Luciano Maiani: Quantum Electro Dynamics, QED. Basic

Transcription

1 Luciano Maiani:. Lezione Fermi 10 Quantum Electro Dynamics, QED. Basic 1. Fiels, waves an particles 2. Complex Fiels an phases 3. Lowest orer QED processes 4. Loops an ivergent corrections 5. Two preictions of Dirac 6. The Shelter Islan Conference, Como meeting,

2 Newton s action at a istance: 1. Fiels, waves a particles the Sun acts on the Earth (or proton on the electron), force irecte from E to S, inversely proportional to the square of the istance the Earth respones to the instantaneous position of the Sun (???) A better picture: Sun creates a fiel a test boy place in the fiel in a given point feels the action of the fiel at that point if the Sun moves, the new position is transmitte to the fiel in some time not faster than the spee of light in vacuum: if Sun evaporates, we feel the effect ony 8 minutes later This is action at contact, hic et nunc. Faraay an Maxwell evelop the concept of fiel for electromagnetic phenomena where it is inispensable: there is nothing in the North to align a magnetic compass, but only a magnetic fiel at any point of the Earth! 2

3 waves an particles in the fiel Maxwell iscovere that perturbations of the fiel propagate as waves in fact he fins: velocity of propagation= velocity of light these waves are light to escribe propagation we have to assume that the fiel at one point (x) is φ(x) couple to the fiel in a point nearby (x1): the Action (i.e. the Lagrange function) must contain terms which ten to equalize the fiel in x1 to the fiel in x. x Terms of the form: [φ(x)-φ(x1)] 2 to make them non vanishing when x1 is very close to x, we ivie by the istance (there are four irections in space time in which we coul take the ifference, so four erivatives, µ=0, 1, 2, 3, time is 0, sum over repate inices): (x 1 ) (x) if we move φ(x), e.g. by an oscillating charge, φ(x1) will follow, with some elay, in orer to minimize the Action; φ(x2) will follow φ(x1),...an so (x) µ (x); µ the isturbance propagates in space an time, with a velocity etermine by the strenght of the coupling of the fiels in nearby points. If the fiel is a quantum fiel, the state of the fiel is escribe by iscrete Planck-Einstein quanta, i.e. particles of efinite mass which propagate with some momentum p an energy E, with: E 2 p 2 = m 2 (x 1 ) (x) ] µ (x)@ µ (x) x2 x1 φ(x1) φ(x2)

4 We have consiere complex fiels to escribe charge particles take a fiel represente by a single complex quantity at each point, φ(x) φ*(x) observables will be real quantities, like e.g. J=φ(x) φ*(x), which is insensitive to the phase of φ(x) in formal language, if we transform φ(x) e iα φ(x), φ*(x) e -iα φ*(x), then J is invariant: J J this is true also if we choose the phase ifferently in ifferent points: α(x) α(x1) The invariance uner phase transformations of the observables which are taken in ifferent points is esirable: why shoul the observer in Tokyo take the same phase choice as the one in Roma? Einstein woul say: no spooky action at a istance! However, there are observables which epen upon fiels in ifferent points, like [φ(x)-φ(x1)]* [φ(x)-φ(x1)]: what happens to them? (x 1 ) (x)! ei (x 1) (x 1 ) e i (x) (x) = e i (x) (x 1 ) (x) (x) [ + i (x)] there is an irregular term 2. Complex fiel an phases = e i (x) (x 1 )+ (x)+i (x) (x) maybe aing another irregular term, the irregularities will cancel!?? = A little simplification: e i (x 1) = e i{[ (x 1) e i (x) e i (x) = e i (x) [1 + i (x)] (x)]+ (x)}= 4

