Noether’s Certainty Theorem

Werner Heisenberg – He may not know where he is going, but he knows exactly where he is
Amalie “Emmy” Noether – Shown here demonstrating her theorem of conservation of awesomeness

By now, most people have heard of Heisenberg’s Uncertainty principle. It states that, no matter how carefully you measure, there will be some pairs of values that cannot be known with as much precision as you like. Instead, there is a fundamental limit on how much we can know simultaneously about, for example, the position and momentum of a particle. This statement has become a darling of some philosophy majors, since it speaks to human knowledge and its limits, and also, perhaps, because takes a little wind out of the sails of all those snooty physicists who think they know everything. However, there is a much less well known, but connected principle, that vastly increases what we know about the universe and our ability to make predictions. In fact, as much as the uncertainty principle casts a pall over the our conceit that  we can understand everything, there is a “certainty principle” (that goes by a different name, a woman’s name in fact) that provides a foundation for the whole enterprise of science. First, though, an explanation of uncertainty.

There is a simple way to  understand the uncertainty principle.  Imagine trying to determine the position and momentum of an electron. To see where it is and how fast it is going, you need to bounce something off of it. Just like when you see something, say, a basketball, with your eyes under normal conditions, you are watching the rebound of the light particles that fell on it. Let’s say you decide to shine a single particle of light, a photon, on the electron. Light comes it a continuous spectrum of energies (what we call colors, when in the visible range), so you might first try to use a photon with a small amount of energy, so as not to disturb the momentum of the particle very much.  However, a low energy photon has a long wavelength. This limits the resolution of the position, since you cannot resolve anything with greater precision than the wavelength of light you are using to see it. If you instead use a photon with a small wavelength, to better observe the position, the photon will have greater energy and interfere with the measurement of the momentum. There is an inherent trade-off between your knowledge of the position and your knowledge of the momentum, since any measurement you preform will disturb the system, to some extent. You might try to improve you technique and be as careful as you can, but there is a fundamental limit: The more you know about the position, the less you can know about the momentum.

An alternate explanation, one I find to be a more lucid approach is to remember that all particles, matter and photon alike, have properties of waves. To describe an electron with a definite momentum, you would use a sine wave of a certain frequency. Since the wave extends forever in all directions, you know nothing about the position of the electron if it has only one possible momentum. To formulate a localized electron, one needs to add together several momentum states, represented by sine waves of different wavelengths, to end up with a “wave packet.” For the electron’s position to be perfectly known, you would need an infinite number of sine waves, and therefore destroy all information about the momentum of the particle. (Here is a good animation of building up an increasingly localized state by adding more and more momentum states).  Expressing a position state as the sum of momentum states (and vice versa) is the same as preforming the mathematical operation called a Fourier transform. Like playing a orchestral symphony using only tuning forks, Fourier’s principle states that any arbitrary waveform can be produced by adding together a sufficient number of pure tones in the right proportions. This is not just of academic interest; the fact that your .mp3 player can hold your entire music collection depends on the ability of song files to be compressed by expressing them in the shorthand of Fourier transforms.

In fact, Heisenberg’s Uncertainty isn’t just one principle; it applies to several pairs of observable quantities, that, like position and momentum, are “conjugates variables,” by virtue of being Fourier transforms of each other.  Time and Energy are also conjugates of each other, as are the angle of an object and its angular momentum.

But there is a much less widely known principle based on conjugate variables that nevertheless has a huge impact on the way we understand the universe.

I wrote in the previous post about Conservation Laws. It is hard to overstate how important these laws are to solving problems in physics (and math). In a universe where objects are buffeted by so many unpredictable forces, knowing that some value is unchanging is like a life-preserver in the maelstrom. You just look at some value at particular point in time, and, come what may, that value stays the same at all future times. For example, the velocity of a falling object can be calculated by equating the potential energy lost with the kinetic energy gained, so that total mechanical energy is conserved. A figure skater spins faster when she pulls in her arms, in order to conserve angular momentum. During a collision, intense, complicated forces act over the brief instants the objects are in contact, but you know that the linear momentum stays the same.

Conservation laws are no accident; they emanate from fundamental symmetries in nature. Because space looks the same even after translating a certain distance, linear momentum is conserved. The fact that total energy stays the same is a result of the law of physics being invariant under translations in time. Angular momentum conservation comes about because of rotational symmetry. The common effect is that a symmetry in one variable leads to a conservation law in its conjugate variable. This “certainty principle,” if I might coin the term, is central to the way be view the pursuit of science, is more commonly know as Noether’s Theorem, named for the woman who proved it in 1915.

If you have seen a picture of Amalie Emmy Noether at all, it was almost certainly surrounded by photographs of men. Noether is often taken as the “token” women in a historically male dominated field. Noether showed mathematically how symmetries in the laws of nature lead directly to conserved quantities.

Modern physics relies on Noether’s work, too. At CERN, conservation laws for things like electric charge, color, and isospin help make sense of the morass of particle tracks that come shooting out of colliders.

As Richard Feynman discusses  in his famous lectures on physics, the whole basis for science is that it makes testable predictions based on universal laws. If the laws changed every time you try an experiment, or where different depending on where you are on Earth, then it would be pointless to spends so much time simply collecting data that had no relvenece to other times and places. However, Noether’s Theorem tells us that the laws of nature are the same, regardless of translations in space or time. So much of what makes the universe so wonderful and comprehensible can be attributed to Noether’s fearful symmetry.

Further Reading:

Emmy Noether’s Wonderful Theorem by Dwight E. Neuenschwander

The Feynman Lectures on Physics by Richard Feynman

The Quantum Universe by Brian Cox and Jeff Forshaw

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Author: lnemzer

Assistant Professor Nova Southeastern University

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