# Several stationary points of the action functional

In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points (for a fixed choice of boundary conditions)? What's the significance of these other solutions?

There can be more than one stationary classical solution to an action principle with pertinent boundary conditions. E.g. because of instantons or gauge symmetry.

Example: A rotating rigid body with moment of inertia $I$. The action is $$\tag{1} S~=~\int_{t_i}^{t_f}\! dt ~L , \qquad L ~:=~\frac{I}{2}\dot{\theta}^2,$$ with Dirichlet boundary conditions (BC) $$\tag{2} \theta(t_i)-\theta_i~\in~2\pi\mathbb{Z} \qquad\text{and}\qquad\theta(t_f)~-\theta_f\in~2\pi\mathbb{Z} .$$ It is possible to satisfy the EOM $\ddot{\theta}\approx 0$ and BC (2) in infinitely many ways corresponding to different number of windings.

Instanton solutions are an important feature of a quantum field theory. On the other hand, gauge ambiguities are typically removed by gauge-fixing.

As an example, consider the real scalar field with action,

$$S=\int \mathrm{d}^4 x \, \left( \partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2\phi^2\right)$$

The principle of stationary action insists $\delta S = 0$, and the fields which allow this are those which satisfy the associated Euler-Lagrange equations, which are simply,

$$(\square+m^2)\phi=0$$

As the OP noted, there are many solutions to the equations of motion; this is not an issue. For the purposes of canonical quantization, we usually expand the field as a plane wave, and promote the Fourier coefficients to operators, etc. A theory may have other solutions, and in many cases it is interesting to study these, c.f. solitons.