Quantum Field Theory

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AcknowledgementsThese lecture notes are far from original. My primary contribution has been to borrow,steal and assimilate the best discussions and explanations I could find from the vastliterature on the subject. I inherited the course from Nick Manton, whose notes form thebackbone of the lectures. I have also relied heavily on the sources listed at the beginning,most notably the book by Peskin and Schroeder. In several places, for example thediscussion of scalar Yukawa theory, I followed the lectures of Sidney Coleman, usingthe notes written by Brian Hill and a beautiful abridged version of these notes due toMichael Luke.My thanks to the many who helped in various ways during the preparation of thiscourse, including Joe Conlon, Nick Dorey, Marie Ericsson, Eyo Ita, Ian Drummond,Jerome Gauntlett, Matt Headrick, Ron Horgan, Nick Manton, Hugh Osborn and JenniSmillie. My thanks also to the students for their sharp questions and sharp eyes inspotting typos. I am supported by the Royal Society.– 4 –
0. Introduction“There are no real one-particle systems in nature, not even few-particlesystems. The existence of virtual pairs and of pair fluctuations shows thatthe days of fixed particle numbers are over.”Viki WeisskopfThe concept of wave-particle duality tells us that the properties of electrons andphotons are fundamentally very similar. Despite obvious differences in their mass andcharge, under the right circumstances both suffer wave-like diffraction and both canpack a particle-like punch.Yet the appearance of these objects in classical physics is very different. Electronsand other matter particles are postulated to be elementary constituents of Nature. Incontrast, light is a derived concept: it arises as a ripple of the electromagnetic field. Ifphotons and particles are truely to be placed on equal footing, how should we reconcilethis difference in the quantum world? Should we view the particle as fundamental,with the electromagnetic field arising only in some classical limit from a collection ofquantum photons? Or should we instead view the field as fundamental, with the photonappearing only when we correctly treat the field in a manner consistent with quantumtheory? And, if this latter view is correct, should we also introduce an “electron field”,whose ripples give rise to particles with mass and charge? But why then didn’t Faraday,Maxwell and other classical physicists find it useful to introduce the concept of matterfields, analogous to the electromagnetic field?The purpose of this course is to answer these questions. We shall see that the secondviewpoint above is the most useful: the field is primary and particles are derivedconcepts, appearing only after quantization. We will show how photons arise from thequantization of the electromagnetic field and how massive, charged particles such aselectrons arise from the quantization of matter fields. We will learn that in order todescribe the fundamental laws of Nature, we must not only introduce electron fields,but also quark fields, neutrino fields, gluon fields, W and Z-boson fields, Higgs fieldsand a whole slew of others. There is a field associated to each type of fundamentalparticle that appears in Nature.Why Quantum Field Theory?In classical physics, the primary reason for introducing the concept of the field is toconstruct laws of Nature that are local. The old laws of Coulomb and Newton involve“action at a distance”. This means that the force felt by an electron (or planet) changes– 1 –
Page 5: 4.7.2 Some Useful Formulae: Inner a
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 and 52: which confirms (2.120). So we learn
Page 53 and 54: 3. Interacting FieldsThe free field
Page 55 and 56: 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
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where we’ve made use of the prope
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which, after direct substitution, t
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for any 2-component spinor ξ which
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HelicityThe helicity operator is th
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Proof:2∑s=1u s (⃗p) ū s (⃗p)
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Claim: The field commutation relati
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The δ (3) term is familiar and eas
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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
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The denominators in each term are d
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Finally, we can also compute the sc
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We could again try to take the non-
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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
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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
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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
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6.7 AfterwordIn this course, we hav
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Quantum Field Theory

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