The physics of your everyday life is governed by the Standard Model, a quantum field theory or gauge theory that has survived decades of experimental checks.
“Quantum field theory is, of course, the description of nature as we know it, with the exception of gravity. Apart from gravity, quantum field theory appears to be all there is.”
Leonard Susskind | Advanced Quantum Mechanics Lecture 6, ~0m | Stanford, YouTube
“We’ve studied two different theories a little bit... quantum electrodynamics and quantum chromodynamics. Both of them are gauge theories. In fact, just about all of nature as we know it, in one way or another, is controlled by gauge theories of different kinds.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 5, ~0m | Stanford, YouTube
Are there unanswered questions in physics? Of course. We are at the mercy of our own technological and computational inadequacies. That said, one must navigate the experimentally sound to understand why these shortcomings do not tarnish the beauty of what has already been uncovered.
In simplistic terms, the Standard Model tells us that the smallest things of which everything is made, have a very deep connection to something with which we are all familiar: Geometry.
In more complicated terms - in terms you are unlikely to understand just yet - the Standard Model tells us that everything around you is built out of a system of space and time filling quantum fields, which break down into or are constructed out of, tiny elementary particles. These elementary particles, which bind together at infinitesimal scales to form atoms and ultimately every type of macroscopic object that surrounds you, are actually waves.
Confusing right? Everything is constructed out of quantum mechanical wave functions. Even more confusingly, these fields are continuously acted upon by mathematical objects called symmetry operators - whatever that means for now - a process which truly embodies this topic’s kinship with geometry.
We are going to teach you the gritty details of the Standard Model and the nuance of its geometric implications in one hour using simple language, schematic mathematical detail, repetition, and assistance from one of the best to ever do it: Dr. Leonard Susskind. Some teach string theory. He helped found it. We weave some of his most insightful lecture segments, along with snippets from a few others, into our broader storyline.
“All you need are fields. That should puzzle you a little bit. How can it be… I’m telling you… and I’m not going to back off of this later… this podium right in front of me is made of fields. Little vibrating numbers, at every point of space, are what make up this podium, are what make up me, are what make up you. But we feel solid, how could it be that something as tangible and solid as ourselves, or the earth, can be made up of vibrations, or oscillations, or gradients in fields filling space. Well the answer of course is quantum mechanics. Quantum mechanics is what you need, and when you add quantum mechanics to fields, you make quantum field theory.”
Sean Carroll | Particles, Fields and The Future of Physics, 23m 30s | Fermilab, YouTube
“Okay so we've talked about a lot of things so far... fields... quanta of fields... the relationship between fields and particles, or the relationship between fields and their quanta. The quanta are the discrete indivisible units of energy that quantum mechanics implies for waves, and waves of course, are fields. Fields and waves are more or less the same thing.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 1, ~0m | Stanford, YouTube
“Waves and particles are somehow the same thing, and if you want to penetrate that more deeply you got to learn some quantum mechanics. And a place to do it is in my lectures which don't cost anything, they're on the internet.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 1, 50m 35s | Stanford, YouTube
What is known, is virtually inaccessible to the general public.
For what reason? Below are a few biggies.
The deepest insights are conveyed not with words, but with complicated mathematics.
We view the objects around us as singular things vs. things made of many smaller pieces.
The significance of motion and its connections to mathematics go unnoticed.
“To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... if you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
The meaty takeaways from physics are math equations. No way around it. And this is no grade school algebra - more like a decade of complicated coursework and branches of mathematics you didn’t know to exist. Group theory is a particularly important one that we will tackle together. It forms the backbone of the Standard Model.
Despite the chalkboards of physics classrooms being decorated with mathematics, few realize just how interconnected their content and histories are, whether we look to the work of mathematicians like Hamilton and Lagrange guiding the physics community, or physicists like Newton and Einstein building out entire branches of mathematics (calculus and hyperbolic geometry respectively).
“I first heard this remark from my friend Dick Feynman many years ago, that the difference between mathematicians and physicists is (like) the difference between masturbation and sex. I think what he meant is, physicists are interacting with a partner, namely the physical world, and mathematicians are interacting with... (laughter from the class)."
