There is simplicity and complexity in the approach that leads to the standard model.  Simply stated one finds symmetries and exploits them to deliver the structure. While the overview is compelling and understandable, one can easily find that the details are difficult to sort through.  The following comments are meant to try and state and give some examples.  There is plenty of deeper subtlety that remains unexplored such as anomalies, renormalization and a host of other issues. The famous model depicting the earth supported on the back of a turtle purportedly prompted one student to ask what supports the turtle. The answer given by the professor was “Its turtles all the way down”.  I suspect that we have come only a little ways since then and it is probably still my best answer to the most ambitious students.

 

The general structure generated by identifying a group such as SU(2) isospin leads us to a representation (column vectors and matrices).  Remember that if the notion of a quantum state is compelling then the projection of the abstract ideas of a group onto a real system should be realized by a description in terms of these states. A representation is this implementation. You find the states of a system and write the way rotations or any other group mixes up the states to give you the new states.  If the system is linear (addition works) then a specific transformation should be the merely all of the complex number that link the states.  Because there are alternate choices for the way a state is described (momentum states or position states). The representations vary based on the states you choose. To from fundamental particles, we look for groups of states that mix but remain isolated from other states.  For SU(2) of spin theses states are the multiplets of total spin J2. Therefore we expect a spin ˝ particle remains always as a spin ˝ but that it can exist in many states of orientation. We label the orientation with Jz .  We can also find another particle with spin 1 or spin 3/2 or spin 2.  Each will remain a member of their separate multiplet. The multiplet is labeled by the total spin and the members of the multiplet will be labeled by the z projection.  Similarly the SU(2) of weak isospin groups states into pairs (e,ν ), (u,d). The electron and neutrino multiplet remain distinct from the up and down quark multiplet but electron and neutrino are mixed by the SU(2) transformation.

 

For SU(2), there are three generators: τx τy τz.  The relationship between generators and the transformation is based on the requirement that the transformations on the states must be unitary. Therefore they should have the form

 

 

 

There are linear combinations of the generators that are often more sensible. For SU(2) a second set of equivalent generators is  τ+ τ- τz and  τ2.  Plus and minus labels τ+ τ- identify the raising and lowering operators that change one state in the multiplet to the other. The z operator τz is the observable within a multiplet. And the magnitude τ2, labels the multiplet and doesn’t change.  Remember because τ+ τ- τz do not commute, their observables cannot be simultaneously measured, only one can be chosen to label the states.  In physics there may be no direct reference to this structure.  The multiplets may be provided as (u,d) with the SU(2) structure only partially clarified by the way they are written. Often no explicit discussion of the overall state labels is given. A student might be reassured by the comment that the magnitude of the isospin is ˝ with two components of τz being u and d.

 

The matrix or transformation that is not included in the SU(2) group are those members of U(2) with determinant not equal to 1.

The removal of these types of group members from consideration is denoted by the S in SU(2).  The impact of these members on states is to change the phase of each member of a multiplet by the exact same phase. This of course is simply multiplying the entire wave function by a phase and is therefore the same as the U(1) transformation.

 

U(N)=SU(N)xU(1);        Any U(N) group member can be obtained by performing the combined operations of a SU(N) transformation and a U(1) transformation.

 

 

U(1) is the simplest group of transformations. There is one generator for U(1) and the group leads to a simple structure for the states. While U(1) is a good starting point to examine the impacts of symmetry and groups on the way one might build a theory, it lacks much of the structure that more complex groups will require.

 

There are two separate uses of symmetry that help build the standard model: Global symmetry and Gauge symmetry.

GLOBAL (equally at all locations)

  1. Symmetries where the related transformations do not change the physics, hardly change the physics or do not change some of the physics:  We will restrict our considerations to global transformation where the transformation is applied equally at all locations. This is the standard application of a transformation (e.g. rotation through some angle and/or multiplying the wf everywhere by a common phase.)
    1. Symmetries that do not change the physics are exact global symmetries. The Lorentz transformation is a good example of an exact global symmetry. Charge conservation, Baryon number conservation, Lepton number conservation are all due to exact global symmetries of the standard model.
    2. Symmetries that do not change the physics much or change the physics in only a few cases, are approximate global symmetries. CP is an approximate symmetry. It applies in all but the smallest number of processes.  SU(3) of flavor is an approximate global symmetry. The u, d, s quark can be considered to be approximately the same in analyzing the quark structure of the low mass hadrons. 
    3. Symmetries that do not change some of the physics: The SU(3) of flavor is such an example. Up and down quarks have very different electric charges. The flavor symmetry applies only when considering the strong interaction. It is not an overall fundamental symmetry.

