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by clicking the link the description. Physicists study how the universe works
at the most fundamental level. They’re like Sherlock Holmes. They’ve just come to the
scene of the crime which is the universe. And they’re looking high and low trying to find
clues that can tell them what happened, how things came to be, but also how it all works.
If you ask a physicist, what make the universe tick? What’s at the core of physics theories?
One of the common answers you’ll hear is symmetry. Symmetry lies at the heart of physics,
and it appears to be a fundamental property of the universe. And it leads to something quite
profound. You see, “Every symmetry leads to a conserved quantity” – For example,
it leads to the law of conservation of energy – the idea that energy overall is neither created
nor destroyed, it only changes form -- similarly, conservation of momentum and electric charge.
And it even leads to fundamental forces, such as the electromagentic force.
So, what is symmetry? How does it play a role in physics? How does it lead to some of the most
fundamental rules the universe seems to obey? The answer will probably shake your
idea about what physics really is. And it’s coming up right now…
What is symmetry? Symmetry is about actions – actions that in some sense don’t change
anything. This might sound like a stupid concept. If you don’t change anything,
then what’s the point of doing anything? Well, let’s look at that.
You might have unknowingly studied symmetry in one of your elementary school classes, for example if
you played with mirrors and geometrical objects. Consider an equilateral triangle.
This is a triangle where all sides and angles are equal. In this case we can
put a mirror from one corner to the middle of the opposite side. If we do this and look into
the mirror, the reflection of the image will complete the triangle.
Each side is a mirror image of the other side. This is a symmetry of the equilateral triangle!
But we do more. There is another operation we can do, which recovers
the original triangle. Consider rotating the triangle by 120 degrees. Again, the triangle
will look identical to what it was before. Now, the problem is that we can’t see the
change. So to help us understand the symmetry of this triangle, let us write the letter A in it.
This will help us identify the permutations. Let’s start from the beginning,
the initial position is one permutation. Then we can mirror it, this gives us another permutation.
Then we can rotate by 120 degrees, this gives us the 3rd permutation. Again, we can consider the
reflection giving us the 4th permutation. We can rotation with 120 degrees again to
get the 5th permutation and consider the reflection to get the 6th permutation.
Now note the next time we rotate with 120 degrees we end up where we started.
Conclusively there are 6 permutations we can do with this triangle. Now you’re probably asking,
why are you wasting time on elementary school math? Well, let me reveal the secret of this exercise.
What we just talked about is a simple example of something much more complex, namely 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 also say it’s of order 6 as there
are 6 permutations, also called group elements. As another example, if we considered the
symmetries of a square, the group theory behind it would be the dihedral group of degree 4.
Going back to our triangle, notice how we were able to change between the elements,
or permutations, using two different operations, rotation, and reflection.
We can move between all elements using combinations of these operations.
We have thus two ways of describing this group. Either by considering the list of all elements,
or by the operation we performed, in this case reflections or rotations. In mathematics
we call the latter, “generators,” because from the generators of the group we can produce
all the elements of the group. You might ask, well what’s the
problem with having just a list of elements? Well, it’s fine for a simple group. But consider
how many elements there would be if we did this for a hexagon, which is a more complex dihedral
group of degree 6. Just like with the triangle, the generators are reflections and rotations,
though the rotation angle is now only 60 degrees. The point is, it’s a lot shorter and simpler to
describe a group in terms of the generators than a potentially long list of elements.
And to make matters worse. Groups can have an infinite number of elements, making it impossible
to describe them without generators. Before moving on, let us consider the
importance of group theory, because this is where the magic begins.
I said in the beginning that symmetry is about actions that in some sense don’t change anything.
What I meant with actions is applying the generators to a group. And as we have seen,
the result of applying a generator doesn’t change anything visible.
They key here is that this is not the same as nothing changed. The object has been transformed,
but it looks the same. This is key; we can do something,
but in the triangle, it holds its shape., It remains the same and in general we stay within the group elements.
Overall, what I want to impart is that symmetries give us rules for how to transform something
while conserving a quantity. In the case of the triangle,
that conserved quantity is its shape, and the transformation is rotation and reflection.
This leads us to what is called Noether’s theorem developed by Emmy Noether around 1915.
This theorem states something profound about nature. In its simplified form,
it says that “For every symmetry there is a corresponding conservation law.”
This is huge! This directly connects symmetries with conserved quantities.
For example, time translation symmetry, which is the idea that the laws of physics are the
same today as they were yesterday, as they will be tomorrow, leads to the principle
of conservation of energy. And space translation symmetry,
which is the idea that the laws of physics are the same here as they are 10 meters away,
or on the moon or anywhere else in the universe, leads to the law of conservation of momentum.
Similarly the fundamental forces of nature also arise from certain symmetries.
So we looked at a triangles and polygons earlier. But what happens
if we take the limit of having a polygon with an infinite number of edges? We get a circle.
A circle of some radius, r, can be described on a 2D xy-plane using polar coordinates
by these equations: If we make the simplification
that the radius is 1, we simply get the following, Two equations which are functions of the angle ϕ.
This is what we get using real numbers , but it limits us mathematically. If we allow
ourselves to use complex numbers to represent the circle, we can write it with just one equation:
Where r is the radius, e is Euler’s number representing a constant, and i is the square root
of minus one. It’s called an imaginary number. Since we chose the radius to be 1,
we can simplify this to just this. This allows us to write one
complex equation that achieves the same mathematically as two real equations.
This single complex number Z can describe any point on the circle of radius 1.
All we had to do was change one axis to be imaginary. So the circle is on a complex plane.
This shows how complex numbers can be powerful math tools.
Notice the point on the circle is completely determined by the angle phi.
