Introduction to Quantum Electrodynamics. Properties of Dirac Spinors Description of photons Feynman rules for QED Simple examples Spins and traces. Summary – Spin1/2. u (1) , u (2) , v (1) , v (2) need not be pure spin states, but their sum is still “complete”.
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Properties of Dirac Spinors
Description of photons
Feynman rules for QED
Simple examples
Spins and traces
Much of the content of these slides is acknowledged to come from
“Introduction to Elementary Particles”
By David Griffiths, Wiley, 1987
u(1), u(2), v(1), v(2) need not be pure spin
states, but their sum is still “complete”.
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Label:
Label theirspins s1, … sn.
Label internal lines with 4momenta q1, … qn
Directions:
Arrows on external indicate Fermion or antiFermion
Arrows on internal lines preserve flow
Arrows on internal lines preserve flow
External photon arrows point in direction of motion
Internal photon arrows do not matter
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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)
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NOTE: qj2 = mj2c2 for internal lines.
NOTE also: use of the ”slash – q” a 4 x 4 matrix rather than a 4vector!
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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.
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p1 , s1
p3 , s3
e 
e 
u (s3 )(3) ige u(s1 )(1)
q
u (s4 )(4) ige u(s2 )(2)


p2 , s2
p4 , s4
NOTE u(k) is short
for u(sk )(pk)
Example – e eScatteringUse index “”
BUT use index “”
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where
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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
and (usually) sum over final spins and average over initial ones.
whereis a 4 x 4 matrix (in this case,).
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4 x 4 matrix
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4 x 4 matrix
WHY ?
scalar
Scalar
NO u’s !!
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4 x 4 matrix
scalar
?
4 x 4 matrix
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where m is mass of electron and M is mass of .
Average over
4 initial spins
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Spin average over initial states
4 g
0
0
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e 
e 

Evaluate CrossSection for e ScatteringAssume so that we can ignore the recoil of the, and therefore p1 = p3 = p and p2 = p4 = 0
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p2 , s2
p4 , s4
p2 , s2
p4 , s4
+
Time
p1 , s1
p3 , s3
p1 , s1
p3 , s3
Fermion line (backwards)
g in
g out
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where q = p1p3, etc..
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