Quantum Field Theory

The Grand Unification epoch: \(10^{-43}\) to \(10^{-35} \text{s}\)

Thus begins the Grand Unification epoch. That phrase has historical roots: when physicists worked on the unification of the three microscopic forces, they called the fruit of their labor "Grand Unified Theories" (GUT), but that unification is not more "grand" as the one including gravitation.

The Planck epoch was the wont of highly speculative physics, full of promising ideas but far from any hope of verifiable predictions or even theoretical consistency. With the GU epoch, we are on firmer ground. There’s no verified theory as of yet, but scores of viable attempts, based on the tried and true methods that made the standard model so successful.

We need to go on a detour — I won’t promise it will be quick, but hopefully it may be enlightening and lift some of the veil that obscures the profund beauty behind the complexity of the mathematical apparatus.

Symmetry as a generating principle

First things first: I said that post-compactification, branes and strings would recede into the background, their body "curled up" into the extra dimensions, leaving only the traditional quantum fields of QFT to be seen. We could say that these fields represent "elementary particles", although it would be more correct to say that the particles are excited eigenstates of the fields. What are those particles? To raise ourselves up to the particle content of the likely GUTs, we need to start from the particles that we do know to exist: those of the Standard Model.

Actually, let’s fast forward a bit more, to the early 1930s, when quantum field theory was nascent. Things seemed simple at the time: the microscopic universe was thought to be composed of atoms and light. Atoms are made of electrons (negatively charged) around a nucleus, and that nucleus is composed of nucleons, some positively charged (protons), some neutral (neutrons). Light, it had been known since Einstein’s discovery of the photo-electric effect, is not only an electromagnetic wave, it can be represented as a quantum particle: the photon. Even the more exotic radiation like X-rays or gamma rays have been shown to be photons outside of the visible light spectrum.

Might these four building blocks give birth to all the matter in our universe? The Presocratic dream of pinning down the "four elements" had never seemed so close. However the picture had to get more complex before it got simpler again (and even more beautiful).

Antimatter: One prediction of Dirac’s work on relativistic quantum mechanics (the ground for quantum field theory) was that each particle had an antiparticle, with exactly opposite charges (but same mass). Anti-electrons (called positrons), antiprotons and antineutrons then join the party (the photon is its own anti-particle), as do antiparticles for all fields we’ll mention going forward.
The neutrino: Studying experimental results about beta decay, Enrico Fermi hypothesized that a new particle was needed to preserve the conservation laws from relativity. It had to be electrically neutral and nearly massless: the neutrino was born. Despite behaving a lot like a photon (moving at light speed, electrically neutral), the neutrino had a spin 1/2, like the electron, proton and neutron (photon has spin 1). Spin-1/2 particles would be called fermions going forward.
Quarks: The theory of the strong nuclear force was developed by Hideki Yukawa, who made the successful prediction that it had a mediator: pion particles that are constantly exchanged between nucleons in their nucleus straitjacket. Troublingly however, a whole bestiary of particles were found while trying to verify this theory: some were force-mediating particles like the pions (they were called mesons), others were matter particles (spin 1/2 fermions) like the nucleons; they were called baryons. Together they form the group of hadrons. Intense work to organize that zoo led to the realization by Gell-Mann and Zweig that all hadrons known at the time could be describe as composite of even smaller particles, which they called quarks. Three of them were theorized at the time: up, down and strange. The up and down quarks make up the nucleons from our daily experience.
Generations: that’s still not all. Not only were new particles discovered all the time, strange "echos" of existing ones were too: particles that shared all the same characteristics and quantum numbers, except for mass: they were heavier. The first one was the muon, a heavier electron analog observed in the 1930s. Later it was found the muon had its very own neutrino. With that in mind, it was easy to observe that the strange quark shared all the characteristics of the down quark, only it was heavier. With the discovery of the charm quark, a heavier copy of the up quark, it was clear this was not a coincidence, and the concept of second generation was coined. To the original group of [up & down quarks, electron, neutrino] would then correspond [charm & strange quarks, muon, muon-neutrino]. In following decades, a complete third generation was found too: [top & bottom quarks, tau, tau-neutrino].

This covers all the known matter content of the universe. But to get a more complete picture, we need to understand the forces, and their mediating particles, better. That is the great achievement of the standard model.

