The Principle of Least Action
Nature is lazy. Scientific paradigms and "ultimate" visions of the universe come and go, but the idea of "least action" has remained remarkably unperturbed. From Newton's classical mechanics to Einstein's general relativity to Schrödinger's quantum field theory, every major theory has been reformulated with this single principle, by Euler, Hilbert, and Feynman, respectively. The reliability of this framework suggests a form for whatever the next major paradigm shift may be.
Action is a strange quantity, in the units of energy multiplied by time. A principle of least action does not explicitly specify what will happen, like an equation of motion does, but simply asserts that the action will be the least of any conceivable actions. In some sense, the universe is maximally efficient. To be precise, the action integrated over any interval of time is always minimal. Euler and Lagrange discovered that not only is this principle true, but one can derive all of Newtonian physics from it. The Newtonian worldview was often characterized as "clockwork," both because clockwork was an apt contemporary technology, and because of the crucially absolute measurement of time.
In Einstein's relativity, absolute time was no longer possible, and totally new equations of motion had to be written. Now we have to deal with four-dimensional spacetime, rather than the familiar three-dimensional space and the special dimension of time. At speeds much less than the speed of light, a first-order approximation can transform Einstein's equations into Newton's, yet the resemblance is hardly obvious. However, the principle of least action remains much the same, but with a difference that intuitively connects to the essence of Einstein's insight: instead of just integrating over time, we must integrate over space too, with the speed of light serving as a constant exchange rate between spatial and temporal units. The essence of relativity is not its well-known consequences—time dilation, length dilation, or even E=mc^2. Rather, it is the more intuitive idea that space and time are simply different ways of looking at the same thing. Much more complicated mathematics is needed to derive Einstein's equations from this principle, but the legendary mathematician David Hilbert was able to do it. Maxwell's theory of electromagnetism, too, can be derived from the least action principle by a generalization of operators. Even more remarkably, combining the least-action tweaks that lead to Einstein's and Maxwell's equations respectively produces modern relativistic electromagnetism.
By this point you may be imagining that practically any physical theory can be formulated using the principle of least action. But in fact, many cannot —for instance, an early attempt at quantum electrodynamics, put forth by Paul Dirac. Such theories tend to have other issues that preclude their practical use; under many conditions, Dirac's theory prescribed infinite energies (clearly a dramatic difference from experiment). Quantum electrodynamics was later "fixed" by Feynman, a feat for which he won the Nobel Prize. In his Nobel lecture, he mentioned that the initial confirmation he was on the right track was that his version, unlike Dirac's, could be formulated as a principle of least action.
I believe it's reasonable to expect it will be possible to explain the next major physical theory using the least action framework, whatever it may be. Perhaps it will benefit us as scientists to explore our theories within this framework, rather than attempting to guess at once the explicit equations, and leaving the inevitable least action derivation as an exercise for some enterprising mathematician to work out.
The essential idea of least action transcends even the deepest of theoretical physics, and enters the domain of metaphysics. Claude Shannon derived a formula to quantify uncertainty, which von Neumann pointed out was identical to the formula used in thermodynamics to compute entropy. Edwin Jaynes put forth an interpretation of thermodynamics, in which entropy simply is uncertainty, nothing more and nothing less. Although the formal mathematical underpinnings remain controversial, I find it very worthwhile, at least as an intuitive explanation. Jaynes' followers propose a profound connection between action and information, such that the principle of least action and the laws of thermodynamics both derive from basic symmetries of logic itself. We need only accept that all conceivable universes are equally likely, a principle of least information. Under this assumption, we can imagine a smooth spectrum from metaphysics to physics, from the omniverse to the multiverse to the universe, where the fundamental axis is information, and the only fundamental law is that you can never assume more than you know.
Starting from nothingness, or the absence of information, there is a flowering of possible refinements toward specifying a universe, one of which leads to our Big Bang, to particles, fields and spacetime, and ultimately to intelligent life. The reason that we had a long period of stellar, planetary and biological evolution is that this is the path to intelligent life, which required the least action. Imagine how much action it would take to create intelligence directly from nothing! Universes without intelligent life might require even less action, but there is nobody in those universes to wonder where they came from.
At least for me, the least action perspective explains all known physics as well as the origin of our universe, and that sure is deep and beautiful.
