This video is sponsored by Square Space. Stay tuned to the end to find out about a very
special offer for Arvin Ash viewers. Classical mechanics as formulated by Isaac
Newton and others before him, was the way that the universe was thought to work. But
over decades and centuries, scientists found more and more problems.
Then Quantum theory came along and solved many of those problems. For example, the photoelectric
effect where only certain frequency of light resulted in ejection of electrons was solved
by Einstein in 1905 who showed that photons are not pure waves, but come in packets of
energy proportional to their frequency. Other issues such as why the acceleration
of electrons around a nucleus does not give off a constant stream of photons was solved
by the Niels Bohr model of the atom. But quantum theory came with a very high price.
The highest price was probably determinism - it introduced randomness to the nature of
reality. Events were not deterministic anymore but probabilistic. Fortunately we don’t
see this in our macro world because it gets smoothed out, but at micro scales there is
randomness everywhere. Another price we paid was Continuity – forces
and particles at micro scales did not move or behave smoothly. Things occurred in chunks
of energy. You can measure a particle here and the same particle there, but you cannot
presume a smooth continuous trajectory from here to there. So our smooth deterministic
classical world-view was shattered by the discrete, probabilistic quantum world-view.
In the last two videos, I talked about the difficult and, so far, unsuccessful effort
to combine Einstein’s classical theory of general relativity with quantum mechanics.
But this lack of success has not been a big issue for making predictions because at very
small scales, general relativity can be largely ignored because gravity has a negligible effect.
But small things do move at very high speeds. And so to describe them at velocities near
or at the speed of light, Einstein’s prior theory, Special relativity has to apply.
That integration of special relativity with quantum mechanics and electromagnetism is
called Quantum electrodynamics, or QED. This is the most accurate theory man has ever come
up with. Not only that but it is also one of the most
important concepts that led to the best description of the nature of the universe, and that is
quantum field theory. What is QED and how does it work? That’s
coming up right now…
What QED does is replace the classical theory of electromagnetism, which involves continuous electromagnetic fields and puts it in terms
of discrete quantities - a quantum formulation. The idea is that small things such as electrons
and other subatomic particles can go very fast, and when they do, special relativity
has to apply. Many scientists over decades helped develop
the theory of QED. But In 1928 the first big step towards this unification was taken by
British physicist Paul Dirac (1902-1984), when he published the Dirac equation which
blended quantum mechanics and special relativity. Notice how similar this equation is to the
Schrodinger equation. Psi which is the wave function is present
in both equations and indicates that this is a quantum equation.
Dirac made a Hamiltonian, which is the sum of all the energy, in terms of space-time.
And this signifies how special relativity is integrated into the equation.
Notice the terms MC^2, and p the momentum on the left side.
This comes from special relativity. It represents energy. This equation says that
the rate of change of the wave function with respect to space and time, or rather space-time,
depends on the energy of the fermion. It is like the Schrodinger equation, except that
it evolves in space-time. Schrodinger equation treats time and space
as independent coordinates, but Dirac integrated space-time into the equation as shown by the
terms x and t as the coordinates of the wave function. X represents the 3 spatial dimensions,
and t of course is time. One of the remarkable things about this equation
is that there are some formulations that result in negative solutions.
Negative solutions are not uncommon in math. Typically, we just throw them out as being
nonsense. But Dirac was a man of conviction. He believed
his equation was so accurate that negative solutions had to be considered as a reflection
of reality. His equation was predicting anti-matter. No
one had ever thought of that before. Dirac took the bold position of insisting that antimatter
had to exist based simply on the math of his equation, and no experimental data. Notice
the negative and positive exponent solution for the function psi.
That's where the idea of antimatter and matter comes from.
Sure enough, 4 years after Dirac published his equation, in 1932 the positron, which
is the antimatter equivalent of the electron, was indeed discovered by American physicist
Carl Anderson. Today we know that not only the electron,
but all elementary particles have an antimatter equivalent.
Later in the 20th century, three scientists began formulating a quantum theory of electromagnetism
based on Dirac’s equation, and formulated the central concepts of QED. These scientists
were Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. They were awarded 1965
Nobel prize in physics for their efforts. The result was the following. Here’s what
they come up with. This represents the Lagrangian or the main equation for QED. The Lagrangian
contains all the energy terms of the theory, in other words, it contains all information
relevant for the theory. And from this Lagrangian, you can obtain the
behavior of the electromagnetic forces. Don’t let this formulation intimidate you. I will
explain intuitively what it means. Here we can identify the psi and psi bar representing
the wave function of the spin half field like the electron-positron field. Psi is the particle,
psi bar is the antiparticle. The Lagrangian describes all the energies
of the system. It shows that the fermion field represented by psi and psi bar, is coupled
to the photon field which is contained in A sub mu, which is part of D sub mu.
The i, gamma mu, D sub mu contains information about how fermions like electrons interact
with photons, in other words, how they interact with the electromagnetic field.
And this is important for deriving the Feynman rules, which we will get to in a second.
The m is the mass of the particle like it could be the electron mass.
The last term contains the electromagnetic field tensor, F_μν, which contains the photon
field and the relevant information about the electromagnetic force. This term basically
contains Maxwell’s equations. The difference between F sub mu nu and F superscript
mu nu is a flipping of the electric field in opposite directions.
So the formulation here gives us all of the expected properties of the electromagnetic
forces in the system. Now that you’ve seen the math, let’s show
what happens more intuitively. In the 19th century, it was thought that two
electrons near each other would simply repel each other without ever touching, similar
to the way that two like poles of a magnet repel each other without ever touching.
Paul Dirac and others showed that what is really happening is that there is no continuous
field, but rather the forces are mediated by the exchange of discrete photons. They
are emitting and absorbing photons which results in attraction and repulsion.
