Particle Physics WS 2012/13 ( )

Transcription

1 Particle Physics WS /3 (3..) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 6, 3.

2 How to describe a free particle? i> initial state x (t,x) V(x) f> final state. Non-relativistic particles Schrödinger Equation. Relativistic Spin particles Klein-Gordan Equation 3. Relativistic Spin/ particle Dirac Equation

3 Dirac Equation Dirac Ansatz: find equation which is linear in and t Hψ (α p + β m)ψ i ψ still need to fulfill relativistic energy momentum relation t (solution ψ of Dirac eq. must be solutions to Kl. Gordan) t Φ Φ m Φ E p + m t ψ H ψ -(-i α x x iα i α y y z + βm) (-i α z x x i α i α y y z + βm)ψ z [(α + α x x y +α y z z β m ) + (α x α y + α y α x ) + (α x y xα z + α z α x ) x + (α x β + βα x ) + (α x yβ + βα y ) + (α y zβ + βα z ) ] ψ z + (α z yα z + α z α y ) y z α x α y α z β α i α j + α j α i if i j α i β + βα i 3

4 Pauli-Dirac Representation α x α y α z β α i α j + α j α i if i j α i β + βα i Lowest order solution: 4x4 matrices, thus ψ is 4 component vector (Dirac spinor) ψ one choice of matrices: Dirac-Pauli representation I ; σ x ; σ y i i ; σ z ψ ψ ψ 3 ψ 4 β I I ; α j σ j σ j ; 4

5 Covariant form of Dirac Equation x -β (-iα x x (iβα x x i α i α y y z + βm)ψ i ψ z t + iβ α + iβ α y y z m)ψ -i β ψ z t γ μ (β, βα μ ) μ ( t, x, y, z ) (iγ μ μ m )ψ covariant form of Dirac equation 5

6 Description of free Spin+/ particles: Content of Today Follow the same recipe as for Schrödinger and Klein-Gordan Equation: massage the Dirac equation to have the form of the continuity equation to read off the density and the current. ρ t + j Determine the solution of the Dirac Equation (4 dim matrices 4 energy eigenvectors) (choose representation of solutions for particles and for antiparticles all with positive energies) Compute the normalization of the wave function Proof that Dirac Equation describes Spin ½ particles Use Helicity to distinguish degenerated particle and antiparticle solutions Description of intermediate particles Next time: Description of (3 prongue) vertices 6

7 Adjoint Equation Working with matrix equation, we now must consider hermitian rather than complex, conjugated equation Dirac: (iγ μ μ m )ψ (iγ t + iγk x k - m )ψ hermitian conjugated ( dagger : T* ) (AB) T B T A T Dirac : -i t ψ γ - i x k ψ γ k - m ψ γ μ (β, βα μ ) x γ -i t ψ γ + i x k ψ γ k - m ψ -i t ψ γ γ - i x k ψ γ γ k - m ψ γ σ x ; σ y σ z i i ; adjoint spinor: ψ ψ γ -i t ψγ - i x k ψγ k - m ψ β I I ; α j σ j σ j ; i μ ψγ μ + m ψ adjoint equation 7

8 Probability Density and Currents x ψ (iγ μ μ m )ψ i μ ψγ μ + m ψ x ψ iψγ μ μ m ψψ + i( μ ψ)γ μ ψ+ m ψψ i μ (ψγ μ ψ) Continuity equation: μ j μ with j μ (ρ, j) j μ ψγ μ ψ ρ ψγ ψ ψ ψ 4 4 i ψ i ψi i ψ i > Historically the positive density ρ was the original motivation for Dirac s work Charge current for an electron: 8

9 Solution of Dirac Eq: case particle at rest Easiest case: free particle at rest Ansatz: ψ u(e, p) ei(px Et) u(e, p) Φ Φ Φ 3 Φ 4 (iγ μ μ m ) i(px Et) u(e,p) e (γ E γ γ p y γ - m) p i(px Et) (γ E m ) u(e,p) e i(px Et) u(e,p) e E Φ Φ Φ 3 Φ 4 m Φ Φ Φ 3 Φ 4 solutions with and with E-m Ψ Ψ 3 e imt e +imt Solution: E +m > Ψ Ψ 4 E -m < e imt e +imt 9

