Everything you’ve ever held is buzzing with activity—electrons, atoms, and photons are all vibrating, interacting with each other, and generally just making a mess of things, and creating what you can see, smell, taste, touch, and hear. People have always wanted to understand what they interact with, and there’s a great tendency towards what philosophers call reductionism, where scientists ask what something is made of, what the things that create that thing are made of, so on and so forth ad nauseum until eventually people arrive at the thing, the most fundamental object in the universe. You can see this sort of reductionism everywhere in physics if you know where to look—it’s particularly present in the kinds of physics you see on TV, namely high energy physics (string theory in particular).

Reductionism has been successful to a point. We would never know about the atom’s structure if people never asked what the atom was made out of—people would keep assuming it’s an unbreakable sphere, or worse, like Ernst Mach, nonexistent. However, we can never know how something behaves simply by knowing what it’s made of. I can look at a sandwich with condiments, vegetables, and meat, and say “ah-ha, I know exactly how this will taste because I know what all of its ingredients taste like!” Sure, you can know how ingredients taste, but do you know how they’ll work together? Do you know how, say, juices from a bit of chicken will interact with some tomatoes until you take a bite? Of course not. Even if you know about how a sandwich tastes, that’s still not giving you all the necessary information about that sandwich.¹

Going on with the food metaphor, I’ll give you an example: The other day, my mom and dad made some really good hamburgers—they went all out. My mom made some caramelized onions, and a really good, spicy barbecue sauce; my dad made homemade buns and grilled the burgers. When I first sunk my teeth into one of those burgers, I was amazed how the sauce was, sadly, not as spicy as it was on its own, or how the savory bacon and sweet caramelized onions had a sort of perfect harmony unknown to man—in short, while all the ingredients are lovely on their own, together they created the kind of burger that would make a 1950s film noir detective shed not one, but two tears.

Everything—at least when it comes to behavior, motion, and all that stuff—is more than what composes it.² Take for example a living organism, which is, technically speaking, just a bunch of cells—what makes life possible, more than the existence of chemicals, are those chemicals interacting with each other. Life didn’t simply start because one day a molecule decided to prank god—life is an emergent process, started from billions of years’ worth of chemistry, physics, and chance.

Much in that same way, it’s impossible know how a material behaves simply by knowing what it’s made of. If I scream “molybdenum disulfide” are any of you really going to know its optical or electrical properties? I’d certainly hope not. (If you can do that, what the hell are you doing reading this? Go get a Nobel prize or something.) We need some actual physics to explain what’s going on, like its optical or electrical properties—of course, since physics isn’t a monolithic entity, there are different branches to look at: you have high-energy physics, which studies particles at extremely high energies (that doesn’t fit the bill); you have optics, which looks at how works, which is good enough, but doesn’t explain material properties; then you have condensed matter physics, which looks at how matter is assembled, and explains/predicts its properties based off how it comes together—if we’re trying to predict how a material behaves, I think that’s our best bet.

Condensed matter physics is the single largest branch of physics, and there are tons of scientists studying it. This is often credited to condensed matter physics being really useful—because studying solids in the early 20th century lead to our electronics and photonics industries, paving the way for our modern era—but, in my opinion, condensed matter is also huge because of just how much stuff there is to cover. Because of how easy it is to study solids, liquids, and other phases of matter relative to, say, cosmic waves, it’s easy to find out new things with experiments, and easy to come up with new theories given all that data.

Believe it or not, condensed matter physics is actually quite fractured. There are a lot of different little subfields that share common themes and techniques, all interwoven together. There are a lot of people studying superconductivity, for example, but there are also tons of people studying systems where electrons strongly interact with each other, then you have people studying semiconductor devices, or how electrons flow through materials, you have people studying liquid crystals, molecular solids, organic electronics, and so much more—many fields of physics are like this, but condensed matter is so split apart because there is just so much stuff to cover.

In order to understand anything other than the bare minimum about a material, we need to understand how its atoms and electrons interact, how photons interact with that material, and more—there’s a lot of stuff to take into account. If you want to learn everything there is to know about a material, you could, hypothetically, write down a long Hamiltonian—a statement of a system’s kinetic and potential energy—and solve tons of equations for a bunch of particles at the same time, be my guest—though be warned that no one has done that in the past 100 years. If you want to do calculations for a realistic system, you need to calculate the properties of at least \(10^{23}\) atoms all interacting with each other.

So, there’s much ado about nothing. We can’t realistically know the properties of a full system of interacting particles, but we need to know how they all interact in order to understand the system! What are we gonna do? There’s gotta be a secret solution around here. Simple, siis—work smarter, not harder. We must accept two things: first, as lousy physicists, we must accept defeat, and acknowledge what we’re trying to do is hard; second, as discussed with sandwiches earlier, more is different³—knowing what makes a system doesn’t really tell us everything about it.

So, since more is different, we need a few ways to calculate what goes on in a many-body system—there are a few methods, with some of them more theoretically interesting than others. I’m going to talk about a lot of things in this series. My next post will be about ab initio (literally “from the beginning”) methods—because I think they’re a good jumping-off point for the many-body problem since they’re focused on doing things from the ground up—and there’s a lot of interesting physics to talk about. After I get all the ab initio stuff out of the way, I’ll talk about some of the other, more abstract methods that physicists use.