Quantum Field Theory

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4.7.2 Some Useful Formulae: Inner and Outer Products 1035. Quantizing the Dirac Field 1065.1 A Glimpse at the Spin-Statistics Theorem 1065.1.1 The Hamiltonian 1075.2 Fermionic Quantization 1095.2.1 Fermi-Dirac Statistics 1105.3 Dirac’s Hole Interpretation 1105.4 Propagators 1125.5 The Feynman Propagator 1145.6 Yukawa Theory 1155.6.1 An Example: Putting Spin on Nucleon Scattering 1155.7 Feynman Rules for Fermions 1175.7.1 Examples 1185.7.2 The Yukawa Potential Revisited 1215.7.3 Pseudo-Scalar Coupling 1226. Quantum Electrodynamics 1246.1 Maxwell’s Equations 1246.1.1 Gauge Symmetry 1256.2 The Quantization of the Electromagnetic Field 1286.2.1 Coulomb Gauge 1286.2.2 Lorentz Gauge 1316.3 Coupling to Matter 1366.3.1 Coupling to Fermions 1366.3.2 Coupling to Scalars 1386.4 QED 1396.4.1 Naive Feynman Rules 1416.5 Feynman Rules 1436.5.1 Charged Scalars 1446.6 Scattering in QED 1446.6.1 The Coulomb Potential 1476.7 Afterword 149– 3 –
Page 6 and 7: AcknowledgementsThese lecture notes
Page 8 and 9: immediately if a distant proton (or
Page 10 and 11: and, to agree with experiment, shou
Page 12 and 13: to about the TeV . This is precisel
Page 14 and 15: where the official name for L is th
Page 16 and 17: The initial data required on a Cauc
Page 18 and 19: The definition of a Lorentz invaria
Page 20 and 21: where A is the area bounding V and
Page 22 and 23: where Γ ρµν is some function of
Page 24 and 25: 1.3.4 Internal SymmetriesThe above
Page 26 and 27: An Example: A Real Scalar FieldFor
Page 28 and 29: from all the others. Free field the
Page 30 and 31: Claim: The commutation relations fo
Page 32 and 33: We can deal with the infinity in (2
Page 34 and 35: But there’s a problem. E is infin
Page 36 and 37: We interpret the state |⃗p〉 as
Page 38 and 39: But is this Lorentz invariant? It
Page 40 and 41: Since the classical field ψ is not
Page 42 and 43: = i 2∫d 3 y (∇ y [φ(y), π(x)]
Page 44 and 45: 2.7 PropagatorsWe could ask a diffe
Page 46 and 47: ∫=d 3 p(2π) 3 12E ⃗pe −ip·(
Page 48 and 49: this is equivalent to saying that |
Page 50 and 51: which, on one-particle states, give
Page 52 and 53:
One can also consider interactions
Page 54 and 55:
• [λ 4 ] = 0: this term is small
Page 56 and 57:
The scalar Yukawa theory has a slig
Page 58 and 59:
where U(t, t 0 ) is a unitary time
Page 60 and 61:
the interaction picture and follow
Page 62 and 63:
ψ † ∼ b † + c. To get non-ze
Page 64 and 65:
A similar discussion holds for comp
Page 66 and 67:
momentum conservation, |⃗p 1 | =
Page 68 and 69:
3.5 Examples of Scattering Amplitud
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p 1/p 1/p 2p 2+p 2p 1p /p / 21Figur
Page 72 and 73:
3.5.1 Mandelstam VariablesWe see th
Page 74 and 75:
Now we understand the profile of th
Page 76 and 77:
vertex rather than (−iλ/4!). To
Page 78 and 79:
where ω = E m −E n . This gives
Page 80 and 81:
that A = −g for a single meson de
Page 82 and 83:
and 〈Ω| Ω〉 = 1. Note that thi
Page 84 and 85:
to an exponential,〈0|S |0〉 = ex
Page 86 and 87:
So what’s the point of all of thi
Page 88 and 89:
Note that an antisymmetric 4 × 4 m
Page 90 and 91:
One can construct many other repres
Page 92 and 93:
and the spinor rotation matrix beco
Page 94 and 95:
so thatS[Λ] † = exp ( 12 Ω ρσ
Page 96 and 97:
4.3 The Dirac EquationThe equation
Page 98 and 99:
where we have introduced some new n
Page 100 and 101:
4.4.3 ParityThe spinors ψ ± are r
Page 102 and 103:
We’re now armed with new terms in
Page 104 and 105:
4.6 Symmetries and Conserved Curren
Page 106 and 107:
Here the second transformation foll
Page 108 and 109:
4.7.1 Some ExamplesConsider the pos
Page 110 and 111:
We have analogous results for the n
Page 112 and 113:
5. Quantizing the Dirac FieldWe wou
Page 114 and 115:
which means that H = ∫ d 3 x H ag
Page 116 and 117:
The calculation of the Hamiltonian
Page 118 and 119:
an electron, the two can annihilate
Page 120 and 121:
• At least away from singularitie
Page 122 and 123:
where, as usual, all fields are in
Page 124 and 125:
• Each internal line gets a facto
Page 126 and 127:
p,s/p,s/p,s/p,s/+q,r / /q,rq,r / /q
Page 128 and 129:
and v(⃗p). In the non-relativisti
Page 130 and 131:
6. Quantum ElectrodynamicsIn this s
Page 132 and 133:
So what are we to make of this? We
Page 134 and 135:
6.2 The Quantization of the Electro
Page 136 and 137:
We now write ⃗ A in the usual mod
Page 138 and 139:
and we can make the usual expansion
Page 140 and 141:
• We could try to impose the cond
Page 142 and 143:
PropagatorsFinally, it’s a simple
Page 144 and 145:
charge. For QED, the theory of ligh
Page 146 and 147:
In Coulomb gauge ∇ · ⃗A = 0, t
Page 148 and 149:
The D 00 piece of the propagator do
Page 150 and 151:
6.5.1 Charged Scalars“Pauli asked
Page 152 and 153:
scattering process was one of the c
Page 154 and 155:
The overall + sign comes from treat
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Quantum Field Theory

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