I realize that it is highly subjective to label one principle "the most beautiful in all of physics." If you ask ten different people you will likely get ten different answers, or at least three or four different answers. That said, I bet certain principles are more likely to come up than others. Here I will discuss a few together with some of the justifications that could be given for claiming them as "the most beautiful" in physics. The list of principles and their justifications is by no means exhaustive (on both counts) and input from others is appreciated. Here they are, in no particular order:
Conservation of Energy, Momentum, Angular Momentum, Electrical Charge, etc.Conservation laws, in general, are very useful in that they tell us that certain quantities don't change. For instance, if you calculate the combined momentum of two objects before a collision you can be sure that the
total momentum of the bodies after the collision will be the same provided there are no
external forces acting on the system. This applies even if the nature of the bodies has changed considerably, for instance two cars deforming and breaking into pieces during a collision. Or say an electron and positron collide and annihilate, producing two photons. Momentum is still conserved even though the nature of the particles involved changes completely during the collision.
Conservation laws allow us to calculate the outcome of many processes (or at least severely restrict the range of possible outcomes) without having to understand the internal details of the process at all.
The ultimate origin of conservation laws is intimately related to the next principle...
Symmetry in Physical LawSymmetry is the idea that there are operations that can be performed upon something, such that afterward the thing looks the same as it did before. Symmetry in physical law is then the idea that the mathematical equations of physics are invariant under certain groups of transformations. (Here
group is a technical term for a set of algebraic objects which obey certain laws of combination.
Group theory is the natural way to study symmetries mathematically.)
Symmetry in physical law appears at first glance an apparently empty concept. After all, the form of the laws doesn't change after the symmetry operation, so what does that tell us? Well, it turns out that, so far,
every result of physics is derivable from a symmetry principle. This is truly incredible, and I think the implications of this have yet to filter through to the undergrad level, although the result itself has been known for many years.
Because of a mathematical result called Noether's theorem every (continuous) symmetry of physical law is directly related to a conservation law. For instance:
- The conservation of energy is a result of the symmetry of physical law under translations in time.
- The conservation of momentum is due to the symmetry of physical law under translations in space. This, incidentally, explains why energy and momentum are so closely related in relativity: it's just because space and time are so closely related in relativity.
- The conservation of angular momentum is due to the symmetry of physical law under rotations. The tricky bit here is that you can tell when you are rotating, but not that you have rotated. The rotations considered must be constant in time in order to get the right symmetry group.
- The symmetry of physical laws under "boosts" (changes of velocity of the reference frame) results in the conservation of a quantity which, as far as I'm aware, lacks a popular name but is nonetheless conserved. It equates to the law that the center of mass of a system moves uniformly in a straight line (in the absence of forces).
- The conservation of electric charge is related to an abstract (i.e., "purely mathematical") symmetry operation called U(1) gauge symmetry. It is not a symmetry like translations or rotations in space. It is a transformation of an abstract quantity called the phase of the wavefunction - a quantum mechanical thing that is hard to describe in words (although the mathematics is unambiguous). A truly remarkable thing happens when you try to quantize a theory involving a gauge symmetry, which will be discussed under the Quantum Field Theory heading.
- Other gauge transformations based on the SU(3)xSU(2)xU(1) symmetry group of the standard model result in conservation laws for quantities which are less well known, such as weak hypercharge and color charge etc.
The question is: is nature symmetric under
every symmetry group? Evidently not. The Galilean group of transformations under boosts was replaced in favor of the Lorentz group by Einstein (more precisely, by Lorentz, Minkowski and Poincare based on Einstein's insights). Other symmetry groups have been tried and found wanting, or only approximately correct.
A famous example of this is the once posited SU(2) iso-spin symmetry of hadrons. This resulted in a theory of the strong nuclear force that is only approximately accurate. However, as this theory is considerably simpler than the currently accepted theory (quantum chromodynamics) it is still sometimes used today as an approximation in various situations.
So the question is, how do we decide which symmetry groups to use in the description of nature? Answer: experiment.
Stay tuned for a follow up post including the action principle (my personal pick for most beautiful in all of physic), quantum field theory and possibly other things...
or they can be written as the product of factors like
where the r's are the roots of the polynomial and A is just a (constant) scaling factor out front.
and we simply need to find the constant A. To do this we simply need to evaluate both sides at a particular value of x. It could get difficult to do this though unless we pick a nice value for x. Let's try x=0. But both sides are zero when x is zero so this is no good! Solution: divide both sides by x and take the limit as x goes to zero. This is a common trick really, and it works a treat here.
Now we know A. All that remains is to put it together. But notice that there is one term in the denominator in A for each term in the product from sin(x)/x. We can take a typical term and simplify it like this 
. So the product ends up being:
