Superconductivity and Quantum Field Theory

Last year, the discovery of a supposed room temperature superconductor blew everyone's minds. That didn't work out, but the math behind superconductors is very interesting and a grand tour of 20th century physics. Let’s find out how it works, with equations. Even the Higgs mechanism, of Higgs boson (or "god particle") fame, makes an appearance, but in a different way than usual.

Here is what we need:

1 - From quantum mechanics to quantum field theory (QFT)

2 - Fields

3 - Lagrangians

4 - Symmetries

5 - Spontaneous symmetry breaking

6 - Gauge fields

7 - The Higgs mechanism

8 - Theories of superconductivity

9 - The Meissner effect

The Schr?dinger Equation

First, superconductivity is a quantum mechanical effect. The fundamental equation in quantum mechanics is the Schr?dinger equation. The Schr?dinger equation itself doesn’t tell you anything about what a particular quantum mechanical particle (like an electron) does - it is effectively just a “scaffolding” equation which requires you to replace its components with other equations. If you want to see how an electron behaves in the orbit of an atom, for example, you pick an equation that describes the electron’s movement, and if you plug it into the Schr?dinger equation and it “solves” the equation, then you know it’s a description allowed by the underlying physics.

But the Schr?dinger equation doesn’t work for our purpose here because of a big drawback: it effectively tracks just one particle. Below is the equation: the x in the equation is the position of one particle.

But superconductivity is about the interaction of many particles (as it turns out, pairs of electrons). So we need something like the Schr?dinger equation, but something that allows us to put a solution for many interacting particles into it, and see if it fits. This step takes us from quantum mechanics into quantum field theory.

Fields

We do that by “promoting” the x in the Schr?dinger equation towards giving us information about all positions in a universe, rather than just the position of one particle. Remember that the essence of quantum mechanics is that it is “quantized” - nature is discrete, not continuous, and allows only certain values of particles, and merely puts probabilities on those. For example, an electron around an atomic nucleus can only be in certain orbits around it, which is the reason why atoms only emit light in very particular frequencies, when excited. Those orbits all represent movement equations that solve the Schr?dinger equation. So the Schr?dinger equation itself isn’t “quantized”. Instead, it just allows for solutions that are: its solutions are waves, and waves have defined, hard, discrete (not continuous) frequencies. So now we use the same recipe, but for every position in space.

In a quantum field theory (QFT) equation, the x from the Schr?dinger equation goes from being the location in one wave or of one particle, to all locations in space. At every x, we store a value. That’s why it’s called field theory: if you have a value for every x in space, you are describing a field. To be more precise, each location x in QFT becomes not just a three-dimensional position, but a four-dimensional spacetime location: it is a vector of x, y, z and time t. (It is four-dimensional, not three-dimensional, because QFT is relativistic: it accurately respects that nothing can be faster than light, so we have to adjust both time and space when considering where we are. By always talking about four-dimensional spacetime positions that include negative time as a coordinate, we can do that.) If we pretend for a moment that we live in a 2-dimensional world (coordinates x and y), then the image below shows three different kinds of fields that we will encounter shortly: a real scalar field (store one number at each 2D field location), a complex scalar field (store one complex number, i.e., a number that consists of a real and an imaginary component, at each 2D field location), and a vector field (store one 3D direction at each 2D field location).

A real scalar field just has a value but no direction in each location:

A complex scalar field has a two-dimensional direction in each location (because one complex number has a real part, and an imaginary part):

A vector field has a three-dimensional direction and length in each location:

Lagrangians & Equations of Motion

So what is the equivalent to the Schr?dinger equation (from quantum mechanics) in quantum field theory? That is where we come across the so-called Lagrangian. A Lagrangian is actually a concept from classical (non-relativistic, non-quantum) mechanics: it describes a particular system you’ve set up (like, a weight dangling off a spring), and you can now start plugging equations into it to see if those solve your Lagrangian, and thus accurately describe the evolution of your physical system over time. More precisely, a physical system’s Lagrangian describes the math of the evolution of the system’s total energy. But the system’s equations of motion can get derived from the Lagrangian directly. In other words, by setting up an equation of the system's total energy, we can derive all of the system's behavior from that equation.

