Noether's Theorem and The Symmetries of Reality

Thank you to Brilliant for supporting PBS Digital Studios. Conservation laws are among the most important tools in physics. They feel as fundamental as you can get.

And yet they're wrong, or at least they're only right sometimes. These laws are consequences of a much deeper,. .

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. more fundamental principle, Noether's Theorem. Conservation laws are the cheat codes of physics.

They make it possible to solve physics problems. . .

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that would otherwise be painfully difficult, or even impossible. More than cheat codes, conservation laws are close to the source code. They emerge from profound and simple truths about the basis of reality.

They emerge from the fundamental symmetries of nature. The connection between conservation laws and symmetry,. .

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. is encapsulated in Noether's theorem. But before we dive into this extremely elegant idea,.

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. . let's talk about the seeming paradox that inspired it,.

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. . and the genius who discovered it.

When it was published in 1915, Einstein's general theory of relativity. . .

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opened as many questions as it answered. Among them is the fact that energy is not always conserved in general relativity. The simplest example of this is the case of cosmological redshift.

As the universe expands, light traveling through that expanding space is stretched out. Its wavelength increases, and so the energy of each photon drops. Where does the energy from redshifted photons go?

In 1915, the expansion of the universe hadn't yet been discovered,. . .

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but the failure of energy conservation. . .

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was still clear from the math of general relativity. Two of the "greats" of the era, David Hilbert and Felix Klein,. .

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. sought the help of a young mathematician, Emmy Noether,. .

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. to understand the seeming paradox. She discovered why the law of conservation of energy broke down in general relativity.

The law was not fundamental after all. She realized that all conservation laws arise from a more fundamental relationship,. .

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. which we call Noether's theorem. A simple expression of Noether's theorem is this: .

. . For every continuous symmetry of the universe, there exists a conserved quantity.

Let's unpack this. First, what do we mean by symmetry? Actually, first, what don't we mean?

We say a face is symmetric if it looks the same under a mirror reflection. Snowflakes are symmetric under sixty degree rotation. Playing cards, under 180 degree rotation.

But these are what we call "discrete symmetries",. . .

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single flips around one axis, or rotation by specific amounts. Noether's theorem applies to continuous symmetries. Something is "continuously symmetric".

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. . If it stays the same for any size shift in a given coordinate.

For example, a long road in the middle of nowhere. . .

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is continuously symmetric under spatial translations in the direction of the road. A perfect sphere is continuously symmetric under rotational translations. In both cases, the environment stays the same for shifts along a symmetric coordinate.

In the case of Noether's theorem, when we say the environment stays the same,. . .

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we mean the equations that give the laws of motion for the system. For example: moving along a perfectly flat road, the downward force of gravity stays constant. We have symmetry to spatial translation, and Noether's theorem tells us.

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. . there's a corresponding conserved quantity.

That quantity is Momentum. If two cars collide on that road,. .

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. the sum of their combined momentum stays the same. But what if the road is hilly?

Momentum doesn't appear to be conserved. It can be lost or gained to the gravitational field. This is because the direction of the gravitational field changes with respect to the road.

It's not symmetric to translations along the road. On the other hand,. .

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. the gravitational field across the whole stretch of road. .

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. doesn't change from one point in time to the next. The system is symmetric to time translations.

It doesn't matter when the collision happens, the results are the same. Noether's theorem reveals that this time translation symmetry. .

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. gives us energy conservation. And the last classic example: .

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. . if the factors driving the laws of motion are symmetric under rotation,.

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. . for example, the spherically symmetric gravitational field.

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. . experienced by a satellite orbiting the Earth,.

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. . then, Noether's theorem predicts another conserved quantity: .

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. . Angular momentum.

By revealing the underlying source of conservation laws,. . .

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Noether's theorem handily explains when, and why, they're broken. That includes the apparent breaking of conservation of energy in general relativity. See, Einstein's description of gravity reveals the dimensions of space and time.

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. . to be dynamic and changeable.

If the very nature of space can change over time,. . .

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then, continuous time symmetry is broken. That's the case with the expanding universe. Energy can be lost, in the case of cosmological redshift,.

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. . and it can be created from nowhere, in the case of dark energy.

The law of conservation of energy is fundamental in Newtonian mechanics. . .

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in which space and time are unvarying and eternal,. . .

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but in Einstein's universe, energy conservation is only valid as a special case. It only applies for parts of the universe where we can approximate space. .

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. as unchanging over time. However, it's possible to use Noether's theorem.

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. . to come up with an analogous quantity that is conserved.

That quantity is the rather esoteric Landau-Lifshitz Pseudotensor. It "saves" energy conservation, by incorporating. .

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. the entire universe's gravitational potential energy. .

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. to offset the seeming gains or losses to redshift and dark energy. This quantity is, however, controversial, in its application and in its interpretation.

It's also worth an entire episode, so I'll leave it alone, for now. Despite its profound implications,. .

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. the math behind Noether's theorem is surprisingly straightforward. It falls like magic out of another deep law of the universe:.

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. . the Principle of Least Action,.

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. . which states that the universe will always "choose" the path between two states.

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. . that minimizes the change in the action.

This is a rather abstract quantity,. . .

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that measures the effort involved in moving between two states. . .

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over some time interval. It's a generalization of Fermat's principle,. .

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. which states that light will always take the path between two points. .

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. that minimizes the travel time. The Principle of Least Action extends Fermat's principle.

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. . to any object moving on any path, or indeed any system,.

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. . quantum-mechanical to cosmological, evolving between two states.

The Principle of Least Action can be used to derive the laws of motion,. . .