Leonard Susskind | Classical Mechanics Lecture 7, ~1h 38m | Stanford, YouTube
Shortened quote below
“Why does mathematics work? Sometimes I think to myself, how could it not work. The world has some coherence to it... things don't just randomly happen... how do you describe things that don't randomly happen? If they don't randomly happen, you have to have some kind of quantitative framework for explaining what happens. Here's what I would say: Not, why does mathematics work, but why do we need mathematics to explain physics? Why is it so hard to explain physics in the English language? The intuitions and the concepts that we evolved with are not sufficient to understand them. In trying to explain it to people who don't have the mathematical background, we get stuck.”
Leonard Susskind | Why does mathematics work? | Udacity, YouTube
In fact, when it comes to really small stuff, we can’t separate the two subjects. When we zoom in and analyze the smallest structures of which everything is made, we find that we can only describe those objects with mathematics. Words don’t work, though we give them names.
Matter is made of atoms. Atoms are made of elementary particles like quarks and electrons. And those objects are, strictly speaking, quantum mechanical wave functions that satisfy a certain differential wave equation - the Dirac equation.
“Question: Roger (Penrose), how accurately does math describe the physical world?
Answer: Well it is extraordinarily precise, but I think people often find it puzzling that something abstract as mathematics could really describe reality as we understand it…. I mean reality... you think of something like a chair or something you know something made of solid stuff, and then you say... well, what's our best scientific understanding of what that is? Well, you say, it's made of fibers and cells, and so on... and these are made of molecules, and those molecules are made of atoms, those atoms are made out of nuclei and electrons going around, and you say well… what's a nucleus? Then you say... well it's protons and neutrons, and they're held together by things called gluons… and neutrons and protons are made of things called quarks, and so on. And then you say, well what is an electron? And what's a quark? And at that stage, the best you can do is to describe some mathematical structure... you say, they're things that satisfy the Dirac equation, or something like that… which you can't understand what that means, without mathematics.”
Roger Penrose | Is Mathematics Invented or Discovered? | Closer to the Truth, YouTube | Note: Shortened quote
There is a very interesting phenomenon that occurs naturally in both physics and mathematics. It has to do with complicated objects which can be represented as a bunch of simpler objects. There are different kinds depending on the object and the particulars. If you see words like decomposition or expansion or series, you know you’re barreling towards an example.
Quantum fields are big complicated objects. You could say that they are constructed out of lots of elementary particles, or you could say that they break down into lots of elementary particles. In math speak this is called Fourier decomposition.
Think about the implications of that for a moment: We often take for granted that everything around us, including us, is comprised of systems of tons of tiny pieces in perpetual motion.
“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.”
Richard Feynman | The Feynman Lectures on Physics Chapter 1: Atoms in Motion | CalTech
You can’t see all of this complexity because you are so darn big. Big things lose sight of the smaller details and rely on macroscopic constructs to interpret the behaviors of systems of tons of tiny objects.
An empty room is filled with tons of particles floating in the air, but it is approximately empty to you. The particles are too small for your attention. When all of the particles in the air start to move and vibrate more rapidly, you might say the room has gotten hotter - a convenient macroscopic construct to summarize the notion of tons of particles now jiggling faster than they were before.
Particles of one kind of configuration enter your nose and you’ll smell one thing. Particles of a different kind of configuration enter your nose and you’ll smell a different thing. Smell is your way of interpreting various kinds of particle configurations.
If the air is made to expand (i.e. particles in the air move further apart) and compress (i.e. particles in the air pack closer together) in some particular pattern, the particles will vibrate your ear drum in a particular way and you'll hear one thing. If the air is made to expand and compress in a different kind of pattern, you’ll hear a different thing.
Water appears to be a single continuous phenomenon. Zoom in closely and you will find it to be made of H20 molecules moving around, pushing and pulling on each other in ways which roll up to the aggregate motion or flow of macroscopic water. Each molecule is constructed of two hydrogen atoms and an oxygen atom bound together, and each individual atom is constructed of electrons and quarks.