Here is a very simple example for that. In classical mechanics, Newton’s second law states that the force on an object is equal to its mass times its acceleration: F = m x a. That is an equation of motion, because it would describe how an object moves. You can derive it straightforwardly from a Lagrangian, by using something called the Euler-Lagrange equation. The image below shows how.

Back to quantum field theory. So, a Lagrangian describes the state of a physical system, more precisely its energy. The equation below is the Lagrangian for the so-called “free scalar field”. A scalar field simply means: we have one value for each spacetime position. (A scalar is a vector with one component - i.e., one value.) The “free” part means that this field doesn’t interact with anything. But as short as this equation looks, this is really a completely valid field theory. It looks much more complicated than it is. That Φ is the field. Like in the Schr?dinger equation, this Lagrangian is just a “scaffolding”: we’re simply stating what would constitute a valid field, but now we have to actually plug in an equation for such a field. So the Φ will be an equation that we’re going to pick to describe our field, and when we put it into the Lagrangian, it will need to solve the equation. The ? is the partial derivative operator: it means you take the derivative of the field equation Φ. The μ on the partial derivative operator has a very particular meaning (Einstein himself came up with it): it means, run μ from 0 to 3, take the derivative in each of the 4 directions of spacetime (x, y, z and t), and add it all up.

For example, let's say we just have a one-dimensional field in the x-direction, and Φ(x,t) = x * t. Then that partial derivative operator would just work out to: ?μ Φ(x,t) = ?μ (x * t) = ?x (x * t) + ?t (x * t) = t + x. You can see how the μ gets replaced with all spacetime directions, and each derivative gets added up.

So this Lagrangian describes a free scalar field. Which means, just like we did with Newton’s second law above, we can take this Lagrangian, run it through the Euler-Lagrange equation, and we get an equation of motion - for the field. Below is that equation of motion for the free scalar field: it is called the Klein-Gordon equation.

The Klein-Gordon equation is the equivalent of F = ma for free, scalar, real (one-number) particles. Why particles? Right now, that is more or less just an interpretation: we simply interpret any excitation of this field (any non-zero value) as a particle traveling somewhere.

Quantization

But, none of this is actually quantum mechanical, nor quantum field theory. If you look at the Klein-Gordon equation and its Lagrangian, there is nothing in there that seems to force it to allow only discrete wave solutions. We need to turn this from a classical equation into a quantum mechanical one. Because particles are no longer dots in space, but smeared out equations (waves, or functions), we need to replace “hard” values like a particle’s position and momentum by operators. Where classical equations deal with numbers and arithmetic operators that act on them, quantum mechanical equations deal with functions and functional operators that act on them.

So we use the same trick that enabled the derivation of the Schr?dinger equation: we replace fields by operators. In Schr?dinger, that meant: replace momentum p by - i h ?/?x. Here, it means: replace the field Φ by operator-Φ, and putting other constraints onto the field. Then the solutions that the field admits will turn out to be waves rippling across the field, and those waves’ parameters will be the “quantized” nature of the field. In fact, those field excitations are now much more obviously particles, just like particles in our universe show up as waves too.

What does an actual permitted solution for a free scalar field look like? Here is a solution for the one-dimensional, free, scalar, real field: the expectation value of the quantum field operator at position x, for a particle with momentum k. This by the way is a great example of the famous uncertainty principle: this equation exactly nails down the particle’s momentum k (meaning, you can pick a discrete number and fill it in). But you have no idea of the particle’s location x: it’s completely smeared out through space x, it’s literally everywhere (all x work, when you plug them into the equation).

Now, important for our next step: that square root term is the particle's energy.

Try k = 0: you see a particle at rest with momentum 0 and with mass m. Go back all the way, to our field Lagrangian: the mass m shows up in front of the square of the field is that mass m. Turns out this works well: the square field term in the Lagrangian describes the field’s particle’s mass.

That's actually it, that’s our universe: all particles in the universe, and all forces, are excitations in these kinds of fields, more specifically, quantum fields. The "quantum" part means that these fields' equations only permit excitations that are multiples of a particular "quanta" of energy, not just any random number. There are several types of quantum fields overlapping and interacting in our universe. Each field has its own Lagrangian, but the Lagrangians have cross-terms that couple the fields to each other.

