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A Chain of Springs

In this note, we will explore a chain of springs, which is a simple example of a system that can be described by a field. Analyzing this system will help us understand field theories and eventually lead to the Klein-Gordon equation.

Table of Contents​

Case 1: Two Springs​

Let's start with a simple case: two springs connected to a wall and to each other. Suppose the mass-springs both have mass and spring constant , and the spring connecting them has a spring constant .

Denote the displacement of the first spring as and the displacement of the second spring as . The displacement of the spring connecting them is then .

The kinetic energy of the system is simply given by the sum of the kinetic energies of the two masses:

The potential energy of the system is given by the sum of the potential energies of the two springs, plus the potential energy of the spring connecting them:

The Lagrangian of the system is then given by :

Since we have two generalized coordinates, we will need to use the Euler-Lagrange equations twice to find the equations of motion.

First, the Euler-Lagrange equation for is:

Similarly, the Euler-Lagrange equation for is:

We can arrange these equations into a matrix form:

As always with matrix equations, we can solve this system by finding the eigenvalues and eigenvectors of the matrix. The motivation for this is that given a matrix , if , it is not immediately clear what is. However, it does appear to resemble an exponential differential equation like , which has the solution . In that case, if we replace with , it becomes an exponential differential equation, and we can solve it.

If we denote the matrix as , and the vector as , we can write the equation as:

In our case, we have a second-order differential equation. The eigenvectors of the coefficient matrix, are called the normal modes or eigenmodes of the system.

I shall employ the following fast trick to find eigenvalues - the determinant of the matrix is equal to the product of the eigenvalues, and the trace is equal to the sum of the eigenvalues. Thus, if is half the trace and is the determinant, the eigenvalues are .

The eigenvalue-eigenvector pairs turn out to be:

In the first case, the eigenvalue is , and the eigenvector is . This means that , which is the case where the two masses move together.

In the second case, the eigenvalue is , and the eigenvector is . This means that , which is the case where the two masses move in opposite directions.

For either case, if the eigenvalue is , then plugging this back into the differential equation yields:

This is analogous to the simple harmonic oscillator equation; the solution to this is:

Therefore, our two normal modes are:

Our general solution is then a linear combination of these two modes:

Case 2: Infinite Springs​

Now, let's consider the case where we have an infinite number of springs connected to each other.

Imagine a chain of masses connected by springs attached to the ground. They oscillate up and down (assume gravity does not exist; the upward-downward direction is arbitrary). Instead of thikning about the th spring, we can think about the spring at some position . The height of the spring at position is .

This is where we start to see the connection to fields. Instead of thinking about the position of each mass, we think about the displacement of the spring at each position. This is a field - a function of position and time.

We will now apply a fundamental shift in our thinking. Since kinetic energy is a quantity for a single particle, it does not make sense to talk about the kinetic energy in this case. Instead, we will think about the kinetic energy density - the kinetic energy per unit length:

Similarly, the potential energy density is the potential energy per unit length:

We will make some changes to simplify the problem:

  1. Let for simplicity.
  2. Apply a change of variables: . This is a simple scaling of the field.

With these changes, the new energy densities are:

Lastly, we will define (this is a new variable, no longer the mass of the particles):

Just like with energies, the Lagrangian has a density as well. The Lagrangian density is the Lagrangian per unit length:

Substituting the expressions for and , we get:

To convert from densities to total quantities, we integrate over the length of the chain:

In order to derive the equations of motion, we will use the Euler-Lagrange equation for fields, a generalization of the Euler-Lagrange equation for particles. We shall derive it from Hamilton's principle to find the equations of motion. Since the action is the integral of the Lagrangian over time, the action for the field is:

As before, we will vary the function a bit and set the variation to zero. In this case, .

The key is that unlike our particle case, the Lagrangian has more quantities to vary. More specifically, it depends on , , and . As such, we will need to include all of these in the chain rule when varying the Lagrangian.

Below is the expansion of the change in the Lagrangian when varying . The terms in green are the new terms that did not appear in the particle case.

Just like before, the first term vanishes when integrated over all spacetime, and we are left with:

This must hold for all , so the part inside the parentheses must vanish:

Euler-Lagrange Equation for x-t Fields: For a field , the Euler-Lagrange equation is:

Plugging in the Lagrangian density, we get:

One solution to this equation is, as one might expect, a wave equation:

The constants and must satisfy . (You can verify this by plugging in the solution into the equation of motion.)

The general solution is a superposition of these waves, each with different wavenumbers.

It is often useful to write this in terms of complex exponentials. Additionally, since is continuous, the most general solution is an integral over all possible values. Therefore, the most general solution is:

This is the solution to the wave equation, and it describes the motion of the chain of springs.

Case 3: Klein-Gordon Equation​

Next, we will add two new dimensions to the system, and .

The Lagrangian density for this system is:

Nothing has changed significantly; there are just more terms in the Lagrangian density.

