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.

 

Classical theory

  • Point particles
  • Interactions
    • Fields: ultimately we need to introduce fields in order to treat classical forces.  Since light is an oscillation of these fields we have a concrete example an entity that is only field energy and so these, to me, become a basic ingredient.
  • Newton’s Laws
  • Space and time

 

 

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. Newton’s first law defines a world where the interactions are somehow turned off.  This is an important approach and typically one first tries to isolate abstractions of reality that contain only some features. Hopefully one can find essential features. An interaction is defined as the difference between “non-interacting” realities and those where the interactions are present.  Newton first describes what the world would be like if there were no forces (1st law) and then attributes variations from this behavior (accelerations) as due to forces.

 

 

Waves

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.

 

  • Classical waves are a manifestation of the behavior of a conglomeration of fundamental objects moving according to fundamental rules.  Waves are not fundamental.  Wave theory in this regard is like thermodynamics.
  • Sound
  • Waves are:
    • Local
    • Spread out
    • Add via interference rules (constructive and destructive)

 

Space time

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)

  • Distance
  • Time
  • Translation, rotation
  • Boost
    • Galilean velocity transformation
    • Relativistic velocity transformation
  • Stretch, curve space

 

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.

 

Practical tools

  • Matrix representation of rotation
  • Vector notation and vectors
  • Tensors
  • Expanding a charge distribution (tensors)
  • Vector contraction
  • Einstein notation
  • Relativity and four vectors