Even more elegant: remember how a field is a thing with a number for each spacetime location. Instead of a number, we can also store two or more numbers at each location. Turns out: the Higgs field has one number, the electron field two, the photon field three (more or less). Or rather, the Higgs field is a scalar field, the electron field is a spinor field (described by the Dirac equation), and the photon field is a vector field. Each of these fields has their own Lagrangian.

The Standard Model of physics describes all known particles and forces in the world: the electromagnetic force (which includes photons and thus light), the strong force (which binds protons and neutrons together in atoms), and the weak force (which causes beta decay and makes the sun burn). It's missing gravity which has its own theory (general relativity). Here is the full Lagrangian of the Standard Model. It has Lagrangians for all particles and all interactions. Φ is the Higgs field, A the electromagnetic potential, one of the e is the electron field, etc. The strong nuclear force and the weak force and their associated particles are in there too, with their own Lagrangians and coupling terms. Now that you know how to read a Lagrangian, you can decipher some of the components of the Standard Model. You see lots of derivative operators, all the different fields, and the small-cap coupling constants that tie the different fields together.

This is the halfway point to superconductivity. In short: particles and forces are described by fields, fields and solutions are quantized, and a Lagrangian encodes each field.

Symmetry

We need three more tools: symmetry, symmetry breaking, and gauge fields. We start with symmetry. That is literally your elementary school symmetry: if you flip a rectangle by its axes, it remains the same. If you rotate it, it doesn’t. If you rotate a square by 90 degrees, it remains the same, but not if you rotate it by 91 degrees. But you can rotate a circle by any amount, and it will stay the same.

This is hugely important in our universe. The simple reason is that it would just be weird if nature had preferences - i.e., asymmetries. The laws of the universe should be independent from the context from which you’re viewing them. (Otherwise you could make quasi-religious statements: there would have to be “special” directions or locations.) Come to think of it, it is actually supremely strange that humans can even distinguish left from right, and that human bodies actually have systematic asymmetries (our heart is on our left side). Why would one direction or side be so special that you could distinguish it systematically from the other? Similarly, DNA is systematically right-handed, which is also a systematic asymmetry. It does turn out that one of the four fundamental forces of physics, the weak force, has a systematic asymmetry: electrons created in the subatomic process known as β decay are always 'left-handed', and sometimes people speculate whether all these asymmetries are connected.

But anyway, we generally want our mathematical equations that describe nature to have symmetries, because if a theory was inherently asymmetric, you would have to explain where the asymmetry comes from. If there is one overarching principle to all laws of physics, it is that nature appears to strive for simplicity: no more than necessary, and always seeking out the lowest energy state possible.

That has an unexpected consequence. There is a mathematical theorem called Noether’s theorem which states that if a Lagrangian has a symmetry, then that symmetry must come with some quantity that is being conserved. For example, everyone knows that energy is always conserved. We might think that this is one of the universe’s foundational principles. It is not: the foundational principle is that the Lagrangian of a system that is symmetric in TIME must imply the conservation of energy.

Meaning: take any Lagrangian describing a physical system, replace time t with t + d, and the formulas will work out to be the same - they’re time-invariant. And if you then follow Noether’s theorem, that theorem then proves that the system has constant energy. This is, astoundingly, a purely mathematical principle that nature follows exactly. Energy conservation is not a fundamental principle of nature - it's just that any system that is symmetric in time must conserve energy.

In the same vein, Lagrangians that are invariant to shifting them “left or right” in space have “translational invariance” - and their conserved quantity is momentum p. Lagrangians that are invariant to rotations have rotational invariance - and they preserve angular momentum L. Nature wants to be symmetrical, and as a result we have conserved quantities.

Ground States

But back to superconductivity. One of the defining characteristics of superconductors is that they only work below certain temperatures: once you fall below that, electrical resistance disappears, and current flows smoothly. (That was the whole discussion about the supposed superconductor LK-99: it was rumored to function even at room temperature.) An important observation is that this potentially sounds strange given what we just said, that nature wants to be symmetrical: apparently here nature isn’t - apparently the laws of physics do change below a certain temperature in a superconductor? Indeed: in a superconductor, symmetry is “broken”. Which in and of itself is a hugely important principle of our universe: If a Lagrangian describes the state of a physical system, the symmetry of that system shows up in the Lagrangian, and that’s where it can be broken.

