Quantum Field Theory

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One can also consider interactions between particles. Obviously these are only importantfor n particle states with n ≥ 2. We therefore expect them to arise from additionsto the Lagrangian of the form∆L = ψ ⋆ (⃗x) ψ ⋆ (⃗x) ψ(⃗x) ψ(⃗x) (2.126)which, in the quantum theory, is an operator which destroys two particles before creatingtwo new ones. Such terms in the Lagrangian will indeed lead to inter-particle forces,both in the non-relativistic and relativistic setting. In the next section we explore thesetypes of interaction in detail for relativistic theories.– 46 –
3. Interacting <strong>Field</strong>sThe free field theories that we’ve discussed so far are very special: we can determinetheir spectrum, but nothing interesting then happens. They have particle excitations,but these particles don’t interact with each other.Here we’ll start to examine more complicated theories that include interaction terms.These will take the form of higher order terms in the Lagrangian. We’ll start by askingwhat kind of small perturbations we can add to the theory. For example, consider theLagrangian for a real scalar field,L = 1 2 ∂ µφ ∂ µ φ − 1 2 m2 φ 2 − ∑ n≥3λ nn! φn (3.1)The coefficients λ n are called coupling constants. What restrictions do we have on λ nto ensure that the additional terms are small perturbations? You might think that weneed simply make “λ n ≪ 1”. But this isn’t quite right. To see why this is the case, let’sdo some dimensional analysis. Firstly, note that the action has dimensions of angularmomentum or, equivalently, the same dimensions as . Since we’ve set = 1, usingthe convention described in the introduction, we have [S] = 0. With S = ∫ d 4 x L, and[d 4 x] = −4, the Lagrangian density must therefore have[L] = 4 (3.2)What does this mean for the Lagrangian (3.1)? Since [∂ µ ] = 1, we can read off themass dimensions of all the factors to find,[φ] = 1 , [m] = 1 , [λ n ] = 4 − n (3.3)So now we see why we can’t simply say we need λ n ≪ 1, because this statement onlymakes sense for dimensionless quantities. The various terms, parameterized by λ n , fallinto three different categories• [λ 3 ] = 1: For this term, the dimensionless parameter is λ 3 /E, where E hasdimensions of mass. Typically in quantum field theory, E is the energy scale ofthe process of interest. This means that λ 3 φ 3 /3! is a small perturbation at highenergies E ≫ λ 3 , but a large perturbation at low energies E ≪ λ 3 . Terms thatwe add to the Lagrangian with this behavior are called relevant because they’remost relevant at low energies (which, after all, is where most of the physics we seelies). In a relativistic theory, E > m, so we can always make this perturbationsmall by taking λ 3 ≪ m.– 47 –
Page 5 and 6: 4.7.2 Some Useful Formulae: Inner a
Page 7 and 8: 0. Introduction“There are no real
Page 9 and 10: At distances shorter than this, the
Page 11 and 12: which allows us to express all dime
Page 13 and 14: 1. Classical Field TheoryIn this fi
Page 15 and 16: and the potential energy of the fie
Page 17 and 18: 1.2 Lorentz InvarianceThe laws of N
Page 19 and 20: Example 3: Maxwell’s EquationsUnd
Page 21 and 22: (where the sign in the field transf
Page 23 and 24: from which we see thatδφ = −ω
Page 25 and 26: Another Cute TrickThere is a quick
Page 27 and 28: 2. Free Fields“The career of a yo
Page 29 and 30: Substituting into the above express
Page 31 and 32: Hmmmm. We’ve found a delta-functi
Page 33 and 34: 2.3.2 The Casimir Effect“I mentio
Page 35 and 36: This is still infinite in the limit
Page 37 and 38: This space is known as a Fock space
Page 39 and 40: From this result we can figure out
Page 41 and 42: 2.6 The Heisenberg PictureAlthough
Page 43 and 44: But what about arbitrary spacetime
Page 45 and 46: where T stands for time ordering, p
Page 47 and 48: Im(p 0)Im(p 0)−E p+ E pRe(p 0)−
Page 49 and 50: We may expand ψ(⃗x) as a Fourier
Page 51: which confirms (2.120). So we learn
Page 55 and 56: means that for experiments at small
Page 57 and 58: The interaction picture is a hybrid
Page 59 and 60: Actually these last two terms doubl
Page 61 and 62: • Obviously we can’t cope with
Page 63 and 64: where the ± signs on φ ± make li
Page 65 and 66: Now, using Wick’s theorem we see
Page 67 and 68: solid lines to denote its charge; w
Page 69 and 70: p 1p 1/p 1p 1/p 2p 2/p 2p 2/Figure
Page 71 and 72: Notice that the momentum dependence
Page 73 and 74: 3.5.2 The Yukawa PotentialSo far we
Page 75 and 76: 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
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The δ (3) term is familiar and eas
Page 117 and 118:
Let’s pause our discussion to mak
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In what follows we will often drop
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5.6 Yukawa TheoryThe interaction be
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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
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We could again try to take the non-
Page 131 and 132:
These remain true even in the prese
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line can be reached by a gauge tran
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which means that ⃗ ξ is perpendi
Page 137 and 138:
6.2.2 Lorentz GaugeWe could try to
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The minus signs here are odd to say
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a 1,2 †⃗p, while |φ〉 contain
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where we’ve introduced a coupling
Page 145 and 146:
Let’s now consider a complex scal
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6.4.1 Naive Feynman RulesWe want to
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which is the claimed result. You ca
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Electron ScatteringElectron scatter
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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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