Is Symmetry Fundamental to Reality? Gauge Theory has an Answer

BACKGROUND VIDEOS:

CHAPTERS:
00:00 Symmetry - root of physics
01:31 What is symmetry?
03:24 Intro to Group Theory
06:04 Noether's Theorem
07:17 U(1) symmetry simplified
09:43 Dirac equation transformation
11:10 How QED comes from U(1) symmetry
12:47 U(1) SU(2) SU(3) explained simply
15:32 Symmetry is the foundation of the universe
15:54 Further study on Wondrium

SUMMARY:
If you ask a physicist, what is at the core of physics, you will hear symmetry. What is symmetry? Gauge theory explained simply.

Symmetry is about actions that don't change anything. If we take an equilateral triangle, and put a mirror from one corner to the middle of the opposite side, we will see that the whole triangle. This is a symmetry of the equilateral triangle. Similarly we can rotate the triangle by 120 degrees, and it will look identical to what it was before.

What we just did is a simple example of something more complex - group theory. Group theory is the math behind the symmetries.The mathematics behind the symmetries of the equilateral triangle is called the dihedral GROUP of degree 3, where 3 refers to the triangle having three corners. We can change the elements, or permutations, using two different operations, rotation, and reflection. These two operations are called generators. The result of applying a generator doesn’t change anything visible. This is symmetry.

Symmetries give us rules for how to transform something while conserving a quantity. For the triangle, that conserved quantity is its shape, and the generators are rotation and reflection.

This leads us to Noether’s theorem which states that “For every symmetry there is a corresponding conservation law.” This directly connects symmetries with conserved quantities.

What happens if we take the limit of a polygon with an infinite number of edges? We get a circle. A circle of some radius, r, can be described on a 2D plane using polar coordinates by two equations. If we use complex numbers to represent the circle, we can write it with just ONE equation. This allows us to write one complex equation that achieves the same mathematically as two real equations.

It turns out that there’s also a symmetry group associated with this circle of complex numbers with a radius or magnitude of 1. It is called the U(1) group. The elements of the group are all the infinite possible angles phi around the circle.

Quantum mechanics is built on complex numbers. We can apply the symmetry with the simple transformation of moving around the circle. Do described the movement of fermions, we can use the Dirac equation. It describes any matter particle, like an electron, with some mass m moving in space. It does not describe any forces.

If U(1) symmetry exists, it would mean that if we applied our transformation, the Lagrangian would not change. The problem is that the Lagrangian DOES change when we apply this transformation, so this tells us that no U(1) symmetry exists.

However, if we modify the equation, by adding a new quantum field to the theory, a gauge field, we can get a symmetry. Another name for a gauge field is a force. Our theory works, and obeys U(1) symmetry transformations if we add some new terms to the equation. It turns out that this new term describes the electromagnetic force. The entire theory of Quantum Electrodynamics can be derived by the new transformed equation.

So by taking a theory for fermions (Dirac equation) and demanding a U(1) transformation we got the theory of electromagnetism. Similarly, the standard model is constructed to respect three symmetries or special unitary groups. And each group leads to a symmetry resulting in a conservation law and a fundamental force.

The U(1) group gives us conservation of electric charge, and is associated with the electromagnetic force. The SU(2) group gives us conservation of weak isospin, or weak charge, and is associated with the weak force. The SU(3) group leads to conservation of color charge and is associated with the strong force. It leads to the theory of quantum chromodynamics.

In addition, the number of generators corresponds to the number of bosons involved with each force. U(1) has one generator and one photon. SU(2) has 3 generators and 3 W+, W-, and Z. SU(3) has 8 generators and 8 different gluons.
#gaugesymmetry
#grouptheory
#noetherstheorem
Symmetries seem to be the foundation of the laws of physics. Why this is the case is something no one knows.

