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

2. L I claim the expression serves this purpose

3. LL L LL L L L

4. You can (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 K-G + L DIRAC L (r,t) describes e + e  objects describes photons 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?

5. 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  antiparticle (hermitian conjugate) state particle state What does such a PRODUCT of states mean? Recall the “state functions” have coefficients that must satisfy anticommutation relations. They must involve operators!

6. We introduce operators  (p) and  † (p) satisfying either of 2 cases: or Along with a representation of the (empty) vacuum state: | 0  such that:

7. † The Creation Operator †

8. “creates” a free particle of 4-momentum p The complex conjugate of this equation reads: The above expression also tells us: | 0 | 0  which we interpret as:

9. The Annihilation Operator

10. If we demand, in general, the orthonormal states we’ve assumed: 1 0 0 0 0 an operation that makes no contribution to any calculation

11. Then  † (p)|0  = zero =  (p-q)|0  This is how the annihilation operator works: If a state contains a particle with momentum q, it destroys it. The term simply vanishes (makes no contribution to any calculation) if p  q. |p 1 p 2 p 3  =  † (p 1 )  † (p 2 )  † (p 3 ) |0  [  (p-q)   † (p)  (q) ] | 0 | 0 

12. or  † (p)  † (k)|0  =  † (k)  † (p)|0  |pk  = |kp  and if p=k this gives |pp  = |pp  This is perfectly OK! These must be symmetric states BOSONS  † (p)  † (k)|0  =  † (p)  † (k)|0  |pk  =  |kp  and if p=k this must give 0 These are anti-symmetric states FERMONS in contrast

13. The most general state

14.  s s s dk 3 k  g h {  (r,t),  † (r ´,t)} =  3 (r – r) If we insist: that these Dirac particles are fermions Recall the most general DIRAC solution: we can identify (your homework) g as an annihilation operator a(p,s) and h as a creation operator b † (-p,-s) ab†b† a b †

15. Similarly for the photon field (vector potential) [ A(r,t), A † (r ´,t)] =  3 (r – r) If we insist:Bosons! d -s-k-s-k † d -s-k-s-k Remember here there is no separate anti-particle (but 1 particle with 2 helicities). Still, both solutions are needed for mathematical completeness.

16. interactions between Dirac particles (like electrons) and photons appear in the Lagrangian as Now, since It means these interactions involve operator products of ( a † + b ) ( a + b † ) ( d † + d ) creates an electron annihilates an electron annihilates a positron creates a positron creates a photon annihilates a photon giving terms with all these possible combinations: a†b†d†a†b†d† a † ada † ad † bb † d † bb † dbad † bad

17. a†b†d†a†b†d† a † ada † ad † bb † d † bb † dbad † bad What do these mean? In all computations/calculations we’re interested in, we look for amplitudes/matrix elements like:  0|daa † |0  Dressed up by the full integrals to calculate the probability coefficients a†b†d†a†b†d† creates an electron creates a positron creates a photon ee ee  a † ad † annihilates an electron ee ee 

18. a † ad † ee ee  time a † ad ee ee  time

19. Particle Physicists Awarded the Nobel Prize since 1948 1948 Lord Patrick Maynard Stuart Blackett For development of the Wilson cloud chamber 1949 Hideki Yukawa Prediction of the existence of mesons as the mediators of nuclear force 1950 Cecil Frank Powell Development of photographic emulsions to study mesons 1951 Sir John Douglas Cockcroft Ernest Thomas Walton Transmutation of nuclei using artificial particle accelerator 1952 Felix Bloch Edward Mills Purcell Development of precision nuclear magnetic measurements

20. Particle Physicists Awarded the Nobel Prize since 1948 1954 Max Born The statistical interpretation of quantum mechanics wavefunction Walther Bothe Development of coincident measurement techniques 1955 Eugene Willis Lamb Discovery of the fine structure of the hydrogen spectrum Polykarp Kusch Precision determination of the electron’s magnetic moment 1957 Chen Ning Yang & Tsung-Dao Lee Prediction of violation of Parity in elementary particles 1958 Pavel Alekseyevich Čerenkov Il’ja Mikhailovich Frank Igor Yevgenyevich Tamm Discovery and interpretation of the Čerenkov effect

21. Particle Physicists Awarded the Nobel Prize since 1948 1959 Emilio Gino Segre & Owen Chamberlain Discovery of the antiproton 1960 Donald A. Glaser Invention of the bubble chamber. 1961 Robert Hofstadter Discovery of nuclear structure through electron scattering off atomic nuclei 1965 Sin-Itiro Tomonaga, Julian Schwinger, and Richard P. Feynman Fundamental work in quantum electrodynamics 1968 Luis W. Alvarez Discovery of resonance states through bubble chamber analysis techniques 1969 Murray Gell-Mann Classification scheme of elementary particles by quark content

