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by: Dr. Simeon Wiza


Dr. Simeon Wiza
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This 30 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 544 at University of Washington taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/192457/phys-544-university-of-washington in Physics 2 at University of Washington.




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Date Created: 09/09/15
Quantum Electrodynamics Phys 544 Eric Thrane Outline 0L Fig 1 The running of on Foch Space The playground of QFT The Lagrangian Formulation of QM Feynman Diagrams A tool for doing perturbation theory Gauge Invariance and Photons The running of the fine structure constant 2 2 e e 1 a N 4717hc 4n 137 Hilbert Space 0 NonRelativistic QM Hilbert Space States are described by kets Energy eigenkets form a complete basis le you can write any state as a linear combination of energy states Pgtc1E1gtc2E2gtc3E3gt A problem we need a vector space that will allow for the creation of particles pair production and the annihilation of particles with antiparticles Hilbert space is too small Foch Space 0 Relativistic QM Foch Space States are still described by kets In addition to having different energies states can have different numbers of particles in them Pgt c1E1gt c2E1E1 c3E1E2gt We can define operators that add or remove particles atagiven momentum A A t wane Foch Space 13gt We ve expanded our vector space to accommodate antimatter What kind of Hamiltonian can we write down It has to be Lorentz invariant Enter the quantum field TlVacuumgt av I d3k A ik x A l ik x The Fleld x 5f ake 61k 6 3 271 2Eltkgt The job of writing down a Lorentz invariant Hamiltonian is easier if we use fields Fields are just linear combinations of creationannihilation operators They re divided by E 2 so that they transform conveniently under Lorentz boosts The eikX are just left over from a Fourier transform Basically though they re just fancy creationannihilation operators Writing a Hamiltonian Now with our fields handy we can write down a Hamiltonian that might look something like this H f d3x mg V2 m2qu The Hamiltonian looks everywhere in space and tries to remove particles When it finds one it hits it with E2 p2 m2 and then puts back the particle Good thing we divided by E 2 in each of the fields to cancel one of the E s The Lagrangian Formulation Indeed we could write down a Hamiltonian like that but it turns out to be a bit of pain to do calculations In the 1940 s Feynman came up with a new way of thinking about quantum mechanics which makes things easier Fortunately our discussion of fields will be directly applicable to Feynman s Lagrangian formulation of Quantum Mechanics Lagrangian Formalism Often in QM we want to calculate matrix elements of the timeevolution operator to find the probability that some initial state will be measured later as some final state In the Lagrangian formulation you rewrite this matrix element as an integral over paths Hamiltonian Lagrangian operators commuting variables ltw20qagtltw2eflqagt ltwzl lqagt2 e lquot all paths Perturbation Theory Doing the sum over all paths is hard There are many paths and exact solutions are rare 2 eifd4xL allpaths if We can however do the sum over all paths for a free particle which we can then 39 do perturbation theory about 1 Field Lagrangian Before we get into perturbation theory let s write down an example of a Lagrangian Recall our Hamiltonian H f d3x mg V2 m2qu If we transform it into a Lagrangian we get 1 2 1 2 2 L 509145 Em x Lint Field Lagrangian To do perturbation theory we expand the exponential of Lint Ipzlfjl m2a11pamseifd4moilifd4x39M54 Here I have chosen Lint m4 k is the order parameter that we are expanding in The next contribution in the series will go like 3 Feynman Diagrams There is lots of fun math involved in calculating these expansions but Feynman devised a nice bookkeeping method for doing them Once you know the Feynman Rules for a given Lagrangian you just have to plug and chug The first thing you do is draw a cartoon 1 2 1 2 2 A 4 m xZ Forthe Lagrangian Iwrote down illk m ki I E39 lquot1i39l quot l i before called 14 theory the I I l rules are pretty simple Every order of perturbation gets a vertex Every leg corresponds to a 1 particle Here two s are scattering off each other The amplitude for this process is A l Minltuvxski I Ellrliclvun l I I QED Diagrams In QED we have three particles in our Lagrangian photons y e and e39 Speaking of Lagrangians aren t you dying to see the QED Lagrangian A The QED Lagrangian Here it is Everything we know about light and its interactions with matter resides in this Lagrangian 1 L wltza mw ZFWgt2 eWwAM 17 ir01JT FM E 9A BVAM What type of scattering can we achieve with this Lagrangian Y gt e e Can we get a photon to turn into an e e pair 1 L W9 mw ZFW2 eWwAM 17 WOW FM E 9A BVAM 4Momentum Conservation The relativistic nature of spacetime is automatic in QED because we built our Lagrangian out of those convenient fields Interestingly 4momentum is conserved but not by virtual particles 11 CPT Theorem Also interesting is the CPT theorem Almost all Lagrangians that we use in particle physics are invariant under the combined operations of C P and T This means that antiparticles moving forward in time are mathematically equivalent to particles moving backward in time WOW The QED Lagrangian Let s take another look at the QED Lagrangian What symmetry properties does it have T 1 ii ZYO1J L 7171395 mm 102w2 eiTJrWJAM FM 5 BMAV 6VAM Noether s theorem tells us that each symmetry of the Lagrangian corresponds to a conserved quantity Why do we care about symmetry Symmetry been berry berry good to H US Jeff Wilkes Symmetry Conserved Quantity rotation angular momentum translation linear momentum Lorentz transformations 4momentum w gteihp global U1 charge Gauge Symmetry There s a special type of symmetry in QFT called gauge symmetry It s not a true symmetry it s only there because of redundancies in our description of the Lagrangian 1 L 1JiD mw mef DH 5 9M ieAM Enter the covariant derivative Gauge Symmetry You may have already seen that EM is invariant under gauge transformations 1 AM 9 AM gar100 n QED we recast this invariance in terms of group theory We say L is invariant under local U1 transformations m w a e w Gauge Theories Gauge theories are just theories that like QED have some gauge field AM that cleans up the mess after a symmetry operation so that the Lagrangian is left invariant Consequences of QED QED predicts the Coulomb potential Vr ocr One can derive this from looking at scattering amplitudes like these p F a q quotI I 1 quotL ta 393 gt 5 3 V I 1 T 1 gt 39 quot a r f 39 Consequences of QED Remember spinorbit coupling from QM There s an interaction in the Hamiltonian describing bound electrons that goes like LS This is also predicted by the spin behavior of QED fermions We observe it in atomic spectroscopy Consequences of QED e e annihilationproduction is correctly described by QED Classical EM does not tell us how a y could bounce off an atom and make an e e pair QED also predicts the Lamb shift The Running of or The coolest prediction of QED is that or changes as a function of interaction energy Every electron is surrounded by a sea of virtual photons and electron positron pairs The higher the energy of a collision the shorter the distances we probe The vacuum shields the charge of the electron and as we get past the shielding the apparent charge goes up Ol303eV 105 Olcolol Seething with Virtual Particles The Future of QED Startlingly QED tells us that on runs off to infinity as we keep increasing energy This means that perturbation theory our only working tool breaks down at high energies as ei y become more and more strongly coupled Most people take this to mean that QED is replaced at high energies by a more fundamental theory


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