[ArvinAsh]

If you found yourself lost in this video or if you want to brush up on some of the background information, here are some videos I made that will help:

[Grandunifiedcelery]

Thank you so much for the wonderful video as usual. 🤩
Next, please tell us about SU(5) symmetry, SO(10) symmetry, E6 symmetry, , , 🥺

[ryan-cole]

The U(1)×SU(2) group actually combine into a single group called the electroweak symmetry. This symmetry is broken by the higgs field, creating a completely different U(1) group for electromagnetism, sometimes denoted U_em(1) to differentiate it.
The weak force remaims completely broken and doesn't actually have a symmetry group.

[kavjay]

Amazing video, as always. Also, thank you for including some governing equations. Many authors/creators/producers avoid including any mathematical equations because they fear it would intimidate their audience. So, it is refreshing to see some maths equations not only being included, but also being clearly explained. Thank you for respecting our intelligence enough to include some maths. Excellent work. I'm looking forward to your next video

[devamjani]

None of this would have been possible without Emmy Noether, she truly deserves much more recognition.

[sahebchoudhury]

I have no knowledge of these complex maths. But I still like listening to your explanations. Sometimes I get some vague idea and sometimes clear. I learn something new. I wish I knew math well.

[monkieassasin]

Wow, this video was stunning. I did not expect this to be described this well. Your best video I’ve seen by far.

[NNiSYS]

Howdy Arwin! Here again from Perú. Just watched your video. I had to watch it three times to repeat the dopamine rush! Thank you so much for your wonderful CLARITY. There is so much BEAUTY in it! Thank you.

[miguelangelmaypech]

Excelent video, well explained. Can someone solve a question that I have? why does we ask the Lagrangian to fulfill for instance the U(1) symmetry a priori, without knowing that this Will give us the EM interaction? thanks

[anthonycarbone]

This turned out to be more fascinating than I originally thought. Thank You!!!

[picksalot]

I looks to me that symmetrical objects are more "stable" than irregular ones. A force acting on an irregular object tends to reduce those irregularities. Round pebbles in a stream are a good example.

[HighWycombe]

This is a great video to get started on how Symmetry leads to the Standard Model. It provides a learning path, tells you what you have to go away and study more deeply elsewhere if you are going to get to the bottom of this subject. We learn that symmetries lead to conserved quantities, Noether's theorem, generators, Euler's number, then rotation in a complex plane, the symmetry groups U(1), SU(2), SU(3). Most other videos assume that you already know stuff. This is the very best "beginning "video that I've found. I feel orientated.

[Naturamorpho]

This is one of those videos I'll have to watch more than once... great job, as always! Thanks, Arv!

[chemistchemist]

Every process in the universe favors the formation of high symmetry objects. I believe the reason is to use the less possible energy and to use less information to increase the entropy. I have seen these patterns while working on my research project and by studying Claude Shannon's information theory.

[andrewroberts]

This video is incredibly well put together and beautiful. My thanks to dear Mr. Ash! Another masterpiece!

[williambunting]

That was an awesome presentation. Cool to see that symmetry is built on the work of Emmy Noether.

[TitoTheThird]

15:23 Curiously, I've also come across the sequence 1, 3, 8 in Ramanujan's continued fractions related to the three symmetries of the Platonic solids. Note that the ONLY integers n > 1 such that 24/(n^2-1) is also an integer are 5, 3, 2, yielding the aforementioned 1, 3, 8. The integer n=5, of course, figures prominently in the Rogers-Ramanujan continued fraction and icosahedral symmetry. There are analogous continued fractions involving n=3 (for tetrahedral symmetry) and n=2 (for octahedral symmetry.) Hmm, I wonder if there is a connection?

[KaliFissure]

And gravity breaks symmetry, apparently. Only over totality of universe is it symmetric

[Liquifiedpizzas]

Your CG and visualization work has gotten a lot better!

[anthonycarbone]

I find humans desire order and seek order in order to satisfy their desire for order. Symmetry is elegant and shows the underlying universe contains order even though the outcome of this symmetry is pure chaos or disorder. I believe this represents the quest for the unknowable becoming the knowable. Math is the ultimate expression of order and balance and symmetry creates the ingredients to make a complete theory of everything.