Here is an example. We go back to our free, real, scalar field Lagrangian from earlier. But we add a term: Φ^4. Lagrangians are quite “modular”: by adding terms, you’re effectively adding particle interactions. In the simplest, interaction-free Lagrangian, we had the field Φ getting multiplied with itself, and that was it. That Lagrangrian was (interaction-)”free”, so it modeled a field whose particles didn't interact, not even with themselves. Knowing that, we can now guess what the Φ^4 stands for: interactions of the real scalar particles with themselves. This takes us from a free theory to an interaction theory. (That λ in front of the term is just a constant.)

Now take a look at the Lagrangian and see what happens when you replace Φ with -Φ. The answer is, nothing: it will remain the same Lagrangian. Because we only have Φ show up in powers of 2, any sign changes just cancel out, and the Lagrangian remains unchanged. That is a symmetry (called parity: flip the sign, and nothing changes).

Remember that the Lagrangian captures the energy of a physical system. More precisely, it is the difference between kinetic energy T and potential energy V: L = T - V. The kinetic energy above is made up of all the terms that have a spacetime derivative in them (because kinetic energy is about movement). The potential energy are all the other terms. Nature always aspires to be in the “simplest” possible state, and that principle is at work in any physical system in another way too: the system will seek out to be in a ground state that minimizes its potential energy. What would that state look like for our simple Lagrangian here? If we plot the potential for m^2 > 0 and λ > 0, it is a straightforward chart, and it is very obvious that the vacuum ground state is Φ = 0, see below. (See the equation for the potential V in the bottom of the chart: it's just the two terms in the left of the Lagrangian above.) That ground state is the state that the physical system described by the above Lagrangian will always settle into, in steady state and without any outside influence. It's quite logical that this ground state is where the potential is 0 and the field is 0. What else would it be?

Symmetry Breaking

But now consider the same Lagrangian, but with one tiny change: flip the sign in front of the m^2 term to negative. If we follow the same process as above to plot the potential energy, we suddenly have two minima, and they are not at 0. All we did was slightly modify one parameter in the Lagrangian: we made m^2 negative by replacing the sign in front of the mass term. As you can see, if you just plot that function of the potential, we broke the potential's symmetry. Now, if nature wants to minimize the energy in the vacuum and find the ground state to settle into with the lowest possible potential, then it will have suddenly two different field values to choose from, the low points in the valleys either on the left or on the right. But not only that: look at the potential V in those valleys - it is no longer 0. Which means: in a broken symmetry Lagrangian, the vacuum energy is now no longer 0. When the system settles into its lowest energy state, it still has a non-vanishing, constant amplitude when it sits in one of those minima.

We can derive the vacuum state as follows. To find the minima of a function, you just take the function's first derivative and set it to 0. That's very easy to do, and the box below shows how to. Those are exactly the solutions we find in the chart above. (The second derivative tells you if a minimum is a global or local one.)

So we saw that when we break the symmetry of a system, something happens to the system's ground state: the system's lowest energy state now has a non-zero amplitude. But what does that mean from a practical point of view? To understand that, we can place ourselves into the ground state, and calculate a "new" Lagrangian that reflects how the system "looks" from that point of view. The way to do that mathematically is to use a Taylor expansion: that is a simple trick to approximate any function, in a particular point. You do that by taking the function's value in that point, and adding to it the function's first derivative in that point, plus the second derivative, and so on. It makes intuitive sense: these are just higher- and higher-order approximations of what the function looks like in that point. So let's calculate that Taylor expansion of the symmetry-broken Lagrangian, in the location of one of the minima. The box below shows that.

It ends with a grand discovery: if you look for the term that signifies the particle mass in the newly modeled Lagrangian, the mass has changed: that term now has an additional factor of 2 in front of it, compared to the original Lagrangian. It looks like when we break the symmetry of a field, the field's particles take on a different rest mass! This makes sense, because the vacuum state of our field now has this constant, non-vanishing amplitude, so it is only natural that the field's particles have a different mass. Symmetry breaking affects particle mass.