I cannot give a very detailed exposé of the standard model here. If I could briefly communicate one idea, though, it would be this one: symmetry is the source of dynamics.

That is by far the most striking feature of modern physics. It is not intrinsically connected with QFT or the standard model at all, but only with those models has it openly reached its current prominence. It had been a long time in the making though.

First, "rational mechanics" in the eighteenth century had to re-frame Newton’s mechanics under the guise of a variational method. Maupertuis established the principle of least action: that the physical evolution of a system must minimize the quantity of action spent (a quantity linked to the energy of the system). This changed the perspective of mechanics from trying to solve equations of dynamics, to trying to optimize for the "best" motion possible.

Action can be defined as the integral, over a path of the lagrangian, a functional of all the parameters defining a motion (for instance, in classical mechanics, position and velocity).

From that definition of the lagrangian \[\mathcal{S} = \int_t \mathcal{L}(t,\mathbf{x},\dot{\mathbf{x}}) dt\] and the principle of least action \[\delta S =0\] we can derive the Euler-Lagrange equations: \[\frac{\partial \mathcal{L}}{\partial \mathbf{x}} - \frac{d}{dt} (\frac{\partial \mathcal{L}}{\partial \dot{\mathbf{x}}})=0\]

With this approach, we can start from the data of a lagrangian, and deduce the dynamics of the theory from it. It is only a mathematical re-phrasing so far, but of the kind that will open wide horizons.

For instance, it enabled the most fundamental incursion of symmetry, with Noether’s theorem. We all have an idea of what a symmetry is in daily life. What does it mean for a physical theory to have a symmetry.

We can define it as being invariant under a transformation. There are very simple examples: classical physics is invariant under spatial translations. It doesn’t matter where I set the origin of space \(\mathbf{0}\): it can be at the center of the earth, at my feet, in outer space, the dynamics of the system will always be the same. In other words, there is no privileged point in space. The same goes for time: whether \(t=0\) is today, 0 A.D. or 15 billion years ago does not change physical phenomena.

Since we’ve just argued that a theory should be described by its lagrangian, to each symmetry should correspond a transformation that leaves the lagrangian unchanged.

To take a simple example, the free Lagrangian of Newtonian mechanics is: \[\mathcal{L} = \tfrac{1}{2}m\dot{\mathbf{x}}^2\] It doesn’t depend on \(\mathbf{x}\) nor \(t\), so spatial and temporal translation invariances are automatically verified.

What Noether’s theorem says, in plain English, is that every global symmetry of the lagrangian (that is a transformation that doesn’t depend on spacetime coordinates and applies everywhere the same) determines a conserved quantity.

The spatial translation invariance from above yields the conservation of momentum. The invariance by time translation leads directly to the conservation of energy. Noether has showed that the conservation of these quantities is not a posit but a natural consequence of the theory’s symmetries!

The preoccupation with symmetry had then opened the way for the cross-pollination of physics with algebra. Indeed, the kind of global symmetries we’ve mentioned are not isolated: they belong to a structure. For instance, the combination of two translations is still a translation. The kind of structure where transformations can be composed while preserving the symmetry is called a group.

The set of symmetries of classical physics was represented by the Galilean group and encompassed space and time translations, rotations around an axis, and a "boost" (change of Galilean referential). The latter invariance is just another way of stating Galileo’s principle of relativity! In special relativity, mechanics could be re-phrased in the 4D Minkovski space, and the group of interest was the Poincaré group.

The next sea change, as so many times before, was courtesy of Maxwell’s electromagnetic theory. Let’s remind ourselves of Maxwell’s equations: \[\begin{aligned} \nabla \boldsymbol{\cdot} \mathbf{E} & = \rho \\ \nabla \boldsymbol{\cdot}\mathbf{B}& = 0 \\ \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{E} & = -\partial_t \mathbf{B} \\ \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{B} & = \mathbf{j}+\partial_t\mathbf{E} \end{aligned}\]

\(\mathbf{E}\) and \(\mathbf{B}\) are the electric and magnetic fields, and \(\rho\) and \(\mathbf{j}\) are the charge density and the current vector. The corresponding Lagrangian is

\[\mathcal{L}_{EM} = \underbrace{\tfrac{1}{2}(\mathbf{B}^2-\mathbf{E}^2)}_\text{Field energy} - \underbrace{\rho\phi + \mathbf{j}\cdot \mathbf{A}}_\text{Matter-field interaction}\]