GUAGE (different at different locations, counterintuitive way to apply symmetry)

  1. Symmetries that can be extended to the local regime and made to preserve physics by adding in interacting fields.  This process is call a gauge transformation and is the mechanism that is used to introduce the four forces. 
    1. Gravity: Unfortunately gravity has not been successfully integrated fully into the picture. Einstein’s original idea for curved space and time is the real birth of the gauge idea. He was speculating as to what the effects would be if space were curved by different amounts at different places. If he allowed mass/energy to be the source of this locally generated curvature then he could describe gravity. This is a classical theory and the extension of the General Theory of Relativity to the small scale quantum level remains a significant problem. However the impact of gravity is so small that it is often irrelevant. Of course, to those trying to find the correct fundamental view it may be the critical key.
    2. QED: This is the simplest of the forces and results from a local U(1) symmetry. This is the symmetry responsible for charge conservation. The complete details are more complex than this simple picture because the Weak and E&M interactions are intertwined. The U(1) that we have naively chosen to help deliver QED is not the U(1) of the Electroweak interaction. What this means is that the quantity that generates the U(1) transformation is not simply interpreted as an electric charge. Some oher observable is associated with the U(1) generator, Y, hypercharge.
    3. Electroweak: SU(2)LxU(1)- The unification of QED and the weak force was accomplished by identifying two component states that are left handed as one symmetry and the overall phase change of U(1) as another. However the generators or quantities of interest for the U(1) states in not the charge but the hyper charge Y. One can then build a (uL,dL) pair that can be characterized by a value of hypercharge and weak isospin (with left handed particles).  The actual familiar structure of particles and interacting fields does not drop directly out of the gauged symmetries.  In order to get all the know characteristics (e.g. massive W+, W-, Zo, and massless photons) a strange mixing mechanism called spontaneous symmetry breaking needs to be added.  The Higgs particle (new particle) is introduced that makes the starting vacuum state change to a nonsymmetrical state. Physics on the earth’s surface has a broken symmetry where the gravity of earth picks out a special direction (down).  Higgs particles present all around us make the environment different than what would be put in as the fundamental vacuum of the theory.  Spontaneous symmetry breaking of the Higgs mixes the SU(2)LxU(1) to provide the necessary particle states.
    4. QCD: SU(3) – The strong interaction is based on assuming that there are triplet states of the quarks labeled by color. There are 8 generators of SU(3). One can write these as 6 raising and lowering operators, U±, V±, T± plus two multiplet labeling (measurable) operators T3 and Λ8. This structure is a bit more complicated than SU(2). The two observables for the states can be represented on a plane using the x(T3) axis and y(Λ8) axis for the observables. The T± move you only along x. The T’s then look very similar to SU(2). The U’s and V’s move you both in x and in y at the same time. Despite the fact that there are two state labels, the fundamental representation for the group has three objects.  The three objects that form the fundamental representation (R,G,B). Form a triangle on the constructed observable x-y plane. Then all the U±, V±, T± will move you around the multiplet but not to any other states.

 

An important application of the exact global symmetries is the realization that they lead to a certain number of conserved quantities and associated currents that describe how the conserved stuff moves around in space so that there is never a loss or gain (Noether’s Theorem).  The conserved quantities can be identified as the generators. Therefore if treated as exact global symmetries:

 

 

This only applies to continuous symmetries (parameters are continuous). Parity, Charge conjugation and Time reversal are not continuous. So although one finds consequences if these symmetries are required, conserved quantities and associated currents are not guaranteed.

 

ASIDE: *** The simplicity of U(1) can sometimes lead to misunderstandings when applied to more complex groups. For example identifying U(1) with charge or Baryon number leads to a single generator for the group, B, Q respectively.  The total charge or Baryon number for a composite system is simply obtained by adding the values of all constituents. The charges can be labeled +, -.  SU(3) is the symmetry of the strong interaction. Here the charges are R, G, B and adding charges is not a simple sum. There are the associated charges . The sum of R + G + B can equal zero.  As the group complexity increases so does the addition rule for charge addition.

 

Translation in space and time are also interesting to consider in this light. Since all the generators commute there is no mixing of the components of the momentum vectors. A state can be labeled by px, py, pz, and E. Each simply adds a phase factor to the wave function.  A general translation to xo, yo, zo to a plane wave is obtained by multiplying by exp(ipx*xo+ipy*yo+ipz*zo). So momentum and energy are simpler than angular momentum.   Of course the full blown treatment of the space-time symmetries combines translations, rotations and boosts (Poincare) so the overall description is a rich structure.

 

One often turns to QED for insight. Here again the current of the U(1) charge symmetry is simpler than the SU(2) weak or SU(3) of QCD. QED delivers one vector potential, and the related electric and magnetic field. QED can be described in terms of one particle the photon.  The current that generates this field is based on a particle moving but remaining the same particle. The weak interaction will require 3 fundamental field particles W+, W-, Zo. The currents that generate the fields allow electrons to change into neutrinos and up quarks to change into down quarks. ***