Elementary particles are incomprehensibly tiny. You are huge - composed of trillions of cells, each of which is composed of trillions of atoms - and lose sight of such intricacies.
Things in our universe move in mathematically meaningful ways. The trajectories that both small things and big things take in space, exhibit clear structure. You take advantage of this all of the time.
Obama wants to throw you a football. You want to catch it. That’d be pretty difficult if the trajectory taken by the ball was a random and chaotic one. You need to be able to predict where it will be in the future. Lucky for your game of catch, the path isn’t arbitrary. Obviously Obama’s arm plays a role in this movie, but gravity always yanks the ball down using the same rules. The path looks like a simple symmetric function that you’ve almost certainly seen before.
The laws of physics prefer that we take certain paths and not others, and make sure we do so. This includes footballs and the tiny things which constitute footballs in aggregate. The tiny things don’t just move in any ol’ way, and their mathematically meaningful motion rolls up and averages out to the purposeful motion of big stuff. We call the motion of big stuff, classical physics, and we call the motion of small stuff, quantum physics.
The structured motion of big stuff under the influence of gravity, like planets, stars, and apples, caught Newton’s attention in the 17th century. He wrote down equations which captured that structure in his book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). They’re called Newton’s equations. He could prove it to you by predicting the future. He could tell you what an object’s trajectory would be and thus the future location of an object.
He tackled one part of the puzzle, matter’s motion in gravity. Despite the puzzle being larger, we always find ourselves coming back to equations of motion. When we get to quantum mechanical wave functions and move away from big classical objects, we still consider equations of motion.
Newton’s equations are differential equations. These are equations which have things called derivatives in them. More on those later - the key takeaways for now:
All objects, classical or quantum, obey differential equations of motion.
These equations encapsulate the motion of objects and when solved, give you their motion.
They are time evolution equations. This is calculus and it is no coincidence that Newton discovered the subject while looking at planets and apples. Calculus gets a bad rep from bad high school teachers. It’s a deeper and more interesting subject than one might realize. One could say that physics is all about solving differential equations.
“Question: Current physics seems to be concerned with sophisticated mathematics like Riemannian manifolds, self adjoint operators, path integrals over anti-commuting spaces, but wouldn’t it be fair to say that most of this higher mathematics boils down to differential equations, even if we are dealing with elementary particles or with the entire universe?
Answer: It almost always has to do with differential equations. Let me take back one phrase: 'Almost'. It 'always' has to do with differential equations, yes. It maybe that other mathematical things are more important than the differential equations in this particular area or that particular area, but it always has to do with differential equations.”
Leonard Susskind | Physics Is Differential Equations | Udacity, YouTube
Things are normally kicked off in classical physics. We are big classical objects surrounded by big classical objects, so it is often easier to understand. There are roughly four objects to worry about classically. Gravity or the classical gravitational field, light or the classical electromagnetic field, and two kinds of matter, charged and uncharged. Charged matter interacts with light and uncharged matter does not. All matter interacts with gravity.
Gravity is confusingly, space and time itself. One might imagine gravity as an infinitely expansive ocean, with light and matter sitting in it, or on top of it. Let’s go very matter-centric and just imagine both the gravitational and electromagnetic fields as overlapping, simultaneous seas that host us, the matter objects that swim in them.
Matter, light, and gravity, all move, and can affect each other's movement. Physics is all about motion. As fish move through the ocean, they cause the ocean to ripple and flow. If you jiggle the gravitational or electromagnetic fields, you get gravitational or electromagnetic waves.
Make the electromagnetic field wiggle at one rate and you might see color or visible light waves. Make it wiggle a touch faster and you might see another color. Make it wiggle more slowly and you might see nothing. Your eyes only have the chemistry to react with light waves in a narrow frequency window. When the electromagnetic field wiggles very rapidly, we might call its oscillations x-rays if the frequency ends up in a certain range. When it wiggles very slowly, we might find ourselves with radio waves.
