The standard model

of particle physics is the most successful, most

accurate physical theory ever developed, describing

with stunning accuracy the fundamental quantum

building blocks of our universe. But even more stunning is how it

was discovered, by peering deep into the symmetries of reality. As far as we can

tell, mathematics is the language in which

the universe is written. Our laws of physics are

equations of motion tuned by the fundamental constants. Previously, we’ve talked

a bit about the symmetries of these equations and how they

lead us to conserved quantities like energy and momentum. But that’s just the tip of

the theoretical iceberg. These symmetries can be portals

into entirely new aspects of reality. The most amazing example of

this is the standard model of particle physics. Today, I’m going to

open the first portal of the standard model

and show you the origin of the electromagnetic field. To appreciate the

theoretical whirlwind that is the standard model,

we need to introduce the idea of a gauge theory. In simple terms,

a gauge theory is one that has mathematical

parameters or degrees of freedom that can be

changed without affecting the predictions of the theory. An example would be

a ball rolling down a hill under a constant

gravitational acceleration. The speed of the ball at

the bottom of the hill depends on its

change in altitude. But it doesn’t matter what we

define to be altitude zero– the bottom of the

hill, sea level, even the center of the earth–

for the equations of motion of the ball, the altitude

zero point is irrelevant. It’s what we call a gauge

freedom or a gauge symmetry. And we say that the

equations of motion are invariant to that parameter. That’s a pretty basic example. But it turns out that

these gauge symmetries are an important feature of

most of our physical theories describing the universe. Newton’s laws of

motion and gravity, Maxwell’s equations

for electromagnetism, Einstein’s general

relativity, and of course, the standard model. We’re not quite sure

why this is the case, but it seems to be a trend. A theory that has

these gauge symmetries is called a gauge theory. Today, we’re going to look at

the simplest of the symmetries of the standard model. The standard model is ultimately

based on quantum field theory, but we’re going to use

the Schrodinger equation. That’s the most basic equation

of motion of quantum mechanics. It describes the evolution

of the wave function, which is the mathematical

object that contains all the information about a

particular physical system. We can never see

the underlying wave function of, say, a particle. The best we can do

is make a measurement of physical observables,

like position or momentum. The wave function can

represent different observables and it determines

the distribution of possible results

of measurement of those observables. In this episode, we’ll be

talking about the position wave function. OK. Pay attention to

this bit of math. It’ll be important. The square of the magnitude

of this wave function tells us the

probability distribution of a particle’s position. The position that we observe

when we look at the particle is picked randomly

from that distribution. This step of squaring

the wave function is called the Born rule. And this innocuous seeming step

introduces a simple symmetry that has profound implications. Let’s see what happens when

we square the wave function. The function is an oscillation

in quantum possibility, moving through space and time. It’s no simple wave. It’s literally complex in

