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## Modern Physics for Frommies III

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**Fromm Institute for Lifelong Learning, University of San**Francisco Modern Physics for Frommies III A Universe of Leptons, Quarks and Bosons; the Standard Model of Elementary Particles Lecture 3 Modern Physics III Lecture 2**Agenda**• Administrative Matters • Marrying Relativity and Quantum Mechanics • Quantum Field Theories • Relativistic Wave Equations • Fermions and Bosons • Particles and Antiparticles • Second or Field Quantization • Quantum Electrodynamics (QED) • Weak Interactions • Strong Interactions • Experiment – Atlas movies Modern Physics III Lecture 2**AdministrativeMatters**• Error in URL for Modern Physics I • Should be: http://modphysicsfrommiies.wiki.usfca.edu Note double i • First Physics and Astronomy Colloquium: • Wednesday, 2 February 2011, 1145-1250 in HR 232 • Professor E. Commins, UC Berkeley, Eighty Five Years of Electron Spin. • Full schedule of colloquia will be provided as soon as it is available. It will also be posted in Fromm Hall. Modern Physics III Lecture 2**Marrying Relativity and Quantum Mechanics**1st Try, Klein Gordon Equation: Schrödinger Equation is just a statement of energy conservation Kinetic Potential Total The simplest physical system is that of an isolated free particle, i.e. V = 0. The left and right sides of this expression transform differently under Lorentz transformations. We have to fix this some how. Modern Physics III Lecture 2**Recall from our discussions of relativity**So we might guess that a relativistic analogue of the Schrödinger equation might look like “simplfying” squaring and rearranging Klein-Gordon Equation Looks similar to the E.M. wave equation with an extra term, but we have some fundamental problems. Modern Physics III Lecture 2**(1) We have introduced “extra” negative energy solutions**Will see that these can be associated with antiparticles (2) The Klein-Gordon equation is 2nd order in time, different from the Schrödinger equation upon which the probability density interpretation of the wave function is based. The square of a negative energy solution of the Kline Gordon equation is not positive definite, i.e. it can be negative. Negative energy solutions with negative probabilities do not sound like components of a healthy theory Modern Physics III Lecture 2**The Dirac Equation:**Want something 1st order in time derivatives like the Schrödinger equation Hamiltonian operator, e.g. for S. eqn. Classical: K+V In the spirit of the relativistic linking of space and time maybe we should try being 1st order in space derivatives as well. Paul Adrian Maurice Dirac 1902-1984 The coefficients ai cannot simply be numbers, equation would not be invariant even under rotation. y cannot be a simple scalar Modern Physics III Lecture 2**Dirac proposed that the above equation be considered a**matrix equation with the wave function y written as a column vector and the coefficients ai and b written as square matrices. For a free e- at rest: Spin up Spin down (+) energy (-) energy Modern Physics III Lecture 2**The Dirac equation produces probabilitiy densities which are**positive definite for all solutions. We still have to figure out an interpretation of the negative energy states In the mean time we can ignore the negative energy solutions and do a lot of relativistic electron calculations. Fermions and Bosons: It can be shown that all Dirac equation solutions are Klein-Gordon solutions, i.e. Lorentz invariant. The reverse is not true. The Dirac equation applies to spin ½ particles, fermions. Other fermions: p, n, q,n.D, etc. Fermi-Dirac statistics a.k.a Pauli exclusion principle Modern Physics III Lecture 2**There is another type of particles, the boson, having**integral spin. Examples: p, g, r, g, G, f Bose-Einstein statistics, i.e. no Pauli exclusion principle We can loosley associate the Dirac equation with fermions and the Klein-Gordon with bosons Now, it’s time to revisit the negative energy solutions and meet the electron’s evil twin. Modern Physics III Lecture 2**Hole Theory – Particles and Antiparticles:**The existence of (-) energy solutions we must do something to prevent electrons from making transitions into those states and cascading down into oblivion. Ultraviolet catastrophes anyone. Dirac (1930): Fill up the negative energy levels and the Pauli principle will block transitions into these states. E A (-) energy electron can absorb radiation and be excited into a (+) state of energy +E leaving a hole in the Dirac sea. electron mc2 0 radiation The hole is the absence of an electron of charge –e and energy –E and is perceived by an observer as a particle of charge +e and energy +E -mc2 hole Modern Physics III Lecture 2**Positively charged electron or positron. e-e+pair**production. E electron Correspondingly, a hole in the Dirac sea , a positron, can act as a trap for a (+) energy electron → e-e+ pair annihilation mc2 0 radiation -mc2 hole Carl Anderson (1932) at Cal Tech observed just such a particle in a cloud chamber exposed to cosmic rays The first antiparticle had been predicted and observed. Modern Physics III Lecture 2**Dirac’s resting place in Westminster Abbey**Modern Physics III Lecture 2**So far we have made a “first cut” at merging relativity**and quantum mechanics but we are still dealing with classical fields. Field some aspect of the properties of a region of space (space-time) that can be quantitatively assessed at every point in that region. Examples of classical fields: Scalar: Temperature of H2O surrounding power plant. Vector: Electric field due to an electron Force / unit charge We need to quantize this field. For E.M. replace smooth continuous field with an assemblage of photons, E = hf New way of describing force at a distance. Modern Physics III Lecture 2**Particle Exchange: Two people glide towards each other on a**frictionless surface. As they pass (1) throws a medicine ball at (2) who catches it. • Recoils against the throwing of the ball and is deflected. • absorbs the momentum of the ball and is deflected in the opposite direction as (1) (1) (2) Redraw this as a space-time