Question

Show that the Euler-Lagrange equation of $$\mathbf{L}[\phi]=\int L\left(\phi, \phi_{\mu}\right) d^{4} x \equiv \int \frac{1}{2}\left[\eta^{\mu v} \partial_{\mu} \phi \partial_{v} \phi-m^{2} \phi^{2}\right] d^{4} x$$ is the Klein-Gordan equation. Verify that \(T^{\mu v}=\partial^{\mu} \phi \partial^{v} \phi-\eta^{\mu v} L\) are the currents associated with the invariance under translations. Show directly that \(T^{\mu v}\) is conserved.

Short answer

The Euler-Lagrange equation derived from the given Lagrangian density is the Klein-Gordon equation. Moreover, the given \( T^{\mu v} \) are verified to be the currents associated with invariance under translations. Lastly, \( T^{\mu v} \) is proven to be conserved under the Noether's theorem.

Step-by-step solution

Derive the Euler-Lagrange equation

Start with the functional \( L[\phi] \) and derive the following Euler-Lagrange equation: \[ 0 = \partial_{\mu} \frac{\partial L}{\partial (\partial_{\mu} \phi)} - \frac{\partial L}{\partial \phi} \]

Substitute the Lagrangian density

Insert the provided Lagrangian density into Euler-Lagrange equation: \[ 0 = \partial_{\mu} \left( \eta^{\mu v} \partial_{v} \phi \right) - m^{2} \phi \] This leads to \( \partial_{\mu} \partial^{\mu} \phi + m^2 \phi = 0 \), which is the Klein-Gordon equation.

Verify the currents for invariance under translations

It is given that \( T^{\mu v} = \partial^{\mu} \phi \partial^{v} \phi - \eta^{\mu v} L \). By applying infinitesimal translation \( x^{\mu} --> x^{\mu} + a^{\mu} \), the Lagrangian is invariant and so the conservation law according to Noether's theorem applies with a current \( j^\mu = \partial^{\mu} \phi \delta \phi - \eta^{\mu v} L \). By expanding \( \delta \phi = - a_{\nu} \partial^{\nu} \phi \), the current \( j^\mu = \partial^{\mu} \phi \partial_{\nu} \phi a^{\nu} - \eta^{\mu v} L a_{\nu} \). Comparing this with the given expression for \( T^{\mu v} \) shows they are equivalent when \( a_{\nu} = \delta_{\nu}^{v} \) thus verify that the given definition of \( T^{\mu v} \) are indeed the currents associated with the invariance under translations.

Prove the conservation of \( T^{\mu v} \)

In order to show directly that \( T^{\mu v} \) is conserved, we can take the divergence: \( \partial_{\mu} T^{\mu v} = 0 \). After performing the derivative calculation, the equation simplifies to \( \partial_{\mu} T^{\mu v} = \partial_{\mu} \left( \partial^{\mu} \phi \partial^{v} \phi - \eta^{\mu v} L \right) = 0 \). This shows that \( T^{\mu v} \) is indeed conserved according to its definition via Noether's theorem.

Key concepts

Klein-Gordon Equation

The Klein-Gordon equation is a fundamental equation in quantum mechanics that describes the behavior of scalar fields. It is a relativistic wave equation that extends the classical wave equation to account for special relativity. The form we commonly encounter in physics is:\[ \partial_\mu \partial^\mu \phi + m^2 \phi = 0 \]Where \( \phi \) represents the scalar field, and \( m \) is the mass of the field. The solutions to this equation describe how these fields propagate through spacetime.

In the context of the Euler-Lagrange equation, inserting the appropriate Lagrangian density into the variational principle helps derive this equation naturally. It emphasizes concepts such as energy conservation and field dynamics, extending Newtonian mechanics.
Understanding the Klein-Gordon equation is crucial for those studying fields in quantum field theory as it lays the foundation for more complex equations and theories.

Noether's Theorem

Noether's theorem is a powerful and fundamental principle in physics. It links symmetries and conservation laws, highlighting that every differentiable symmetry of a physical system's action implies a conservation law. For example, the symmetry of a system under time translation leads to the conservation of energy.

In terms of the Euler-Lagrange framework, Noether's theorem helps find conserved currents by examining the symmetries in the Lagrangian. In our example, considering the translational symmetry, the theorem indicates the existence of conserved currents, represented by \( T^{\mu v} \), which remain unchanged even when we perform translations in space and time.
It is widely applicable in field theory, showing that conserved quantities (like momentum and energy) are directly related to these symmetries. This connection between symmetry and conservation principles is a cornerstone in understanding physical laws.

Lagrangian Density

The Lagrangian density is a field theory analogue of the Lagrangian of classical mechanics. It describes the dynamics of a field and is an integral part of forming equations of motion in field theory through the action principle.
The Lagrangian density for a scalar field typically takes a form like:\[ L(\phi, \phi_{\mu}) = \frac{1}{2} \left[ \eta^{\mu u} \partial_{\mu} \phi \partial_{u} \phi - m^2 \phi^2 \right] \]Here, \( \eta^{\mu u} \) represents the metric tensor in spacetime, ensuring the equation respects the principles of relativity. \( \phi_{\mu} \) refers to the partial derivative with respect to the spacetime coordinates, capturing how the field varies.
The Lagrangian density is integral in deriving equations like the Klein-Gordon equation through the Euler-Lagrange equation. It provides a concise way to express the physical laws governing a field in spacetime. The choice of a particular form for the Lagrangian density can significantly affect the properties and solutions of the associated field equations.