22. Particle Physicists Awarded the Nobel Prize since 1948 1976 Burton Richter and Samuel C. C. Ting Discovery of new heavy flavor (charm) particle 1979 Sheldon L. Glashow, Abdus Salam, and Steven Weinberg Theory of a unified weak and electromagnetic interaction. 1980 James W. Cronin and Val. L. Fitch Discovery of CP violation in the decay of neutral K-mesons 1984 Carlo Rubbia and Simon Van Der Meer Contributions to the discovery of the W and Z field particles. 1988 Leon M. Lederman, Melvin Schwartz, and Jack Steinberger Discovery of the muon neutrino

23. Particle Physicists Awarded the Nobel Prize since 1948 1989 Norman F. Ramsey Work on the hydrogen maser and atomic clocks (founding president of Universities Research Association, which operates Fermilab) 1990 Jerome I. Friedman, Henry W. Kendall and Richard E. Taylor Deep inelastic scattering studies supporting the quark model. 1992 Georges Charpak Invention of the multiwire proportional chamber. 1995 Martin L. Perl Discovery of the tau lepton. Frederick Reine Detection of the neutrino. 1999 Gerardus ‘t Hooft and Martinus J. G. Veltman Renormalization theories of electroweak interactions 2002 Raymond Davis, Jr. and Masatoshi Koshiba The detection of cosmic neutrinos

24. David J. GrossH. David PolitzerFrank Wilczek Kavli Institute for Theoretical Physics, University of California Santa Barbara, CA, USA California Institute of Technology Pasadena, CA, USA Massachusetts Institute of Technology (MIT) Cambridge, MA, USA The Nobel Prize in Physics 2004 "for the discovery of asymptotic freedom in the theory of the strong interaction" b. 1941 b. 1949 b. 1951

25. In Quantum Electrodynamics (QED) All physically are ultimately reducible to this elementary 3-branched process. We can describe/explain ALL electromagnetic processes by patching together copies of this “primitive vertex” ee ee  ee ee p1p1 p3p3 p2p2 p4p4 Two electrons (in momentum states p 1 and p 2 ) enter… …a  is exchanged (one emits/one absorbs)… Our general solution allows waves traveling in BOTH directions Calculations will include both and not distinguish the contributions from either case. …two final state electrons exit. Coulomb repulsion (or “Møller scattering”) Mediated by an exchanged photon!

26. ee ee  time bad † These diagrams can be twisted/turned as long as we preserve the topology (all vertex connections) and describe an equally valid (real, physical) process What does this describe?

27. Bhaba Scattering

28. A few additional notes on ANGULAR MOMENTUM Combined states of individual j 1, j 2 values can be written as a “DIRECT PRODUCT” to represent the new physical state: | j 1 m 1 > | j 2 m 2 > J 0 I 0 0 I 0 J We define operators for such direct product states A 1  B 2 | j 1 m 1 > | j 2 m 2 > = (A| j 1 m 1 >)(B | j 2 m 2 >) then old operators like the MOMENTUM operator take on the new appearance J = J 1  I 2 + I 1  J 2 +

29. So for a fixed j 1, j 2 | j 1 m 1 > | j 2 m 2 > all possible combinations which form the BASIS SET of the matrix representation of the direct product operators How many? How big is this basis? Giving us NEW - dimensional operators acting on new long column vectors

30. 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 J 1 + 0 0 0 J 2 0 0 0 0 0 0 0 0 2j 1 +1 states 2j 2 +1 states We’ve expanded our space into: Obviously we still satisfy ALL angular momentum commutator relations.

31. All angular momentum commutator relations still valid. J 3 is still diagonal. OOPS! RECALL in general the direct product state is a LINEAR COMBINATION of different final momentum states. This is the irreducible 1 x 1 representation for m = j 1 + j 2. Two eigenstates give m=j 1 +j 2 -1 But The best that can be done is to block diagonalize the representation m = j 1 + j 2 only one possible state (singlet) gives this maximum m-value! | j 1 j 1 > | j 2 j 2 > = | j 1 +j 2,j 1 +j 2 > m =  ( j 1 + j 2 ) either | j 1, j 1 -1 >| j 2, j 2 > or | j 1, j 1 >| j 2, j 2 -1 > corresponding to states in the irreducible 2 dimensional representation | j 1 +j2, j1+j2-1> and | j 1 +j 2 -1, j1+j 2 -1 > J 2 is no longer diagonal!

32. This reduces the (2j 1 +1)(2j 2 +2) space into sub-spaces you recognize as spanning the different combinations that result in a particular total m value. These are the degenerate energy states corresponding to fixed m values that quantum mechanically mix within themselves but not across the sub-block boundaries. The raising/lowering operators provide the prescription for filling in entries of the sub-blocks.

35. The sub-blocks, correspond to fixed m values and can’t mix. They are the separate (lower dimensional) representations of Angular momentumSpace Dimensions Irreducible Subspaces 2  2 4  2 3  2 3  3 4  3 4  4 = 1  2  1 = 1  2  2  1 = 1  2  2  2  1 = 1  2  3  2  1 = 1  2  3  3  2  1 = 1  2  3  4  3  2  1