From Newton to Heisenberg - Part 1

From Newton to Heisenberg - Part 1

November 19th 2020

The Power of “natural”

It is a well known story that the apple stole the moon’s spotlight in Newton’s story. The real question Newton posed was “why does an apple fall but the moon doesn’t?”. If you asked if there wasn’t an answer at the time Newton was alive that would’ve solved his curiosity, the answer would be yes. It is crucial to examine the historical context of why Newton came to this question.

Before Newton created his version of physics, Galileo had already laid the foundations upon which Newton built his work. But the Aristotelian school held the scientific hegemony for explaining physical phenomena. So what did Aristotle say about the moon? He separated objects into two groups: terrestrial and celestial. Heavenly objects are perfect shapes like the circle or sphere whereas earthly objects are deformed and thus imperfect. According to his theory, moon should have been a perfect sphere with no dents and potholes. But alas, Galileo had the ingenuity to tilt his telescope upward to see the moon up close. Did he see a perfect sphere? No. It was an ugly rock with numerous craters on its surface. Now, Newton couldn’t help but ask: why doesn’t the moon fall then?

Newton surely stood on the shoulders of giants. The breakthrough already happened when Galileo shifted the attention to “no acceleration” from “at rest”. PreGalilean science considered objects at rest to be natural. This question of what’s natural sets the standards for what should be explained. If stationarity is the natural state of things, any moving object demands an explanation, including the heavenly bodies. Aristotle had to explain why the moon was orbiting around the earth. His version of science at first glance could seem laughable but it is not easy to dispute without the centuries-long accumulation of knowledge. He thought the angels were pushing the planets.

So how did Newton explain the bizarre levitation of the moon? Well, there’s one way where a falling object can never touch the ground. The ground should also fall at the same rate. Sounds stupid, eh? Hear Newton out. We all know that a flying object moves in a parabola due to gravity. We assume everything “falls” to the ground because we’re only seeing objects being thrown in a minuscule scale compared to the size of the Earth. Imagine we threw a ball so strong that it could barely escape the atmosphere. It will still “fall” due to gravity but what happens is that the Earth’s curvature comes into play, which creates the same effect as though the Earth is “falling” as well. The moon is forever falling because someone/something threw the moon at the same rate as the Earth’s curvature is forcing the ground to fall. Newton unified the divided world of celestial and terrestrial bodies into one framework. The moon is no different than an apple in Newton’s eyes.

Fundmental forces

The gravitational force, electromagnetic force, weak nuclear force, and strong nuclear force. Despite the complexity of the universe, these are the only discovered fundamental forces, yet. These four forces are responsible for all physical phenomena notwithstanding how grand or trivial.

Four Forces by Lillian Anna Blouin

\(F=ma\), the simple physical law Newton formulated, tells us somewhat of a more elaborate story: where there’s acceleration, there’s force. This comes back to Galileo’s idea that non-accelerating objects are exempt from interrogation, but any acceleration immediately starts an investigation in search of nothing but one thing: a force.

On the earthly scale, the most important force is the electromagnetic force, which we will be discussing in even more depth as it is the gateway to quantum physics. Universities offer a two-semester sequence of courses on electromagnetism, and even further deep in graduate programs. Out of all four forces, only the electromagnetic force receives such favorable treatment. This is the force that forms everything in the universe.

The gravitational force is all too familiar to all of us since it is easily observable, from a ball falling on the ground to the Earth’s orbit around the Sun. For what it’s worth, the gravitational force is very weak compared with the rest. Think about how the Earth creates gravity to make things drop to the ground but it only takes a magnet behind your iPhone to countervail that and hold it in your car as you navigate the highways.

Nuclear forces, the strong and the weak, are not important to us on a daily basis but it is the most crucial force in the space. It is the force responsible for the solar activities, which in turn provides energy for all living things on Earth.

When one unites, another divides

Newton eliminated the boundary between the heavens and the earth but inadvertently nixed something else: our free will. What \(F=ma\) tells us is that if we know the current velocity, the time difference, and the amount of acceleration, then we can figure out the velocity of a given time in the future. Think about the definition of the acceleration \(a\): \(a = \mathrm{d}v/\mathrm{d}t\). Forget about the real-analytic definition of differentiation in terms of the limit, and instead take the infinitesimal route. If \(\Delta t\) is a time difference that is so small that it’s almost zero, or equivalently “infinitesimal”, then defining \(v(t)\) to be the velocity at time \(t\), \[ a = \dfrac{v(\Delta t) - v(0)}{\Delta t}. \] This tells us that \(v(\Delta t) = v(0) + a \Delta t\), which implies that we once we know the velocity at time 0, the acceleration, and the time difference, we can calculate the velocity of any given time. In other words, Newton viewed the world as a massive machine, operating through time. Once the machine is set in motion, it follows a completely predictable path which allows for no free will.

