Quantum Field Theory

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to about the TeV . This is precisely the regime that is currently being probed by theLarge Hadron Collider (LHC) at CERN. There is a general belief that the frameworkof quantum field theory will continue to hold to energy scales only slightly below thePlanck scale — for example, there are experimental hints that the coupling constantsof electromagnetism, and the weak and strong forces unify at around 10 18 GeV.For comparison, the rough masses of some elementary (and not so elementary) particlesare shown in the table,ParticleneutrinoselectronMuonPionsProton, NeutronTauW,Z BosonsHiggs BosonMass∼ 10 −2 eV0.5 MeV100 MeV140 MeV1 GeV2 GeV80-90 GeV120-200 GeV??– 6 –
1. Classical Field TheoryIn this first section we will discuss various aspects of classical fields. We will cover onlythe bare minimum ground necessary before turning to the quantum theory, and willreturn to classical field theory at several later stages in the course when we need tointroduce new ideas.1.1 The Dynamics of FieldsA field is a quantity defined at every point of space and time (⃗x, t). While classicalparticle mechanics deals with a finite number of generalized coordinates q a (t), indexedby a label a, in field theory we are interested in the dynamics of fieldsφ a (⃗x, t) (1.1)where both a and ⃗x are considered as labels. Thus we are dealing with a system with aninfinite number of degrees of freedom — at least one for each point ⃗x in space. Noticethat the concept of position has been relegated from a dynamical variable in particlemechanics to a mere label in field theory.An Example: The Electromagnetic FieldThe most familiar examples of fields from classical physics are the electric and magneticfields, ⃗ E(⃗x, t) and ⃗ B(⃗x, t). Both of these fields are spatial 3-vectors. In a more sophisticatedtreatement of electromagnetism, we derive these two 3-vectors from a single4-component field A µ (⃗x, t) = (φ, ⃗ A) where µ = 0, 1, 2, 3 shows that this field is a vectorin spacetime. The electric and magnetic fields are given by⃗E = −∇φ − ∂ ⃗ A∂tand ⃗ B = ∇ × ⃗ A (1.2)which ensure that two of Maxwell’s equations, ∇ · ⃗B = 0 and d ⃗ B/dt = −∇ × ⃗ E, holdimmediately as identities.The LagrangianThe dynamics of the field is governed by a Lagrangian which is a function of φ(⃗x, t),˙φ(⃗x, t) and ∇φ(⃗x, t). In all the systems we study in this course, the Lagrangian is ofthe form,∫L(t) =d 3 x L(φ a , ∂ µ φ a ) (1.3)– 7 –
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: which allows us to express all dime
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
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
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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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