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

F = g g F = 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 =

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

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

give two independent equations OR 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:

[-( A-A)][-( A-A)] summing over ,  FF 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!

L L L = 0 FmnFmn FmnFmn So with and next we get Note: 0 in the Lorentz Gauge The Klein Gordon equation for massless photons!

 / = e+iq/ħc +iq  / = e+iJ · /ħ U(1) : SO(3) : q = -e for electrons q = +e for positrons q = +2e/3 for up quarks +iq The ± sign is just a convention, as in rotations:  / = e+iJ · /ħ SO(3) :

 (D )/ = e-iq/ħcD   / =  [e+iq/ħc] = e+iq/ħc D= + i A We can generalize our procedures into a PRESCRIPTION to be followed, noting the difference between LOCAL and GLOBAL transformations are due to derivatives:   / =  [e+iq/ħc] = e+iq/ħc  for U(1) this is a 1×1 unitary matrix (just a number) the extra term that gets introduced If we replace every derivative  in the original free particle Lagrangian with the “co-variant derivative” D= + i A g ħc then the gauge transformation of A will cancel the term that appears through  (D )/ = e-iq/ħcD i.e. restores the invariance of L

The free particle Dirac Lagrangian can be made U(1) invariant only by introducing a charged current introducing a VECTOR FIELD particle which couples to that charge The conserved quantity “discovered” was ELECTRIC CHARGE. The particle coupling to CHARGE was interpreted as the PHOTON. CHARGE is the source of the PHOTON FIELDS through which Dirac particles interact. This is believed to be the underlying principal of the fundamental electromagnetic force: VECTOR PARTICLES mediate interaction forces.

SU(2) Iso-spin multiplets Are there HIGHER symmetries? SU(2) spin-multiplets just one of many ANGULAR MOMENTUM representations Dirac matrices and Dirac spinors already keep this space separate. SU(2) Iso-spin multiplets already expanded into SU(3) and higher as we generalized isospin to include concepts like hypercharge, strangeness, and charm. Are these all some kind of charge?

Is UP or DOWN some kind of CHARGE that generates fields? that couples to a force carrying vector particle? Is there some kind of fundamental ISOSPIN FORCE? The theorists Yang & Mills extended the U(1) formalism that explained e&m forces in an effort to explore ISOSPIN.

L L L 1 satisfying one, 2 satisfying the other. Imagine 2 possible states: the flip sides of some spin-½ (2-component) property Spin, ISO-spin, or even something NEW L Sum of 2 Dirac lagrangians Applying the Euler-Lagrange equations results in 2 independent Dirac equations 1 satisfying one, 2 satisfying the other. Written more compactly as spinors and its adjoint Note: 1 and 2 each already 4-component Dirac spinors L “mass matrix” where If m1=m2 then M=mI and L which “looks” just like the 1-particle Dirac Lagrangian

But NOW  represents a 2-element column vector and we can explore an additional invariance under  The most general SU(2) matrix is of course U = ei( /2)·  where   Pauli matrices  Following the success of U(1) Yang-Mills promoted the obvious global phase transformation to a LOCAL invariance, writing g ħc U = ei ( /2)·    where   (x)  q ħc U = ei (x) compare to the U(1) transformation:

= (U) + U() Like before, the Dirac Lagrangian (as it stands) is NOT invariant under this transformation = (U) + U() The fix again is to replace  by a “covariant derivative” 3 vector fields are needed to span the space of this transformation operator

so that the (D)' = D term remains invariant Then, assuming an appropriate GAUGE transformation of the G fields is possible: so that the (D)' = D term remains invariant To figure out the necessary transformation property of the Gauge fields we’ll use the fact that then

 ' D' in other words the transformed which means in particular U U †

U U † U U†Gi i 2 which would look similar to the if could pull through U or U† this would just be which would look similar to the gauge transformations under U(1) Why can’t we? Let’s focus on this term: OK to commute! Not OK! which we can just write as i 2 U U†Gi

= RT(=g/2) U U† i i 2 2 recalling that You will show for homework that i 2 i 2 = RT(=g/2) U U† a 3-dim space(-like) rotation applied to the i/2 matrices So i,j count over the iso-space generators (Pauli matrices 1,2,3)  counts over the spacetime coordinates (ct, x, y, z)

U U † RG- Since Now remember the i are linearly independent matching like terms we find:  RG-  fields are “rotated” …and shifted by a gradient (a gauge shift)

L=iħcgmDmc2 =iħcgmmc2 - (gg m )·Gm The resulting Lagrangian (so far) L=iħcgmDmc2   =iħcgmmc2 - (gg m )·Gm  2 3 separate 4-vector fields (like A) where we’ve introduced 3 new vector fields  G = (G1, G2, G3 ) each with its own free Lagrangian (kinetic energy) term   = - F ·F 1 4 ? but not quite the same as before since THIS is not an invariant

=  (RijG j- i ) -  (RijG j- i ) F i' = Gi'-Gi' =  (RijG j- i ) -  (RijG j- i )  = ( Rij )G j +Rij (G j) -  i - ( Rij )G j - Rij(G j) +   i RR((x)) for a local transformation = Rij (G j - G j ) + ( Rij -  Rij )G j F i 

F i = Gi-Gi - G×G Actually with 3 vector fields there IS another anti-symmetric term possible   G×G and, with it, the more general   2g ħc F i = Gi-Gi - G×G  restores invariance!

L=[iħcgmmc2 ]- F F-(gg m )·Gm So NOW for our newly proposed SU(2) theory we have     L=[iħcgmmc2 ]- F F-(gg m )·Gm 1 4  2 describing two equal mass Dirac particle states in interaction with 3 massless vector fields Gm Think of something like the -fields, A   This followed just by insisting on local SU(2) invariance! In the Quantum Mechanical view:   Dirac particles generate 3 currents, J = (g   )  2 These particles carry a “charge” g, which is the source for the 3 “gauge” fields

The full product has nothing smaller than quadratic G terms Furthermore with: linear linear quadratic  The full product has nothing smaller than quadratic G terms (KE terms of free particles) plus cubic and quartic terms (interaction terms describing VERTICES of gauge particles with themselves!!)

plus “self-interaction” terms: field-current interaction 3 like this: one for each Gi plus “self-interaction” terms: These gauge particles (“force carriers”) are NOT neutral! (like s are with respect to electric charge)

In general NON-ABELAIN GAUGE THEORIES: introduce more interactions (vertices) for SU(2) we saw both 3 and 4 particle interaction vertices have (still) massless gauge particles (like the photon!) the gauge field particles posses “charge” just like the fundamental Dirac states not electric charge - we’re trying to think of NEW forces

i.e., the strong force does not couple to flavors. The YANG-MILLS was built on the premise that there existed 2 elementary Dirac (spin-½) particles of ~equal mass serving as sources for the force fields through which they interact NO SUCH PAIRS EXIST proton/neutron isospin states were the inspiration, but there is NO massless vector (spin 1) iso-triplet (isospin 1) of known particle states r-mesons? 770 MeV/c2 p,n,r now recognized as COMPOSITE particles isospin of up,dn quarks generalized into SU(3)  SU(4) The strong force must be independent of FLAVOR up charm top down strange bottom i.e., the strong force does not couple to flavors. SO WHERE DOES THE STRONG FORCE COME FROM?

We WILL find these ideas resurrected in: SU(3) color symmerty of strong interactions SU(2) electro-weak symmetry