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The Dirac Equation

The Dirac Equation. 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 x m

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The Dirac Equation

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  1. The Dirac Equation I acknowledge much input from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

  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.  Two extreme cases: Brian Meadows, U. Cincinnati

  3. Massless Particles: 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. P = 0: 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 ie(- 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 ´ (1- 2)-1/2. • 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 4 component spinors • Photons wave-functions are plane-wave solutions • Plug into Maxwell’s equations and obtain p m pm= 0(i.e. m=0) • Plug into the Lorentz condition and obtain pm e m = 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. Begin New Lecture Here Brian Meadows, U. Cincinnati

  24. 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 lines indicate Fermion or anti-Fermion Arrows on internal lines preserve flow External photon arrows point in direction of motion Internal photon arrows do not matter Brian Meadows, U. Cincinnati

  25. 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

  26. 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

  27. 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  function. Result will include factor • Erase the  function and you are left with • Anti-symmetrize (“-” sign between diagrams with swapped Fermions) Brian Meadows, U. Cincinnati

  28. p1 , s1 p3 , s3 e - e - u (s1)(3) ige u(s3)(1) q u (s2)(4) 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 Time Brian Meadows, U. Cincinnati

  29. 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: Time Brian Meadows, U. Cincinnati

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