Quantum Field Theory

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and v(⃗p). In the non-relativistic limit, p → (m, ⃗p), and( √p )· σξu(⃗p) = √ → √ ( )ξmp · ¯σξ ξv(⃗p) =( √p · σξ− √ p · ¯σξ)→ √ m(ξ−ξ)(5.49)In this limit, the spinor contractions in the amplitude for ψψ → ψψ scattering (5.45)become ū s′ · u s = 2mδ ss′ and the amplitude isq,rp,s/p,s/q,r/ /= −i(−iλ) 2 (2m)( δs ′s )δ r′ r(⃗p − ⃗p ′ ) + µ − δ s′r δ r′ s2 (⃗p − ⃗q ′ ) + µ 2(5.50)The δ symbols tell us that spin is conserved in the non-relativistic limit, while themomentum dependence is the same as in the bosonic case, telling us that once againthe particles feel an attractive Yukawa potential,U(⃗r) = − λ2 e −µr4πr(5.51)Repeating the calculation for ψ ¯ψ → ψ ¯ψ, there are two minus signs which cancel eachother. The first is the extra overall minus sign in the scattering amplitude (5.46),due to the fermionic nature of the particles. The second minus sign comes from thenon-relativistic limit of the spinor contraction for anti-particles in (5.46), which is¯v s′ · v s = −2mδ ss′ . These two signs cancel, giving us once again an attractive Yukawapotential (5.51).5.7.3 Pseudo-Scalar CouplingRather than the standard Yukawa coupling, we could instead considerThis still preserves parity if φ is a pseudoscalar, i.e.L Yuk = −λφ ¯ψγ 5 ψ (5.52)P : φ(⃗x, t) → −φ(−⃗x, t) (5.53)We can compute in this theory very simply: the Feynman rule for the interaction vertexis now changed to a factor of −iλγ 5 . For example, the Feynman diagrams for ψψ → ψψscattering are again given by Figure 25, with the amplitude now( )[ūA = (−iλ) 2 s ′ (⃗p ′ )γ 5 u s (⃗p)] [ū r′ (⃗q ′ )γ 5 u r (⃗q)] (⃗p ′ )γ 5 u r (⃗q)] [ū− [ūs′ r′ (⃗q ′ )γ 5 u s (⃗p)](p − p ′ ) 2 − µ 2 (p − q ′ ) 2 − µ 2– 122 –
We could again try to take the non-relativistic limit for this amplitude. But thistime, things work a little differently. Using the expressions for the spinors (5.49), wehave ū s′ γ 5 u s → 0 in the non-relativistic limit. To find the non-relativistic amplitude,we must go to next to leading order. One can easily check that ū s′ (⃗p ′ )γ 5 u s (⃗p) →m ξ s′ T (⃗p−⃗p ′ )·⃗σξ s . So, in the non-relativistic limit, the leading order amplitude arisingfrom pseudoscalar exchange is given by a spin-spin coupling,q,rp,s/p,s/q,r/ /→ +im(−iλ) 2 [ξs′ T (⃗p − ⃗p ′ ) · ⃗σξ s ] [ξ r′ T (⃗p − ⃗p ′ ) · ⃗σξ r ](⃗p − ⃗p ′ ) 2 + µ 2 (5.54)– 123 –
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4.7.2 Some Useful Formulae: Inner a
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0. Introduction“There are no real
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At distances shorter than this, the
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which allows us to express all dime
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1. Classical Field TheoryIn this fi
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and the potential energy of the fie
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1.2 Lorentz InvarianceThe laws of N
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Example 3: Maxwell’s EquationsUnd
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(where the sign in the field transf
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from which we see thatδφ = −ω
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Another Cute TrickThere is a quick
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2. Free Fields“The career of a yo
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Substituting into the above express
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Hmmmm. We’ve found a delta-functi
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2.3.2 The Casimir Effect“I mentio
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This is still infinite in the limit
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This space is known as a Fock space
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From this result we can figure out
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2.6 The Heisenberg PictureAlthough
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But what about arbitrary spacetime
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where T stands for time ordering, p
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Im(p 0)Im(p 0)−E p+ E pRe(p 0)−
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We may expand ψ(⃗x) as a Fourier
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which confirms (2.120). So we learn
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3. Interacting FieldsThe free field
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means that for experiments at small
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The interaction picture is a hybrid
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Actually these last two terms doubl
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• Obviously we can’t cope with
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where the ± signs on φ ± make li
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Now, using Wick’s theorem we see
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solid lines to denote its charge; w
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p 1p 1/p 1p 1/p 2p 2/p 2p 2/Figure
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Notice that the momentum dependence
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3.5.2 The Yukawa PotentialSo far we
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where we’ve introduced the dimens
Page 77 and 78: • We do not consider diagrams wit
Page 79 and 80: is the probability for the transiti
Page 81 and 82: meaning that the flux is given in t
Page 83 and 84: But the last term vanishes. This fo
Page 85 and 86: the S-matrix elements, where we wer
Page 87 and 88: 4. The Dirac Equation“A great dea
Page 89 and 90: where we have suppressed the matrix
Page 91 and 92: 4.1.1 SpinorsThe S µν are 4 × 4
Page 93 and 94: which can be anti-hermitian if all
Page 95 and 96: so the requirement (4.44) becomes
Page 97 and 98: 4.4 Chiral SpinorsWhen we’ve need
Page 99 and 100: 4.4.2 γ 5The Lorentz group matrice
Page 101 and 102: 4.4.4 Chiral InteractionsLet’s no
Page 103 and 104: where we’ve made use of the prope
Page 105 and 106: which, after direct substitution, t
Page 107 and 108: for any 2-component spinor ξ which
Page 109 and 110: HelicityThe helicity operator is th
Page 111 and 112: Proof:2∑s=1u s (⃗p) ū s (⃗p)
Page 113 and 114: Claim: The field commutation relati
Page 115 and 116: The δ (3) term is familiar and eas
Page 117 and 118: Let’s pause our discussion to mak
Page 119 and 120: In what follows we will often drop
Page 121 and 122: 5.6 Yukawa TheoryThe interaction be
Page 123 and 124: where we’ve put the φ propagator
Page 125 and 126: The denominators in each term are d
Page 127: Finally, we can also compute the sc
Page 131 and 132: These remain true even in the prese
Page 133 and 134: line can be reached by a gauge tran
Page 135 and 136: which means that ⃗ ξ is perpendi
Page 137 and 138: 6.2.2 Lorentz GaugeWe could try to
Page 139 and 140: The minus signs here are odd to say
Page 141 and 142: a 1,2 †⃗p, while |φ〉 contain
Page 143 and 144: where we’ve introduced a coupling
Page 145 and 146: Let’s now consider a complex scal
Page 147 and 148: 6.4.1 Naive Feynman RulesWe want to
Page 149 and 150: which is the claimed result. You ca
Page 151 and 152: Electron ScatteringElectron scatter
Page 153 and 154: momenta, we have the amplitudes giv
Page 155: 6.7 AfterwordIn this course, we hav
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Quantum Field Theory

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