L Dirac = i ħ c  g m     mc 2 

1. Look at the FREE PARTICLE Dirac Lagrangian LDirac=iħcgm  mc2 Dirac matrices Dirac spinors (Iso-vectors, hypercharge) Which is OBVIOUSLY invariant under the transformation ei (a simple phase change) because  e-i and in all  pairings this added phase cancels! This is just an SU(1) transformation, sometimes called a“GLOBAL GAUGE TRANSFORMATION.”

2. What if we GENERALIZEthis? Introduce more flexibility to the transformation? Extend to: but still enforce UNITARITY? eia(x) LOCAL GAUGE TRANSFORMATION Is the Lagrangian still invariant? LDirac=iħcgm  mc2 (ei(x)) = i((x)) + ei(x)() So: L'Dirac = -ħc((x))gm +iħce-i(x)gm()e+i(x) mc2

3. L'Dirac = -ħc((x))gm+iħcgm() mc2 LDirac For convenience (and to make subsequent steps obvious) define: -ħc q (x)  (x) then this is re-written as L'Dirac = +qgm()+LDirac recognize this???? the current of the charge carrying particle described by  as it appears in our current-field interaction term

4. L'Dirac = +qgm()+LDirac If we are going to demand the complete Lagrangian be invariant under even such a LOCAL gauge transformation, it forces us to ADD to the “free” Dirac Lagrangian something that can ABSORB (account for) that extra term, i.e., we must assume the full Lagrangian HAS TO include a current-field interaction: L=[iħcgmmc2]-(qgm)Am defines its transformation and that under the same local gauge transformation

5. L=[iħcgmmc2]-(qgm)Am • We introduced the same interaction term 3 weeks back • following electrodynamic arguments (Jackson) • the form of the current density is correctly reproduced • the transformation rule • Am' = Am +  l • is exactly (check your notes!) • the rule for GAUGE TRANSFORMATIONS • already introduced in e&m! The search for a “new”conserved quantum number shows that for an SU(1)-invariant Lagrangian, the freeDirac Lagrangain is “INCOMPLETE.” If we chose to allow gauge invariance, it forces to introduce a vector field (here that means Am) that “couples” to.

6. The FULL Lagrangian also needs a term describing the free particles of the GAUGE FIELD (the photon we demand the electron interact with). But wait! We’ve already introduced the Klein-Gordon equation for a massless particle, the result, the solution A = 0 was the photon field, A Of course NOW we want the Lagrangian term that recreates that! Furthermore we now demand that now be in a form that is both Lorentz and SU(3) invariant!

7. We will find it convenient to express this term in terms of the ANTI-SYMMETRIC electromagnetic field tensor More ELECTRODYNAMICS: The Electromagnetic Field Tensor but A=(V,A) and J=(c, vx, vy, vz) do! • E, B do not form 4-vectors • E, B are expressible in terms of  and A the energy of em-fields is expressed in terms of E2, B2 • F = A-A transforms as a Lorentz tensor! = Ex since = Bz since

8. Actually the definition you first learned: In general = -Ex = -Bz Fik = -Fki= = 0 While vectors, like J transform as “tensors” simply transform as

9. x' = Lx or x = L-1x' Under Lorentz transformations

10. So, simply by the chain rule: and similarly:

11. (also xyzyzxzxy) both can be re-written with (with the same for xyz) All 4 statements can be summarized in

12. The remaining 2 Maxwell Equations: are summarized by ijk = xyz, xz0, z0x, 0xy Where here I have used the “covariant form” F= g gF=

13. To include the energy of em-fields (carried by the virtual photons) in our Lagrangian we write: L=[iħcgmmc2]- F F-(qgm)Am 1 2 But need to check: is this still invariant under the SU(1) transformation?  (A+) (A+) = A+   A  = 

14. 1 2 1 16 “The Fundamental Particles and Their Interactions”, Rolnick (Addison-Wesley, 1994) L Heaviside -Lorentz units “Introduction to Elementary Particles”, Griffiths (John Wiley & Sons, 1987) L Gaussian cgs units

15. L L L L The prescriptions and give two independent equations OR summing over ALL variables (fields) gives the full equation WITH interactions Starting from L (and summing over , ) with L FmnFmn Let’s look at the new term:

16. summing over ,  FmnFmn survive when =, = and when =, =  ,  now fixed, not summed [-(A-A)][-( A-A)] • A = ggA = ggA 1 sum over (but non-zero only when =, = ) FmnFmn where  since this tensor is anti-symmetric!