Fields and Action

As I mentioned earlier, the principle of least action is one of the most fundamental principles of physics, governing all known laws of physics.

Without it we would have no reason to believe in energy conservation or even the existence of solutions to the equations we write down. We will also base our study of fields on an action principle. The action principles that govern fields are generalizations of those for particles. Our plan is to examine in parallel the action principles that govern fields and those that govern particles, comparing them as we go. To simplify this comparison, we will first restate the action principle for nonrelativistic particles in the language of fields.

4.2.1 Nonrelativistic Particles Redux

I want to briefly go back to the theory of nonrelativistic particles, not because I am really interested in slow particles, but because the mathematics has some similarity with the theory of fields. In fact, in a certain formal sense it is a field theory of a simple kind—a field theory in a world whose spacetime has zero space dimensions, and as always, one time dimension.

To see how this works, let’s consider a particle that moves along the x axis.

Ordinarily we would describe the motion of the particle by a trajectory x(t).

However, with no change in the content of the theory, we might change the notation and call the position of the particle ϕ. Instead of x(t), the trajectory would be described by ϕ(t).

If we were to re-wire the meaning of the symbol ϕ(t), that is, if we use it to represent a scalar field, it would become a special case of ϕ(t, Xⁱ)—a special

case in which there are no dimensions of space. In other words, a particle theory in one dimension of space has the same mathematical structure as a scalar field theory in zero dimensions of space. Physicists sometimes refer to the theory of a single particle as a field theory in (0+1)-dimensions, the one dimension being time.

Fig. 4.1 illustrates the motion of a nonrelativistic particle. Notice that we use the horizontal axis for time, just to emphasize that t is the independent parameter. On the vertical axis we plot the position of the particle at time t, calling it ϕ(t). The curve ϕ(t) represents the history of the particle’s motion. It tells you what the position ϕ is at each moment of time. As the diagram shows, ϕ can be negative or positive. We characterize this trajectory using the principle of least action.

As you recall, action is defined as the integral of some Lagrangian , from an initial time a to a final time b:

Figure 4.1: Nonrelativistic Particle Trajectory.

For nonrelativistic particles, the Lagrangian is simple; it’s the kinetic energy minus the potential energy. Kinetic energy is usually expressed as , but in our new notation we would write or instead of v for velocity. With this notation, the kinetic energy becomes , or . We’ll simplify things a little by setting the mass m equal to 1. Thus the kinetic energy is

What about potential energy? In our example, potential energy is just a function of position—in other words it’s a function of ϕ, which we’ll call V(ϕ).

Subtracting V(ϕ) from the kinetic energy gives us the Lagrangian

and the action integral becomes

As we know from classical mechanics, the Euler-Lagrange equation tells us how to minimize the action integral and therefore provides the equation of motion for the particle.¹ For this example, the Euler-Lagrange equation is

and our task is to apply this equation to the Lagrangian of Eq. 4.1. Let’s start by writing down the derivative of with respect to :

Next, the Euler-Lagrange equation instructs us to take the time derivative of this result:

This completes the left side of the Euler-Lagrange equation. Now for the right side. Referring once more to Eq. 4.1, we find that

Finally, setting the left side equal to the right side gives us

This equation should be familiar. It’s just Newton’s equation for the motion of a particle. The right side is force, and the left side is acceleration. This would be Newton’s second law F = ma had we not set the mass to 1.

The Euler-Lagrange equations provide the solution to the problem of finding the trajectory that a particle follows between two fixed points a and b.

They’re equivalent to finding the trajectory of least action that connects the two fixed end points.²

As you know, there’s another way to think of this. You can divide the time axis into a lot of little pieces by drawing lots of closely spaced vertical lines on Fig. 4.1. Instead of thinking of the action as an integral, just think of it as a sum of terms. What do those terms depend on? They depend on the value of ϕ(t) and its derivatives at each time. In other words, the total action is simply a function of many values of ϕ(t). How do you minimize a function of ϕ? You differentiate with respect to ϕ. That’s what the Euler-Lagrange equations accomplish. Another way to say it is that they’re the solution to the problem of moving these points around until you find the trajectory that minimizes the action.