Now that we’re well acquainted with the characters in the play that is Particle Physics (the fundamental particles of the Standard Model), we can begin to delve into the real meat of what it is Physicists are trying to do. Simply stated – Physicists are interested in telling the future. It sounds farfetched, but the crux of Physics lies in building mathematical models which provide the user with the power to make predictions about the evolution of dynamic systems based on information about the system’s initial state. For a long time, the primary objective of Physics could be stated in the form of a single problem:
1) Given the present position and momentum of the constituents of some system acted upon by a known external force, determine the position and momentum of the system for all later times.
This sounds like a tall order – maybe even an impossible task. The universe is an infinite collection of systems. To understand every single one of them would require an infinite number of different analyses, right? Wrong. Just like the physicist was able to simplify the problem of thinking about the wealth of different kinds of matter by breaking down the problem to its fundamental parts – the Standard Model particles – the Physicists is here able to generalize the problem of understanding the evolution of a system into a single guiding principle – The Principle of Least Action. (The mathematical structure of the standard model can even be traced back to this principle!).
The Principle of Least action states: A physical system will evolve in time in such a way as to extremize its action. This probably sounds like a lot of mumbo jumbo. What is “Action” and what does it mean to extremize it? To answer that question, consider a different one: if you wanted to go from your house to your favorite restaurant, what path would you take? Now, perhaps you prefer the scenic route, but more likely than not you’d choose the path which takes the least amount of time. How exactly would you do this? Well, you’d probably whip out your GPS and let it guide you through the most direct path, recalibrating systematically to optimally avoid any traffic build up or road hazards.
This is perfectly analogous to what I mean when I say a system extremizes its action. Just as when we travel from our home to our favorite restaurant, we choose the path for which we are forced to spend the least amount of time, a particle traveling from point A to point B will choose the path for which it is forced to spend the least action. Where we let physical obstructions like buildings and traffic determine the shape of our path, the particle uses information about its underlying space – the forcefields that it is immersed in and constraints on its motion – to guide its trajectory.
Amazingly, the analogy between our chosen path and the path of a physical system is more than just an analogy when it comes to light – it’s an exact match. For particles of light – photons – the path of least action is exactly equivalent to the path of least time. That is, if we want to know how a ray of light will move from point A to point B, we need only determine the path between those two points which minimizes the travel time – break out the GPS!
Practically speaking, to compute the action for a path we rely on a Mathematical object called the Lagrangian. In particular, the Lagrangian of a physical system is equal to the difference between its Kinetic Energy, that is the energy acquired by an object in motion, and its Potential Energy, that is the energy of a distribution of objects on account of their interactions through forces like gravitation and electricity. Think of the Lagrangian as being analogous to the information analyzed by your GPS, with the Kinetic Energy being associated with the available streetways and the Potential Energy being associated with obstructions like traffic. The operative variable is the difference between these two entities because the optimal Action will be the one which strikes the perfect balance between taking the most direct path (Kinetic Energy) and avoiding the costliest obstructions (Potential Energy).
Once Physicists came up with the Principle of Least Action, they essentially felt that was it – Physics was “solved”. Perhaps this sentiment is best summarized in a quote by the famous Physicists and Mathematician Pierre Simon Laplace, he references Newton’s laws, but the Principle of Least Action and Newton’s Laws are perfectly equivalent mathematically. Laplace wrote, “An individual being equipped with unlimited calculating powers and given complete knowledge of the dispositions of all particles at some instant of time, could use Newton's equations to predict the future, and to retrodict with equal certainty the past, of the whole universe.”