\(\phi\) and \(\mathbf{A}\) are respectively the electrical potential and the magnetic (vector) potential. They are defined directly from the fields as:

\[\begin{aligned} \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{A} &=\mathbf{B} \\ -\boldsymbol{\nabla} \phi &= \mathbf{E}+\partial_t \mathbf{A} \end{aligned}\]

Their existence is guaranteed by Maxwell’s equations together with differential calculus (divergence of a curl is \(0\); curl of a gradient is \(\mathbf{0}\)). However they are not unique.

Clearly for a global transformation that just adds a constant term: \[(\phi,\mathbf{A}) \rightarrow (\phi + \phi_0, \; \mathbf{A}+\mathbf{A}_0)\] the electromagnetic field, which is a derivative of the potentials, is unaffected, and therefore the whole dynamics of the system as well.

To this global symmetry corresponds this conservation law:

\[\partial_t \rho + \boldsymbol{\nabla} \cdot \mathbf{j}=0\]

That is the conservation of electric charge! Noether’s theorem works perfectly to derive this basic principle.

Nothing so far is too different from previous examples: a global symmetry yields a conservation law. But a remarkable fact about Maxwell’s theory is that it’s stable under local transformations. For any function \(f(\mathbf(x),t)\) that is twice derivable (i.e. sufficiently "smooth"), Maxwell’s equation is invariant under the following transformation: \[(\phi,\mathbf{A}) \rightarrow (\phi - \partial_t f, \; \mathbf{A}+\boldsymbol{\nabla} f)\]

This type of local symmetry is called gauge invariance (because originally the choice of potential among all the possible ones helped "gauge" the system).

This looks like it could be mere chance: we just "happened" to find a gauge invariance in Maxwell’s theory, which he had not especially noticed himself — great! But there might be something more. Digging in, we can easily prove that Maxwell’s theory is the simplest field theory verifying this gauge invariance.

It’s starting to look as if the symmetry, far from being an incidental feature of the theory, is instead the very root of the theory.

Quantum Electrodynamics: the original quantum field theory

To go to the next step, we need to switch to quantum field theory. Namely Quantum Electrodynamics (QED), the QFT version of Maxwell’s electromagnetism.

QFT is based on the relativistic version of quantum mechanics whose chief architect was Paul Dirac.

The original version of quantum mechanics was not relativistic, as we can see looking at Schrödinger’s equation: \[-\frac{1}{2m}\Delta \psi = i \partial_t \psi\] On the left-hand side we have a laplacian operator, so a second-derivative of spatial parameters, and on the right-hand side a first-derivative of time. That the four dimensions are treated differently is a good indicator that the theory is not relativistic (that is, not invariant under Poincaré transformations.

Dirac used a beautiful "factorization" of the Klein-Gordon equation (an early relavistic quantum field equation), to derive Dirac’s equation, which turned out to perfectly describe the behavior of an electron:

\[i\gamma^\mu \partial_\mu \psi = m \psi\]

In relatvistic notation, the \(\mu\) index runs over the four dimensions of space and time, and the conventional raising/lowering rules apply. The \(\gamma^\mu\) matrices are defined by the following equalities: \[\{ \gamma^\mu,\gamma^\nu \} \hat{=} \gamma^\mu \gamma^\nu+\gamma^\nu\gamma^\mu=- 2\eta^{\mu\nu} I\]

\(\eta\) is the metric of the Minkowski space, so just \(\text{diag}(-1,1,1,1)\). This definition of the gamma matrices actually yields all sorts of algebraic insights, which we can’t delve into here. Suffice to say they define a Clifford algebra, a notion we’ll come back on later.

The smallest matrices that can fulfil these constraints are \(4\times 4\). So the Dirac quantum field is not just a complex function of space and time, like the Schrödinger field, it is a vector-like object with four components, called a spinor. However by the way it transforms and behaves, a spinor is not a relativistic four-vector (those are visible thanks to their \(\mu\) index, conventionally).

