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