Constructing SU(N) Yang-Mills Theory

From U(1) to SU(N): A New Kind of Phase

Let's begin by recalling the structure of QED. We started with a free Dirac fermion $\psi$, described by the Lagrangian $\mathcal{L}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi$. This has a global symmetry under the U(1) transformation $\psi (x)\to e^{i\alpha }\psi (x)$, where $\alpha$ is a constant phase.

Now, consider a theory with $N$ different types of fermions that are indistinguishable except for a new kind of "charge" or "color." We can group these $N$ fermion fields into an N-component column vector, or multiplet:

A simple Lagrangian for these $N$ free, degenerate fermions is:

This Lagrangian is invariant under a global transformation that "rotates" the different fermion types into each other:

where $U$ is any constant $N\times N$ unitary matrix with determinant 1, i.e., $U\in \text{SU(N)}$. Because $U$ is constant, it commutes with the derivative ${\partial }_{\mu }$, and the Lagrangian remains invariant as $\bar{\psi }\to \bar{\psi }{U}^{†}$, so $\bar{\psi }{\partial }_{\mu }\psi \to \bar{\psi }{U}^{†}U{\partial }_{\mu }\psi =\bar{\psi }{\partial }_{\mu }\psi$.

The Challenge of Local Non-Abelian Symmetry

Following the logic that led us to QED, we now elevate this global symmetry to a local gauge symmetry. We demand that the Lagrangian be invariant even when the transformation matrix $U$ depends on the spacetime coordinate $x$:

Just as in the U(1) case, the derivative term immediately presents a problem. When we transform the derivative of the field, the product rule introduces an unwanted term:

The Lagrangian is no longer invariant because of the new term involving ${\partial }_{\mu }U(x)$. Our simple derivative is not "covariant"—it does not transform in the same way as the field $\psi$ itself. The core of our problem is that for a non-abelian group like SU(N), matrices do not commute in general, which makes the structure more complex than the simple phase factors of U(1).

This distinction between a global transformation (like rotating a sphere as a whole) and a local one (where the rotation varies from point to point) is visualized below. The local transformation distorts the grid, just as the ${\partial }_{\mu }U(x)$ term "distorts" the Lagrangian's invariance.

The Solution: The Non-Abelian Covariant Derivative

To solve this, we must introduce a covariant derivative, $D_{\mu }$, which is designed precisely to absorb the unwanted term. We require that $D_{\mu }\psi$ transforms "like $\psi$":

To achieve this, we generalize the covariant derivative from QED. Instead of a single gauge field, we introduce a set of them, packaged into a matrix-valued field called the gauge potential, $A_{\mu }$. The covariant derivative is defined as:

Here, $g$ is the coupling constant, analogous to the electric charge $e$. Since $\psi$ is an N-component vector, $A_{\mu }$ must be an $N\times N$ matrix. As we are building a theory with SU(N) symmetry, it is natural to express $A_{\mu }$ as a linear combination of the generators of the SU(N) Lie algebra, $T^a$:

Here, $a$ runs from 1 to $N^2-1$, the dimension of the SU(N) group. We have therefore introduced $N^2-1$ new vector fields, $A_{\mu }^a(x)$. These are the non-abelian gauge fields. For SU(3), the gauge group of the strong force, these are the eight gluon fields.

The following video provides an excellent introduction to this generalization from abelian to non-abelian gauge theory, reviews the necessary group theory, and constructs the covariant derivative.

Transformation of the Gauge Field

With the covariant derivative defined, we can now determine how the gauge potential $A_{\mu }$ itself must transform to satisfy our requirement that $(D_{\mu }\psi {)}^{\prime }=U(D_{\mu }\psi )$. By substituting the definition of $D_{\mu }$ and demanding this equality holds, we can solve for the transformation rule of $A_{\mu }$. This is a central calculation in gauge theory.

The resources below walk through this derivation in detail. While the video provides a dynamic explanation, the text from Ramond's textbook offers a classic, rigorous presentation you can study at your own pace.