Such a “nomological determinism” argument was first published by Pierre Simon Laplace in 1812 by the name of “Laplace’s Demon”. Laplace asserts that if a demon knows the precise location and momentum of every atom existing in the universe, then their past and future values are naturally followed by the physical law. Therefore, the demon is in full knowledge of the physical system’s past, present, and future. In this universe, there’s no place for free will to play a role. Descartes solves this issue of free will by circumscribing the limits of the physical laws. In essence, Descartes claims that the physical laws do not apply to human minds. In so doing, Descartes might be making the same mistake as when Aristotle divided the heavens from earth.

MIT's stair-climbing robot Cheetah.

As we have seen, \(F = ma\) has a far greater implication than just describing how the objects move. It tells use that everything is, in some sense, moving algorithmically. If we have \(v(0)\) and \(a\), we know \(v(1)\). If we know \(v(1)\), we know \(v(2)\) and so forth. This is exactly like the robot designed to climb up a staircase. The difficulty lies in teaching the robot how to climb one step up. Once the robot has mastered moving up a step, the rest is all set.

Charles Babbage's Difference Engine

The way we see the world is very much in line with what Newton taught us: everything operates algorithmically accordingly to a law. At the time and perhaps even at present, the difference between a living creature and an inanimate object was generally assumed to be that a living creature operates automatically but an inanimate object doesn’t. \(F=ma\) unveiled the uncomfortable truth that there’s no such distinction. If that’s the case, why can’t we devise a “machine” that runs automatically? This was what Charles Babbage had thought in the early 19th century. Babbage devised an apparatus called ‘the Difference Engine’ that was supposed to be the calculator of the time but wasn’t implemented due to technological limitations. It is credited to be the first model of what we now call computers.

But is it really that predictable?

Although Newtonian mechanics seems to regard the world as all too predictable, things are not that simple. One of the motions that we learn in (maybe) high school is the oscillation of a single pendulum. There is a reason why we never go beyond one pendulum: because it’s not solvable.

Chaotic trajectory exhibited by a double pendulum

A double pendulum is a physical object where a pendulum is attached to the bottom of another. The movement of a double pendulum was already written out hundreds of years ago by Isaac Newton. The weird thing is that a slight change in its initial condition amplifies into an unpredictable outcome.

This kind of chaotic behavior was also observed by Edward Norton Lorenz, an American meterologist, when he was solving a system of differential equations that modeled the earth’s atmosphere. The following simplified system of differential equations is now known as the Lorenz system: \[ \begin{aligned} \dot{x} &= \sigma(y-x),\\
\dot{y} &= x(\rho-z)-y,\\
\dot{z} &= xy - \beta z. \end{aligned} \] The rumor has it that Mr. Lorenz was working on his computer program to solve the differential equations, only to realize that he needed to compute it all over again. Unlike modern computer programs, during Mr. Lorenz’s time, a computer program was written on a punch card, which could be a great ordeal. Mr. Lorenz decided to truncate the last few digits of the initial conditions to save time, and shortened \(x=0.0123165\) into \(x=0.0123\), only to realize the results diverged drastically after a certain point.

The diverging solution paths of the Lorenz system (x-coordinate). Black is the original and red is when the initial value is truncated.

As would any levelheaded scientist, Mr. Lorenz was first skeptical of his own programming. However, observing the unchanging pattern of the computer simulation that he produced, Mr. Lorenz came to believe that the nature operates in a way that “two states differing by imperceptible amounts may eventually evolve into two considerably different states”.

Physicists have realized that even the most sophisticated of devices produce some degree of measurement error, and therefore, measurements are inherently erroneous. In a chaotic system, such an infinitesimal error can exponentially amplify itself. This tells us that, despite the deterministic nature of the Lorenz system, the chaotic behavior makes it inevitably impossible to make a forecast. The YouTube video shows the three-dimensional movement of the Lorenz attractors. If we follow the trajectory, it sometimes remains in one swirl but randomly jumps to the other side. One might think “we only need to find the ‘line’ which delineates the two swirls”. Woe is me! That is not just difficult, but rather impossible because of the embedded fractal structure. Mr. Lorenz revealed that the deterministic predictability is an illusion.

The Mandelbrot set is a set of complex values for which \(z_{n+1}=z_n^2+c,\;z_0=0\) does not diverge. This is a one-liner but look what it generates. Can you carve out the exact boundary between the Mandelbrot set and the rest? It looks possible when zoomed all the way out, but as we zoom in, we quickly realize that a simple mathematical definition has created a pattern that endlessly self-replicates itself. The boundary in such *fractals* is forever unreachable.

Zooming into the Mandelbrot set

Skeptics have thought that \(F=ma\) is too simplistic to explain our sophisticated universe. However, what the Lorenz system unveiled was the hard reality that a deterministic world can be infinitely unpredictable, and that a complex pattern does not need an equally complex cause. What if Mother Nature works that way? Humankind would be doomed to a life of ignorance and oblivion.

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