Tue Mar 17 2026 14:02:34 GMT+0000 (Coordinated Universal Time)

Construction of the Yang-Mills Lagrangian

Explains how to construct the Yang-Mills Lagrangian.

Reference: (Peskin & Schroeder, 1995)

In the Construction of the QED Lagrangian, we required the Lagrangian to be invariant under local gauge transformations, specifically $U(1)$ transformations

\psi(x) \mapsto e^{i\alpha(x)} \psi(x)

Since the QED Lagrangian was constructed (almost) uniquely from such a requirement, it is expected that a Lagrangian invariant under transformations belonging to a more general group can be constructed in a similar manner.

Specifically, we treat the field $\psi(x)$ as an $n$ -plet and require the Lagrangian to be invariant under transformations by an $n$ -dimensional unitary representation of a Lie group:

\psi(x) \mapsto V(x)\psi(x)

Generally, the elements of the connected component containing the identity of a Lie group can be expressed using generators $t^a$ as

V(x) = e^{i\alpha^a(x) t^a} = 1 + i\alpha^a(x) t^a + \mathcal{O}(\alpha^2)

These generators form a Lie algebra, whose structure is characterized by the commutation relations:

[t^a, t^b] = i f^{abc} t^c

The constants $f^{abc}$ are called the structure constants.

From here, the same arguments used for the QED Lagrangian can be applied. The covariant derivative is defined as

D_\mu = \partial_\mu - i g A_\mu^a t^a

where a gauge field $A_\mu^a$ is introduced for each independent generator. Under local gauge transformations, it transforms as

A_\mu^a \mapsto A_\mu^a + \frac{1}{g} \partial_\mu \alpha^a + f^{abc} A_\mu^b \alpha^c

The last term originates from the non-commutativity of the group and does not appear in the case of abelian groups like in QED.

The field strength tensor is given by

F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c

This also contains a term originating from the non-commutativity. As a result, $F_{\mu\nu}^a$ is not gauge invariant itself, but the kinetic term $(F_{\mu\nu}^a)^2$ is gauge invariant.

Constructing a renormalizable Lagrangian that is invariant under parity and time reversal using the quantities above yields:

\mathcal{L}_{\text{YM}} = \bar{\psi}(i\cancel{D} - m)\psi - \frac{1}{4}(F_{\mu\nu}^a)^2

Such a theory is called a Yang-Mills theory. In QED, interactions exist only between the gauge field and the Dirac field, but in non-abelian Yang-Mills theories, there are also self-interaction terms of the gauge fields proportional to $f^{abc}$ .

References

Peskin, M. E., & Schroeder, D. V. (1995). An Introduction to quantum field theory. Addison-Wesley.