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Previous posts such as this ask about types of stationary point in Hamilton's Principle. There is, however, another aspect to discuss: the question as to whether the extremal path is unique.

One geometric way to envisage this is to assume that multiple paths are simultaneously extremal. I believe that this is an explanation for lenses, but I have not seen lenses explained as multiple classical solutions to Hamilton's Principle. (The multiple paths being the 360 degrees of rays between source and focus, etc also demonstrable through Fermat's principle.)

One can generalise lenses, but also consider a simpler case. Let the surface of a sphere be the action (phase space) surface which is minimized in classical paths. Thus (ignore antipodals here) between two points $A$ and $B$ the geodesic is the unique classical path. In quantum form the WKB approximation would no doubt have constructive maxima on this path.

However if the sphere has a disk (containing that geodesic) cut out, the shortest path now has exactly two choices: around one or the other rim from $A$ to $B$. Presumably WKB would maximize the quantum paths on these two (although I havent proved this). If so then classically we have a quantum-like phenomenon: a particle has a choice in going from $A$ to $B$. Experimentalists might see this and wonder whether the particle went from $A$ to $B$ via the LHS, the RHS or both....

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Roy Simpson
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2 Answers2

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[Another comment to answer transplant]

It seems like you're asking about a classical analog to the superselection sectors of quantum mechanics. One situation where this occurs in classical mechanics is when considering particle motion of a manifold with non-trivial topology - i.e. with holes and handles. In such cases there can be more than one extremal path from A to B. An example is an arcade pin-ball machine, if you're familiar with those.

Also, when you talk about optics it is important to keep in mind that there are two different regimes, those of wave optics and geometric optics. In the second case one has well-defined "trajectories" and you can find extremal trajectories. Not so in the first case.

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Dear Roy, for a chosen initial configuration $x_i(t_i)$ and final configuration $x_f(t_f)$, there can exist more than one local extrema of the action. However, this point is irrelevant in classical physics. At every moment, including the initial moment $t_i$, the particle also has a well-defined velocity $\dot x_i(t_i)$, and the principle of least action is just a way to derive the differential equations that determine the motion. For a given $x(t_i)$ and $\dot x(t_i)$, the evolution will be inevitably unique.

In your case, the initial velocity either says that the particle will avoid the removed disk from the left side, or from the right side, or it will hit the missing disk (and perhaps gets reflected from it) - unless you include some potential that repels the particle from the disk, there is no reason why the particle should be obliged to avoid it. So no issue of the type you mention exists in classical physics.

Quantum mechanics

The situation is different in quantum mechanics. All trajectories contribute and as the double-slit experiment shows, there may be interference between many classical histories. The interference pattern in the double-slit experiment may be obtained by adding the neighborhood of "two classical trajectories" only - these trajectories are piecewise linear and go through the two slits.

Quantum tunneling may also be phrased in terms of a contribution of complexified trajectories in the complexified time - that are also local extrema of the action.

A much more interesting situation arises in quantum field theories. The vacuum - that lasts eternally - is a global minimum of the action. However, in the Euclidean spacetime, they may also exist other local minima. Because they're local minima, they also solve the classical equations of motion.

In quantum field theory, such solutions are called instantons because they are localized both in space and in the Euclidean time (near one instant of time). They contribute to probabilities of various processes because one must sum over all histories, including those that get mapped to the instanton if one uses the Euclidean formalism (via the Wick rotation). In particular, they produce the 't Hooft interaction in gauge theories - a product of all fermions in the theory (a fact that is derived from the fermionic zero modes on the background of the instanton).

Instantons have to be stable (minima of the Euclidean action, not maxima) and they're typically protected by a topological charge by which they differ from the vacuum configuration (some "winding number" or "homotopy"). There can also exist unstable solutions - the saddle points that are minima with respect to most directions but they are maxima with respect to a finite number of directions in the configuration space. Such saddle-like mixed extrema are called sphalerons.

Luboš Motl
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