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# The Dirac Equation - PowerPoint PPT Presentation

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

I acknowledge much input from

“Introduction to Elementary Particles”

By David Griffiths, Wiley, 1987

• 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:

• 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

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

• 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:

• Choose one factor:

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

• Solutions at zero momentum:

Negative energy

Positive energy

• 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

• Evaluate the wave-functions using:

• To obtain

Could Flip sign of E and p

 E+mc2

• 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)

• 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)

• 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

• Define the adjoint Dirac spinor as

• Clearly it flips the last two coordinates.

• Then is invariant:

• Because

• Define

• Then the following transform in the indicated ways:

Note that each is a linear combination of 16 products of y components

• 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

Maxwell’s equations

Equation of continuity:

• In homogenous ,linear, isotropic, medium with

• Conductivity

• Dielectric constant

• Permeability

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

• Substitute into Maxwell’s equations:

• Auxiliary conditions:

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

• Equation of continuity becomes:

Invariant is zero in all frames.

• 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

• Lorentz condition is then

• The Lorentz condition can be further restricted without changing it. We can, for example, choose

• Maxwell’s equations are then

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

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

states, but their sum is still “complete”.

• Photons have two spin projections (s):

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

• For external lines, write factor

• For each vertex write a factor

To obtain a product:

E.g.:

pj , sj

pk , sk

e -

e -

q

u (sk)(k) ige u(s))(j)

• 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 !

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)

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 “”

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)