the mathematical sense. It has two components, one

real and one imaginary. These components oscillate

in sync with each other, but they’re offset, shifted

in phase by a constant amount. Phase is just the

wave’s current state in its up-down oscillation. When we apply the Born

rule, what we’re doing is squaring these two waves

and adding them together. But it turns out that this

value doesn’t depend on phase. The magnitude squared of the

real and imaginary components stays the same, even as those

components move up and down. It’s that magnitude squared

that we can observe. It determines the

particle’s position. The phase itself is

fundamentally unobservable. You can shift

phase by any amount and you wouldn’t change

the resulting position of the particle,

as long as you do the same shift to both the

real and imaginary components. In fact, as long as

you make the same shift across the entire wave

function, all the observables are unchanged. We call this sort of

transformation a global phase shift. And it’s analogous to

transforming our altitude zero point up or down by the

same amount everywhere. The equations of

quantum mechanics have what we call

global phase invariance. Global phase is a gauge

symmetry of the system. Let’s push a little further to

see how far this symmetry goes. This time, we’ll shift the

phase by different amounts at different

locations, while still keeping the real

and imaginary shifts the same at each location. This position

dependent phase shift is called a local phase shift,

instead of a global phase shift. We’ll try this because,

well, we already know that the magnitude squared

of the wave function should still stay the same

under local phase shifts. Let’s see what this

would look like. A global phase shift looks like

this, where all points move by the same amount. However, if we do a

local phase shift, say, only this point here,

only that location changes, as if it were part

of the shifted wave, making a discontinuous spike. If you allow this sort

of local phase shift, you can change each

point in a different way and really mess up

the wave function. That shouldn’t change

our probabilities for the positions

of the particles, but what about observables

besides positions? According to the basic

Schrodinger equation, we just ruined everything. Among other things,

messing with local phase really screws up our prediction

for the particle’s momentum. See, momentum is related to the

average steepness of the wave function. Change the shape of

that wave function with local phase

shifts and you actually break conservation of momentum. Local phase is not

a gauge symmetry of the basic

Schrodinger equation. OK. That was a bust. I guess we’re done here. All right. Wait just a second. Just for funsies, maybe we can

change the Schrodinger equation to find a version that really

is invariant to local phase shifts. To do that, we need to alter

the part of the Schrodinger equation that gives us the

momentum of a particle, the momentum operator. After all, momentum is

what got screwed up. It turns out that we can

add a mathematical term to the momentum operator

that’s specially designed to undo any mess we make to

the phase of the wave function. If we choose this

term correctly, it absorbs any local changes

we make to the phase. And what is that extra term? Well, it’s something we

call a vector potential. I won’t go into that right

now, but the important and absolutely bizarre thing

about this mathematical entity is that it looks like

something very familiar. It looks exactly like the

type of vector potential that you would have

in the presence of an electromagnetic field. So we’ve discovered that

the only way for particles to have local

phase invariance is for us to introduce a new

fundamental field that pervades all of space. And it turns out that

field already exists, and it’s the

electromagnetic field. This is totally crazy. We just rediscovered

electromagnetism by insisting on a gauge symmetry

that we had no right to expect to exist in the first place. But we didn’t just

rediscover the EM field, we learned a ton about it. By discovering how it fits

into the Schrodinger equation, we’ve unlocked its

quantum behavior. And now we know how it interacts

with particles of matter to give them this symmetry. We also learned about the

origin of electric charge, which we now see as a coupling turn. Any particle that has

this kind of charge will interact with

and be affected by the electromagnetic field

and be granted local phase invariance. But the reverse is also true. In order to have this particular

type of local phase invariance, particles must possess

electric charge. By the way, applying

Noether’s theorem tells us there is a

conserved quantity associated with any symmetry. In this case, the symmetry

is local phase invariance and the conserved quantity

is electric charge. At this point, we only need

a couple of extra steps to produce the full

description of electromagnetism in the quantum world. Quantum electrodynamics, or QED. First, we need to upgrade