Plugging this into the Euler-Lagrange equation, we get:

This equation is known as the Klein-Gordon equation. It is a relativistic wave equation that describes scalar particles. We will see in the future how this equation arises in quantum field theory.

We can also clean up the notation a bit. Recall that for fields, the Euler-Lagrange equation is:

First, notice that for all , , , and , the same term appears in the Lagrangian density. We can combine them into a sum over all dimensions :

In special relativity, it is often customary to use the coordinate instead of such that all coordinates have the same dimension. Changing the coordinate to does not change the physics, but it does make the equations easier to write. Often, we use a system called natural units in theoretical physics, where , to simplify equations.

We will also remove the summation symbol and simply write as the index - the summation is implied. This is known as the Einstein summation convention.

Making both of these changes, the Euler-Lagrange equation for fields becomes:

Euler-Lagrange Equation for Fields: Given a field defined by the Lagrangian density , the Euler-Lagrange equation is:

For our Klein-Gordon equation, we know that the equation becomes . In special relativity, we define geometry with the Minkowski metric :

(under the mostly-minus convention) Notice that the signs match the signs of the derivatives in the Klein-Gordon equation. We can write it as:

Using the Einstein summation convention, we can write this as:

Lastly, if one is familiar with how raising and lowering indices work, we can write this as . The operation is known as the d'Alembertian operator, and because it is very common, it is often written simply as or .

Klein-Gordon Equation: The Klein-Gordon equation for a scalar field is:

This is the Klein-Gordon equation in a more compact form, and it is what most textbooks, papers, and researchers use.

It turns out that this equation is very important in quantum field theory. It describes scalar particles, and it is the starting point for many quantum field theories.

Momenta​

Recall that in Lagrangian mechanics, we generalize the concept of momentum to the canonical/conjugate momentum:

In field theory, we generalize this to the canonical/conjugate momentum density:

Summary and Next Steps​

In this note, we explored a chain of springs and derived the Klein-Gordon equation. We saw how the concept of fields arises from a system of particles and how the Lagrangian density is used to describe the system.

Here are the key points to remember:

  • A chain of springs can be described by a field, which is a function of position and time.

  • The kinetic energy, potential energy, and Lagrangian of the system can be written as densities rather than total quantities:

    QuantityDensityIntegral
    Kinetic Energy
    Potential Energy
    Lagrangian

  • For a matrix differential equation, the eigenvalues and eigenvectors are the normal modes of the system. The general solution is an integral over all possible normal modes.

  • Einstein's summation convention simplifies the notation for equations with many indices. It is often used in special relativity and field theory.

  • The Euler-Lagrange equation for fields is a generalization of the Euler-Lagrange equation for particles. It is given by :

    We derived it from Hamilton's principle and used it to find the Klein-Gordon equation.

  • The Klein-Gordon equation is an equation that describes scalar particles. It is given by :

    where , and is the (inverse) Minkowski metric. (The Minkowski metric is self-inverse.)

  • The canonical/conjugate momentum density is defined as .

It is important to see that a field theory is defined by its Lagrangian density. Given an expression for , all the equations of motion, normal modes, and solutions can be derived. Field Theory is not in itself a specific theory; it's more of a class of theories that describe fields.

Different field theories arise from different Lagrangian densities. As mentioned, it turns out that the Klein-Gordon equation describes scalar spin-0 particles, and it is the starting point for many quantum field theories. Other field theories exist for other types of fields, such as spin-1/2 (Dirac, spinor fields) and spin-1 (Proca, vector fields).

We will also see in the future that quantum field theory can arise from the same principles we have discussed here. Just like with the coupled springs, we can construct a set of coupled quantum oscillators that describe particles and fields. Then, we imagine an infinite number of these oscillators, and we arrive at quantum field theory.

References​

Appendix: Eigenvalue Trick​

The trick I used to find the eigenvalues of the matrix is a common one. Here I will provide justification for it.

The usual way to find the eigenvalues of a matrix is to solve the characteristic equation . Our trick is a shortcut to find the eigenvalues without explicitly calculating the characteristic equation. It relies on two properties of the matrix: the trace and the determinant. I will now show why this trick works.

  1. The trace of a matrix is the sum of the eigenvalues.

    This is clear when we realize that the trace does not depend on the basis we choose. If we choose a basis where the matrix is diagonal, the trace is the sum of the diagonal elements, which are the eigenvalues. Thus, the trace is the sum of the eigenvalues.

  2. The determinant of a matrix is the product of the eigenvalues.

    This has a nice geometric interpretation. Recall that the determinant of a matrix is the volume scaling factor of the matrix. If we imagine a volume that is spanned by the eigenvectors of the matrix, applying the matrix scales the volume by the determinant. But it also scales each eigenvector by the corresponding eigenvalue. Thus, the determinant is the product of the eigenvalues.