Quantum Field Theory

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The overall + sign comes from treating the fermions correctly: we saw the same minussign when studying scattering in Yukawa theory. The difference now comes from lookingat the non-relativistic limit. We have ¯vγ 0 v → 2m, giving us the potential betweenopposite charges,U(⃗r) = −e 2 ∫d 3 p(2π) 3 e i⃗p·⃗r|⃗p| 2 = − e24πr(6.99)Reassuringly, we find an attractive force between an electron and positron. The differencefrom the calculation of the Yukawa force comes again from the zeroth componentof the gauge field, this time in the guise of the γ 0 sandwiched between ¯vγ 0 v → 2m,rather than the ¯vv → −2m that we saw in the Yukawa case.The Coulomb Potential for ScalarsThere are many minus signs in the above calculation which somewhat obscure thecrucial one which gives rise to the repulsive force. A careful study reveals the offendingsign to be that which sits in front of the A 0 piece of the photon propagator −iη µν /p 2 .Note that with our signature (+−−−), the propagating A i have the correct sign, whileA 0 comes with the wrong sign. This is simpler to see in the case of scalar QED, wherewe don’t have to worry about the gamma matrices. From the Feynman rules of Section6.5.1, we have the non-relativistic limit of scalar e − e − scattering,p,sq,r/p,s // /q,r= −iη µν (−ie) 2 (p + p′ ) µ (q + q ′ ) ν(p ′ − p) 2 → −i(−ie) 2 (2m) 2−(⃗p − ⃗p ′ ) 2where the non-relativistic limit in the numerator involves (p+p ′ )·(q+q ′ ) ≈ (p+p ′ ) 0 (q+q ′ ) 0 ≈ (2m) 2 and is responsible for selecting the A 0 part of the photon propagator ratherthan the A i piece. This shows that the Coulomb potential for spin 0 particles of thesame charge is again repulsive, just as it is for fermions. For e − e + scattering, theamplitude picks up an extra minus sign because the arrows on the legs of the Feynmanrules in Section 6.5.1 are correlated with the momentum arrows. Flipping the arrowson one pair of legs in the amplitude introduces the relevant minus sign to ensure thatthe non-relativistic potential between e − e + is attractive as expected.– 148 –
6.7 AfterwordIn this course, we have laid the foundational framework for quantum field theory. Mostof the developments that we’ve seen were already in place by the middle of the 1930s,pioneered by people such as Jordan, Dirac, Heisenberg, Pauli and Weisskopf 5 .Yet by the end of the 1930s, physicists were ready to give up on quantum field theory.The difficulty lies in the next terms in perturbation theory. These are the terms thatcorrespond to Feynamn diagrams with loops in them, which we have scrupulouslyavoided computing in this course. The reason we’ve avoided them is because theytypically give infinity! And, after ten years of trying, and failing, to make sense of this,the general feeling was that one should do something else. This from Dirac in 1937,Because of its extreme complexity, most physicists will be glad to see theend of QEDBut the leading figures of the day gave up too soon. It took a new generation of postwarphysicists — people like Schwinger, Feynman, Tomonaga and Dyson — to return toquantum field theory and tame the infinities. The story of how they did that will betold in next term’s course.5 For more details on the history of quantum field theory, see the excellent book “QED and theMen who Made it” by Sam Schweber.– 149 –
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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
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• We do not consider diagrams wit
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is the probability for the transiti
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meaning that the flux is given in t
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But the last term vanishes. This fo
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the S-matrix elements, where we wer
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4. The Dirac Equation“A great dea
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where we have suppressed the matrix
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4.1.1 SpinorsThe S µν are 4 × 4
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which can be anti-hermitian if all
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so the requirement (4.44) becomes
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4.4 Chiral SpinorsWhen we’ve need
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4.4.2 γ 5The Lorentz group matrice
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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 and 128: Finally, we can also compute the sc
Page 129 and 130: We could again try to take the non-
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: momenta, we have the amplitudes giv
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Quantum Field Theory

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