We will take that a step further and look at yet another Lagrangian: that of a two-component field. These could be two particles that interact with each other. See the two Lagrangians in the screenshots below. The first one is the one we’ve been looking at, for the real scalar field. The second one has two fields, but otherwise exactly the same terms for each field: the kinetic energy term for each field Φ1 and Φ2 (with the two derivative operators), the potential energy mass term (with a different mass for each field’s particle), and the field’s self-interaction term (with Φ^4). It has one extra term: the two fields interacting with each other (which you can tell from the fact that it’s also in the order of a fourth power, i.e., like the field self-interacting terms).

What is the vacuum or ground state of this system? We now have two field variables, Φ1 and Φ2, so we have to plot the potential energy (all the terms without a derivative in front of them) on a two-dimensional grid. The image below shows that potential. This is called a “Mexican hat” potential, and you can see why: if we want to minimize the energy, we can pick any of the infinite number of positions that put us into the “valley” of the Mexican hat. Previously, the system's ground state was just the (0,0) point. But that point is now just a local minimum. The global minima - i.e., the possible ground states - are all in that circular valley. All of those positions - each a combination of a field value for Φ1 and Φ2 - plainly occupy the lowest points in the graph, i.e., the points of lowest energy.

Now we ask again: if nature now picks one of these minima (in the valley of the Mexican hat), what does the field “look like” from that vantage point? To find out mathematically, we again use a Taylor expansion: to model any kind of function f(x) at any location x0, we can take the function f, calculate its zeroth, first, second, etc. derivative and add them all together. The image below does exactly that for the two-component field from above.

We end up with an extremely interesting Lagrangian. It is extremely interesting because: the mass of the second particle has disappeared. You can see a mass parameter m in front of the quadratic term for Φ1’ = Φ1 - Φ10 (i.e., the field as viewed from right around the minimum Φ10, aka the Mexican hat valley), which is exactly where we would look for mass terms in the Lagrangian. But there is no quadratic term for Φ2’. So this extremely simple math makes the quadratic term for field Φ2 disappear, all by itself. We knew from the previous example that symmetry breaking affects particle mass. But here, the mass of one of the particles disappears entirely!

Intuitively, that is because of how the potential energy looks: if you’re sitting in an energy minimum in the Mexican hat, you can go in two directions. In one direction, you’re going around the hat, and the potential isn’t changing. It is “costless” to move - so there is no mass. In the other direction, you’re climbing up a potential energy hill. It “costs” to move - so there is a mass. So if you put a two-component real scalar field into a state of broken symmetry, the field goes from having two particles with a mass, to one particle with mass and another massless particle.

This was long a matter of great concern for physicists, because this is really an extremely general result. We made no obvious corner-case assumptions here, we just broke the symmetry of a two-particle field. So it looks like interacting fields in broken symmetries automatically produce massless particles. The concern is: we don’t see those massless particles in nature. Any massless field in nature should be everywhere, because it is so easy for its particles to travel (they are massless). But the only massless particle in nature that we know of is the photon, and photons are indeed absolutely everywhere (light, x-rays, gamma rays etc.) But the photon was already accounted for in the theory of quantum electrodynamics. So when physicists tried to use this kind of field theory to explain the other forces of nature - the strong force and the weak force - they would have expected to see these massless particles (called Goldstone bosons), and they weren’t there.

But it does turn out that similar field theories can indeed explain those other forces. It was Higgs who solved that problem with his famous Higgs boson. The key to solving this is another important foundational piece of 20th century physics: gauge fields. It is the last step we need to explain superconductivity.

Gauge Fields

We go back to symmetry. To reiterate, symmetry is important in nature because you don’t want an arbitrary coordinate system that nature needs to pick - the laws of physics should be independent from your context. First, here is a simple example for what’s called global symmetry: if a field has so-called U(1) symmetry, then it means that the system remains unchanged when multiplied with a “phase factor” - i.e., when you rotate the field. Below is the Lagrangian for a similar field to the one we’ve seen above: here, a scalar, free (non-interacting) field; but this time a complex field, not a real one. Complex numbers are those that have two parts, a real part and an imaginary part. So you can think of the field Φ now as a two-dimensional vector, because it has two independent coordinates - the real coordinate and the imaginary coordinate.

