Quantum Invariance & The Origin of The Standard Model

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Matt O'Dowd (1:02)

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 could 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. Okay, 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 wave 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.

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College football is back. Sir Hilton called on me, the superstition concierge, to make your fan rituals a reality. Need a room to match your lucky number? We got you. Want to make sure our team doesn't wash your lucky jersey? Oh, that smells lucky.