Noether current for a local gauge transformation for the Klein-Gordon Lagrangian?

The Noether current is a powerful concept in theoretical physics, particularly in the context of symmetries and conservation laws. When we talk about local gauge transformations in the context of the Klein-Gordon Lagrangian, we are delving into the relationship between symmetries and the physical quantities that remain conserved as a result of those symmetries. Let's break this down step by step.

The Klein-Gordon Lagrangian

The Klein-Gordon Lagrangian describes a scalar field, typically represented as:

• L = ∂μφ∂μφ* - m²φφ*

Here, φ is the scalar field, m is the mass of the particle associated with the field, and ∂μ represents the spacetime derivative. This Lagrangian is invariant under global phase transformations, where φ transforms as:

• φ → e^{iα}φ

with α being a constant phase. This invariance leads to the conservation of a current associated with this symmetry, according to Noether's theorem.

Local Gauge Transformations

Now, when we consider local gauge transformations, the phase α becomes a function of spacetime, α(x). This means that the transformation now depends on the position in spacetime:

• φ → e^{iα(x)}φ

In this case, the Lagrangian is no longer invariant under this transformation unless we modify it. To maintain invariance, we introduce a gauge field, typically denoted as Aμ, which couples to the field φ. The modified Lagrangian can be expressed as:

• L = (Dμφ)*(Dμφ) - m²φφ*

where Dμ = ∂μ + ieAμ is the covariant derivative, and e is the charge associated with the field.

Deriving the Noether Current

To find the Noether current associated with this local gauge transformation, we start by considering the infinitesimal transformation:

• φ → (1 + iα(x))φ

We can calculate the change in the Lagrangian due to this transformation. The variation of the Lagrangian can be expressed as:

• δL = ∂L/∂(∂μφ) δ(∂μφ) + ∂L/∂(∂μφ*) δ(∂μφ*) + ∂L/∂φ δφ + ∂L/∂φ* δφ*

Using the Euler-Lagrange equations, we can derive the Noether current Jμ associated with the gauge symmetry. The general form of the Noether current for a local gauge transformation is given by:

• Jμ = i(φ*(Dμφ) - (Dμφ*)φ)

Conservation of the Noether Current

According to Noether's theorem, if the action is invariant under a continuous symmetry transformation, then the corresponding Noether current is conserved. This means that:

• ∂μJμ = 0

This conservation law implies that the physical quantity associated with the Noether current, which can be interpreted as charge in the context of gauge theories, remains constant over time in a closed system.

Summary

In summary, the Noether current for a local gauge transformation of the Klein-Gordon Lagrangian is derived from the requirement of invariance under local phase transformations. By introducing a gauge field and modifying the Lagrangian accordingly, we can derive a conserved current that reflects the underlying symmetry of the system. This interplay between symmetries and conservation laws is a cornerstone of modern theoretical physics, particularly in quantum field theory.