The opposite is also true, the ocean can move, like a strong current for instance, and cause the fish to move, or work to prevent movement. Fish can affect the motion or dynamics of the thing which it considers space, and the surrounding space can affect the motion of the fish. Radio towers wiggle the electromagnetic field in certain ways, electromagnetic waves propagate out which affect the matter in your car in certain ways, and ta-da, we’re jamming. We use this setup to send signals around.
Because these two seas can affect our motion, we seem them as force fields. A field is something which takes on a value at every point in space. A force field is a field that takes on a value at every point of space and exerts force on stuff. Forces accelerate us. To accelerate means, to change our velocity, i.e. the speed at which we are moving and/or the direction in which we are traveling. Force fields are like our puppeteers. Gravity for instance, keeps the moon circling earth, the earth circling the sun, and the sun circling the black hole at the center of the Milky Way galaxy (strictly speaking, this also stems from the combined gravitational field of all of the stars orbiting about in the Milky Way galaxy). It also keeps us strapped down to earth, but not much else. Light makes magnets stick to refrigerators, or electrons circle nuclei. Electrons circling nuclei = atoms, so obviously the electromagnetic field’s job as a puppeteer is incredibly important. Life would be tricky without atoms.
Matter interacts with and affects the motion of classical force fields, and classical force fields interact with and affect the motion of matter. When the motion of one thing affects the motion of another thing, we say that their equations of motion are coupled. We end up with ~four sets of classical equations of motion which reflect the interactions between the objects, one for the electromagnetic field (Maxwell's equations), one for the gravitational field (Einstein's equations), one for matter under the influence of gravity (Newton's equations), and one for charged matter under the influence of the electromagnetic field (Lorentz adjustment to Newton's equations).
Quantum physics comes along and says hey, there is actually a collection of space (and time) filling quantum fields. They are infinitely expansive but decompose into tiny quanta that you might see hanging around particular places. A field takes on a value at every point of space. If its value is zero, no particles. If the field is excited, particles.
Those particles are roughly, probability waves. Get a lot of quantum waves moving around and you can get classical stuff where the probabilistic nature of quantum mechanics is averaged over. One kind of wave results in classical matter. The other rolls up to classical force fields. A classical light wave is not the same thing as a quantum one, but is a wave because it is a collection of quantum ones.
They, or the fields which they constitute, have equations of motion. Depending on the context, sometimes it's called the Schrödinger equation, sometimes it’s called the Dirac equation, sometimes it’s called the Klein-Gordon equation, or sometimes it's just called generally, the Hamiltonian. Other times we talk about an object called the Lagrangian, which generates the Hamiltonian. Don’t worry about all of the variations for now.
In quantum physics, all derivatives become operators, or concrete actions on wave functions (or fields) which do stuff to them, like shift them over in space or shift them over in time. The time evolution equation becomes the time evolution operator. These linear operators are also called symmetries. Symmetry operators are the mathematical objects one chats in group theory and that’s where we find ourselves exploring some very profound corners of geometry.
“A very important concept in particle physics, are the field equations of course. The equations of motion of the fields describing the particles. We’ve described boson fields, fermion fields, the electromagnetic field… and all of these fields, satisfy equations. The form of the equations is more or less that of wave equations, and it involves derivatives of the field with respect to space, with respect to time, and the field itself. It involves the undifferentiated field itself and derivatives of the field. How do we code these equations? We could write down the equation of motion for the electromagnetic field. We could write it down for the electron field, all of the fields in the system, and that would just be the equations of motion. But in standard classical mechanical system, field theory, even quantum field theory, the equations of motion always come from a Lagrangian. It is a very important concept in classical mechanics, in quantum mechanics, and in quantum field theory. A Lagrangian is an object that you do certain things to, to generate all of the equations of motion of the system, not just one by one the equations of motion for each field, but it contains in one simple expression, one condensed, compact expression, all of the equations of motion of a system of fields, or a system of degrees of freedom in general.”
Dr. Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 9, ~0m | Stanford, YouTube