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from the equations of classical mechanics. . .

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to Feynman's "path integral" formulation of quantum mechanics. In a sense, the principle is axiomatic. It's a founding assumption behind these derivations,.

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. . and it's also the founding assumption behind Noether's theorem.

However, as a founding assumption,. . .

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that is as close to as fundamental as we can get. Noether's theorem allows us to figure out. .

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. the true conserved quantities for any system that is evolving. .

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. according to the principle of least action,. .

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. as long as we can identify that system's symmetries. This is useful in cosmology, but it's also useful in quantum physics.

The general nature of the theorem means we can apply it. . .

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to, not just the symmetries in the dimensions of space and time,. . .

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but also to more abstract coordinates. For example: another conserved quantity in Physics is electric charge. Noether's theorem tells us that there should be a corresponding symmetry of Nature.

And, there is. That symmetry is the phase of the quantum field. You can rotate the complex phase of an oscillation in a quantum field, by any amount,.

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. . and the observable properties of that field, like its particles, don't change.

This symmetry leads to the conservation of electric charge and electric current. This quantum symmetry is just the simplest. .

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. of a large number of symmetries exhibited by quantum fields,. .

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. the so-called "Gauge Symmetries". They predict a rich family of conserved charges.

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. . that govern the interactions of the particles of the standard model.

For example: the color charge of quantum chromodynamics. . .

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describes the strong interaction between quarks and gluons. The entire standard model of particle physics is what we call a Gauge Theory. It's founded on the fundamental symmetries of quantum fields.

It's going to take a few episodes to explain exactly what these symmetries really are,. . .

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and how they lead to the family of particles and interactions that make up our universe,. . .

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and we'll jump into that before too long. Emmy Noether was one of the greatest mathematicians of the Golden Age of modern physics,. .

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. yet she gained little public recognition in her time,. .

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. and is still only known to the more eager students of math and physics. During her life, she was repeatedly refused any paid academic position, due to her gender,.

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. . until late in her career.

Hermann Weyl, also a giant in the mathematical foundation of quantum mechanics,. . .

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said, in her memorial address: . . .

"I was ashamed to occupy such a preferred position. . .

" ". . .

beside her, whom I knew to be my superior as a mathematician in many respects. " Einstein also called her a genius. Her contributions to mathematics, particularly abstract algebra, redefined entire fields.

Let's be thankful that she took a moment to offer a little of her genius to Physics,. . .

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taking us a big step closer to understanding the fundamental workings of the universe,. . .

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through the continuous symmetries of space-time. Thank you to Brilliant for supporting PBS Digital Studios. Brilliant is a math and science learning site.

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. . that encourages members to learn through problem-solving.

If you want to learn more about Brilliant,. . .

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you can go to: www. brilliant. org/spacetime As always, thanks to all of our Patreon supporters that help keep us explaining the universe.

And this week, an extra huge shout out to Big Bang supporter Fabrice Eap. Fabrice, your continuously time-symmetric contributions are fundamental to the conservation of space-time. So, thank you.

Last week we talked about the incredible wealth of data dropped by the Gaia mission. You guys had a ton to say. Super User asks how Gaia compares to the upcoming LSST.

So the Large Synoptic Survey Telescope will indeed be revolutionary. It does have some overlap with Gaia, in that It'll measure light curves,. .

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. so changes in brightness over time for billions of objects across the entire sky. But Gaia is really optimized for extreme astrometry.

Phenomenally accurate position data that actually allows measurements of velocities. LSST won't do as well in that regard,. .

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. in part because it's on the ground, in the Atacama Andes in Chile, rather than in space. But LSST will see much further and fainter than Gaia.

It has a 35 square meter collecting area, compared to Gaia's 0. 7 square meters. It'll find more small solar system objects,.

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. . detect the effects of weak gravitational lensing in distant galaxies,.

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. . and see more distant supernovae,.

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. . monitoring more distant quasars for their flickering.

In general, it'll do better in what we call the time domain,. . .

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by scanning the entire sky every few nights,. . .

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and it'll keep coming back, and back, and back,. . .

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to measure change in brightnesses at high time resolution over ten years. Simon Clarkstone asks how Gaia can take such precise measurements,. .

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. surely its camera doesn't have pixels this small. It's good insight, Simon.

Gaia's precision is ± 0. 3 mili-arc-seconds. As we mentioned, the width of a hair at a thousand kilometers.

Yet, its pixels are only 60 milli-arc-seconds on their shortest dimension. It can do better than that, off the bat, by analyzing the way the light from each star. .

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. spreads into neighboring pixels, by fitting the so called Point Spread Function. But its extreme precision is possible because it comes back to every field.

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. . many times over the five years of its operation.

Each time stars will have moved slightly,. . .

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but they'll have moved in extremely predictable ways. By feeding the laws of motion to the stars' five-year trajectory,. .

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. there's only a very narrow trajectory of extremely precise locations that ends up working. Nebularium asks whether "white dwarfs" is the correct plural,.

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. . rather than "white dwarves", with a V.

Well, to quote Tolkien's 1937 foreword to The Hobbit,. . .

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in English, the only correct plural of dwarf is dwarfs, and the adjective is dwarfish. In this story, dwarves and dwarvish are used, but only when speaking of the ancient people . .

. to whom Thorin Oakenshield and his companions belonged. You could also check appendix F in Lord of the Rings.

Seriously, people, on SpaceTime. . .

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we only ask that you start with a passing familiarity with quantum physics. . .

. and the etymological foundations of the languages of Middle-Earth. Please do your assigned readings.