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Unit 6

The Beauty of Symmetry

6.6 Physics

PLEASE BE DISCRETE

The idea of symmetry plays an important role in physics. It primarily manifests as the concept of invariance, the idea that certain quantities or properties do not change under certain actions. A famous invariant is energy; the energy of a closed system does not change. This is more commonly understood to mean that energy is neither created nor destroyed. Conservation laws abound in physics and have an important connection to symmetry. To get an idea of the ubiquitous role of symmetry in physics, we have to consider both discrete and continuous symmetries.

Up until this point, we have really only considered discrete symmetries, such as those of the equilateral triangle. An object with discrete symmetries has motions that leave it invariant that cannot be smoothly turned into one another. In other words, with our triangle, both 120° and 240° rotations leave it invariant, but all of the rotations between 120° and 240° are not symmetries, so we cannot smoothly change one symmetry into another. Continuous symmetry, on the other hand, can be seen in the rotation of a circle; every rotation can be smoothly turned into every other rotation while the circle remains invariant. Both discrete and continuous symmetries are important in physics. Let's first look at the discrete type.

An important discrete symmetry in physics is actually made up of three situations that are thought to remain invariant. The first of these is the conjecture that the universe would behave the same if every particle were interchanged with its anti-particle. For instance, if we were made of antiprotons and positrons instead of protons and electrons, we would not be able to observe any difference. This swapping is called a charge-conjugation transformation, and like permuting pancakes, can be thought of as a motion that leaves the initial system invariant. It is referred to as C-symmetry.

The second idea is that the universe would behave the same if left and right were interchanged. This is known as a parity inversion and is basically what we observe in a mirror. The idea that the mirror universe behaves no differently than our own is known as P-symmetry. Unfortunately, both Cand P-symmetries do not always hold. There are certain situations in which inverting charge, orientation, or both, leads to a different outcome than if the inversions had not occurred.

The third discrete symmetry is that of time reversal. Now, over long scales, this idea is nonsense; of course the future is distinctly different from the past. However, if one considers two billiard balls colliding (ignoring friction), the incoming speeds and angles of the balls are the same as the outgoing, so if this collision were run backwards in time, like rewinding a videotape, we would not be able to tell. This is called T-symmetry, and it is less general than both C- and P-symmetries. In fact, T-symmetry is, by itself, not really true, but something fascinating happens when all three symmetries, C, P, and T, are considered together.

Each of the C-, P-, and T-symmetries acting alone, or even any two of them acting as a pair, do not leave a physical system invariant. However, these "broken" symmetries tend to cancel each other out when all three are taken together in what is known as CPT-symmetry. Basically, what this means is that if every particle were swapped out with its anti-particle, and all coordinates were inverted, and time was run backwards, the universe would behave no differently than it does now.

CPT-symmetry is a fundamental prediction of the Standard Model of Particle Physics. The standard model predicts what kind of particles should and do exist, as well as their properties such as mass, charge, and spin. This model has been remarkably accurate in its ability to explain the interactions of all particles observed so far, not only for protons, neutrons, and electrons, but also morefundamental particles such as quarks and neutrinos.

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CONTINUITY AND CONSERVATION

Let's now turn our attention to continuous symmetries. Recall that something possessing continuous symmetries can remain invariant while one symmetry is turned into another. Space itself, or more precisely, spacetime (the combination of both space and time) possesses such continuous symmetries. For instance, if two billiard balls collide in one location, I expect the result would be no different than if they had collided a foot to the left, or a centimeter to the left, or a micron to the left. The collision remains invariant under translations of any magnitude in spacetime. This is a continuous symmetry.

A remarkable theorem, proved at the beginning of the 20th century by Emmy Noether, has the physical consequence that for every continuous symmetry in spacetime, a quantity is conserved. Noether was a German mathematician and theoretical physicist who made fundamental contributions in both physics and algebra and greatly expanded the role of women mathematicians in Germany. In 1933, despite her accomplishments, she was forced to flee Nazi Germany because she was Jewish. Once safely in the United States, she taught at Bryn Mawr and also lectured at Princeton's Institute for Advanced Studies until her death in 1935.

According to Noether's theorem, every quantity that we consider to be conserved corresponds to an underlying symmetry of spacetime. For instance, a common notion in physics is that momentum is conserved, such as in the collision of two billiard balls. This conserved quantity stems from the fact that spacetime has continuous spatial translational symmetry, or put in other terms, "every location is just as good as every other location."

Conservation of energy, the law that helps roller coasters to function, stems from the idea that if we conduct an experiment at one particular time and then conduct the same experiment under the same conditions ten minutes later, we should not expect to find a different result. Conducting an experiment ten minutes later is what we call a translation in time. It's basically like when we imagined picking up our sine wave frieze motif and shifting it to the right, except that now we are shifting an event forward in time. Noether's theorem implies that this time-translational invariance is what gives rise to the law of conservation of energy.

Finally, the fact that spacetime is invariant under continuous rotations gives rise to another important conserved physical quantity. If we ignore things such as stars, planets, people, and dust and simply focus on perfectly featureless spacetime, we would find that every direction is just as good as (i.e., equivalent to) every other direction. This is rotational invariance and it, like the space and time translations, is both continuous and gives rise to its own conserved quantity, angular momentum in this case. Conservation of angular momentum, for example, is why a figure skater can spin faster if she pulls in her arms.

Noether's theorem is yet another example of how symmetries show profound connections among seemingly disparate ideas. It is this mysterious power to bring some rhyme and reason to our world that compels mathematicians to study the structure of symmetries and groups. There are undoubtedly many surprising connections left to uncover via the power of group theory.

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