The Dirac Equation. Origin of the Equation. In QM, observables have corresponding operators, e.g. Relativistically, we can identify p i ~ Schroedinger equation (nonrelativistic): KleinGordon equation (relativistic): Both differential equations are linear & second order in x m
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
I acknowledge much input from
“Introduction to Elementary Particles”
By David Griffiths, Wiley, 1987
p i~
Two extreme cases:
Brian Meadows, U. Cincinnati
Weyl Equation
Brian Meadows, U. Cincinnati
When
Brian Meadows, U. Cincinnati
{, } = 2g
Brian Meadows, U. Cincinnati
Negative energy
Positive energy
Brian Meadows, U. Cincinnati
(x) = ae(i/~)x¢ pu(p) [ x´ (ct, r ) and p´ (E/c, p ) ]
( p  mc) u = 0 (Algebraic, NOT a differential equation)
Brian Meadows, U. Cincinnati
Could Flip sign of E and p
E+mc2
Brian Meadows, U. Cincinnati
can be applied to u(1), u(2), v(1), v(2)
Brian Meadows, U. Cincinnati
Brian Meadows, U. Cincinnati
Brian Meadows, U. Cincinnati
Brian Meadows, U. Cincinnati
Note that each is a linear combination of 16 products of y components
Brian Meadows, U. Cincinnati
(Think of g0 as reversing the sign of the terms withpzorpx+ipyrelative to the other two terms)
P flips sign of
pseudoscalar
Brian Meadows, U. Cincinnati
Maxwell’s equations
Equation of continuity:
Brian Meadows, U. Cincinnati
Brian Meadows, U. Cincinnati
Invariant is zero in all frames.
Brian Meadows, U. Cincinnati
Therefore, A is a 4vector
Brian Meadows, U. Cincinnati
Brian Meadows, U. Cincinnati
4 component spinors
where one choice, for photons traveling in the z direction, is.
Transverse polarization
Brian Meadows, U. Cincinnati
u(1), u(2), v(1), v(2) need not be pure spin
states, but their sum is still “complete”.
Brian Meadows, U. Cincinnati
Label:
Label theirspins s1, … sn.
Label internal lines with 4momenta q1, … qn
Directions:
Arrows on external lines indicate Fermion or antiFermion
Arrows on internal lines preserve flow
External photon arrows point in direction of motion
Internal photon arrows do not matter
Brian Meadows, U. Cincinnati
Always follow a Fermion line
To obtain a product:
(adjointspinor)()(spinor)
E.g.:
pj , sj
pk , sk
e 
e 
q
u (sk)(k) ige u(s))(j)
Brian Meadows, U. Cincinnati
NOTE: qj2 = mj2c2 for internal lines.
NOTE also: use of the slash – q essentially the component along !
Brian Meadows, U. Cincinnati
Now conserve momentum (at each vertex)
5. Include a d function to conserve momentum at each vertex.
where the k's are the 4momenta entering the vertex
6. Integrate over all internal 4momenta qj. I.e. write a factor
For each internal line.
Brian Meadows, U. Cincinnati
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 eScatteringUse index “”
BUT use index “”
Brian Meadows, U. Cincinnati
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 – ee ee (Moller Scattering)Brian Meadows, U. Cincinnati