Rationale for the Principle of Least Action

Theoretical physicists use Hamilton's principle, according to which the effect always takes an extreme value, to derive the laws of motion. Richard Feynman applied this principle to quantum field theories using the path integral method, according to which a particle takes all possible paths, with each path weighted by its effect and integrated over all paths. But how can this fundamental role of the physical quantity “effect” be explained?

The fact that the equations of motion in all areas of physics can be derived from the principle of least action is highly remarkable. The natural philosophical tradition of thought saw this as a universally valid principle of nature to always run optimally. Max Planck also interpreted the principle of the smallest effect as an indication of the purposefulness of nature. Contemporary physicists avoid such a metaphysical superstructures. However, this leaves them speechless when the question arises as to why the principle of extreme effect is so powerful.

We want to try to find a physical reason why the physical quantity „effect" plays such a decisive role in the foundation of physics. Where could the search start?

We have found a hot lead in an unexpected place. While working on relativistic mechanics, we modified the well-known twin paradox slightly and came across an unorthodox idea.

Thought experiment

Let's imagine a huge planet on which two twins arrange to meet for an experiment. One twin has walked around the planet at the equator and determined the circumference of the planet to be 180,000,000 meters using a rod whose length we would call 1 meter. The other twin is now given the task of determining the circumference of the planet by using a rocket to orbit the planet just above the ground along the equator. Both twins also have identical clocks. The space traveler climbs into his rocket and accelerates it to 3/5 of the speed of light c. After he has reached the final speed and flies directly past his brother, both twins press the start button on their clocks. When the rocket passes the twin on the ground again, both twins press the stop button. The twin on the ground reads a time on his watch, which we call 1 second. The twin in the rocket, on the other hand, only reads 0.8 seconds on his watch due to relativistic time dilation. According to his measurement, the distance traveled along the equator was only 144,000,000 meters due to relativistic length contraction. However, he agrees with his twin on the speed of his rocket: the astronaut determines his speed as v = 144,000,000 meters / 0.8 seconds = 180,000,000 meters / second, while his brother comes to the same conclusion with v = 180,000,000 meters / 1 second.

At the moment the rocket passes by, the two twins are obviously in almost the same place and in the same present. However, the two twins disagree about the duration of time and distance traveled in between. Consequently, the present cannot be determined by spatio-temporal coordinates. Einstein had already come to this conclusion with his considerations on the non-simultaneity of coincident events for observers moving in opposite directions.

However, there is one physical quantity that has undergone the same change for both twins between the two encounters: The effect ΔS = E * Δt. For the twin on the ground, ΔS = m₀c² * Δt, for the twin in the rocket is ΔS = mc² * Δt' = γ * m_oc² * Δt / γ = m₀c² * Δt, where γ = (1 - (v/c)²)^-0,5  is the Lorentz factor. However, for this consideration to apply to twins that do not have the same rest mass m₀, we must use the relative change ΔS / S_o rather than the absolute change ΔS.

The thought experiment thus leads to the following result: During the transition from the present G1 to G2, the effect value S for each material particle and the macroscopic objects composed of them changes by a certain ratio S₂/S₁. Accordingly, it is not time that characterizes the present, but the parameter effect. So it is not time that flows equally for all physical entities, but the relative change of the effect value! This would make it immediately clear why all physical equations of motion can be derived from the principle of extremal action: All macroscopic objects take the path in space and time that gets them from A to B in the nearest possible present.