SU(N) Gauge Theory Construction

Hello! Welcome to the first lesson in our new module, "Non-Abelian Gauge Theories and QCD." In the previous module, we explored the powerful path integral formalism, culminating in understanding its deep connection to statistical mechanics. Now, we return to the operator formalism to build the theoretical core of the Standard Model of particle physics.

Our journey so far has included a detailed study of Quantum Electrodynamics (QED), the theory of light and matter. The foundational principle of QED is local U(1) gauge invariance. You'll recall how we started with a global phase symmetry in the Dirac Lagrangian, promoted it to a local, spacetime-dependent symmetry, and found that this required introducing a covariant derivative and a gauge field—the photon. The dynamics of this photon field were then described by the gauge-invariant Maxwell Lagrangian, built from the field strength tensor $F_{μν}$ .

Today, we will generalize this entire procedure. We'll move beyond the simple, commutative (Abelian) U(1) group to more complex, non-commutative (non-Abelian) symmetry groups like SU(N). This generalization is the gateway to understanding the strong and weak nuclear forces.

Our learning outcome for this lesson is to construct a non-Abelian gauge theory with SU(N) symmetry, deriving the covariant derivative and the field strength tensor.

From Global to Local SU(N) Symmetry

Let's begin by constructing a theory with a global non-Abelian symmetry. Instead of a single Dirac field, imagine a set of $N$ fields that we group into a column vector, or "multiplet":

ψ (x) = ψ_{1} (x) ψ_{2} (x) ⋮ ψ_{N} (x)

The Dirac Lagrangian for this multiplet is $L = ψ ˉ (i γ^{μ} \partial_{μ} - m) ψ$ , where $ψ ˉ = ψ^{†} γ^{0}$ . This Lagrangian is invariant under a global transformation $ψ (x) \to U ψ (x)$ , where $U$ is any constant $N \times N$ unitary matrix with determinant 1, i.e., $U \in SU(N)$ .

Any such matrix $U$ can be written as:

U = e^{i α^{a} T^{a}}

Here, the $α^{a}$ are $N^{2} - 1$ real parameters, and the $T^{a}$ are the $N^{2} - 1$ generators of the SU(N) Lie algebra. For SU(2), these are proportional to the Pauli matrices; for SU(3), they are the Gell-Mann matrices. These generators are Hermitian, traceless, and, most importantly, they do not commute. Their commutation relations define the Lie algebra:

[T^{a}, T^{b}] = i f^{ab c} T^{c}

The real numbers $f^{ab c}$ are called the structure constants of the group. This non-zero commutator is the essential feature that distinguishes non-Abelian theories from the Abelian U(1) theory of QED.

Now, we "gauge" this symmetry by demanding invariance under local transformations, where the parameters $α^{a}$ depend on the spacetime coordinate $x$ :

ψ (x) \to U (x) ψ (x) with U (x) = e^{i α^{a} (x) T^{a}}

The term $ψ ˉ m ψ$ remains invariant, but the derivative term $ψ ˉ i γ^{μ} \partial_{μ} ψ$ does not. Under the local transformation, the derivative becomes:

\partial_{μ} ψ (x) \to \partial_{μ} (U (x) ψ (x)) = (\partial_{μ} U (x)) ψ (x) + U (x) \partial_{μ} ψ (x)

The extra term $(\partial_{μ} U) ψ$ breaks the invariance. Just as in QED, we must replace the ordinary derivative $\partial_{μ}$ with a covariant derivative $D_{μ}$ .

The Non-Abelian Covariant Derivative

We define the covariant derivative $D_{μ}$ by the property that it transforms "covariantly" with the field it acts on:

D_{μ} ψ (x) \to U (x) (D_{μ} ψ (x))

This ensures that the kinetic term $ψ ˉ i γ^{μ} D_{μ} ψ$ is gauge invariant.

To achieve this, we introduce $N^{2} - 1$ gauge fields $A_{μ}^{a} (x)$ , one for each generator, and package them into a Lie algebra-valued matrix field, the gauge potential:

A_{μ} (x) = A_{μ}^{a} (x) T^{a}

The covariant derivative is then defined as:

D_{μ} = \partial_{μ} - i g A_{μ} = \partial_{μ} - i g A_{μ}^{a} T^{a}

Here, $g$ is the coupling constant, analogous to the electric charge $e$ in QED.

The following video provides a concise derivation of how the gauge field $A_{μ}$ must transform to make this construction work.

Deriving Yang-Mills Theory And Quantum Chromodynamics (QCD) | Quantum Field Theory

This video from Dietterich Labs walks through the steps of promoting the global SU(N) symmetry to a local one, introducing the covariant derivative and deriving the transformation rule for the new gauge fields.

