Review some of the elements of classical theory, discuss the way waves appear in a classical theory and highlight some mathematics and notation that will be useful.
It is always difficult to find the correct starting point. One interesting aspect of particle physics is that it leads to a different picture of what may be the most important elements of a theory. Some of the comments below are meant to stimulate thought and are not meant to be definitive.
The goal is to try and ask questions at this point of what
constitutes a fundamental classical theory.
Often when studying physics the student is struck by key ideas. I was excited by the idea that objects
naturally tend to keep moving at constant velocity. It seemed to be a profound
observation that added considerable clarity to how things behave. On the other hand, I devoted very little time
to considering what a point particle was or what it implied. I just accepted this element or viewpoint. For most introductory problems the question
of the nature of the smallest bits of matter are irrelevant and whether they
are small extended objects or really some dimensionless object is not very
significant. Even studying E&M as an
undergraduate, where a point particle will have infinite field energy because
the radius extends to zero, the point like nature was inconsequential. One of the major worries however for particle
physicists in the 70s (graduate school) was the problem of
Renormalization. How does one remove or
separate out those parts of the calculations that lead to infinite
answers. It seemed to me that something
must be wrong if calculations lead to infinities. While this may indeed be true I had no
problem accepting point charges as fine elements of a classical E&M
theory. I was willing to accept the fact
that these entities existed and that one need not concern themselves with the
energy required to create them. I had
basically accepted a renormalized theory very early on in my studies. So it is
worthwhile asking ourselves: what are the contents of the theory we now use to
understand our world? It also helps to
consider the abstractions that become important.
Quantum mechanics will develop a new framework for the description of matter and fields. Often the new framework is developed using the familiar concepts of waves and particles of classical physics. Of course, QM being the more fundamental theory, the best way of understanding would be have an intuition at the quantum level as to how matter and energy behave and then extrapolate that to the large scale level. This of course is not really possible because very few people have quantum experience as undergraduates. So we review how classical waves behave. We try to understand in what context they appear classically. The we can use classical wave behavior as a guide but at the same time realize that classical waves and quantum wave-like behavior are not at all the same thing. Classical waves are due to a medium of things that do not behave at all like a wave. Quantum wave-like behavior is just how things behave and all matter will act according to these rules.
One important ingredient in our theory will be the way we view space and time. For classical theories one basically postulates that our universe exists in something called space that has three dimensions and that events are marked by an absolute quantity called time. Einstein (general relativity) blurred the separation by making space and time a consequence of the evolution of our universe. He also eliminated the separate role that time played as a parameter and linked it with space as a special fourth dimension (special relativity)
Time is absolute (classical)
Distance is absolute (classical)
Review some of the mathematical tools to deal with space and time and also to describe other quantities such as velocity, momentum, force … These quantities will share common structure. What is the structure and how do we define the structure?
Scalars: temperature, charge, mass, speed
Vectors: position, velocity, Electric field
Tensors: moment of inertia, stress, strain
Elements will have different structure. As a matter of fact the structure can change depending on the theory. For a fully relativistic theory the electric and magnetic fields need to be combined into a second rank tensor and the neither field retains is character as a vector.