This "USELESS" Equation is The Mathematical Basis of ALL MATTER!

Support us and talk to Arvin on Patreon: https://www.patreon.com/arvinash BACKGROUND REFERENCE VIDEOS: Quantum Field Theory Explained: https://youtu.be/eoStndCzFhg Quantum Electrodynamics Explained: https://youtu.be/PutOOpAkjQ4 How Symmetries Work: https://youtu.be/paQLJKtiAEE CHAPTERS 0:00 Model the universe starting with nothing 0:54 What's a quantum field? 2:12 The Dirac Lagrangian 4:39 Gauge principle: demanding U1 symmetry 6:49 Demanding local symmetry 8:10 Photon field allows equation to obey local symmetry 10:17 Quantum Electrodynamics (QED) results SUMMARY The Dirac equation holds the secrets to everything that we can see in the universe. In this video, I show how with this equation, we can start with a completely empty universe, make just one assumption, ask just one simple question, and from that derive the entire theory of matter. We build a universe starting with a completely empty spacetime. I introduce the gauge principle, which is a physics principle that is the basis for all fundamental interactions in quantum field theory. This is the theory that describes all matter and all forces in the universe, except gravity. So now imagine an empty universe with nothing in it. Let’s now add a field that permeates the entire universe. A quantum field is not made of anything. It is a mathematical concept. You can think of it simply as properties in spacetime. We can model such a field with the Dirac Lagrangian, which is an equation describing all fermion fields, including our electron field. The Lagrangian simply describes the difference between a system’s kinetic and potential energy. It captures how particles move in spacetime. You'll notice an imaginary term. It is there because we are describing wave functions which have complex components. The “i” is necessary to describe them mathematically. It is a mathematical convenience. The “i” doesn’t represent anything physical. What is fascinating about this equation is how useless it is, yet it’s fundamental. It’s useless because it doesn’t describe anything measurable. It just describes a field with no interactions. Without interactions we can’t measure anything. But it’s fundamental because this is the best thing we have that correctly describes how matter works. #symmetry #quantummechanics The equation describes the electron field with two excitations, that of an electron and that of the antielectron. An excitation is simply like a bundle of energy in the field corresponding to the energy of the particle. And this field spans all of spacetime. So very simply put, this is describing an electron with a mass m which can move in spacetime. To make this equation come to life we have to introduce the gauge principle. To do that we have to make a demand to this equation. The demand we’re going to make is that this equation must have a symmetry like that of a circle with a radius of one. Basically, what we are saying is that if we rotate the entire field in any direction, it should act the same. Nothing should change. After doing the math, we find that the equation obeys the symmetry of our circle globally. In other words, if we rotate the whole field spanning the entire universe by any degree, it won’t change. This is called U1 symmetry, which this Lagrangian obeys. But what if we demand that this symmetry be valid locally at some arbitrary spacetime coordinate x. In other words, we are asking whether the equation changes if we rotate the field at some confined location, rather than globally. It turns out that the solution to regain symmetry is to introduce a new field, a force field - the photon field! If we go through the steps of the math, we can demonstrate that we can have local symmetry of the electron field if we introduce a new force field mediated by photons. And we now have an interacting theory where the extra term we were forced to add is the QED interaction. This is the basis Quantum ElectroDynamics, which is the theory of the electromagnetic force. It shows us how electromagnetism interacts with anything that has an electric charge. This is THE interaction of electromagnetism responsible for everything you see. because you need photons to see with your eyes. You feel electromagnetic interactions when something touches your skin. Electromagnetism is the way you experience the world, and it is how we measure basically anything. Without this interaction, the universe wouldn’t be much of anything to us.