the Schrodinger equation to the Dirac equation so

it works with Einstein’s special relativity. And we talked about

that in this episode. Then, we need to apply quantum

principles to our field, like considering its

internal or self energy and allowing quantized

oscillations in the field itself. Those oscillations in our

new electromagnetic field turn out to be the photon. But what about all those

fundamental particles without electric charge? Neutral particles

like neutrinos. To understand those, we’ll need

to go beyond the Schrodinger equation and to explore

new gauge symmetries. It turns out that

local phase invariance is just the simplest

of the larger suite of gauge symmetries

of the standard model. Those symmetries are obtusely

named, U1, SU2 and SU3, and they predict the

fields that give rise to electromagnetism, the weak

and the strong nuclear forces, respectively. The fields that arise from

these gauge symmetries are called gauge

fields, and they all have their associated

oscillations, their associated particles. These are, the gauge bosons,

the photon for electromagnetism, the W and Z bosons for

the weak interaction, and the gluon for the

strong interaction. Together, they govern

the interactions of the metaparticles

of the standard model. And we’ll come back to

that in future episodes. Perhaps the greatest

mystery here is not the nature

of the quantum field nor the connection

between symmetry and the fundamental

forces, perhaps it’s the fact that by pure

exploration of mathematics, delving many layers

of abstraction deeper than our

capacity for intuition, we are led to true discoveries

about physical reality. And following those

mathematical labyrinths reveals physical theory with

stunning predictive power, like the standard model

of particle physics. Mathematics truly seems to

be the language in which the universe is written. We should be amazed that

we can learn that language and through it,

comprehend the underlying nature of space time. Last time on Space

Time Journal Club, we looked at a new result

potentially detecting a particle beyond the standard

model, the sterile neutrino. Let’s see what you had to say. Sebastian Elytron says

that the standard model won’t change for unverified

discoveries, it has standards. Totally agreed. And no one, not even

the researchers, are claiming the actual

discovery of this new particle, yet. April put it well in her

response to your comment, a 6.1 sigma level

of significance doesn’t mean for sure that

sterile neutrinos exist. It could mean there’s some

other physical process that we don’t understand yet. Regardless, it will definitely

mean something happened. Will it be something

interesting? That remains to be seen. Richard Brockman and

badly drawn turtle point out the danger of

combining multiple experiments to increase the significance

of your results. If you have enough

experiments to choose from, you can just select from those

with the high significance until it takes you over the

five sigma significance hurdle. In the case of the

mini bird experiment, they combined their 4.8

sigma fermion result with the 3.6 sigma result from

the only other substantially similar experiment that has

ever been performed, that one at Los Alamos. So there’s some

justification because there weren’t any very

similar experiments with lower significance. On the other hand, it seems

like they didn’t integrate the constraints to sterile

neutrinos derived from the ice cube experiment or from the

cosmic microwave background radiation. It’s much harder to

incorporate those because the experiments

were so different. Still, it’s a worthwhile

note of caution. A few of you point out that if

you build a wall of lead, one likely, you’d think, to

try to stop neutrinos, you would just collapse

into a black hole. I think you’re right. Call off the project,

guys, no more wall.

For the reference of others: https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

You don't "just" change the Schrodinger equation to the one derived from the hamiltonian of a charged particle, you also use the gauge freedom of electromagnetism. The potentials in electromagnetism have a gauge freedom, and that freedom is exactly what you use to cancel out the effect of local phase invariance.

First example entirely wrong. Sum total of gravity at center of earth is zero.

@1:24 can someone explain how his assertion is correct that it wouldn't matter if it was at the center of the Earth?

I mean, if the center of the Earth was the place where the entirety of Earth's mass lived in a single point, sure. But if situated exactly at the center of the Earth, wouldn't we be subjected to gravity pulling <b>away</b> from us in all directions, and thus we would be weightless??

But the formal axiomatic vibration within phase invariant nine dimensional non linear one space phase matrix would equal more than one!!! Even a theorist would know that!!! Just ask me if you want to know more haha

I hope they call the Grand Unified Equation, the Buzzinga Equation.

In the U(1)xSU(2)xSU(3) group of the Standard Model, the U(1) part is NOT the electromagnetic field!

U(1)xSU(2) are BOTH parts of the electroweak interaction which can't simply be separated into em and weak. This only happens after symmetry breaking (below the electroweak energy scale). Only then we get the U(1) group of the electromagnetic force – but it's not the same as in the Standard Model lagrangian.

It's a bit confusing because the groups are the same so it's quite a common mistake. But the unterlying mechanism is more complex, so be cautious.

So your telling me that by understanding gauge theory as a freedom baseline like the representations of altitude, as well as realizing that the universe contains continuous symmetries like gravity with corresponding energy conservations, we ended up finding fields and their corresponding vibrations like the electron and photon in the EM field which introduced all the electromagnetic, strong, and weak nuclear forces, which came about through the wonderful layers of abstracted realities created with math.

