Imagine a toy city, seen from afar. Now imagine that some of the buildings have Lego-shaped castellations, others have Lego-shaped holes in the walls, and there are a few loose Lego bricks lying around. All this evidence leads us to guess that the whole toy city is made up of Lego bricks. When we get up close, we see that our guess is correct.
By a similar blend of evidence and theorizing, John Dalton, around 1800, came up with the Atomic Theory — the theory that says that matter in all its variety (whether grass, mud, cricket balls, etc.) is made from a finite number of fundamental tiny building blocks.
Even earlier, Isaac Newton, in 1687, postulated that everything in physics could be explained by particles acted upon by forces. In very complicated scenarios it might be necessary to have a near infinite number of particles. For example, when Newton calculated the shape of the Moon’s orbit around the Earth he imagined the Earth and the Moon subdivided into tiny volume-elements (‘particles’), then determined the gravitational attraction between a particle in the Moon and a particle in the body of the Earth, and then determined the net Earth-Moon attraction by summing over all such pairs. He had to be careful that he didn’t miss any pairs out, or count some pairs more than once. It was a difficult task: “the only problem which gave me a headache”. He found the agreement between his predicted orbit and the strength of gravity on Earth to “answer pretty nearly”. This is one of the most remarkable theoretical confirmations ever made.
These three examples (Lego, the Atomic Theory, and Newtonian Mechanics) lead us to wonder whether every problem, however complicated, may always be broken down into the interactions between elemental tiny components.
Finally, we come to Margaret Thatcher. One of her most famous quotations, in 1987, was “There is no such thing as society”. This quotation caused such a storm that her press office took the unusual step of offering an explanation. It seems that Mrs Thatcher had meant that society is abstract, not real, and so the forward march of progress must be due solely to the motivations and actions of individuals.
We now know that all these theories are wrong — wrong if they assume that the given problem can always be reduced to a sum over elemental parts. In physics, despite the enormous success of Newtonian Mechanics (viz., the calculation of the Moon’s orbit), it is an astonishing discovery to learn that rather few problems can be solved in this way. We have all heard the saying “the whole is more than the sum of its parts” but this is only the beginning of correcting the wrongness. A society is more than just individuals; it is made up of entities that do exist, are real, and are influential (for example, the pub where my talk is held, the famous old university nearby, and so on).
In physics, it is the Principle of Least Action that teaches us a radically different approach. It shows us that it is not just a case of new properties emerging when all the parts are considered together; rather, it is the realization that (in all but a few exceptionally simple cases) the system cannot be considered as a collection of simple elemental components. Thus, instead of particles, we might have to have as our fundamental elements: lever arm, pendulum, capacitor, electromagnet, flexing beam, suspension bridge, planets, black hole, binary star, hydrogen atom, a flowing river, a spinning top, and so on. The components are neither simple, nor universal, but are system-dependent, that is they have to be re-formulated for each new scenario.
We have lost elemental simplicity — what have we gained in its place? What we gain is a universal Principle instead of a universal particle. The telling ingredients are not particles and forces, they are energies. The energies come in two types: kinetic (the energy of motion) and potential (the energy of configuration). However, the Principle of Least Action leads to even greater insight: rather than kinetic and potential energies, the true dichotomy is between ‘individual component’ and ‘super-structure’ energies. Finally, the Principle postulates that the individual energies and the super-structure energies always act in opposition to each other (one increases at the expense of the other) yet, through time, Nature takes the path where the difference between them is as small as possible.
How do we know that this more complicated, less intuitive Principle, is correct? For three reasons:
- It answers to many more scenarios —not just the usual Newtonian, but also the relativistic and quantum-mechanical domains.
- It works whether applied to a system that is stationary or moving (I dub it the ‘Harrison’s Chronometer’ of physics).
- It gives us new physical insights —especially into the nature of energy, but also into the nature of space and time.
As Einstein said (paraphrased), the method must be as simple as possible, but not simpler.
Returning to Margaret Thatcher, maybe the Principle of Least Action is hinting that, while there will always be a tension between the needs of individuals and the needs of society, for a stable society, that tension must be the minimum possible.
Featured image credit: Atoms by PIRO4D. CC0 public domain via Pixabay.