Look how similar the Lagrangian is to that of the scalar, free, real field - it still has the twice-differentiation, and it has a quadratic field term with the particle mass in front. The only difference is that you’re now multiplying with a complex-conjugated version of the field, Φ*. Complex conjugation just means you’re flipping the sign on the imaginary part of the complex number. Now, what happens when we substitute this entire field Φ with a version of itself that we multiply with a phase factor (i.e., a term that rotates a complex number by an angle theta)? See the derivation below: the Lagrangian remains exactly the same. So the complex, scalar, free field has global U(1) symmetry built in.

However, we want an even stronger condition: local symmetry. U(1) is a global symmetry: you rotate the entire field, and its energy equation doesn’t change.

But what if you want that phase factor (the rotation angle) to be different at each location in the field? Now that would be a really strong symmetry: even more flexibility, and even less coordinate-system-fixing for nature.

If we’re talking about local coordinate systems, we can’t continue to just look at “scalar” fields, because at any location in the field, those just store one value (see above) - and a single number doesn’t have any direction. We need to be looking at fields that have a direction, i.e., vector fields. In a vector field (also see above), each location in the field stores a vector, i.e., a pointer in a particular direction. The particular symmetry we want is that we can rotate each of those vectors independently, and still retain an overall field symmetry. So in each location in space, we want to have a completely independent coordinate system. That sounds completely impossible, but it is exactly what gauge fields do. You just need one trick to make it happen.

In the example above, we multiplied our field Φ with a phase factor to rotate it. Then we plugged it back into the Lagrangian, which turned out to remain the same. So now we generalize that even more. What we want to define is a field transformation G that says Φ’ = G Φ. Remember how we can view Φ as a two-dimensional vector, because it’s a complex number in each field location. When you multiply a vector with a matrix, you get a rotated vector. So the operator G in this equation will be a matrix. We want to have a matrix G that can depend freely on each location in space - so that at each location in space, we can have a completely independent coordinate system, or rotation. What we want to know now: what would a location-dependent G have to look like?

When starting again with the Lagrangian for a complex, scalar, free field from above, the problem is the derivative ??? Φ: remember that that means “take the derivative of the field Φ into all directions ?? (running from 0 to 3, i.e., all space directions and time) and add that up”. So if we replace Φ with Φ’ = G Φ everywhere, then will ??? Φ’ = ??? (G Φ) = G ??? Φ? I.e., can we just pull out the matrix G from that derivative? For global symmetry yes - the phase factor in our example above doesn’t depend on any coordinates, so it’s just a constant factor, and constants just flow straight through derivatives without affecting them. But now we want G to depend on local coordinates, because we want the freedom to rotate the field differently in every spacetime location. Then it doesn’t work anymore, because then you need to take the derivative also of G in all directions, and moreover you have to use the product rule for derivatives, so ??? (G Φ) = (??? G) Φ + G ??? Φ. That sure looks very different and doesn’t seem like it would leave the Lagrangian unchanged.

The trick is to define a new derivative operator D?? in the following way: D?? Φ’ = G D?? Φ. Then, the transformation matrix G passes right through that derivative operator. That means we can have G depend on the field location, and it would STILL work - the new derivative operator wouldn’t affect G. It turns out that if we define such a new derivative operator and do some math, it HAS to be of the following form: D?? = ??? - i g A??. In other words, we cannot get away without introducing something new to the equation: another field A. What is A? It is an entirely new field: a vector field that “couples” to the scalar field Φ with a coupling constant g. (That derivative operator D is called the covariant derivative.) The image below shows what happens when you define the operator D and put it into the Lagrangian. As you can see, we successfully recover the original Lagrangian - meaning, it is actually the case that we now have a “different rotation at each location in space”-invariant Lagrangian.

Intuitively, what is happening here is this: because we introduced the requirement that the field’s Lagrangian not change however we rotate differently in each local point of it, we need to also introduce another field that’s counteracting all those local changes, in order for the whole thing to remain the same. So, we can’t get to local symmetry if we don’t ALSO introduce this new field A. This is “gauge invariance”: a Lagrangian for a field that has an independent coordinate system in each of its locations. If you want to make a Lagrangian gauge-invariant (i.e., giving it complete freedom to have any kind of locally rotated coordinate system), you need a new, “compensating” field A. This is not even really physics, it is math. There is just no other way to write down the field equation for a locally symmetric, gauge-invariant field.