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. We can write the transformation of
the U(1) group formally as follows, Where n is the generator, meaning n represents
how much we rotate around the circle. Now, where have you seen complex numbers
in physics? In quantum mechanics. It’s built on complex numbers.
Let’s try to apply the symmetry of just moving around a circle with this simple transformation.
To describe fermions, which are matter particles, we need some equation of motion
that describes their behavior. To do this we can use the Dirac equation.
This is a scary looking equation, but don’t be intimidated by this. You don’t have to understand
the math. I just want you to see the equation so you can see conceptually what happens to the math
when we do these transformations.
L is the Lagrangian. A Lagrangian is just the difference between
kinetic and potential energy of a particle. Psi as the wavefunction for a matter particle,
and psi bar is the wave function for the equivalent antimatter particle.
The wavefunction, if you recall, when squared tells us the probability of finding the particle
in a given location when measured. The first part in the parenthesis
describes the particle’s movement in spacetime, and m is the mass.
Ok so this equation describes some matter particle, like an electron,
with some mass m moving in space. You might say, ok so what? I don’t see any
forces here. This is where group theory comes in. If U(1) symmetry exists, it would mean that if we
applied our transformation this equation like so: where n is -1 for matter particles, and 1
for antimatter particles, the Lagrangian would not change .
The problem is that when we do this, the Lagrangian DOES change, so this
tells us that there's no U(1) symmetry, and we wind up with a broken theory.
However, if we modify the equation, by adding a new quantum field to the theory,
a so-called gauge field, we can get a symmetry. Another name for a gauge field is a force.
So when a force is added to the equation, we find that a symmetry exists. Our theory then works,
and obeys U(1) symmetry transformations. So what is this force that we just added?
It turns out that this new term that was added to the equation describes the electromagnetic force.
The “e” in this case is not euler’s number, it is the electric charge, and A_μ represents the photon
field. So this tells us that photons are the force particles that mediate the electromagnetic force.
So while we had electrons in the initial Dirac equation with no interaction, what we have here
is an equation with electron to electron interactions, mediated by the photon field.
The full equation looks like this: This additional term in front contains
what’s called the electromagnetic field strength tensor, which is
also defined from the photon field. This term essentially contains all the Maxwell equations.
Notice what happened here. We took a theory for fermions and demanded a U(1) transformation.
Adding the additional mathematical terms to make the symmetry work, gave us the electromagnetic
force mediated between fermions by photons. This is the theory of electromagnetic force,
also known as quantum electrodynamics, or QED. We ended up with the entire theory for
electromagnetism just by demanding one relatively simple symmetry.
Because of Noether’s theorem, we know something is conserved. In this case of the electromagnetic
force, electric charge is conserved. As it turns out the standard model
is constructed to respect these symmetries or special unitary groups, U(1), SU(2) and SU(3). Sometimes if you ask a
mathematician, they will write the entire standard model in terms of these groups.
When scientists say they are studying Group theory, they
are basically studying the mathematical symmetry transformations of objects or abstract things.
It turns out each group leads to a symmetry resulting in a conservation
law and a fundamental force. As we saw the U(1) group gives us
conservation of electric charge. It’s a symmetry associated with the electromagnetic force.
The SU(2) group gives us conservation of weak isospin, or weak charge. This symmetry
is associated with the weak force. And as you might guess. 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 other words, the three symmetry groups
that the standard model respects gives us three of the fundamental forces of our universe.
We can do a similar mathematical exercises as we did for U(1) for the SU(2) and SU(3) groups,
but the math is much more complex. The U(1) group is like a one dimension sphere,
or circle, but you need 2 dimensions to visualize it because a circle transforms on a 2D plane.
Similarly the SU(2) group is like a 3 dimensional sphere
that transforms in four dimensions, which we can’t visualize
And SU(3) is symmetry is even more difficult to visualize,
because it would be an eight dimensional sphere that requires 9 dimensions to visualize it.
Now, you might think this is pretty cool, but it gets even cooler. Remember that the
generators are like the rules for how to transform, while respecting the symmetry?
The U(1) group has 1 generator because there’s just one rotation around a circle.
Now ask yourself, how many bosons are responsible for mediating the electromagnetic force,
associated with this group? Well, it’s just one single particle, the photon.
It turns out, each generator of a gauge group corresponds to a force mediating particle.
So with the SU(2) group which results in the weak force, there are 3 generators since
there are three ways to transform a 3D sphere. Well , it is mediated by a total of 3 bosons,
the W+, W- and the Z bosons. Likewise, the SU(3) group has
8 generators since we are dealing with an 8D sphere. So it should come as no surprise then
that the strong force is mediated by 8 different gluons of different combinations of color charges.
Symmetries seem to be the foundation of the laws of physics. Why this should be the case
is something no one can really answer. Most people subjectively find beauty in symmetries.
Even people with more symmetrical faces are considered more attractive.
Maybe the universe itself finds beauty in symmetry too.
If you want to go further into the details of symmetry, and also how it is fundamentally
at the root of physics, I highly recommend a lecture available on Wondrium, today’s sponsor,
called, “The Power of Symmetry.” It’s part of a 12 chapter course called,
“The Higss Boson and beyond” taught by someone you might be familiar with because I interviewed him
on this channel last year, and is one of my favorite science educators, Sean Carroll. In an
effort to explain our search for the Higgs Boson, Sean reviews everything from quantum field theory,
to particle physics, to symmetry. It’s a must watch for physics enthusiasts.
You’ll find Sean and many other fantastic educators on Wondrium. That’s why I have
been a member of Wondrium for years. I couldn’t recommend them more. And that’s why my testimonial
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