The result of this derivation, using the convention $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$, is the non-abelian gauge transformation law for the gauge potential:

This transformation is more complex than its abelian counterpart ($A_{\mu }\to A_{\mu }-{\partial }_{\mu }\alpha$). It has two parts:

1. A homogeneous term: $U{A}_{\mu }{U}^{†}$. This term looks like a rotation of the gauge field in the group space. A field that transforms this way is said to be in the adjoint representation of the group.
2. An inhomogeneous term: $\frac{i}{g}({\partial }_{\mu }U)U^{†}$. This is analogous to the derivative term in the U(1) transformation and is essential for cancelling the unwanted term from the derivative of $\psi$.

Because of this inhomogeneous term, the gauge field $A_{\mu }$ does not transform in a simple way (like in the adjoint representation). This "spoiled" transformation property is a defining characteristic of a gauge connection. The following video segment further analyzes this crucial point.

The Principle of Local Gauge Invariance

What we have just accomplished is fundamental. By starting with a global symmetry and demanding it holds locally, we were forced to introduce new fields—the gauge fields—that mediate an interaction. Their sole purpose is to "compensate" for the local variation of the phase, allowing the derivative to be covariant. This beautiful and powerful idea is the principle of local gauge invariance.

The entire procedure can be summarized as follows:

1. Start with a matter Lagrangian invariant under a global non-abelian symmetry group $G$ (e.g., SU(N)).
2. Localize the symmetry by making the transformation group element $U$ a function of spacetime, $U(x)$. The ordinary derivative ${\partial }_{\mu }$ breaks the invariance.
3. Introduce a covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ by adding a new Lie-algebra-valued gauge potential $A_{\mu }=A_{\mu }^a{T}^a$.
4. Demand that $D_{\mu }$ transforms covariantly, which fixes the transformation rule for the gauge potential $A_{\mu }$.

This procedure gives us a new Lagrangian for the matter fields that is locally gauge-invariant:

Notice that this process has automatically generated an interaction term, $g\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi$, describing how the matter fields $\psi$ couple to the gauge fields $A_{\mu }$.

Conclusion

In this lesson, we have taken the crucial first step in building the modern framework for fundamental forces. We have generalized the principle of local gauge invariance from the abelian group U(1) to non-abelian groups SU(N). This generalization is not a mere mathematical exercise; it is the source of the immense richness and complexity of the Standard Model.

Key Takeaways:

• Starting with a Lagrangian with global SU(N) symmetry, promoting the symmetry to be local (spacetime-dependent) breaks the invariance of the kinetic term.
• To restore invariance, the ordinary derivative ${\partial }_{\mu }$ must be replaced by a covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$.
• This requires the introduction of a matrix-valued gauge potential $A_{\mu }=A_{\mu }^a{T}^a$, which consists of one vector field for each generator of the Lie algebra. For SU(N), this means $N^2-1$ gauge fields.
• The gauge potential transforms in a specific non-homogeneous way: $A_{\mu }^{\prime }=U{A}_{\mu }{U}^{†}+\frac{i}{g}({\partial }_{\mu }U)U^{†}$. This rule ensures that the covariant derivative of a matter field transforms simply.
• This constructive process, known as the gauge principle, automatically generates the interaction term between matter and the newly introduced gauge fields.

We have successfully introduced the gauge fields and determined their coupling to matter. However, these fields do not yet have their own dynamics; they have no kinetic term in the Lagrangian. Our work is not yet complete.

Preview of the Next Lesson:

In the next lesson, we will give the gauge fields "a life of their own." We will construct a kinetic term for the $A_{\mu }$ fields that is itself invariant under the non-abelian gauge transformations. This will lead us to derive the non-abelian field strength tensor $F_{\mu \nu }$ and the Yang-Mills Lagrangian, ${\mathcal{L}}_{\text{YM}}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{\mu \nu a}$. A surprising feature will emerge: unlike the abelian photons, the non-abelian gauge bosons will interact with themselves.