5 An Abelian Gauge Theory using erivatives: let us consier a vector fiel (one vector Aµ(x) at each point x) an consier the transformation of: uner the combine transformations: we fin: (x 1 ) (x)! e i (x) [ (x 1 ) (x) + i µ (x)! e i (x) [@ µ (x)+i@ µ (x) µ (x) ia µ (x) (x)! (x)! e i (x) µ (x) ia µ (x) (x) (x); A µ (x)! A µ (x)! e i (x) [@ µ (x)+i@ µ (x) (x) ia µ (x) (x) i@ µ (x) (x)] = = e i (x) [@ µ (x) ia µ (x) (x)] (!!!) we can form invariant combinations with the covariant erivative [@ µ (x)+ia µ (x) (x)][@ µ (x) ia µ (x) (x)] (scalar fiel) (x) µ [@ µ (x) ia µ (x) (x)] (Dirac fiel) 5

6 Electro Dynamics as a Gauge Theory It is known that the Electric an Magnetic fiels of the Maxwell equations can be expresse in terms of a vector potential Aµ(x) an that Electric an Magnetic fiels remain unchange if one as a erivative to the vector potential: the transformation we have just introuce We can ientify the vector fiel an the transformations just introuce with the vector potential of Maxwell theory. The theory thus constructe with the covariant erivative is invariant uner Abelian (phase) transformations, ifferent in each space point. This is an Abelian Gauge Theory; the electron interacts with the vector fiel via the Action: e Aµ(x)J µ (x), J µ = µ e is a normalization constant ientifie with the electron charge.

7 Action: e Jµ A µ The stories of QED electrons enter in the Action via the electromagnetic current: electron an photon fiels inuce creation an annihilation of electrons/positrons/ photons, accoring to the scheme: [a e +ikx + be ikx ]; [ae ikx + b e +ikx ] J : initial! final e! e (a a); e +! e + (bb ) e e +! 0(ba); 0! e e + (a b ) the story of a process is mae by lines of electrons/positrons going from one vertex to another, possibly starting from sources an ening to etectors; in each vertex a 3-boy process takes places accoring to the table; J µ = µ A [ce ikx + c e +ikx ]: 0! ;! 0 photon lines also go from one vertex to another or in (fro sources) or out (to etectors) only connecte iagrams matter (not mae by two or more iscnnecte parts) each iagram (story) gives an amplitue, sum over stories, make the A 2 at the en, to get the probability. 7

8 3. Lowest orer QED processes + Note: arrows follow the flow of the charge: positrons appear as if they were electrons running back in time Compton scattering (cross section compute by Klein an Nishina, 1928) same for e+ Pair prouction on nuclei by photon (a) or electron (b) (a) (b) - Moeller scattering (note the minus sign ue to Fermi statistics) same for e+ + Bhaba scattering (1936) 8

9 Total photo-absorption in Carbon an Lea J. Beringer et al. (Particle Data Group), Phys. Rev. D86, (2012). Experimental Methos an Colliers/ Passage of Particles through Matter The control of photon absorption has many applications most important is the calculation of the ose absorbe by patients irraiate for cancer therapy, to minimize sie harmful effects. 9

10 Tree Diagrams x y tree iagram = no close loops suppose we have incoming an outgoing particles with efinite momenta (sources an etectors are very far..) accoring to instructions, we must sum the amplitues corresponing to all values of the coorinates where interactions happen (vertices) summing over each coorinate implies that momentum is conserve at each vertex as a consequence, external momenta are conserve: Ptot,in = Ptot,out for a tree iagram, in aition, the external momenta fix the values of the internal momenta, so that there is nothing more to sum. p+k=p +k k k k k p p+k=p +k p p p-k =p +k p 10

