Tru Physics

U(1) Gauge Theory

—

by

Tru Physics
in

Introduction

U(1) gauge theory forms the cornerstone of quantum electrodynamics (QED), which describes the interactions between charged particles and electromagnetic fields. The “U” in “U(1)” stands for “unitary”, while the “1” signifies that the transformation involves a 1-dimensional unitary group. In essence, U(1) gauge theory is a mathematical framework that incorporates the principle of local gauge invariance into the quantum mechanical description of electromagnetic interactions.

Local Gauge Invariance

Local gauge invariance, a central concept in U(1) gauge theory, asserts that the physics of a system remains unchanged under a local transformation – that is, a transformation that can vary from point to point in space and time. In U(1) gauge theory, this local transformation is a phase shift of the wave function.

Consider a wave function \Psi(x), where x denotes space and time coordinates. The local U(1) gauge transformation is expressed as:

\Psi(x) \rightarrow e^{i\theta(x)}\Psi(x)

where \theta(x) is a real-valued function of space and time. The theory dictates that the physics described by \Psi(x) and e^{i\theta(x)}\Psi(x) should be identical.

The Gauge Field and Gauge Covariance

To ensure that the physics remains unchanged under local U(1) transformations, we introduce a complex vector field called the gauge field (also known as the electromagnetic potential). The gauge field transforms as follows under a U(1) gauge transformation:

A_{\mu}(x) \rightarrow A_{\mu}(x) + \dfrac{1}{e}\partial_{\mu}\theta(x)

where A_{\mu}(x) is the gauge field, e is the charge of the field, \mu is the spacetime index, and \theta(x) is the function determining the transformation.

The covariant derivative D_{\mu}, which replaces the ordinary derivative in the equations of motion, ensures the equations are invariant under U(1) gauge transformations:

D_{\mu} = \partial_{\mu} + ieA_{\mu}(x)

Maxwell’s Equations and the Lagrangian

In U(1) gauge theory, the dynamics of the electromagnetic field are described by Maxwell’s equations. These can be derived from a Lagrangian, which is also invariant under U(1) gauge transformations. The Lagrangian density for QED is:

\mathcal{L} = -\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar{\Psi}(i\gamma^{\mu}D_{\mu} - m)\Psi

where:

F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}

is the electromagnetic field tensor, \bar{\Psi} and \Psi are the Dirac spinor fields, \gamma^{\mu} are the gamma matrices, and m is the mass of the electron.

Conclusion

The U(1) gauge theory lays the foundation for the modern understanding of electromagnetic interactions in the framework of quantum mechanics. The theoretical framework it provides underpins all of quantum electrodynamics and has been extended to describe the weak and strong nuclear forces in the form of the SU(2) and SU(3) gauge theories, respectively. The unification of these theories into the Standard Model represents one of the most successful and tested theories in the realm of particle physics.

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Want to study more? Visit our Index here!

Comments

Have something to add? Leave a comment!