The lagrangian that yields Dirac’s equation as its Euler-Lagrange equation is:

\[\mathcal{L}_{Dirac}=\bar{\psi} (i\gamma^\mu\partial_\mu - m) \psi\]

To get the QED lagrangian, we only need to take the Dirac lagrangian applied to a free electron, and add the Maxwell lagrangian we’ve seen before. There’s jsut some "translation" to be done since we’re using relativistic 4D notations now:

The 4-current \(j_\mu\) is the 4-vector \((\rho, \mathbf{j})\).
The 4-potential \(A_\mu\) is the 4-vector \((\phi, \mathbf{A})\).

We also define a kinetic field tensor defined as: \[F_{\mu\nu} \hat{=} \partial_{[\mu}A_{\nu]} \hat{=} \partial_\mu A_\nu - \partial_\nu A_\mu\] Taking first derivatives is analogous to the way we go from potential to field in classical electromagnetism, and indeed this anti-symmetric \((0,2)\) tensor, with its 6 degrees of freedom, contains the same information as the two 3-vectors \(\mathbf{E}\) and \(\mathbf{B}\).

We can obtain the QED lagrangian by adding to Dirac’s free electron lagrangian the terms from Maxwell’s classical lagrangian.

\[\mathcal{L}_{QED}=\underbrace{\bar{\psi} (i\gamma^\mu\partial_\mu - m) \psi}_\text{Electron (Dirac free field)} -\underbrace{\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu}}_\text{Photon (Maxwell free field)} - \underbrace{e j^\mu A_\mu}_\text{Electron-photon interaction}\]

It’s part of the great beauty and plasticity of the lagrangian framework that we can just do that: a simple addition. Taming the complexity of the calculations that ensue is another story, but lagrangian manipulation is an intuitive and powerful framework.

Now let’s come back to our gauge invariance. We could re-trace our steps and apply the same global gauge invariance as in the classical version: \(A_\mu \rightarrow A_\mu + A^0_\mu\) (i.e. adding a constant 4-vector to the potential). The result of applying Noether’s theorem is unsurprisingly that \(j^\mu\) is conserved (\(\partial_\mu j^\mu\)=0).

But there is another global gauge invariance: rotations in the complex plane:

\[\psi \rightarrow e^{i \theta_0} \psi\]

The conserved quantity according to Noether’s theorem is:

\[\bar{\psi} \gamma^\mu \psi\]

We will identify this to the electric current \(\tfrac{j^\mu}{e}\), allowing to re-write the lagrangian:

Now, does local gauge invariance work? Let’s now assume that \(\theta\) is not a constant but a function over spacetime points:

\[\psi \rightarrow e^{i \theta(x_\mu)} \psi\]

Unfortunately, we can see that the first term (the free electron) is not invariant under this transformation. That is because spacetime derivative \(\partial_\mu\) does not transform like \(\psi\) (i.e. it’s not just rotated, there is an extra term that appears):

\[\partial_\mu \psi \rightarrow e^{i \theta} (\partial_\mu + i \partial_\mu \theta)\psi\]

But we still have a lever to action: the lagrangian doesn’t have only \(\psi\) as an input, it had \(A_\mu\). If we imposed that vector’s transformation, we could retrieve the full lagrangian’s invariance. This can be done by imposing:

\[A_\mu \rightarrow e^{i \theta} A_\mu + \tfrac{1}{e} \partial_\mu \theta\]

The pesky extra term from the free electron will cancel with the one from the interaction term. We can now use this fact to define the covariant derivative, which transforms in "safe" way:

\[\begin{aligned} D_\mu \psi \; \hat{=} \; & (\partial_\mu - i e A_\mu) \psi \\ \rightarrow & (\partial_\mu + i \partial_\mu \theta) e^{i \theta}\psi - ie(A_\mu + \tfrac{1}{e} \partial_\mu \theta ) e^{i \theta}\psi \\ =& e^{i \theta} D_\mu \psi \end{aligned}\]

We can even compute the commutator of between covariant derivatives (the commutator between regular derivatives if \(0\)):

\[[D_\mu \psi,D_\nu \psi] = ieF_{\mu \nu}\psi\]

All of this allows us to rewrite the lagrangian entirely in terms of this covariant derivative.

\[\mathcal{L}_{QED}=\underbrace{\bar{\psi} (i\gamma^\mu D_\mu - m) \psi}_\text{Covariant Dirac field} +\underbrace{\tfrac{1}{4e^2}[D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]}_\text{Covariant gauge field}\]

Using the mathematical techniques developed for Quantum Field Theory, one can describes the physics of electricity, magnetism and light with an astounding accuracy, that was entirely unthinkable a few decades earlier. QED turned out to be the stepping stone onto which the whole Standard Model was built.