10 General solution of Dirac Eq. (iγ μ μ m ) u(e,p) (γ E γ γ p y γ - m) u(e,p) I I E σp σp I I m u(e,p) E m I σp σp (E + m) u(e,p) take into account substructure of the matrix, choose Ansatz u u A u B E m I σp σp (E + m) u A u B (E-m) u A (σp) u B (σp) u A () u B σp + i i p y + + ip y ip y u A ip y u + ip y p B u B z ip y u + ip y p A z

11 General solution of Dirac Eq. u A ip y u + ip y p B u B z ip y u + ip y p A z explizit solutions by making an arbitrary choice for u A, u B (motivated by particle-at-rest-solution) u A () u A () u B (3) u B (4) u N u N u 3 N 3 u 4 N 4 For all 4 solutions (per construction): E p + m Compair with p solutions u, u correspond to E > particle at rest solution u 3, u 4 correspond to E < particle at rest solution

12 Antiparticle solution Ψ 3 u 3 e i(px Et) N 3 e i(px Et) Ψ 4 u 4 e i(px Et) N 4 e E - E p p E < particles i(px Et) u 4 (E,p)e i px Et v (E, p ) e i(px Et) (iγ μ μ m )ψ (γ μ p μ m) u u 3 (E,p)e i px Et v (E, p ) e i(px Et) v N 4 v N 3 Will understand later why 4 and 3 E > antiparticles (iγ μ μ m )ψ ( γ μ p μ m) v (γ μ p μ + m) v

13 Particle and Antiparticle Spinors ) Ψ i u i (E,p ) u N i(px Et) e u N u 3 N 3 u 4 N 4 ) Ψ i v i (E,p ) i(px Et) e E + p + m E p + m v N 4 v N 3 v 3 N v 4 N E + p + m E p + m For calculation can choose different spinor systems, all give same result: {u, u, u 3, u 4 }, {v, v, v 3, v 4 }, {u, u, v, v } will be default used in the following 3

14 Wave function normalization Wave functions have to be normalized to E: (see continuity eq.) ρ dv ψ ψ dv ψ T! ψ dv E Example: Ψ u (E,p ) i(px Et) e ρ u u N N ( + +p y + ) N +E m N E N E + m Same result for N i with u, u, v, v 4

15 Summary of Solutions Ψ i u i (E,p ) u E + m i(px Et) e u E + m satisfy (γ μ p μ m) u Ψ i v i (E,p ) e i(px Et) satisfy (γ μ p μ + m) v v E + m v E + m These solutions have positive energies: E p + m 5

16 Description of free Spin+/ particles: Content of Today Follow the same recipe as for Schrödinger and Klein-Gordan Equation: massage the Dirac equation to have the form of the continuity equation to read off the density and the current. ρ + j t Determine the solution of the Dirac Equation (choose representation of solutions for particles and for antiparticles all with positive energies) Compute the normalization of the wave function Proof that Dirac Equation describes Spin ½ particles Use Helicity to distinguish degenerated particle and antiparticle solutions Description of intermediate particles Description of (3 prongue) vertices 6

17 Angular Momentum Conservation First attempt to resolve ambiguity in dim sup-space: Operator of any conserved quantum number commute with H: Hψ (α p + β m)ψ L x y z z y Check rel. angular momentum L x Test [H, L x ] : x p L y z x x z L z x y y x [H, L x ] (α + α x x y + α y z + βm)(y z z z y ) (y z z y ) (α x + α x y + α y z + βm) z (α α y z z y ) -i (α x p) [H, L] -i (α p) x L is not a conserved QN x (analog for [H, L y ], [H, L z ] ) some angular momentum is missing 7

