einstein’s Field Equation can only be solved for the very simplest cases.

We can solve Schrodinger’s Equation for two particles by hand, for three particles with a supercomputer. But anything more than three particles is so fantastically complicated that the equation can only be solved approximately — all that wonderful accuracy gone to waste. We have an exact solution for the hydrogen molecule (2 electrons), but for anything as complicated as a single molecule of water (10 electrons) we have only approximations and quantum heuristics.

We can solve Einstein’s Equations for situations that are perfectly symmetric. A sphere is easy. A spinning sphere is really, really difficult.

But any realistic situation in astronomy becomes so complicated that we don’t even have an algorithm that would let a computer go to work on the problem. Big Bang cosmology is based on the Cosmological Principle, which says that the universe is the same everywhere. We make that assumption not because the evidence for it is solid, but because we can’t solve Einsten’s Equations for realistic distributions of matter.

Since Newton, we physicists have taken it for granted that mathematical theory provides a quick and elegant way to understand something — much easier than doing each particular experiment and measuring the outcome.

The best theories that we have aren’t like that. A computer the size of the universe couldn’t solve the equations faster than the universe is generating the answers.

To Einstein and Schrodinger, God said, “You want a theory of everything? You want to understand how the Universe unfolds — OK, here’s the trick that I use. Here’s the soul of my magic. Here in these equations is the way I generate the future from the present. Have at it!”

Technical note: To solve an equation can mean two different things.

• An analytic solution is an equation that you can derive using symbols. For example, the solution for an object moving in a uniform gravitational field without friction is a parabola, and you can get the equation for the parabola from the equations of motion.
• A numerical solution is often possible when no analytic solution exists. For example, computers can accurately trace the course of a space probe by advancing in tiny increments, one millisecond at a time. By making the time increment progressively shorter, it’s possible to get more and more accuracy by using more and more computing power.

The equations of QM and of GR both have analytic solutions in the simplest cases. For slightly more complicated cases, they both have numerical solutions suitable for today’s computers. For situations that are yet a little more complicated, the numerical algorithms become intractable. This is to say that to solve (for example) the Schrodinger equation for the ground state of a water molecule would require a computer larger than the entire universe.

When we have a general-purpose quantum computer, this statement will become obsolete.