In summary: we started with a scalar, complex field that describes some kind of particle. We want to make it gauge-invariant, and the only way to do that is to introduce another field, A. That raises the question: what is that new field A? It turns out: this is the electromagnetic field - i.e., the field that describes photons. So, simply by starting with a simple field and making it gauge-invariant, the theory of electromagnetism pops out. The image below shows how to create the Lagrangian for a charged, gauge-invariant field with electromagnetic forces. It is the same Lagrangian of the complex, free, scalar field made gauge-invariant by introducing the covariant derivative D - but then we add another term, which is intended to capture the potential energy from the new field A’s self-interactions. (Remember, the Lagrangian is very modular, we can just add terms for new field interactions that we want to capture to make it more precise.) If you try to multiply this field with a local phase factor, as we did in the derivation above, you will see that this Lagrangian also doesn’t change - it is gauge-invariant.

It is almost strange that a field with local rotations can get defined, but it is possible and works out well. This all still seems very abstract, because we started with a weird complex scalar field that describes some simplistic particle for which we don’t really know what it is. But here is how this exact principle showed up in reality: in 1928, Paul Dirac figured out how to write down an equation that describes the behavior of electrons, the famous Dirac equation. That equation is a Lagrangian that worked perfectly in describing the electron field, whose excitations turn out to be electrons. But it wasn’t a gauge-invariant equation, it just had simple global symmetry, not the local symmetry of gauge invariance. But if you follow the above process and introduce a covariant derivative to make it gauge-invariant, you have to introduce a new field A, as we saw above - entirely because math says so. And it then turns out that that field A is the electromagnetic potential field - i.e., the field that describes the photon. Incredibly, the coupling constant g that seemingly came out of nowhere in our derivation above turns out to be the electromagnetic charge q. So, nature, in the process of making itself gauge-invariant and because it already has the electron flying around, creates the electromagnetic field to enable itself to have local symmetry. And that is why electrons interact with the electromagnetic field, and the strength of that interaction is determined by the electron charge q. It is an incredible closing-of-the-loop of how these seemingly disparate theories hang together. The image below shows the derivation of that. The Dirac equation follows directly from the Schr?dinger equation by making it compatible with the laws of special relativity - and if you then make the equation gauge-invariant, you get electromagnetism for free. Both of these steps are astounding because all they do is impose symmetries: special relativity imposes the symmetries of spacetime, and gauge invariance imposes the symmetry of being able to not have any fixed coordinate system. By the way, look how similar that final QED Lagrangian looks to our Lagrangians above: all the same terms - a kinetic energy term, a mass term, an interaction term between the two fields (the electron field and the photon field), and a self-interaction term for the photon field.

Broken Symmetry In Gauge Fields

Now we have one last step towards superconductivity. Remember how broken symmetries in field theories gave physicists pause because they seemingly required massless particles to show up, and we didn’t see those anywhere. That happened when you broke the symmetry of a Lagrangian for any particular field theory: simply by breaking the symmetry in the equation, you suddenly go from two particles with mass, to just one particle with mass and another massless particle. Higgs had the idea to try something seemingly very simple: when we broke the symmetry of the Lagrangian above, it wasn’t a gauge-invariant Lagrangian. So what happens if you take a gauge-invariant Lagrangian, and you break the symmetry there? The recipe is the same as above: (1) write down the Lagrangian for the field (here, the gauge-invariant Lagrangian); (2) flip the sign in front of the mass term to create a symmetry-broken Lagrangian; (3) calculate the new vacuum ground state of lowest energy for that Lagrangian by taking the first derivative; (4) pick out one vacuum ground state location and use a Taylor expansion on the Lagrangian around that ground state to calculate what the field looks like from that perspective; (5) look at the terms that show up in that final Lagrangian and read off the particles’ masses.

When you do that, you encounter another surprise: before the symmetry breaking, we have two massive scalar particles, plus the massless vector particle (the photon of the electromagnetic field) - and after the symmetry breaking, we have one massive scalar particle, and the photon field has become massive!