Even more, it can be the best and simplest example of how symmetry is the new ur-principle, the true queen of fundamental physics. Let’s recap quickly the steps we went through.

We found the QFT form of the global invariance evident in classical electrodynamics, and quickly found that it still held.
Since local gauge invariance was also a feature of Maxwell’s theory, we made sure that both \(\psi\) (the electron) and \(A_\mu\) (the force) had the appropriate ways to transform under a gauge symmetry.
We found that the lagrangian could be expressed in terms of a QED-covariant derivative: all we had to do was take the free Dirac lagrangian, and replace ordinary derivatives with the QED-covariant ones (to be precise, the ordinary ones are Lorentz-invariant: compatible with special relativity, while the new ones are both Lorentz- and QED-gauge invariant: compatible with both SR and Maxwell’s theory).

The gauge symmetry that we’ve found is not any symmetry, it corresponds to the group \(U(1)\) (the unitary group of order 1).

\[U(1)= \{ e^{i\theta}, \theta \in \mathbb{R} \}\]

It is a Lie group. As a reminder, a group is a set characterized by its elements (here, complex numbers) and an operation (here, multiplication). To be a group it must verify these properties:

It is not empty.
It is stable (if \(a\) and \(b\) are group members, so is \(a \cdot b\)
It has a neutral element \(e\) (\(e\cdot a =a\)).
Every element has an inverse (\(a \cdot a^{-1} = a^{-1} \cdot a = e\)).
The operation is associative (\(a\cdot (b \cdot c) = (a \cdot b ) \cdot c\))

A Lie group is a group that also happens to be a differentiable manifold. In simpler terms, it means that :

A Lie group is "smooth" and continuous ("between" two elements, we can find other elements.
Locally (if we "zoom in"), it looks linear, like a Euclidean space.

Think about another group: the group of rotations by a right angle in a plane. It’s a group with just four elements: rotations by \(90^{\circ}\), \(180^{\circ}\),\(270^{\circ}\),and no rotation at all. It’s clearly a legitimate group, but not continuous. There is a "gap" between the elements.

\(U(1)\), the group of all rotations, verifies the characteristics of a Lie group: between a rotations by angles \(\alpha\) and \(\beta\) there’s an infinity of intermediate angles. And locally all rotations can be linearized since \(e^{i\theta} ~_0 1+i \theta\).

From the dynamics of the Dirac and Maxwell equations, we’ve evidenced the \(U(1)\) symmetry. That’s the traditional approach. The modern approach flips the table and asks: what if we started from the symmetries, and deduced the dynamics?

The steps would be:

Find the simple quantum-mechanical lagrangian that preserves \(SL(3,1)\) (the Poincaré-Lorentz group of relativistic spacetime transformations): that’s the Diract lagrangian.
Find the simplest way to make it locally \(U(1)\)-invariant: thereby emerges the 4-vector potential and the covariant derivative.
What about the kinetic gauge field tensor term? Without it, the lagrangian is already invariant enough for our stated goals, so why do we need it?
Without this term, the Euler-Lagrange equation applied to \(A_\mu\) is entirely trivial \(j^\mu=0\). It describes a world with a potential but no charges.
Since we want to interpret \(A\mu\) as an electromagnetic potential, and since we know there are electric charges, it cannot be. The next simplest term involving \(A_\mu\) giving a non-trivial dynamics is the one that we’ve encountered, \(F^{\mu \nu}F_{\mu \nu}\), or \([D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]\).

By imposing these constraints (relativistic invariance, \(U(1)\) invariance, existence of charges), we have mathematically developed the simplest possible lagrangian. Of course, we’ve used with savvy our knowledge of the physical world to guide our steps, but this was still an abstract progression.

The audacious leap of faith of the standard model is that we can continue this process, and by imposing more complex symmetries, keep answering more and more of our long-held questions.