Well gosh, I need to step up my math game to get a bigger high on this

I love It ! Always a pleasure !I'm learning to Freak out "A Molecule !hahaha Don't try this At Home.

Great video. You took a notoriously difficult subject and made it accessible. This is in the spirit of Einstein who wrote a book for the masses about special relativity. I hope this is a trend in the world that continues. Everybody eats!

This is an excellent series.

"Wall bad!" Mexico is a leader in physics, isn't it?

Not the language in which the Universe was written. That implies intent. It would be more appropriate to say “Math is the language in which the Universe is translated.”

is that tiptoe effect from a strange camera angle?

we need to work on extending life to 200+ years just so i have enough time to understand this stuff…

2:47 Mr. Space Time, what perspective are you aligning that wavefunction thing from? That's slightly off.

5:58 Except that middle one doesn't touch the probability line!!!!!!!!

I wouldn't say I'm OCD, but your text on the Schrodinger Equation and Electric Charge were squint … wait … oh no! I'm OCD.

Let's build a black hole and let the aliens pay for it!

Are Mathematics real? By that, I mean do we invent Maths, or do we discover them?

in terms of momentum and its conservation, what is charge? And why don't neutrinos, photons, etc have it?

But without a wall, how do we keep the neutrinos out?

I love this channel so much. I usually get lost around 1/3 of the video but I enjoy the rest even more

I love this channel, so little liberal propaganda…. just that bit about the wall at the end… and that was only dog whistle.

Yet here in Australia schools think physics can be taught without maths… what a blasphemy

what I didn't understand is why a problem in the conservation of momentum gave rise to EMF 😐

I can somewhat guess about the momentum operator. It maybe wrong but still we have functions in number symmetry that can do the same. Take for eg the distance function. If we assign the function a constant value(take for example 2) and then slide or stretch the number line by basic additive or multiplicative properties the function still remains the same. It provides us the distance(2). Just a wild guess

Before we understand atoms, how can we know the universe?

Science says 1 electron and 1 proton able to form a hydrogen atom.

That is impossible. The 2 particles attract each other with the strongest attraction force in nature F=Ke x pe/R^2.

Therefore they must stick together under that force. The electron is impossible to circle/wave/cloud around the proton to form a stable atom.

If the standard model of atomic structure is wrong, the rest of science will be all wrong.

Scientists invented a force called strong force, it holds protons together.

They forgot to invent another force, that keeps electron away from proton.

The attraction force between electron and proton is 10^33 times stronger than 2 neodymium magnets.

What force can separate proton and electron?

See how stupid is the standard model?

Uncertainty principle, wave-particle duality, wave function collapse, electron shell, orbital, quantum state, energy level, electron hole, electron cloud, all imaginary, made up word puzzles that have nothing to do with reality.

The standard model is like a monkey story.

So sad. You got to follow that shit to get a degree. You got to teach that shit to become a teacher.

This is quickly becoming one of my favorite channels on YouTube. this is the kind of stuff that I wanted to learn in school before they decided that I wasn't normal and started putting me on medication that made me not normal #Ritalinbandwagon

This is the question I have. Are we just fooling ourselves? Math is infinite. There are infinity just in-between 0-1 let alone 1-2, so on and so on… In infinity you will find everything and anything has to happen.. So why are we so amazed that we can describe some of the universe with it.. Just a thought..

The more things change, the more they stay the same.

Quantum ElectoDynamics = Quod Erat Demonstratum. Q.E.D.

Click bait

Mexico will pay for a lightyear thick wall, and it will only take them one lightyear to pay for it. That'll solve all the USA's crime, human trafficking, and drug abuse problems in less than half a parsec.

You should definitely build that light-year-thick wall of lead — and you should make the neutrinos pay for it.

This was one of the most informative videos you've ever posted. It was very simple in my opinion and actually inspiring how maths can lead to actual physical discoveries.

One of the best explanations of Guage symmetry I've ever come across.

