So we identified some groups of particles and asked if they should be grouped together. Yes they seem to share some common feature.  We characterize this commonality by imagining an operation that takes one of the members of these common particles and changes it to another (group operation).  We would also like to have a label that distinguishes these common particles.

 

Label è  singlet, doublet, triplet   (Casmir operators).

Certain members of the group may not change the particle but leave it the same.  This is a property of some elements. These as we will see can be used as labels. Other members of the group change from one to the other.

 

 

SO(3)  rotations in three dimensions

 

the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3 (real vector space with and inner product; length and angle are defined). Our notionn of what a rotation is connected with our understanding of how Euclidean space can be changed. By definition, a rotation about the origin is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space. {RHS vs LHS is preserved}. A transformation that preserves length but reverses orientation is sometimes called an improper rotation.

 

The notion of reversing the handedness is known as parity.  The standard example of parity inversion is “mirror world”.  [ASIDE: the direction perpendicular to the mirror is reversed. Zè-Z. ]

 

The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation. These properties give the set of all rotations the mathematical structure of a group. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth, so that it is actually a Lie group. The rotation group is often denoted SO(3). (Manifold structure is linked to the paramenters that define the rotaions.  There are 2-independent parameneters that identify every rotation

 

SO(3) is the simple 3x3 orthogonal matrices.

  • Simple:determinant = 1, proper rotations
  • 3x3
  • orthogonal:  AT A=1; A transposed = AT is the inverse of A

 

Length preserving and angle preserving transformations on a R3.

 

SO(3) is the group of rotations on 3-d Euclidean space. There are three independent parameters for ever rotation. One way to say this to imagine an axis defined by a unit vector. Two angles can specify the axis (θ, φ). Then a third angle can be used to determine the rotation about this axis.  Another way is to use the THREE Euler angles.

There are potentially 9 independent parameters in a matrix. Rotations limit the type of matrices so that three independent parameters are sufficient.

Thus you can make up a basis of elements Rx, Ry Rz and parameters that define how much of each element to employ and then reach any other member of the group.

 

surjective homomorphism from SU(2) to the rotation group SO(3)

    • for every SO(3), there is at least one SU(2). There may be multiple SU(2) elements that map to an SO(3) element (surjective).
    • Kernel { + I, − I} means that SU(2) element I*A and SU(2) element –I*A map to the same SO(3) element.

 

Returning to our discussion of translation we can define the operator

 

 

 

In three dimensions we can construct a group where where we find all of the translations in 3-dimentsions R3.

 

 

 

 

This group is abelian and has no finite dimensional representations.

{ASIDE: There are no finite representations for translations.  Thus you get

eipx where p is the derivative and in a sense represents an infinite matrix since it has a value for each x.}

 

 

An alternative representation would be E, Lx , Ly , Lz

 

 

\ \frac{1}{-\hbar^2}L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}

 

Wednesday ------------

There were many observed particles and theorists started to group them together to see if there were common features.

Step 1

P N                  è  appear to be very similar spin, mass and nuclear binding

      è  appear to be very similar spin, mass and nuclear binding

η

Particles come in singlets doublets, triplets, quadruplets.

 

What has similar structure?  States built up from spin ½ states.

  • Spin ½ state is a doublet
  • Two spin ½ states can couple to zero which is a singlet
  • Two spin ½ states can  couple to one which is a triplet
  • Three spin ½ states can couple to 3/2 quadruplet (or ½ )

 

The basic building block is a spin ½ state.

This structure can be understood as an SU(2) structure.

  • The fundamental representation of SU(2) is the doublet.
  • The irreducible representations of SU(2) are the singlet, doublet, triplet … behavior

 

The above structure is SU(2) flavor for the up, down. This is referred to as Isospin.

 

Lets get a bit bolder.

 

Mesons odd parity, zero spin, m<1 GeV

 

 (140, 500, 1000, 500)

Should these particle be grouped?

If so,

Octet and a singlet.

 

Baryons

 

Isopsin 1/2, S=0      

Isospin 1, S=-1        

Isospin 1/2, S=-2    

Isospin 0, S=-1       

Eight baryons

 

 

Octet

 

Isospin 3/2, S=0       

Isospin 1, S=-1        

Isospin 1/2, S=-2     

Isospin 0, S=-3        

Ten Baryons

 

 

 

Examining the particles as members of a diagram physicists wondered how does one get an octet and a decuplet ?   The answer was group theory and in particular SU(3). 

 

What is missing form these figures ?   Anwer:  the fundamental representation. But where is the fundamental representation which is a triangle ?

 

*******************x

remember

            specific quantum state labeled by some observables.   It will be important to have a notation that allows us to specify some special vectors.  The arguments within the KET can signify eigenvalues which then define the vector as a state which is a solution to the eigenequations for an observable or a set of observables.

 

     

 

If one can measure the energy and the angular momentum for a quantum system then the observable energy and the z-component of angular momentum might suffice to define the state.  Therefore

 

and the eigenvalues of energy and angular momentum could label certain specific quantum states.

 

 

A general state might then be

a linear combination of some of these specific states. (Any sum of vectors must indeed be a vector).

 

If the eigenvalues are continuous then the sum becomes an integral

 

To find the relevant labels we need to find the operators.

 

Operators will be viewed as a group if they satisfy the group rules:

 

 

Translations, rotations in three dimensions, rotations around an axis, boosts

 

TRANSLATION

 

One can imagine the transformation of a function such that f(x), g(x), h(x) are all the same function but at different locations. f(x) can be translated so as to obtain g(x) and h(x).

 

 

 

Using the Taylor series we can see that

 

 

using the expansion of ex

 

 

the expression

 

 

also

 

The state f(x) is moved as shown by the state labeled h(x) in the above figure.

 

 

 

To be clear we can consider two similar operations. The first shown above we refer to as translation. It is the operation of defining a new function h(x) that is the old function f(x) moved to a new position. h(x)=f(x-c).  In order to define the new function h(x) we can evaluate the old function f at the location x-c (previous location). So we need to find f(x-c) starting at x and moving by an amount c backwards.  h(x) is the translated f(x). 

 

Where here the evolution is in x so we are finding the value of the function at x+ε knowing the value at x.

 

 

 

Representation

  • Vector space,
  • rule
    • D(element) that maps the element of the group to an operator on the vector space
    • For every element U in the group there is a linear operator D(U).

 

SU(N)   group that behaves as NxN unitary matrices

  •   (, complex conjugate-transpose)
  • det U =1