Quantum Chromodynamics: the platonic ideal of Yang-Mills gauge theories

The standard model is composed of two theories: quantum chromodynamics (the theory of the strong nuclear force) and the electroweak model (the unified theory of the electromagnetic and weak nuclear forces).

Remember that quarks were discovered when the numbers of particles similar to the good ol’ protons and neutrons had ballooned beyond all reason. It seemed like more hadrons were discovered with each week that passed. Surely, God, logos or the Universe could not allow that complexity! That’s when Murray Gell-Mann and others developed the “eightfold way”, a system of classification of these pesky particles that was the seed of the idea there might be a sub-structure there. When the existence of quarks was hypothesised, nothing was known about the interaction that bound them together without hadrons. Since quarks in baryons went by groups of three, a new quantum charge was imagined, called color. A proton for instance, would assemble three quarks of different colours: red, blue and yellow. A pion (made of a quark and an antiquark) would assemble opposite charges, for instance a blue quark and an “anti-blue” anti-quark.

There’s no space here to get into the exact details why, but careful consideration of the various hadrons showed that quarks must be governed by \(SU(3)\) symmetry. That is a Lie group that generalizes the simple \(U(1)\) that we already know. Its fundamental representation (one of the ways to mathematically embody the underlying symmetry) is as the set of complex \(3\times 3\) matrices that are unitary (that’s the “U”) and of positive determinant (that’s the “S” for “special).

\[SU(3) = \{ U \in GL(3,\mathbb{C}) \; | \; U^\dagger U = I \text{ and } \text{det}(U)=1 \}\]

So, we have matrices, but what do they act on? In electromagnetism, we had \(U(1)\), whose members were scalars (complex numbers). The \(U(1)\) transformation just multiplied the Dirac spinors (representing the electron) by a complex function of spacetime. Now the \(SU(3)\) transformation multiplies something by a matrix of complex numbers. We know that multiplying a matrix of size 3 by a vector with 3 components yields another vector with 3 components. It looks like the fundamental object of this new theory then ought to be a triplet. This is consistent with the fact we’ve just posited there might be three different color charges. A triplet would then contain blue, red and yellow quarks:

\[\mathbf{q} = \begin{pmatrix} q_b \\ q_r \\ q_y \end{pmatrix}\]

Note that a triplet contain the same type of quark: it can be a triplet of three up quarks, of three charm quarks, of three strange anti-quarks, but there can be no mixing. Let’s remember that each component like \(q_b\) is a full-blown Dirac spinor, mathematically the same type of object as an electron \(\psi\).

We can use this triplet language to write the interaction-free QCD lagrangian:

\[\mathcal{L}_{QCD}^{free}=\bar{\mathbf{q}} (i\gamma^\mu \partial_\mu - m) \mathbf{q}\]

It’s the same as the Dirac free lagrangian but with triplets of spinors replacing the single spinors. In fact, using \(c\) as an index running over colours:

\[\bar{\mathbf{q}}\mathbf{q} = \sum_c{\bar{q_c} q_c }= \bar{q_b} q_b+ \bar{q_r} q_r+ \bar{q_y} q_y\]

So plugging this into the lagrangian, we get:

\[\mathcal{L}_{QCD}^{free}=\bar{q_b} (i\gamma^\mu \partial_\mu - m) q_b + \bar{q_r} (i\gamma^\mu \partial_\mu - m) q_r + \bar{q_y} (i\gamma^\mu \partial_\mu - m) q_y\]

This simply means that the free QCD lagrangian describes three regular particles, independent from each other since the interaction terms are not in there yet. The triplet formalism works as we expect so far.

Before going further we need to move from Lie groups to Lie algebras. The fact that Lie groups are locally Euclidean (linear and “flat”) is not an afterthought, but an absolutely crucial fact. It means that if we focus only on infinitesimal transformations, the Lie group elements are correctly approximated by the elements of an algebra (essentially a vector space with some multiplication structure). The linearity of the vector space will make calculations infinitely easier! . The Lie algebra that corresponds to \(SU(3)\) is usually written as \(\mathfrak{su}(3)\). There are lots of beautiful and clean mathematical ways to develop the theory of Lie algebras, but we’ll stick to a high-level “physicist” introduction. When Lie groups are represented by sets of matrices, as in the fundamental representation, we can prove that all of their elements \(U\) are the exponential of another matrix \(M\). That’s how we define the Lie algebra!

