Klein-Gordon Field Introduction

Previously, we introduced the ideas of field theory and the need for it. In this section, we will focus on the Klein-Gordon field, which is a fundamental example of a scalar field in both classical and quantum field theory.

Table of Contents

• Table of Contents
• Introduction
• Plane Wave Solutions
  • Superpositions of Solutions
• U(1) Symmetries and Conservation of Charge

Introduction

The Klein-Gordon field, as previously shown, is a scalar field that is derived from a set of infinite coupled harmonic oscillators. It is a relativistic field that obeys locality, and has wavelike solutions. In quantum mechanics, it originates from the canonical quantization of the energy-momentum relation

To begin, consider the Lagrangian density of the field, which is the difference between the kinetic and potential energy densities. These are

where is the spring constant associated with the harmonic oscillator. In quantum field theory, it is often defined as . Although we are not analyzing the quantized version of the field in this section, we will nevertheless use this definition for . In this respect, as is affected by , the "mass" describes the stiffness of the oscillators in the field.

Thus, the Lagrangian density is

To promote this to all four spacetime dimensions, we just need to add the other spatial derivatives;

Finally, we can rewrite this in a more compact form with relativistic notation:

where

is the Minkowski metric.

Anyways, with a Lagrangian density, we can now apply the Euler-Lagrange equation to derive the equations of motion for the Klein-Gordon field. For fields, the Euler-Lagrange equation is

In two spacetime dimensions, we need to sum over (i.e. and ). Let's calculate the individual derivatives. First,

Second (with ),

Lastly (with ),

Adding all of these together gives

Just like last time, we can upgrade to all dimensions by simply adding the other spatial derivatives:

And in relativistic notation this is

The operator is known as the d'Alembertian operator (or box operator) in Minkowski spacetime. It is often denoted as .

Plane Wave Solutions

The simplest solution to this equation is a plane wave of the form

We can see this solution satisfies the Klein-Gordon equation by substituting it back into the equation and verifying that both sides are equal. Its second derivatives are

So

From this we can see that the left-hand side must be zero. In other words,

This equation is known as the dispersion relation, the reason for which will become clear soon. We can also add a scaling factor and a phase factor to our plane wave solution:

where is the amplitude and is the phase shift. Using the cosine addition rule

we can also equivalently state this as

and if we introduce the constants

then

Another way to write this is to use complex exponentials through Euler's formula. This gives

where is a complex amplitude. If we are using a complexified Klein-Gordon field, we can forego the real part and write

Superpositions of Solutions

In field theory, we are often interested in superpositions of solutions to the Klein-Gordon equation. If we have two solutions and , then any linear combination of these solutions is also a solution:

where and are constants. This property is a consequence of the linearity of the Klein-Gordon equation. We can prove this to be the case.

Begin with two fields and that satisfy the Klein-Gordon equation:

Then, plug in the linear combination

to yield

This means that we can have a linear combinations of waves with different frequencies and momenta, and still obtain a valid solution to the Klein-Gordon equation. We can even construct an integral over a continuous spectrum of frequencies:

Let's, once again, promote to four spacetime dimensions. Given a plane wave, the exponential now has all three spatial coordinates, each with a different wavenumber;

If we let , then we can write

where

is the wavenumber 4-vector.

The dispersion relation in four dimensions is

U(1) Symmetries and Conservation of Charge

When we complexify the Klein-Gordon field, we can introduce a new symmetry to the field. To begin, we note that the complexified Lagrangian is

For a complex field, we can either work with its real and imaginary part separately, or work with the field and its complex conjugate. Often, the latter is more insightful as it handles the entire field as a single entity. We have two separate Euler-Lagrange equations for each component of the field;

We can introduce a transformation , where is a real constant. The term is a global phase factor and a member of the Lie group . The complex conjugate transforms the opposite way to .

When we apply this transformation, the Lagrangian modifies to

In other words, the Lagrangian is invariant under the global phase transformation. When a continuous group preserves the Lagrangian, it is called a symmetry (or automorphism) of the theory. Under Noether's theorem, this must correspond to a conserved current.

To see what this quantity is, apply the Klein-Gordon equations to and , and multiply each by the other:

Subtracting these gives

We can add and subtract and use the reverse of the product rule and get

We can interpret the quantity

as a conserved current 4-vector associated with this theory. Note the inclusion of to ensure that the current is real, as it must be for a physical observable. Per the equation, obeys a continuity equation

The same symmetry is also present in the Dirac field, so it also has a conserved current, which incidentally is the electric current density. As such, we say that symmetry leads to the conservation of electric charge.