Lecture 7 From Dirac equation to Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2

Transcription

1 Lecture 7 From Dirac equation to Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1

2 Dirac equation* The Dirac equation - the wave-equation for free relativistic fermions follows the requirements : 1) that the wave-equation as in case of the Schrödinger equation should be of 1st order in / t / x 0 2) to allow for a continuity equation with a positive density ψ*ψ: 3) relativistic covariance (with respect to Lorentz transformations) then requires that the wave-equation also has to be of 1 st order in the spacial derivatives / x k (k = 1, 2, 3), i.e.: (1) This equation can be rewritten in covariant notation: * cf. Lecture 10 WS2010/2011 2

3 Dirac equation The covariant form of the Dirac equation: (2) then involves with the four-vector coefficients (3) Further four-vectors are given by: 4-momentum 4-coordinate (4) covariant derivative electromagnetic 4-potential 4-current 3

4 Dirac equation Scalar products are Lorentz invariant, e.g. the invariant mass (5) with where we have employed the pseudometric (Lorentz invariant) tensor: (6) Thus we have: (7) 4

5 Dirac equation Including the interaction with vector fields A μ implies: (8) Then the Dirac equation reads: (9) The (anti-commutator) algebra of the γ-matrices has to follow: (10) with the properties: (11) By counting the number of boundary conditions (cf. Eq. (6) for α k, WS2010/2011) the γ-matrices have to be 4x4 matrices and consequently the wavefunctions Ψ(x)must have 4 components k, β in Lecture 10 5

6 Dirac equation The solution of the Dirac equation are plane waves with positive and negative energies separate the four components wave vector ψ into two vectors with 2 components ϕ, χ for spin up and down (relative to the z-direction = direction of motion): (12) Then we get: (13) using (14) which reads explicitly: (15) 6

7 Dirac equation: fermions I. Consider the positive energy The solution of the coupled equation (15) reads: (16) where σ k (k=1,2,3) are the Pauli matrices. Since the components are two-vectors, we may expand them as (17) N is the normalization factor spin up spin down 7

8 Dirac equation: fermions Then (18) In matrix notation: (19) The solutions of the Dirac equation then read explicitly for fermions with spin up and spin down : (20) 8

9 Dirac equation: anti-fermions II. Consider the negative energy states = anti-fermions Using we obtain for the anti-fermion components with spin up and down : (21) Accordingly free (anti-)fermions are fully defined by the spinors specified above! 8

10 Dirac equation: normalization Constrain for the normalization: Commonly one uses 2 ways of normalization: 1) as in Bjorken, Drell *: (22) thus, (23) in the rest-frame (E=m): * used in Lecture 10 WS2010/2011 2) the normalization used here (e.g. as Aitchison, Hey): (24) 8

11 Dirac spinors Free (anti-)fermions are fully defined by the spinors specified above (with normalization (24)): 1) Spinors with positive energy (fermions): (25) spin up : spin down 2) Spinors with negative energy (anti-fermions): (26) Wave vector ψ : fermions anti-fermions (27) 8

12 Dirac equation: positive and negative energy states Interpretation of the solutions with positive and negative energies: 1) Dirac (1930): particle-hole picture particles E > 0: particles E < 0: hole states =anti-paticles Dirac sea 2) Feynman picture: E<0, e<0 anti-particles: travelling back in time anti-particles=holes Emission of an antiparticle with 4-momentum p μ is equivalent to the absorption of a particle with 4-momentum - p μ Absorption of an antiparticle with 4-momentum p μ is equivalent to the emission of a particle with 4-momentum - p μ

13 π + scattering 1) π + - scattering on a time dependent electromagnetic potential V(t)~e -iωt Interaction by electromagnetic potential absorbtion at time t of the photon of energy hω Matrix element: at time t : π + -meson absorbs the photon of energy and increases its energy hω 13

14 π - scattering 2) π - - scattering π - - scattering π + - scattering with positive energy with negative energy Matrix element: Energy of π is equal to the energy of π + -meson 14

15 π + π - -pair production 3) π + π - - pair production/creation Matrix element: The sum of π and π + meson energies is equal to the energy of the absorbed photon 15

16 π + π - -pair annihilation 4) π + π - - pair annihilation Matrix element: The energy of π and π + mesons is equal to the energy of the produced photon 16

17 Dirac equation: Green functions The Dirac equation for electrons in an electromagnetic field can be obtained from the free Dirac equation (2) by the substitution (minimal coupling) (28) (29) Notation: (30) In order to see how to solve the inhomogenuous Dirac equation (30) for electrons in an electromagnetic field let s first consider the example from electrostatics - solution of Poisson equation: (31) Here ρ(x) is the free charge density. For a pointlike charge, i.e. the static Coulomb potential - solution of (31) - is known: (32) (33) 17

18 Dirac equation: Green functions For a continuous charge distribution ρ(x), the solution of (31) is then obtained by summing the potentials for all particles: (34) The Poisson equation may be solved also using a Green's function which is obtained by solving the point source equation: (35) Then (36) Using (36), the Poisson eq. (31) can be re-written as (37) 18

19 Dirac equation: Green functions The solution of eq. (36) is the spatial Green s function: (38) To solve the Dirac eq. (30) one defines the Green function K(x,x ) (where x,x are 4-vectors) by the requirement (39) Thus, the solution of the inhomogenuous Dirac eq. (30) reads (40) Green function = integration kernel Indeed: 19

20 Electron propagator The general solution of the inhomogenuous Dirac eq. (30) reads (41) homogenuous solution of free Dirac equation inhomogenuous solution of Dirac equation with electromagnetic potential Since the coupling constant is weak, one can use the perturbation theory: (42) 20

21 Electron propagator The explicit form of the Green s function can be written as a Fourier transform (42) Substitute (42) in the eq. for the Green s function (39) (43) Multiply (43) by and one gets Electron propagator: for (44) Note: propagator (44) is defined only for virtual electrons, since for real electrons 21

22 Electron propagator Thus, the Green s function is (45) for positive energy states: In (45) integral over p 0 has 2 poles: The integral in (45) can be evaluated by the method of residues by closing the contour in the lower(upper) half of the p 0 -plane 22

23 Electron propagator Method of residues: 1) For the integral is equal to 2π i times the residue of the integrand at the poles: (46) (47) 2) For (48) 23

24 Electron propagator The integral (45) can be evaluated also by integration along Re(p 0 ) line, however, by shifting the poles by an infinitesimal positive value ε (ε 0): (49) Thus, the electron propagator reads: (50) 24

25 Electron propagator The name propagator is also used for the Green s function since K(x,x ) describes the propagation of the particles from x to x : The wavefunction at the final space time point x w.f. of a free particle with positive energy = free plane waves: (51) The wavefunction at space-time point x: (52) Indeed, for t > t using eq.(47) (53) 25

26 Electron propagator A wave function of positive energy will spread only forward in time and not backward in time, i.e. for t < t one gets: (54) Thus, for the wave function with positive energy (k 0 > 0): =0 from Dirac eq. (55) In a similar way one can show that for negative energy (k 0 < 0) by virtue of K(x-x ) the wave function only propagates backward in time. 26