Watch from 11:06 to 14:56. The first part (11:06-12:09) sets up the globally invariant Dirac Lagrangian. The crucial part is the derivation that follows (12:09-14:56), where the SU(N) covariant derivative is defined and the transformation property of the gauge field matrix A_μ is derived.

As derived in the video, for the covariant derivative to have the required property, the gauge potential $A_{μ}$ must transform as:

A_{μ} (x) \to A_{μ}^{'} (x) = U (x) A_{μ} (x) U^{†} (x) + \frac{i}{g} (\partial_{μ} U (x)) U^{†} (x)

Compare this to the U(1) transformation $A_{μ} \to A_{μ} - \partial_{μ} α$ . The non-Abelian rule has a similar derivative term but also includes a "rotation" term $U A_{μ} U^{†}$ , because the gauge field itself now carries a group index (it's a matrix) and must be rotated in the internal symmetry space.

The Non-Abelian Field Strength Tensor

Having constructed an invariant matter Lagrangian, we now need a kinetic term for the gauge fields themselves, $L_{gauge}$ , which must be both Lorentz invariant and gauge invariant. For QED, this was simply $- \frac{1}{4} F_{μν} F^{μν}$ , where $F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}$ .

Does this simple form work for a non-Abelian theory? Let's explore this question. The following video provides excellent intuition by first trying this "naive" generalization and carefully analyzing why it fails.

The Strong Nuclear Force as a Gauge Theory, Part 4: The Field Strength Tensor

This video by Richard Behiel takes a more pedagogical approach. It first motivates the need for a field strength tensor and then shows precisely why the simple form from QED is not gauge invariant in a non-Abelian theory.

Watch from the beginning to 21:21. Focus on understanding the 'shape of the failure'—that is, the leftover terms that prevent the simple ∂μGν - ∂νGμ from transforming nicely. This failure is a direct result of the non-commuting nature of the gauge fields and the derivative term in their transformation rule.

The failure of the simple approach points to the need for an additional term. The elegant and general way to find the correct gauge-invariant field strength is to compute the commutator of two covariant derivatives. This object is guaranteed to transform covariantly:

[D_{μ}, D_{ν}] \to [U D_{μ} U^{†}, U D_{ν} U^{†}] = U [D_{μ}, D_{ν}] U^{†}

Let's perform the calculation:

[D_{μ}, D_{ν}] = [\partial_{μ} - i g A_{μ}, \partial_{ν} - i g A_{ν}] = [\partial_{μ}, \partial_{ν}] - i g [\partial_{μ}, A_{ν}] - i g [A_{μ}, \partial_{ν}] + (i g)^{2} [A_{μ}, A_{ν}] = 0 - i g (\partial_{μ} A_{ν}) + i g (\partial_{ν} A_{μ}) - g^{2} [A_{μ}, A_{ν}] = - i g (\partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}])

This motivates the definition of the non-Abelian field strength tensor $F_{μν}$ (sometimes called $G_{μν}$ ) as:

F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}]

By construction, $[D_{μ}, D_{ν}] = - i g F_{μν}$ , and therefore $F_{μν}$ transforms in the adjoint representation:

F_{μν} (x) \to F_{μν}^{'} (x) = U (x) F_{μν} (x) U^{†} (x)

Notice the crucial new term, $- i g [A_{μ}, A_{ν}]$ . It arises directly from the non-commutativity of the SU(N) group. Since this term is quadratic in the gauge field $A_{μ}$ , it will lead to cubic and quartic self-interaction terms in the Lagrangian. This means that non-Abelian gauge bosons (like gluons) interact with each other, a stark contrast to photons in QED, which do not.

The following resource provides a clean, formal summary of all the objects we have just defined.

2. Yang-Mills Theory

The lecture notes by David Tong are a standard reference in QFT. They present the mathematical framework of Yang-Mills theory with great clarity and precision.

Read the subsections '2.1 Introducing Yang-Mills' and the first part of '2.1.2 Gauge Symmetry' (pages 27-28 and 31). Focus on the definitions of the Lie-algebra valued gauge potential A_μ (Eq. 2.3), the field strength F_μν (Eq. 2.4), the covariant derivative D_μ (Eq. 2.5), and the gauge transformation rules (Eq. 2.11 and 2.12).

The Field Strength in Component Form

So far, we have worked with the compact matrix notation. It is often useful to write the field strength in terms of its components $F_{μν}^{a}$ . We substitute $A_{μ} = A_{μ}^{a} T^{a}$ and $F_{μν} = F_{μν}^{c} T^{c}$ into the definition:

F_{μν}^{c} T^{c} = \partial_{μ} (A_{ν}^{a} T^{a}) - \partial_{ν} (A_{μ}^{a} T^{a}) - i g [A_{μ}^{a} T^{a}, A_{ν}^{b} T^{b}]

Using the Lie algebra relation $[T^{a}, T^{b}] = i f^{ab c} T^{c}$ and equating the coefficients of the generator $T^{c}$ , we find:

F_{μν}^{c} = \partial_{μ} A_{ν}^{c} - \partial_{ν} A_{μ}^{c} + g f^{ab c} A_{μ}^{a} A_{ν}^{b}

This expression makes the structure explicit. The first two terms are identical to the electromagnetic field tensor, while the third term, involving the structure constants, encodes the self-interaction of the gauge fields.