11 k p 4. Diagrams with loops give (sometime) ivergent results if we want to compute higher orer effects to a given process, e.g. Compton scattering, we have keep fixe the external legs of the iagram an a new vertices an new internal lines, we encounter iagrams with loops. p+k q p+k-q p+k k p k p Consier the first iagram, assign the initial momenta an procee to etermine the momenta of the internal lines by using momentum conservation we remain with one unetermine momentum in the loop, q, an we have to sum the amplitue, A(q), over all values of q, up to infinity!!! p+k-q there are cases where the sum gives a finite result, because A(q) goes enough quickly to zero for large q (the box iagram, see) but cases, such as the first in the figure, where the sum gives an infinite result corrections to the lowest orer, which by the way gives a goo approximation to the experimental value, are infinite!!!! what happens? is QED sick? inconsistent? the problem appeare, in the 30s. At that time: calculations were one in a very complicate theory (now we call it ol fashione perturbation theory) there were no precise ata to confront with Dirac thought that rastic changes were neee to make a consistent theory, perhaps as rastic as the passage from the orbit theory of Bohr to quantum mechanics. p-q q p -q k p 11

12 5. Two preictions of the Dirac equation An electron on a close orbit generates a magnetic fiel (H. A. Rowlan s experiment, 1876) as if it were magnetic ipole As Bohr inicate, the natural unit for the magnetic moment associate to an orbital angular momentum L, (= x p), is the Bohr magneton, µb in a magnetic fiel H along z, an atom with one electron in orbit L has an aitional energy E=µB gllz H(= EL). We have introuce the aitional factor gl, the Lane factor, however we know with Bohr that gl=1, as confirme by measurements of the normal Zeeman effect; the spin of the electron was introuce by Uhlenbeck an Gousmith to explain the anomalous Zeeman effect, by an aitional magnetic interaction term, Es=µB gsszh, so that E=EL+Es=µB H(gssz + Lz ) spectroscopic ata suggeste gs 2. I. the electron anomaly In 1928, gs=2 was erive irectly from Dirac s equation: a great success!! since then, for an elementary spin 1/2 particle we efine the magnetic anomaly as the eviation from Dirac s value: a = g 2 2,a e = 0 preicte by Dirac 0 s equation proton an neutron are certainly not elementary an have large anomalies: µ B = e~ 2mc g p = ± = 2,g n = ± = 0 12

13 II. The 2P1/2 an 2S1/2 egeneracy The Bohr formula for the hyrogen atom gives the energy in terms of one integer quantum number, n (cfr. Lezione 3) e 2 1 E n = 2n 2 R B = a single value n correspons to states with ifferent orbital momentum, L=0, 1,..., n-1, all with the same energy; states with L=0 an 1 are enote with the symbols S an P. The Dirac equation introuces the electron spin an removes partially this egeneracy the result is that, for given n, the energy epens only on the total angular momentum, which may be equal to L +1/2 or L-1/2 (only +1/2 for L=0). The states with n=2 may have L=0 (S), J=1/2, or L=1 (P), J=1/2 an 3/2 thus the states 2S1/2 an 2P1/2 have the same n=2 an the same J=1/2, therefore are preicte to be egenerate 2n 2 2 mc n 2 spectroscopic measurements agree in the 30s with equal energy, within errors of percent. ev 13

14 The Shelter Islan Conference, 1947 Julian Schwinger: "It was the first time that people who ha all this physics pent-up in them for five years coul talk to each other without someboy peering over their shoulers an saying, 'Is this cleare?' The birth of moern QED Willis Lamb reporte a measurement of the energy ifference of 2S1/2-2P1/2 since then known as the Lamb shift, corresponing to: ν=1057 MHz, i.e ev. Isior Rabi reporte a precise measurement of the magnetic moment of the electron by P. Kusch an H. M. Foley: ae= (8) (in parenthesis the error on the last igit) R.P. Feynman reporte on his new version of QED, the Feynman iagrams on the train back from Shelter Islan, H. Bethe erive the first calculation of the Lamb shift. Shortly after, Schwinger compute ae=α/(2 π) Willis Lamb Polykarp Kusch 14

15 Eoaro Amali, 1978 The years of reconstruction: first post-war meeting, Como