Fantastic. You will never know how helpful this video was. Thank You.

Yes yes but we are all waiting. Can it mine Bitcoin?

Your videos are boaring man and that's why I dislike them. It seems that you are reading some essay like a news host. Make it interesting like SciShow Space and Kurgesgazt videos. Put some animations and speak in some easier languages instead of showing equations and theories copied from books.

Quantum

Physics as treated by modern scientists is

unable to explain conventional space energies origin. It's not a mirror

it's a distorting mirror. The scientists need absolutely different approaches

to the Quantum to proceed. We'll share a demo video of our quantum device in

operation soon.

Like an idiot, I thought I could work and listen to this at the same time.

no such thing as stunx or should or amazx or not, doesn't matter, ts just toolx

4:20 Hold. j²=-1. So why is there still an imaginary wave when squaring should eliminate that into real-ity?

1:50 isn't g variant to 0 point parameter?

I appreciate the videos. You cover a lot of material in an advanced manner such that the only people really able to make sense of it are somewhat experts already. But the video you made is interesting and I will try to make sense of it over time.

My gage is pretty long, but, it won’t get no bigger round than a beer can✅

Do the extra frequencies created by the local phase shifts count as new things? I mean the fourier transform of the local phase shifted wave would have other frequency components, do they mean anything? And why does the shift break conservation of momentum?

Okay… say it again, but slower…

Playback Speed: .5Can we, please, stop this “observation” BS? It’s misleading and leads to all the crazy theories that it requires a conscious mind to “observe” something. It is an impact by – a collision with – another particle. “Observation” or “measurement” is simply a side effect of that. The fact that we can make any conclusions from it doesn’t change the behavior such collision causes.

Does a photon create around it (A) Electric field, (B) Magnetic field, (C) A&B, (D) Gravitational field ?

How do you take square of 3D wave having vector properties, like direction?

So beauty explanation professor O'Dowd

Great lecture on a not so easy to explain subject.

Mate that was deep that

I am a physics graduate student. These videos help me a lot in grasping the physical content of the dry equations that I encounter everday.

I love these programs/videos, but you have TERRIBLE LINKS between them. This video mentions another episode on "Quantum Field Theory" but there is ZERO link to that episode either inside this video – or in the youtube 'next' sequences on the site page

~~or~~at brilliant.org or ANYWHERE. How do you expect viewers to stay committed to your fine presentations when you cut us loose at the most FUNDAMENTAL internet communication level?!?!?! .. and believe me, I've explored trying to find these conceptual references in your various episodes you keep referring to .. with no luck. What? do you expect quantum entanglement to keep us 'connected' without any practical effort on~~your~~part??? Nice assumption. WRONG! TRY THIS FOR YOURS TRULY: [email protected]Great Explanation! By the way, Why are you standing on tip of your toes?!

"Gauge" is one of these awfully spelled English words. It could at least just be "Gage" (what the hell is that u doing there??), although a more sensible orthography would probably have it as "Gäidsh" or so.

I was lost almost immediately but I have a question. I thought the EM field was a classical concept that went away with QFT. But he mentioned it, so is that true or not?

Hey there friend! I hope you have a good day, or if you’re not, I hope this makes you feel just a wee bit better. Jesus Christ be with you!😊

Honestly starting to take a class in quantum chemistry and realizing this channel can help me get a better intuition on the subject is actually amazing

Thanks for a great series Matt, but I disagree that math is the language that the universe is written in. The Universe has it's own language. Math is our language that we use to map out and to communicate our own observations, which by their nature are very limited.

Perhaps one day we will be able to check our mappings by creating our own universe, but as korzybski pointed out, "The map is not the territory" and while our maps may allow use to tour the territory or terrain and allow use to utilize and modify it, they will never be the territory.

That, of course, is just a point of view (or mapping), but a very helpful one that helps us to consistently explain (or map out) a lot that is otherwise "unexplainable".