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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

The Dirac Equation

I acknowledge much input from

“Introduction to Elementary Particles”

By David Griffiths, Wiley, 1987


Origin of the 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 xm

    • Attempts were made to find first order equations.

       Two extreme cases:

Brian Meadows, U. Cincinnati


Massless particles the weyl equation
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

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P 0 static equation
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

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Dirac matrices
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:

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Dirac equation
Dirac Equation

  • Choose one factor:

  • The solutions  must be four component “Dirac spinors”

  • Solutions at zero momentum:

Negative energy

Positive energy

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Plane wave free particle solutions
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

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Plane wave free particle solutions1
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


Plane wave free particle solutions2
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


Spin states
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)

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Lorentz transformation of dirac spinors
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

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Adjoint dirac spinor
Adjoint Dirac Spinor

  • Define the adjoint Dirac spinor as

  • Clearly it flips the last two coordinates.

  • Then is invariant:

  • Because

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Scalars pseudo scalars vectors axial vectors and tensors
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

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Parity transformation
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

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Photons maxwell s equations
Photons - Maxwell’s Equations

Maxwell’s equations

Equation of continuity:

  • In homogenous ,linear, isotropic, medium with

  • Conductivity

    • Dielectric constant

    • Permeability

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Potentials
Potentials

  • Introduce vector and scalar potentials (A, f):

  • Substitute into Maxwell’s equations:

  • Auxiliary conditions:

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4 current
4-Current

  • Define 4-current (obviously a 4-vector):

  • Equation of continuity becomes:

Invariant is zero in all frames.

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4 potential
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

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Coulomb gauge
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

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Photon wave functions
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

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Summary spin 1 2
Summary – Spin-1/2

u(1), u(2), v(1), v(2) need not be pure spin

states, but their sum is still “complete”.

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Summary photons
Summary - Photons

  • Photons have two spin projections (s):

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Feynman rules for qed
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

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Feynman rules for qed1
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


Feynman rules for qed2
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 !

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Feynman rules for qed3
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)

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Example e e scattering

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

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Example e e e e moller scattering

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