Principle of Least Action
Principle of Least Action
The modification of the kinetic energy and momentum of a particle by a potential energy field is familiar in many practical applications. It is used to steer an electron beam in video displays and to speed up electrons in particle accelerators. I believe, however, that its appearance in the process that minimizes deviations of photons from a direct path is an instance of an application of a universal physical law – the principle of least action. The following description of this principle is based on Richard Feynman's special lecture on the subject [10].
Feynman illustrated the principle as follows. Suppose you have a body in a gravitational field and you throw it up against the field in a vacuum (to avoid the effect of non-conservative forces like lift and drag from the air). In any fixed interval of time, which can be selected arbitrarily, it travels a specific path. At every moment along the path calculate the kinetic energy and subtract the potential energy. When you integrate the result over the fixed interval of time, the sum will be less than for any other path over the same time interval (such as the false path in Figure.2).
Figure 2: Feynman's illustration of the true path of least action (S1) and a false path (S2). By the principal of least action, for the actual path followed by an object in which kinetic and potential energy are interacting, the action S1 is a minimum. For any false path the action will be greater than the minimum. (Credit: California Institute of Technology.)
The sum of the energies is referred to as the 'action' over the path. The finding that the action over the path is a minimum is the principle of least action. Fundamentally it says that in the absence of non-conservative forces like friction and drag, nature does not waste energy but makes sure that in the application of energy the very minimum is used. Nature operates efficiently.
Non-conservative forces occur at the macro level of everyday life. However, at the elementary particle level they are absent. So at this fundamental level, the principle of least action is as important as the law of conservation of energy.
If there is no potential energy field, there is only kinetic energy contributing to the action. The path of least action becomes one of constant speed in a straight line. This is the path of light in the absence of forced deviations. It is also the path of an automobile being driven to minimize fuel consumption.
In discussing the principle of least action, Roger Penrose points out that it emerges from a profusion of theorems generalizing Newton’s laws of motion, developed by mathematicians like Euler, Laplace, Lagrange, Hamilton, and others [11]. The action integral that gives rise to this principle is also known as the Lagrangian, which has the advantage of lending itself to solutions of relativistic problems. As such it has been developed by Feynman as an alternative to wave mechanics, known as the sum over histories or the Feynman path integrals.
This approach extends the randomness in photon paths way beyond that proposed in the eternity theory of spacetime. In the sum over histories, photons take any and every possible path between a source and the final arrival point. A photon can take a path that initially has the opposite direction to the final destination, circle around via any route, and finally arrive at the destination. An integration sums the probability amplitudes of all possible paths to yield the probability of a specific path. It turns out that all of the wild paths cancel each other out, leaving a number of closely related paths with similar probabilities. The result is similar to the outcome of the wave guidance technique suggested above, which is a version of the path over histories approach with variant paths truncated by a guiding wave.
Feynman has used the sum over histories techniques to explain how quantum effects lead to familiar optical phenomena as reflection at a mirror, diffraction, and the principal of least time [9]. There is in fact a close connection between the path of least action and quantum mechanics. Feynman shown that when the action is significantly greater than the Planck constant, the two methods produce the same result [12].
Given its origins, it is not surprising that the principal of least action can be used to derive Newton's law that the acceleration of a body is proportional to the force applied to it divided by its mass. But whereas Newton's law tells us what happens at any point along a body's path, the principle of least action tells us about the nature of the entire path. The principle of least action is, in fact, a more general law that allows other laws to be derived from it.
Choosing a minimum-energy path out of millions of alternatives is more complex than finding the minimum on a fixed path. The latter can be calculated using simple calculus because the path is fixed. Where the path is not fixed, finding the right path requires the calculus of variations. But Feynman points out there is a clue that indicates when a particle is on the right path: a small deviation from the path has very little effect on the action (to the first order it is zero). It can also be shown that maintaining the path of least action over very small sections of the path leads to a complete path that follows the least action principle (Figure 3).
When fundamental particles with kinetic energy are moving through spacetime there are no non-conservative forces. So how can a particle steer through various energy fields into the future in such a way that out of all the billions of paths it could follow it travels the one that ensures that when it reaches the end of its path it will have expended the least energy? I suggest that a guiding wave available from interstitial energy provides the necessary steering.
Figure 3: Feynman argues that if the action for the entire path of a particle from time t1 to t2 is a minimum, the action for a small part such as ab is also a minimum. (Credit: California Institute of Technology.)
When the guiding wave is split into two parts by holes or slits in a screen, the two parts create an interference pattern that steers photons, electrons, or heavier particles like neutrons, to various points on the screen. The way the particles are deposited indicates that the steering wave is probably a form of sinusoidal oscillation. This also appears to be the case when x-rays are deflected by arrays of atoms in a crystal to produce a similar sort of interference pattern. In fact, quantum mechanics is built on the idea that particles can adopt the guise of a sinusoidal wave. So a reasonable estimate for the proposed guiding wave in interstitial energy is that it is sinusoidal wave of potential energy. Furthermore, since waves in general have an energy that is proportional to their frequency, a minimal energy solution to interaction between the wave and a particle is for the potential energy of the wave to equal the kinetic energy of the particle.
The steering of a particle along a least-action path would then proceed as follows. For simplicity, think of the interaction taking place in two dimension: time along the direction of the mean particle path and deviations at right angles to that path. The guiding wave is distributed in the interstitial spaces around the least-action path as a standing wave that moves along with the particle. It varies sinusoidally and is interacting with a particle that is moving forwards but deviates to and fro in relation to the wave peaks. At their peaks and valleys, the guiding wave peaks have positive and negative energy, available to add or subtract to the kinetic energy of the particle. For the particle to deviate from the least-action path requires extra energy, but this deviation may be forced over the interval of one quantum by the lack of any other path direction at the particular point in the lattice that the particle occupies. When this occurs, the needed energy is donated by a positive peak in the guiding wave.
In doing so, the guiding wave develops a deficit in its energy pattern and is primed with a negative portion able to absorb energy from the particle at the next opportunity. As the deficit builds up, so the energy deficit waiting to pull the particle back on track grows. Thus at the next twist in the lattice that offers the opportunity to take back energy from the particle, the wave does so, pulling the particle back towards a least-action path. The extent that deviations grow beyond least-action path will vary, and some particles may well escape. But there will be an envelope around the path, made up of a great many interstitial corridors across which successful correction is occurring. It is the deviating particles prevented from leaving this envelope that emerges in different directions from a sub-millimeter hole in an interposed screen. The method of cross-track correction is complex in that it extends over an area in the lattice even when considered in one cross-dimension. It becomes even more complex as further dimensions are added.
When we return to the macro level and consider rocks and structures moving through space, we recognize again that these objects are made up of fundamental particles that follow the least action principle. As the individual particles attempt move on minimum-action paths, they will experience the randomness of the lattice and will have adjust their relative positions within the object. But the lattice variations are small compared with the macro dimensions of the body. The space between particles is so large that their individual sub-microscopic variations in relative position are not likely to be observed. The body as a whole follows the least action principle. The quantum variations in the spacetime path have a different outcome at the macro level in this case than in the situation where bullets are fired through closely spaced holes.
It is time to examine the effects of gravity on the spacetime lattice, this is approached in the next section via the special theory of relativity.
12/19/2015