diagram Result of the ball exchange ia as if they had exerted a forcr on one another. This is how forces are conveyed in QFTs. t x Modern Physics III Lecture 2**Feynman Diagrams:**Ignore direction of arrows g If you follow the QFT rules, for 2 charges at rest, QFT predicts a force (and thus the field) that you would find for classical E.M. Richard Feynman 1918 -1988 QFT supplies a more refined notion of how fields are generated allowing extension of force fields to regimes where Q.M. and relativity must be taken into account Where did the photon(s) come from? Emitted and absorbed soley to transmit force. Modern Physics III Lecture 2**Hydrogen atom in ground state**Virtual photons: Recall the relativistic energy momentum relation Work out mass-energy conservation for every step in the scattering process. Some of the exchanged g’s energyhas to be accounted for as mass. How much depends on energy of the electrons and how close they approach. OK thanks to DEDt ≥ ħ (Heisenberg Uncertainty Principle) Such particles are said to be off mass-shell or virtual Modern Physics III Lecture 2**Four Fundamental Forces:**Range of force ≈ Compton wavelength of carrier = The short ranges of the 2 nuclear forces make it impossible to deduce the Maxwell-like classical field equations. Description is available only via QFT. Tables are turned for gravity __________________________________________________________ • Proton radius 10-16 m • ∞ for bare quarks if they were ever bare. In reality, color screening redices range to nuclear scale Modern Physics III Lecture 2**Pre 1974**Three quarks for Muster Mark! -James Joyce, Finnegans Wake (1939) Post 1974 Murray Gell-Mann 1969 Nobel Prize Modern Physics III Lecture 2**We can have higher order processes as well, e.g. 2 photon**exchange In principle, to calculate ee → ee in we would have to sum an infinite number of diagrams. Saved: Each vertex contains a factor of a the E.M. coupling constant or the fine structure constant. We can do a perturbation expansion and get away with considering only the leading terms, e.g. electron gyromagnetic ratio required 72 diagrams (up to 7 vertices) and several years of calculation to match experimental accuracy. It matched to 9 decimal places. Modern Physics III Lecture 2**Renormalization: We’re not finished yet. There is a class**of diagrams that could wreck our theory. Compared to the lowest order diagram, this has 2 extra vertices. We would expect this amplitude to be down by 1372 10,000 More to the game than just counting vertices. other QFT rules also The loopede+e-as well as the gare virtual. The g’s energy, fixed by the kinematics of the ee → ee collision can be shared in an infinite number of ways between the looped particles. We have to sum up all these ways of dividing the energy and find that the amplitude for this process is infinite. What to do? Modern Physics III Lecture 2**Feynman, Schwinger and Tomonaga 1965 Nobel Prize**Julian Schwinger Sinichiro Tomonaga 1918 – 1994 1906-1979 Renormalization Think about theoretical predictions of QFT vs. physically observable experimental quantities measured in the lab. Observed ee scattering. QM what goes on between the 2 electrons is not observable by experimenter. Arbitrarily let's call the left hand e the projectile and the other the target Modern Physics III Lecture 2**Think of the process as the exchange of a virtual g and a**target e “dressed” by the vacuum fluctuation loop. When your experiment measures the e as a target you are measuring all possible manifestations of the target, bare and dressed. The bare e is not an observable. Observable is the combination of bare and dressed. Not just the above, we can surround the target e with an arbitrarily complex mess of virtual loops connected by virtual photons. Measure the electron charge experimentally: This is Adjust or renormalize the bare electron charge so that your calculated dressed charge matches the experimental value. Modern Physics III Lecture 2**If for a proposed QFT of any of the forces, the adjustment**of a finite number of parameters, e.g. the charge and mass, renders the calculation of all observables finite; then the theory is renormalizable, workable and is a possible fundamental theory of the force. Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it! - Paul Dirac as late as 1975 Modern Physics III Lecture 2**The shell game that we play ... is technically called**'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate. -Richard Feynman 1985 But it does work and very well! Modern Physics III Lecture 2**What about the other forces?**Weak Interaction: Yes, in fact the weak interaction and the E.M. interaction are unified at a sufficiently high energy Charged current t Neutral current t Modern Physics III Lecture 2**It took a great deal of effort to get a renormalizable**electro-weak QFT Sheldon Glashow Abdus Salam Steven Weinberg 1926 - 1996 1979 Nobel Prize Modern Physics III Lecture 2**Strong Interaction:**A little history: 1911 Rutherford discovers the nucleus 1932 Chadwick discovers the neutron → Nucleus is protons + neutrons What holds the nucleus together? Need a strong but short range force. 1947 Powell et al. discover pion, p in cosmic rays 1935 Yukawa potential P+ p0 Could be mediated by a scalar boson of mass 200 me Muon, m not it. P+ Modern Physics III Lecture 2**In the 1960s both theory and experiment began to **substructure to “elementary” particles Partons: quarks, antiquarks and gluons Hideki Yukawa 1907 -1981 1949 Nobel Prize Update Yukawa’s picture Modern Physics III Lecture 2**Color and Color Charge: Color was originally introduced to**beat the Pauli principle. Arrows are quark spins Sq = 1/2 u u Clearly Pauli blocked u There are also anticolors for the antiquarks u u Add a new number, color u Mrs. Pauli’s favorite son is now happy! Modern Physics III Lecture 2**Theory of strong interactions between quarks is called**quantum chromodynamics (QCD) Mediated by force carriers called gluons Bound states: Note that in all of the above the colors add up to white. Color is not observed in our “usual” particles (p, n, p, K etc.). Gluons: 8 of them carrying both color and anticolor Modern Physics III Lecture 2