for s1 for EMwaves total energy density of

for s=1 for EM-waves total energy density: # of photons the Klein-Gordon equation: wave number vector:

for FREE ELECTRONS (or any Dirac, i. e. spin ½ particle: muons, taus, quarks) ± 1 or 1, 2 electrons positrons u v a spinor satisfying: v u Note for each: i. e. we write with -s -E, -p for u 3, u 4 Notice: now we express all terms of the “physical” (positive) energy of positrons!

The most GENERAL solutions will be LINEAR COMBINATIONS s dk 3 k g s h s linear expansion coefficients where g g(k, s), h h(k, s), Insisting { (r, t), (r´, t)}= { †(r, t), †(r´, t)}=0 (in recognition of the Pauli exclusion principal), or, equivalently: { (r, t), †(r´, t)} =d 3(r – r ) respecting the condition on Fourier conjugate fields: forces the g, h to obey the same basic commutation relation (in the “conjugate” momentum space) the g g(k, s), h h(k, s) “coefficients” cannot simple be numbers!

s dk 3 k g s s dk 3 k † g s h s † h s

We were able to solve Dirac’s (free particle) Equation by looking for solutions of the form: This form automatically satisfied the Klein-Gordon equation. (x) = e-ix p /h u u (p 1 0 cpz E+mc 2 c(px+ipy) E+mc 2 But the appearance of the Dirac spinors means the factoring effort isolated what very special class of particles? ) 0 cpz E-mc 2 c(px-ipy) E-mc 2 1 c(px+ipy) E-mc 2 -cpz E-mc 2 c(px-ipy) E+mc 2 1 1 -cpz E+mc 2 0 0 u(p ) a “spinor” describing either spin up or down components

What about vector (spin 1) particles? The fundamental mediators of forces: the VECTOR BOSONS Again try to look for solutions of the form p /h -i x (x) = (p)e Polarization vector (again characterizing SPIN somehow) but by just returning to the Dirac-factored form of the Klein-Gordon equation, will we learn anything new? What about MASSLESS vector particles? (the photon!) the Klein-Gordon equation: becomes: or =0 2 Where the d’Alembertian operator: 2

=0 2 is a differential equation you have already solved in Mechanics and E & M Classical Electrodynamics J. D. Jackson (Wiley) derives the relativistic (4 -vector) expressions for Maxwells’ equations can both be guaranteed by introducing the scalar V and vector A “potentials” which form a 4 -vector: (V; A) along with the charge and current densities: (c ; J) Then the single relation: completely summarizes:

(of zero gravitational potential energy) Potentials can be changed by a constant for example, the arbitrary assignment or even leaving everything invariant. In solving problems this gives us the flexibility to “adjust” potentials for our convenience 0 The Lorentz Gauge The Coulomb Gauge 0 In the Lorentz Gauge: 0 and a FREE PHOTON satisfies: a “vector particle” with 4 components (V; A) The VECTOR POTENTIAL from E&M is the wave function in quantum mechanics for the free photon!

so continuing (with our assumed form of a solution) (p) like the Dirac u, a polarization vector characterizing spin Substituting into our specialized Klein-Gordon equation: (for massless vector particles) E 2=p 2 c 2 just as it should for a massless particle!

(p) Like we saw with the Dirac u before, has components! How many? 4 The Lorentz gauge constrains but not all of them are independent! A 0 p = 0 p 0 0 - . p = 0 while the Coulomb Gauge A=0 which you should p = 0 recognize as the familiar condition only for on em waves free photons Obviously only 2 of these 3 -dim vectors can be linearly independent such that p = 0 Why can’t we have a basis of 3 distinct polarization directions? We’re trying to describe spin 1 particles! (mspin = -1, 0, 1)

spin 1 particles: mspin = -1 , 0 , +1 antialigned The m=0 imposes a harsher constraint (adding yet another zero to all the constraints on the previous page!) v=c The masslessness of our vector particle implies ? ? ? In the photon’s own frame longitudinal distances collapse. How can you distinguish mspin = 1 ? Furthermore: with no frame traveling faster than c, can never change a ’s spin by changing frames. What 2 independent polarizations are then possible?

The most general solution: s where s = 1, 2 or s = 1 moving forward Notice here: no separate ANTI-PARTICLE (just one kind of particle with 2 spin states) Massless force carriers have no anti-particles. moving backward

Finding a Klein-Gordon Lagrangian or The Klein-Gordon Equation Provided we can identify the appropriate this should be derivable by The Euler-Lagrange Equation L L L

I claim the expression L serves this purpose

L L L L

You can show (and will for homework!) show the Dirac Equation can be derived from: L DIRAC (r, t) We might expect a realistic Lagrangian that involves systems of particles L(r, t) = L K-G describes photons + L DIRAC describes e+e- objects but each term describes free non-interacting particles L + L INT But what does terms look like? How do we introduce the interactions the experience?

We’ll follow (Jackson) E&M’s lead: A charge interacts with a field through: current-field interactions the fermion (electron) the boson (photon) field from the Dirac expression for J particle state antiparticle (hermitian conjugate) state Recall the “state functions” Have coefficients that must What does such a PRODUCT of states mean? satisfy anticommutation relations. They must involve operators!