So when you break the symmetry of a gauged Lagrangian, you don’t end up with a massless particle - quite the opposite, you suddenly have the previously massless photon field take on mass. That is all that Higgs does in his famous paper of the discovery of the Higgs boson. You can see the whole derivation by typing into ChatGPT: “Show me a gauged Lagrangian of a complex scalar field with self-interactions” and then “Break the symmetry of the Lagrangian”, and you can follow along.

The Superconductivity Lagrangian

This sets up the final step in the mathematical explanation of superconductivity. An important observation is that superconductivity occurs only below a certain temperature: something dramatic happens when the superconductor’s temperature drops below a critical level - electric current moves without resistance, and magnetic fields get expelled from the superconductor. That this all suddenly happens below a certain temperature is a solid giveaway that we are dealing with a broken symmetry: as we saw above, all it took was for one sign in front of a coefficient in a Lagrangian to change, and it dramatically changed its meaning. It seems like we are dealing with something similar in superconductors. Bardeen, Cooper and Schrieffer figured out that the loss of electrical resistance can be explained by the fact that electrons below a critical temperature team up to form Cooper pairs: normally, electrons moving through a conductor scatter off the lattice of ions that make up the conductor, and electrical resistance arises. Also normally, electrons repel each other, because they are negatively charged. But at very low temperatures, electrons suddenly become bound together in pairs. These pairs can move through a crystal lattice without being scattered, and thus with no resistance.

It turns out this can be described with a Lagrangian in field theory. We need a wave function that describes the Cooper pairs, so let’s use a scalar field for that. We are dealing with electromagnetism (somehow, superconductors expel magnetic fields, so they must be involved), so let’s make it gauge-invariant. To make the Lagrangian gauge-invariant, we need a complex field, so let’s upgrade the scalar field from a real to a complex scalar field. That’s as simple as we can make it.

And that is indeed the Ginzburg-Landau Lagrangian, which describes the macroscopic effects of superconductivity. The image below shows how analogous that looks to our simple complex scalar field theory with self-interactions and gauge invariance. Importantly, the coefficients a and b in the Ginzburg-Landau Lagrangian depend on the superconductor temperature!

Putting It All Together

Now to the final mathematical step. We have a gauge-invariant complex field theory. That field theory has a coefficient a in it that depends on temperature. Which means, a can turn negative. That is exactly what happens in a superconductor below its critical temperature. When that happens, we’ve seen the pattern before: it breaks the Lagrangian’s symmetry. And what happens when we break the symmetry of a gauge-invariant Lagrangian? Exactly: the Higgs effect - the gauge field in the Lagrangian takes on a mass. The formerly massless particles making up the gauge field suddenly become massive. What is the gauge field in this Lagrangian? The electromagnetic field. So, the Ginzburg-Landau Lagrangian shows us definitively: when the temperature drops and its symmetry breaks, the photons of the electromagnetic field in the superconductor turn massive. And massive particles can only travel much shorter distances. Light can travel over infinite distances and at light speed simply because photons have no rest mass. If photons take on mass, that ceases to be the case. And that is what explains the Meissner effect! Magnetic fields cannot penetrate superconductors very deeply because their particles, the photons, take on mass inside a superconductor, and that mass slows them down and has them act over much shorter distances. That distance is exactly equivalent to the penetration depth of a magnetic field inside a superconductor, i.e., small.

Summary

So, in summary:

1 - Fields describe nature

2 - Particles are excitations of fields

3 - Lagrangians describe fields

4 - The mass parameter in a Lagrangian describes the mass of the field’s particle

5 - When the symmetry of a scalar field theory breaks, massless particles emerge

6 - But nature is looking for deeper symmetry, so we want gauge-invariant fields

7 - Making a scalar field gauge-invariant requires introducing another field, a massless force field

8 - When the symmetry of a gauge-invariant field breaks, the massless force field particles become massive (the Higgs effect)

9 - The Lagrangian of a superconductor is a gauge-invariant complex scalar field

10 - So when the symmetry of that Lagrangian breaks, the massless force field particles in the superconductor take on mass

11 - Those particles are the photons that make up the electromagnetic field, and massive particles can’t travel very far

12 - That is why magnetic fields can’t travel far inside a superconductor, which is the Meissner effect.