\[\mathfrak{su}(3) = \{ M \text{ such as } U \in SU(3) \text{ and } U = \exp{M} \}\]

Algebras are vector spaces, and this one has to be of finite dimensions (since it is included in the vector spaces of all size-3 matrices, which has 9 dimensions), Therefore it is entirely defined by a basis of generators which we will call \(( \lambda_a )\).

\[\mathfrak{su}(3)= \{ \theta^a \lambda_a , \text{ for } (\theta_a) \in \mathbb{R}^d \}\]

But that’s not all. Products of \(SU(3)\) matrices also belong to \(SU(3)\) (by group stability). If we switch to “algebra view” it means that if \(M\) and \(N\) are part of \(\mathfrak{su}(3)\), then \(\exp(M)\) and \(\exp(N)\) are part of \(SU(3)\) and there is some matrix \(P\) of \(\mathfrak{su}(3)\) such that \(\exp(M)\exp(N) = \exp(P)\).

We so happen to have a formula telling us exactly what a product of matrix exponentials is: \[\exp (M) \exp (N) =\exp ( M + N + [M,N] + \dots )\]

The notation we have used here is the matrix commutator: \([A,B] = AB-BA\). All the following terms also involve commutators. The vector space \(\mathfrak{su}(3)\) is stable by addition so \(M+N\) is part of it. So to ensure that \(M + N + [M,N] + \dots\) is part of \(\mathfrak{su}(3)\), it should be enough to have it that \([A,B]\) is. Since \(\mathfrak{su}(3)\) is a vector space fully generated by its generator basis \(( \lambda_a )\), then that basis must be stable against the action of the commutator:

\[[\lambda_a, \lambda_b] = i f^{abc} \lambda_c\]

The \(f^{abc}\) are called the structure constants. They entirely define the structure of the Lie algebra.

We are now equipped with enough knowledge about Lie groups and algebras to implement the programme we set out to develop: from symmetry shall emerge the dynamics, in the form of a full QCD lagrangian. First we can easily verify that the free lagrangian is invariant under global \(SU(3)\) transformations:

\[\mathbf{q} \rightarrow U_0 \mathbf{q}\]

\(U_0\) is a constant complex matrix from \(SU(3)\). The global invariance is proved by noting that:

\[\mathbf{q} \rightarrow \overline { U_0 \mathbf{q} } = \bar{\mathbf{q}} U_0^\dagger\]

So, by the definition of unitary matrices, terms like \(\bar{\mathbf{q}}\mathbf{q}\) are automatically invariant. And since the matrix is constant, the spatial derivative can just be absorbed (that is: \(\bar{\mathbf{q}}\gamma^\mu \partial_\mu \mathbf{q}\) is also invariant).

However the free lagrangian is not invariant under a local \(SU(3)\) transformation, that is one where the matrix can be different for different spacetime co-ordinates. We know we can write that transformation as:

\[\mathbf{q} \rightarrow U \mathbf{q} = \exp \left( i \frac{\lambda_a}{2} \theta^a \right) \mathbf{q}\]

The \(\theta^a\) are now functions over spacetime. We also know that we can linearise, since we are only looking at infinitesimal transformations.

\[\mathbf{q} \rightarrow \mathbf{q} + i \frac{1}{2} \left( \lambda_a \mathbf{q} \right) \delta\theta^a\]

The transformation of the free lagrangian yields an extra term:

\[\mathcal{L}_{QCD}^{free} \rightarrow \mathcal{L}_{QCD}^{free} - \frac{1}{2} \lambda_a \bar{\mathbf{q}} \gamma^\mu ( \partial_\mu \delta\theta^a ) \mathbf{q}\]

Just like we did for \(U(1)\) we can build a covariant derivative of this form:

\[D_\mu \hat{=} \partial_\mu + i g \frac{\lambda_a}{2} G^a_\mu\]

The covariance is ensured by imposing this transformation to the \(G^a_\mu\):

\[G^a_\mu \rightarrow G^a_\mu + \frac{1}{g} \partial_\mu \delta \theta^a - f^{abc} \delta \theta_c G^b_\mu\]

