The Dirac Equation

1. The Dirac Equation Brian Meadows, U. Cincinnati.

2. Origin of the Equation • In QM, observables have corresponding operators, e.g. • Relativistically, we can identify p i~ • Schroedinger equation (non-relativistic): • Klein-Gordon equation (relativistic): • Both differential equations are linear & second order in xm • Attempts were made to find first order equations. Brian Meadows, U. Cincinnati

3. The Weyl Equation • Dirac sought to find linear, first order equations. • Weyl Equations (relativistic for m=0) are an example: • Factorizing: • The vector s is required to preserve the rank of the equation • The commutation relations ensure that the first equation holds. • They are those of the Pauli matrices and y are 2-component spinors Weyl Equation Brian Meadows, U. Cincinnati

4. Static Equation • For relativistic particle with m>0: When • Clearly this can be written as two linear, first order equations: • When m>0 and p>0 then we write something like Brian Meadows, U. Cincinnati

5. Dirac Matrices • To make this work, we need = and • That the  are 4x4 matrices satisfying the commutation relations: {, } = 2g • Explicitly, they are given by: • where “1” is a 2x2 diagonal matrix andkare Cartesian components of the 2x2 Pauli matrices • Then we can factorize: Brian Meadows, U. Cincinnati

6. Dirac Equation • Choose one factor: • The solutions  must be four component “Dirac spinors” • Solutions at zero momentum: Negative energy Positive energy Brian Meadows, U. Cincinnati

7. Plane Wave (Free Particle) Solutions • Look for solution of the type (x) = ae-(i/~)x¢ pu(p) [ x´ (ct, r ) and p´ (E/c, p ) ] • Introduce into the Dirac equation (i~-mc) = 0: (p-mc)u = 0 (Algebraic, NOT a differential equation) • Evaluate the LHS: • So Brian Meadows, U. Cincinnati

8. Plane Wave (Free Particle) Solutions • Evaluate the wave-functions using: • To obtain Could Flip sign of E and p  E+mc2 Brian Meadows, U. Cincinnati

9. Plane Wave (Free Particle) Solutions • Normalization yyy= 2|E|/c requires that • Label column spinors as • Spin operator • They are NOT eigenstates of S can be applied to u(1), u(2), v(1), v(2) Brian Meadows, U. Cincinnati

10. Spin States • Apply the z-component of the spin-1/2 operator Sz: • Clearly u(1), u(2), v(1), v(2) are not eigenstates unless px=py=0 • In this case, the eigenvalues are +1, -1, -1, +1, respectively. • u(k), v(k) are particle antiparticle pair (k = 1, 2) Brian Meadows, U. Cincinnati

11. Lorentz Transformation of Dirac Spinors • Dirac spinors are NOT 4-vectors. • Transformation y’ = ywhere y’ is in system boosted along x-axis by g. • Clearly • This means that yyy is not an invariant (scalar) since Brian Meadows, U. Cincinnati

12. Adjoint Dirac Spinor • Define the adjoint Dirac spinor as • Clearly it flips the last two coordinates. • Then is invariant: • Because Brian Meadows, U. Cincinnati

13. Scalars, Pseudo-scalars, Vectors, Axial Vectors and Tensors • Define • Then the following transform in the indicated ways: Note that each is a linear combination of 16 products of y components Brian Meadows, U. Cincinnati

14. Parity Transformation • For a Dirac spinor, the operation of parity inversion is (Think of g0 as reversing the sign of the terms withpzorpx+ipyrelative to the other two terms) • Consider parity operation on the quantity • Similar proof for P flips sign of pseudo-scalar Brian Meadows, U. Cincinnati

15. Photons - Maxwell’s Equations Maxwell’s equations Equation of continuity: • In homogenous ,linear, isotropic, medium with • Conductivity • Dielectric constant • Permeability Brian Meadows, U. Cincinnati

16. Potentials • Introduce vector and scalar potentials (A, f): • Substitute into Maxwell’s equations: • Auxiliary conditions: Brian Meadows, U. Cincinnati

17. 4-Current • Define 4-current (obviously a 4-vector): • Equation of continuity becomes: Invariant is zero in all frames. Brian Meadows, U. Cincinnati

18. 4-Potential • It is tempting to define a new vector • Then we would write Maxwell’s equations as: • Has form (scalar) x A = (4-vector) Therefore, A is a 4-vector Brian Meadows, U. Cincinnati

19. Coulomb Gauge • Lorentz condition is then • The Lorentz condition can be further restricted without changing it. We can, for example, choose • Maxwell’s equations are then Brian Meadows, U. Cincinnati

20. Photon Wave-Functions • Photons wave-functions are plane-wave solutions • Plug into Maxwell’s equations and obtain pmpm=0(i.e. m=0) • Plug into the Lorentz condition and obtain pmem=0 • In the Coulomb gauge, A0=0 so that • So the photon wave-functions are where one choice, for photons traveling in the z direction, is. Transverse polarization Brian Meadows, U. Cincinnati

21. Summary – Spin-1/2 u(1), u(2), v(1), v(2) need not be pure spin states, but their sum is still “complete”. Brian Meadows, U. Cincinnati

22. Summary - Photons • Photons have two spin projections (s): Brian Meadows, U. Cincinnati

23. Feynman Rules for QED Label: • Label each external line with 4-momenta p1, … pn. Label theirspins s1, … sn. Label internal lines with 4-momenta q1, … qn Directions: Arrows on external indicate Fermion or anti-Fermion Arrows on internal lines preserve flow Arrows on internal lines preserve flow External photon arrows point in direction of motion Internal photon arrows do not matter Brian Meadows, U. Cincinnati

24. Feynman Rules for QED • For external lines, write factor • For each vertex write a factor Always follow a Fermion line To obtain a product: (adjoint-spinor)()(spinor) E.g.: pj , sj pk , sk e - e - q u (sk)(k) ige u(s))(j) Brian Meadows, U. Cincinnati

25. Feynman Rules for QED • Write a propagator factor for each internal line NOTE: qj2 = mj2c2 for internal lines. NOTE also: use of the slash – q essentially the component along ! Brian Meadows, U. Cincinnati

26. Feynman Rules for QED Now conserve momentum (at each vertex) 5. Include a d function to conserve momentum at each vertex. where the k's are the 4-momenta entering the vertex 6. Integrate over all internal 4-momenta qj. I.e. write a factor For each internal line. • Cancel the d function. Result will include factor • Erase the d function and you are left with • Anti-symmetrize (“-” sign between diagrams with swapped Fermions) Brian Meadows, U. Cincinnati

27. p1 , s1 p3 , s3 e - e - u (s1)(1) ige u(s3)(3) q u (s2)(2) ige u(s2)(2) - - p2 , s2 p4 , s4 Example – e-- e--Scattering • We already wrote down one vertex: Use index “” • The other is similar: BUT use index “” • Leads to Brian Meadows, U. Cincinnati

28. p1 , s1 p1 , s1 p4 , s4 p3 , s3 e - e - e - e - q q e- p3 , s3 e- e- p2 , s2 e- p2 , s2 p4 , s4 Example – e-e- e-e- (Moller Scattering) • One other diagram required in which 3  4 are interchanged (not possible in e-- scattering) • Anti-symmetrization leads to: Brian Meadows, U. Cincinnati