The following video segment provides a rigorous proof that the full field strength tensor we've constructed does indeed have the correct transformation property, and it also unpacks the component form.

The Strong Nuclear Force as a Gauge Theory, Part 4: The Field Strength Tensor

Having seen why the simple approach fails, let's watch the successful construction. This part of Richard Behiel's video presents the full definition of the field strength tensor and proves its gauge covariance in detail.

Watch from 27:31 to the end. The first part (27:31-43:58) is a detailed proof that the full F_μν transforms correctly. The second part (43:58-54:14) shows that the components of F_μν live in the Lie algebra. The final part (54:14-end) explicitly derives the component form F_μν^a, which we just wrote down, and discusses its physical meaning.

Test your understanding!

Consider a gauge theory with symmetry group SU(2). The generators can be taken as $T^{a} = \frac{1}{2} σ^{a}$ (where $σ^{a}$ are the Pauli matrices), and the structure constants are $f^{ab c} = ϵ^{ab c}$ .

Write down the covariant derivative $D_{μ}$ acting on a matter field $ψ$ that transforms as a doublet under SU(2).
Write down the field strength tensor $F_{μν}$ in matrix form.
Write down the component form of the field strength, $F_{μν}^{c}$ , for this SU(2) theory.

Show answer

The covariant derivative is a $2 \times 2$ matrix:
$D_{μ} = \partial_{μ} - i g A_{μ} = \partial_{μ} - i g A_{μ}^{a} T^{a} = \partial_{μ} - i \frac{g}{2} A_{μ}^{a} σ^{a}$
The field strength tensor in matrix form is:
$F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}]$
where $A_{μ} = \frac{1}{2} A_{μ}^{a} σ^{a}$ .
The component form is $F_{μν}^{c} = \partial_{μ} A_{ν}^{c} - \partial_{ν} A_{μ}^{c} + g f^{ab c} A_{μ}^{a} A_{ν}^{b}$ . For SU(2), with $f^{ab c} = ϵ^{ab c}$ , this becomes:
$F_{μν}^{c} = \partial_{μ} A_{ν}^{c} - \partial_{ν} A_{μ}^{c} + g ϵ^{ab c} A_{μ}^{a} A_{ν}^{b}$

Conclusion

In this lesson, we have successfully generalized the gauge principle from the Abelian U(1) symmetry of QED to a generic non-Abelian SU(N) symmetry. This procedure, first developed by Yang and Mills, forms the basis for the modern description of fundamental forces.

Key Takeaways:

Gauging a symmetry is the process of promoting a global symmetry to a local one by introducing a covariant derivative and associated gauge fields.
For an SU(N) symmetry, there are $N^{2} - 1$ gauge fields $A_{μ}^{a}$ , which are packaged into a Lie-algebra valued gauge potential $A_{μ} = A_{μ}^{a} T^{a}$ .
The covariant derivative is $D_{μ} = \partial_{μ} - i g A_{μ}$ . It ensures that the matter Lagrangian remains invariant under local gauge transformations.
The gauge potential $A_{μ}$ transforms inhomogeneously: $A_{μ} \to U A_{μ} U^{†} + \frac{i}{g} (\partial_{μ} U) U^{†}$ .
The field strength tensor $F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}]$ is derived from the commutator of covariant derivatives, $[D_{μ}, D_{ν}] = - i g F_{μν}$ .
The non-Abelian field strength contains a commutator term $- i g [A_{μ}, A_{ν}]$ , which signifies that the gauge bosons interact with themselves.
Under a gauge transformation, the field strength transforms in the adjoint representation: $F_{μν} \to U F_{μν} U^{†}$ .

Preview of the Next Lesson:

We have successfully derived the key building blocks of a non-Abelian gauge theory. In the next lesson, we will use the field strength tensor $F_{μν}$ to construct a gauge-invariant Lagrangian for the gauge fields themselves—the Yang-Mills Lagrangian. We will identify the specific terms in this Lagrangian that describe the self-interactions of the gauge bosons, which are a hallmark of theories like QCD.

Recommended resources

From Global to Local SU(N) Symmetry

The Non-Abelian Covariant Derivative

The Non-Abelian Field Strength Tensor

The Field Strength in Component Form

Conclusion