18 Angular Momentum Conservation σ σ σ σ σ σ σ 3 σ 3 Properties of Pauli matrices: σ σ iσ 3 σ σ -iσ 3 [H, Σ x ] (α + α x x y + α y z + βm) σ z σ - σ (α σ + α x x y + α y z + βm) z [ + [ + [ σ σ σ σ - σ σ σ σ - σ 3 σ 3 σ σ - i(α α y z z [H, Σ] i (α y ) i (α p) σ σ σ x σ σ σ σ σ σ σ σ 3 σ 3 x ] x ] y ] z x p) [H, L + Σ ] J L + Σ is conserved Σ is spin of particle, it does NOT commute with H Eigenvalues of : ±½, thus Dirac Equation describes spin ½ particles! Σ 8

19 Helicity The bonus in the Dirac equation is the extra two folded degeneracy There must be another observable beside E,p which is a conserved quantum number and compute with H, P operators. σ S z ½ Σ z ½ 3 σ 3 u E + m u E + m u,u,v,v are no Eigenstates of S z v E + m However particles moving in z direction are Eigenstates of S z ( p y )! v E + m More general, u,u,v,v are Eigenvalues of spin along direction of flight operator [H,ΣP] s p h -/ sp Helicity observable: h helicity operator: ΣP s s p p h +/ 9

20 Helicity Eigenstates [u,u ] : sub-space of particle spinors [v,v ]: sub-space of antiparticle spinors Freedom of rotation in both sub-spaces resolve for both sub-spaces: 4 common Eigenstates for E, p, h: Σ P a b p p h a b h± sin θ cos φ sin θ sin φ cos θ x φ z θ y u h E + m cos θ/ e iφ sin θ/ p cos θ/ p eiφ sin θ/ u h- E + m sin θ/ e iφ cos θ/ p sin θ/ p eiφ cos θ/ v h E + m p sin θ/ p eiφ cos θ/ sin θ/ e iφ cos θ/ v h- E + m p cos θ/ p eiφ sin θ/ cos θ/ e iφ sin θ/

21 Intermediated Summary Linear Ansatz for Hamiltonian resulted in Dirac equation: Hψ (α p + β m)ψ i t ψ Solution are 4-dim Dirac Spinors. ψ ψ ψ ψ 3 ψ 4 We found three commutating observables: E, p, h (helicity) and have choosen the solutions of Dirac Equation to be particles and antiparticles with helicity + and - and positive energy. Ψ u h+ e i(px Et), Ψ u h- e i(px Et), Ψ 3 v h+ e i(px Et), Ψ 4 v h- e i(px Et) particle solutions antiparticle solutions We have profen that Dirac equation describes S/ particles

22 (see Halzen Martin, Sec. 3.6) Interaction by exchange of Particles Up to now we have: Relativistic description of free spin/ particles Dirac equation + solutions Fermi s golden rule and discussion of phase space W fi π 4 M i (Ei) fi ρ f (E i ) still missing, internal propagators and vertex factors e.g. e - + μ - e - + μ - first order f> second order V fj f> V fi j> i> i> V ji V fi <φ f V φ i > + j< φ f V fj φ j ><φ j V ji φ i > E i E j

23 Interaction by Exchange of Particles Space time picture of a+b c+d First x is emitted by a then it is absorbed by b i> a+b f> c+d j> c+b+x V fi a b M a c+x M(b+x d) E a E b E c E d E x E a E c E x x t d c M fi ab not LI V fi <f V fj j><j V ji i> E i Ej <f V fj j><j V ji i> (E a +E b ) (E c +E b +E x ) <c+x V a> M(a c+x) E a E c E x <d V x+b> M(b+x d) E b E x E d rel. normalization to E particle per volume M fi ab a x c + a x c M fi ba b d b d M fi M a c+x M(b+x d) [ + E x E a E c E x E b E d E x ] 3

24 Interaction by Exchange of Particles M fi M a c+x M(b+x d) [ + E x E a E c E x E b E d E x ] E a +E b E c +E d M a c+x M(b+x d) [ E x E a E c E x E a E c +E x ] E b -E d E c -E a M a c+x M(b+x d) E x E x E a E c E x E x + m x p a p c + m x M a c+x M(b+x d) q m x + q m x is LI! M fi M a c+x M(b+x d) q m x 4