Here is a simplified analogy that I’ve used before. Think of two people on a boat throwing
basketballs at each other. The basketballs are analogous to virtual photon exchange.
The exchange of momentum of the basketballs results in the two boats moving further apart.
This is analogous to how an exchange of virtual photons can confer momentum that makes two
like charges move apart. Similarly the exchange of negative momentum can occur analogous to
the two people throwing boomerangs away from each other. This is analogous to the mechanism
of negative momentum exchange through virtual photons between two dissimilar charges.
With this picture in mind, let’s imagine two charged particles that travel near each
other toward a colliding path. When they get close they will repel each other and take
a path that looks something like this: In classical theory, this would be thought
of as the particles being repelled due to interactions on a continuous field.
So what happens here is that as the two charged particles approach each other, represented
by the lines with the arrows towards each other, an exchange of virtual photons take
place represented by the wavy line in the middle.
This exchange results in two charged particles moving away from each other, represented by
the two lines pointing away from each other. If you understand this, then you have just
learned how to read a Feynman diagram, because that’s what
These brilliant diagrams were developed by American physicist Richard Feynman around
1949.
And they help to intuitively understand the mathematics of QED. In more general terms, the simplest Feynman
diagram can be drawn like this. The direction of the arrow on the, the simplest Feynman. fermion
line tells us if the particle is a matter or the anti-matter particle.
If the arrow goes forward in time (to the right), it is the matter particle, and if
it goes backwards it’s the antimatter particle. So, in this diagram, the lower arrow pointing
to the right represents the matter particle, and the upper arrow represents the antimatter
particle. This diagram can be interpreted as a positron
and an electron annihilating into a photon. But we could also flip this 180 degrees and
then the diagram would show a photon becoming an electron-positron pair. From this simplest
Feynman diagram, you can build all other diagrams for QED.
The lines with the arrows represent fermions and they must be continuous. This is because
of the law of conservation. The same number of fermions coming into a process must also
come out. Note that a photon is a boson, not a fermion.
But you might ask that if a positron and electron come in, and a photon comes out, then how
are things conserved? The reason conservation is still valid is because a matter particle
is considered as +1 fermion and an antimatter particle is considered as -1 fermion. So if
we look on the left side, an electron and positron coming in works out to be +1-1 = 0,
and on the other side there are no fermions, thus we again have 0 coming in and 0 going
out. The rule still works. Now, you have to keep in mind that quantum
theory is such that other events more complicated than the simple exchange of one photon can
occur when two particles come near each other. So for example, two photons can be exchanged.
The Feynman diagram in that case would look like this:
Sometimes a particle can emit a photon and reabsorb it again. The Feynman diagram for
that looks like this: Since photons can carry various amounts of
energy depending on their frequency, it can turn into matter/antimatter pairs – electron/positron,
muon/antimuon, quark/antiquark This can happen during the exchange process, and the diagram
for that occurrence looks like this: The middle ellipse-like figure represents
a matter/antimatter creation and annihilation, resulting in a reconversion back to a photon.
Just about anything can occur in these interactions. So we can have all kinds of diagrams here.
The Key to understanding Feynman diagrams and the brilliance of Richard Feynman is that
he was able to develop an intuitive pictorial representation of highly sophisticated mathematics,
because you see, every one of these Feynman diagrams is really an equation. A rather complicated
equation. So for example, one of the simplest diagrams
is where two electrons come in, two electrons coming out and a photon is being exchanged.
This equation has 7 components – 2 electrons coming in, 2 electrons coming out, a photon
being exchanged, and the vertex where the photon is emitted and the vertex where the
photon is absorbed. Each of these 7 components has an associated element in an equation:
So using these components, we can write the equation that looks like this. This is the
mathematical equivalent of one of the same Feynman diagram.
Note that when two charged particles actually come near each other they exchange many photons,
not just one. And many different types of potential exchanges can occur.
But by simplifying the equation to the exchange of single photon, we can get a pretty good
approximation what is actually happening. And that’s what makes Feynman diagrams so
useful. Multiple components and multiple vertices
and quickly result in very complicated equations. Writing them is not so difficult. But solving
them is a whole different story, and sometimes supercomputers are used to solve them.
We can calculate the probability of any one of the Feynman diagrams. It turns out that
the probability depends mostly on the number of emissions and absorptions. An emission or
an absorption on a Feynman diagram is shown as a vertex. The more vertices, the less likely
it is to occur. Each vertex on a diagram means that it happens 1% as often as a diagram with
one less vertex. So for example a diagram with 3 vertices happens
1% as often as a diagram with 2 vertices. 4 vertices is 0.01% as a diagram with 2 vertices
– 1% of 1%. The lessons from the development of QED were
later used for the development of Quantum Chromodynamics (QCD), which I will talk about
in a future video, and other Quantum Field Theories starting in the 1960’s – like
the Electroweak theory. QED is a quantum field theory.
Electromagnetism was the first force to be expressed in terms of a quantum field theory.
That’s what QED is – it is electromagnetism expressed in terms of quantum fields.
And it is the most accurate theory ever devised by man. It is accurate to withing 12 decimal
points – that's 1 part in a trillion. That’s not just incredible. That astounding.
What you should get from this phenomenal level of accuracy between prediction and measurement
is that Quantum mechanics is the language of reality.
Even though we have to sacrifice some of our intuitions, and perhaps logic, it is simply
the way the universe works. I can’t think of anything in science that we can say with
absolute certainty. But the equations of QED along with the quantum field theories behind
them are about as close to certainty as we are probably ever going to get.
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comments below and I will I try to answer it. I will see you in the next video my friend.