I acknowledge much input from

1
The Dirac Equation
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

3
Massless Particles The Weyl Equation

• Dirac sought to find linear, first order
  equations.
• Weyl Equations (relativistic for m0) 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
4
P 0 Static Equation

• For relativistic particle with mgt0
• When
• Clearly this can be written as two linear, first
  order equations
• When mgt0 and pgt0 then we write something like

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 ?k are
  Cartesian components of the 2x2 Pauli matrices
• Then we can factorize

6
Dirac Equation

• Choose one factor
• The solutions ? must be four component Dirac
  spinors
• Solutions at zero momentum

Negative energy
Positive energy
7
Plane Wave (Free Particle) Solutions

• Look for solution of the type
• ?(x) ae-(i/)x p u(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

8
Plane Wave (Free Particle) Solutions

• Evaluate the wave-functions using
• To obtain

Could Flip sign of E and p
? Emc2
9
Plane Wave (Free Particle) Solutions

• Normalization yyy 2E/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)
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 pxpy0
• In this case, the eigenvalues are 1, -1, -1, 1,
  respectively.
• u(k), v(k) are particle antiparticle pair (k 1,
  2)

11
Lorentz Transformation of Dirac Spinors

• Dirac spinors are NOT 4-vectors.
• Transformation y y where y is in
  system boosted along x-axis by ? (1-? 2)-1/2.
• Clearly
• This means that yyy is not an invariant (scalar)
  since

12
Adjoint Dirac Spinor

• Define the adjoint Dirac spinor as
• Clearly it flips the last two coordinates.
• Then is invariant
• Because

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

14
Parity Transformation

• For a Dirac spinor, the operation of parity
  inversion is
• (Think of g0 as reversing the sign of the terms
  with pz or pxipy relative to the other two
  terms)
• Consider parity operation on the quantity
• Similar proof for

P flips sign of pseudo-scalar
15
Photons - Maxwells Equations

• Maxwells equations
• Equation of continuity

In homogenous , linear, isotropic, medium with
Conductivity ? Dielectric constant ?
Permeability ?
16
Potentials

• Introduce vector and scalar potentials (A, f)
• Substitute into Maxwells equations
• Auxiliary conditions

17
4-Current

• Define 4-current (obviously a 4-vector)
• Equation of continuity becomes

Invariant is zero in all frames.
18
4-Potential

• It is tempting to define a new vector
• Then we would write Maxwells equations as
• Has form (scalar) x A? (4-vector)

Therefore, A ? is a 4-vector
19
Coulomb Gauge

• Lorentz condition is then
• The Lorentz condition can be further restricted
  without changing it. We can, for example, choose
• Maxwells equations are then

20
Photon Wave-Functions
4 component spinors

• Photons wave-functions are plane-wave solutions
• Plug into Maxwells equations and obtain p m pm
  0 (i.e. m0)
• 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
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.
22
Summary - Photons

• Photons have two spin projections (s)

23
Feynman Rules for QED

• Label
• Label each external line with 4-momenta p1, pn.
• Label their spins 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

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)
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 ? !

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 ? function. Result will include
  factor
• Erase the ? function and you are left with
• Anti-symmetrize (- sign between diagrams with
  swapped Fermions)

27
Example e-?- ? e-?- Scattering

• We already wrote down one vertex
• Use index ?
• The other is similar
• BUT use index ?
• Leads to

28
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