With our brand new QCD-covariant derivative in tow, we can define the full lagrangian exactly in the way we did for QED, except that It’s now acting on quark triplets instead of electron singlets.

\[\mathcal{L}_{QCD}=\underbrace{\bar{ \mathbf{q}} (i\gamma^\mu D_\mu - m) \mathbf{q}}_\text{Covariant Dirac field} +\underbrace{\tfrac{1}{4g^2}[D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]}_\text{Covariant gauge field}\]

Based on the covariant derivative definition, we can make the last term explicit in terms of the \(G^a_\mu\) and define a field tensor of order 2, like we did for electromagnetism.

\[\tfrac{-i}{g}[D_\mu,D_\nu] \; \hat{=} \; G_{\mu \nu} \; \hat{=} \; \lambda_a G^a_{\mu \nu}\]

\[G^a_{\mu \nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f^a_{bc} G^b_\mu G^c_\mu\]

We get the following:

\[\mathcal{L}_{QCD}=\underbrace{\bar{ \mathbf{q}} (i\gamma^\mu D_\mu - m) \mathbf{q}}_\text{QCD field} -\underbrace{\tfrac{1}{4}G^{\mu \nu}_a G^a_{\mu \nu}}_\text{Kinetic Gluon field}\]

The last term is very similar to the photon term in QED, except that it’s summing over the index \(a\). Therefore, there are as many force-mediating fields now as there are generators of the Lie algebra. In the case of \(SU(3)\) that number is \(8\). We have then shown that there are eight bosons (force-mediating particles derived from a 4-vector potential and represented by a field tensor) in QCD. They are called gluons.

We can develop the lagrangian even more and make interesting observations.

\[\begin{aligned} \mathcal{L}_{QCD} = & \underbrace{\bar{ \mathbf{q}} (i\gamma^\mu \partial_\mu - m) \mathbf{q}}_\text{Free quark} -\underbrace{\tfrac{1}{4} \partial^{[/mu} G^{\nu]}_a \partial_{[/mu} G_{\nu]}^a }_\text{Free Gluon} \\ &+ \underbrace{ g G^\mu_a \bar{\mathbf{q}} \gamma_\mu \frac{\lambda ^a}{2} \mathbf{q} }_{\text{Quark-Gluon interaction}} \\ & - \underbrace{ \frac{g}{2} f^{a}_{bc} \partial^{[\mu} G^{\nu]}_a G_\mu^b G_\nu ^c}_{\text{ Cubic gluon interaction}} - \underbrace{ \frac{g^2}{4} f^{abc}f_{ade} G_\mu^b G_\nu ^c G_\mu ^d G_\nu ^e }_{\text{ Quartic gluon interaction}} \end{aligned}\]

The QCD field term decomposes into the free quark terms (the free lagrangian) and a gluons-quarks interaction (similar to the electron-photon interaction in QED).The kinetic gluon field also can be decomposed, first to a term that looks exactly like the QED field tensor (except that it’s not gauge-invariant alone) and that we’ll identify with free gluon propagation. The other terms are shaped in a way that when applying Feynman rules, we find that they involve interactions between three gluons, and between four gluons!

This is different from QED: photons could not interact with each other since they have no electric charge. Gluons have no electric charge but they do have a color charge: that is because they can transform elements of the quark triplets: a blue quark can become a red quark by emitting a gluon for instance. While an electron emitting a photon has an effect on its dynamics, but doesn’t change its electric charge. Gluons being color-charged is the reason the cubic and quartic terms are present.

These results are spectacular: by imposing a local \(SU(3)\) gauge symmetry to the Dirac Lagrangian, we get a full-blown theory of strong interactions. All the steps we’ve gone through could have been followed the same for the general case of \(SU(N)\) though. The general case is called the Yang-Mills non-abelian gauge theory ("non-abelian" just means that the elements of the group don’t commute, unlike \(U(1)\) for instance).

The simple Yang-Mills theory for \(SU(N)\) then ultimately gives us:

An \(N\)-uplet of fermions (the particles that are subject to forces, with half-integer spin). Fermions can have mass.
\(N^2-1\) generators that are identified as spin-1 bosons ("force-carrying" particles). These bosons are massless and charge-carrying (i.e. they can mediate the transformation of a fermion particle into another).