If symmetry is going to play an important role in restricting applicable laws of physics, theorists have tried to characterize these transformations. Also the transformations might be able to select out a convenient set of structures that become the obvious “natural elements” of your theory.

 

Here are some well know structure

Scalars: temperature, charge, mass, speed 

Vectors: position, velocity, Electric field

Tensors: moment of inertia, stress, strain

 

  • Describe the structure.
    • What makes a vector a vector and how is it different from tensor?
    • Why do we bother to separate or categorize them?
      • In physics we check units and structure to ensure things make sense. One recognizes that a result that sets a scalar equal to a vector must be wrong. Why?  One

 

 

Elements will have different structure.  [ASIDE: 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.]

In early 1900 people tried to develop a model of the nucleus. Ingredients:

  • proton, neutron è nucleon
  • pion, omega, … è meson
  • delta è first resonant excitation of the nucleon

 

The people refined their models for the protn neutron and the interaction. Ingredients:

  • quarks
  • gluons è Best model for the force

 

However it I impossible to a first principles calculation.

 

???  Which approach is more correct ???

 

How accurate is the first approach in principle. I.e. do you ultimately need the more fundamental approach..

 

Answer boils down to approach the way we deal with a problem involving vectors.  Normally, people choose an axis parallel to the direction of gravity for accelerated motion problems.  This makes the problem easier. One might also choose spherical coordinate for a problem that has spherical symmetry. However, the physics and the solution are independent of the direction of the coordinate axes or the choice of  coordinates.  If enough mesons and bosons are included the calculations can be made as precise a desired.

 

set of states

set of states

meson-bosons

quarks - gluons

coordinates

coordinates

spherical

cartesian

 

Consider a genreal charge distribution

It is very difficult, in general, to find the electric and magnetic fields a point  due to a charge distribution

 

 

Why is this problem so difficult ?

So physicists ask if there are situations that make the problem more tractable.

 

If  is large compared to  [very localized charge distribution at a significant distance from the location where the fields are needed ] one might expect that the details as to how the charge is distributed are unimportant.  The dominant feature will be the total charge.

 

This point of view is mathematically justified by expanding the complete solution in terms of   the distance between a point in the charge distribution and the location where the fields need to be evaluated. The problem then separates into a set of terms where each term is characterized by a specific d dependence and a characterization of the structure of the charge distribution as a multipole.

 

  1. monopole term: charge is point like in nature
  2. dipole term: charge is vector like in nature
  3. HO terms: structure is more complicated and tensors are needed to describe the charge structure that is important.

The details of this procedure can be found in most E&M text books and on the web. For example the following sites describe this expansion nicely.

 

http://en.wikipedia.org/wiki/Spherical_multipole_moments

http://en.wikipedia.org/wiki/Multipole_expansion

 

multipole moments are defined in the above references:

Q_{lm} \ \stackrel{\mathrm{def}}{=}\   
q \left( r^{\prime} \right)^{l} 
\sqrt{\frac{4\pi}{2l+1}}
Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime}).

\Phi(\mathbf{r}) = 
\frac{1}{4\pi\varepsilon} 
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} 
\left( \frac{Q_{lm}}{r^{l+1}} \right)
\sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)

 

 

Usually the student is introduced to this procedure based on the problem expressed in Cartesian coordinates. The more revealing expansion is done in terms of spherical coordinates because the moments can be classified by their transformation properties under rotations.

 

How would one describe the monopole moment in terms of rotational properties?

  {independent è a rotation will not change the monopole moment}

How about the dipole?

  {vector è a rotation will mix the three components}

 

Originally a vector was introduced as a quantity with magnitude and direction. This type of entity was important because so many physics concepts force, position … possess this structure. However another natural way to introduce a vector is to characterize by its transformation characteristics under rotations. There are objects that transform as scalers, vectors, tensors under rotation.

 

To return to the problem of the charge distribution we discovered that we could break a general charge distribution into parts:

  • monopole
  • dipole
  • quadrupole
  • …..

 

Under rotations the form of any element sill change but not mix. 

Consider the monopole as a point it doesn’t change under rotation.

Consider the dipole as a vector. It rotates by pointing in a different direction. The new dipole is simply a transformation from the old dipole only.

This is not a proof but a statement of how these objects transform. The convenient separation can be illustrated by the following:

 

Above we have simply written all the elements of the charge distribution as a column matrix. The length of course goes to infinity.

Because of our choice of elements the rotation matrix takes on a special form.

 

The general matrix reduces to a block diagonal form where much smaller matrices can be used 1, r, z, .  If you can find those ways to break up a general structure such that the transformations on a general structure can be reduced because the elements in the structure don’t mix. Then you call the general structure reducible.  So the next step is to separate the elements.

 

 

If the resulting matrices 1, r, z, can not be further reduces then we say that each of the above equations represents a way for the group to behave. On the monopole all the group elements (ie all possible rotations) just behave as the identity element. For the dipole there are a set of 3x3 matrices and each rotation in the group has an associated matrix. For the quadrupole we need to specifically state that we choose the spherical coordinate expansion where the quadrupole has 5 elements. So now each rotation can be reprepresented as a 5x5 matrix.  In theoretical terms we have found the irreducible representations of the rotation group. Along with finding the matrices we have found a set of structures (monopole, dipole, quad….). 

 

I turns out that these transformations have a convenient mathematical form.

 

 

That is there is an operator  and a set of parameters that can be used to carry out any rotation.  To make sense of the above equation we simply use a Taylor series expansion for the exponential function. For the word operator one substitute the more specific word matrix which is a special case of the general form.  We have found matrix representations for the more general members of the group of rotations which are operators in that they take a state and get a new state.  I can ask you to pick up a book and rotate it. You are operating on the book. You are not a matrix. However mathematically you could represent your actions on the book in the form of a matrix. Thus giving the matrix to a fellow student they would be able to extract which operation you performed ie which rotation you had performed. So operation is a non-specific implementation of a rotation. It include physical realizations, matrix manipulation, any other implementation.

 

The interesting fact is that using the operator J which can be used to generate a rotation has separate the rotation into two convenient parts. A set of parameters and an independent operation. 

 

The structures that we have uncovered can be labeled in several different ways. For our example we could state the size of the matrix needed to perform the rotation. Monopole=1, dipole=3… One can also use the generator J to label the state. In QM this operator is the angular momentum operator. The structures then possess and intrinsic angular momentum. In particle physics we build elementary particles from the irreducible elements of the rotation group and label them by the their angular momentum which is defined as J the generator of the rotations.

 

 

 

 

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)