Physics Hot Topic: Symmetry - The Metaprinciple of Physics
One of the most important results in physics is known as Noether's theorem, obtained by German mathematician and physicist Emmy Noether in 1915. Roughly, it states that whenever a physical system has a symmetry, it also has a corresponding conserved quantity. By 'symmetry' we mean some sort of transformation of the system that leaves it looking the same. For instance, the symmetries of the triangle are rotations by 120 degrees, with and without reflections.
https://www.britannica.com/biography/Emmy-Noether
One of the simplest and most remarkable results that can be found using Noether's theorem is that translational symmetry - "everything looks the same there as it does here" - leads directly to conservation of momentum, one of the most important rules in mechanics. In a similar way, rotational symmetry - "everything looks the same if we rotate it around some axis" - gives conservation of angular momentum, which is the essence of Kepler's second law of planetary motion (here, the axis we rotate around goes through the sun, perpendicular to the plane in which the planet is orbiting). A third powerful example closely related to conservation of momentum comes from temporal symmetry - "everything looks the same in the future as it did in the past" - which gives us conservation of energy, obviously a hugely important theorem.
The formalism in which these results are obtained, called Lagrangian mechanics, is comfortably beyond A-Level, but if your calculus is good you might be able to learn its basics. However, these examples can be grasped at a more intuitive level. Let's consider conservation of momentum. It is obvious that if you roll a ball down a valley it will pick up momentum as it goes down, and lose it as it goes back up on the other side. Our evaluation of its momentum rests on how its local environment looks - is it sloping up or down? If instead we roll it along a level plane, we expect it to maintain its momentum indefinitely. For if it changed there would have to be something about its local environment somewhere that looked different - if not, why would the properties of its motion change? Imagine standing on an infinite plane, and seeing a ball roll towards you from some unknown source. You are at a point A of its trajectory, but you have no way of identifying A, and there is really no way of distinguishing it from another point B. Therefore it must be the case that the ball has the same momentum at A as at B, since if you were instantly transported from A to B nothing would tell you you were in a different place. This is true for any pair of points A and B on its trajectory, so its momentum is constant. The reasoning behind conservation of angular momentum is essentially the same, if you swap translations out for rotations - see if you can get it. It's also a very similar argument for conservation of energy, but with time instead of space - this case is slightly more obscure but in principle deducible by analogy with the momentum case.
This gives a flavour of how understanding the symmetries of a physical system can be extremely powerful. Another case is conservation of electric charge, which follows from a slightly more obscure symmetry - a symmetry of the equations of electromagnetism. Here, symmetry means that if you change all the variables in a particular way, the equation itself remains unchanged. For instance, the equation y=x^2 is unchanged if we swap x to -x, and therefore this is a symmetry of it.
Technically, the symmetry in electromagnetism responsible for conservation of charge is known as U(1) symmetry. Essentially, the equations are unchanged if you multiply a particular variable by any complex number of unit norm (that is, a complex number e^{ia}, where a is real), the set of which is called U(1). Knowing that this symmetry exists in the equations of electromagnetism is a major help in dealing with the theory.
The area of mathematics that deals with symmetries is called group theory, and U(1) is an example of a group. (The basics of group theory are quite accessible, and in fact a part of Further Maths A-Level.) Another example is the group D_6, which describes the symmetries of the triangle, mentioned above. Now, there is a natural way of extending U(1) to groups that deal with complex square matrices, rather than just complex numbers. These groups are called SU(n), where n is the number of rows of the matrices. Remarkably, it turns out that SU(2) is a symmetry of the weak interaction, and SU(3) a symmetry of the strong interaction. This is so remarkable because, even if you had no idea about these forces, if you knew that electromagnetism was a U(1) theory, it would be perfectly natural to then play around with SU(n) theories. Now, almost all of our knowledge about the Standard Model of particle physics, arguably the greatest success of natural science in history, can be condensed into the statement that it is a field theory (don't worry what that means) with U(1)xSU(2)xSU(3) symmetry, known as the Standard Model group.
Now, the Standard Model is certainly not the end of physics, and many people have been trying to work out the deeper theory behind it for many decades. Crucial to this endeavour is finding a mechanism that will produce the Standard Model group. Thus, group theory guides all of these attempts and serves as the great organising and unifying principle in particle physics.
The other angle from which symmetry can be regarded, is as a sort of 'metaprinciple'. This goes back to 1905 and Einstein's landmark paper in which he introduced the special theory of relativity. As you may know, this is the theory that says that moving rods change their length, moving clocks tick more slowly, there is no such thing as simultaneity, and other seemingly nonsensical things. In reality, the theory is immensely rational, if at first glance counterintuitive. Descriptions of events differ according to different observers, but in a very specific way. We can perform 'Lorentz transformations' which take the observables according to one observer to those according to another. Once these transformations are utilised, agreement is recovered. One of the postulates of special relativity is that "the laws of physics are the same in every inertial frame" (inertial just means 'moving at constant velocity'). So physicists have learned to treat special relativity as a 'metatheory', which states that all laws of physics must 'behave nicely' when we apply Lorentz transformations to them. That is, the equations must exhibit a specific symmetry, called Lorentz covariance. Once again we find physics guided by symmetries.
Now, electromagnetism turns out to have the required symmetry, but conspicuously, Newton's law of universal gravitation does not. The essential problem is that Lorentz symmetry requires that "physics is local" - this means that something can only be affected by things in its immediate neighbourhood. For instance, the Sun is capable of heating up the Earth because of the photons that arrive here and hit the atmosphere, oceans and land. However, Newtonian gravity has no such mechanism - instead, some unknown influence somehow instantly causes the Earth to 'feel' the presence of the Sun and act appropriately. (It is interesting to note that Newton was completely aware of this rather glaring logical deficiency in his theory, but as observation vindicated him time after time, and attempts to solve the problem were unsuccessful, it was somewhat forgotten by the physics community.)
This prompted Einstein to develop his greatest achievement - the general theory of relativity. This basically generalises Lorentz transformations and covariance to what is known as general covariance. This enforces locality, and gravity as the observed effect of a curved spacetime is what elegantly results. Now, much like the Standard Model, it is known that Einstein's gravity cannot be the final word, and again, many people have been working for a while to find a so-called theory of quantum gravity. In this, they are guided and constrained by general covariance, just as Beyond Standard Model theorists are guided by the Standard Model group.
In fact, in some unified approaches, most notably string theory, both are guides, as physicists attempt to get out both general relativity and the Standard Model of particle physics from one underlying theory. It is impossible to overstate the importance of symmetry principles in these endeavours. It's also worth noting that symmetries are also just as important in condensed matter physics, the area of physics that studies solids and liquids, and phenomena such as magnetism and superconductivity.
In short, symmetry principles have been central to the activity and results of physics for a century, and the future is certainly symmetric. While they are absolutely indispensable in current research, they are also, as the examples we started with hopefully show, applicable to school-level physics, and are a wonderful way to gain a deeper appreciation of what you're learning. Moreover, anyone hoping to study physics at university will have to get used to using symmetry arguments at some point, so why wait?
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