Quantum Field Theory

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and, to agree with experiment, should choose Bose statistics (no minus sign) for integerspin particles, and Fermi statistics (yes minus sign) for half-integer spin particles. Inquantum field theory, this relationship between spin and statistics is not somethingthat you have to put in by hand. Rather, it is a consequence of the framework.What is Quantum Field Theory?Having told you why QFT is necessary, I should really tell you what it is. The clue is inthe name: it is the quantization of a classical field, the most familiar example of whichis the electromagnetic field. In standard quantum mechanics, we’re taught to take theclassical degrees of freedom and promote them to operators acting on a Hilbert space.The rules for quantizing a field are no different. Thus the basic degrees of freedom inquantum field theory are operator valued functions of space and time. This means thatwe are dealing with an infinite number of degrees of freedom — at least one for everypoint in space. This infinity will come back to bite on several occasions.It will turn out that the possible interactions in quantum field theory are governedby a few basic principles: locality, symmetry and renormalization group flow (thedecoupling of short distance phenomena from physics at larger scales). These ideasmake QFT a very robust framework: given a set of fields there is very often an almostunique way to couple them together.What is Quantum Field Theory Good For?The answer is: almost everything. As I have stressed above, for any relativistic systemit is a necessity. But it is also a very useful tool in non-relativistic systems with manyparticles. Quantum field theory has had a major impact in condensed matter, highenergyphysics, cosmology, quantum gravity and pure mathematics. It is literally thelanguage in which the laws of Nature are written.0.1 Units and ScalesNature presents us with three fundamental dimensionful constants; the speed of light c,Planck’s constant (divided by 2π) and Newton’s constant G. They have dimensions[c] = LT −1[] = L 2 MT −1[G] = L 3 M −1 T −2Throughout this course we will work with “natural” units, defined byc = = 1 (0.1)– 4 –
which allows us to express all dimensionful quantities in terms of a single scale whichwe choose to be mass or, equivalently, energy (since E = mc 2 has become E = m).The usual choice of energy unit is eV , the electron volt or, more often GeV = 10 9 eV orTeV = 10 12 eV . To convert the unit of energy back to a unit of length or time, we needto insert the relevant powers of c and . For example, the length scale λ associated toa mass m is the Compton wavelengthλ = mcWith this conversion factor, the electron mass m e = 10 6 eV translates to a length scaleλ e = 2 × 10 −12 m.Throughout this course we will refer to the dimension of a quantity, meaning themass dimension. If X has dimensions of (mass) d we will write [X] = d. In particular,the surviving natural quantity G has dimensions [G] = −2 and defines a mass scale,G = cM 2 p= 1M 2 p(0.2)where M p ≈ 10 19 GeV is the Planck scale. It corresponds to a length l p ≈ 10 −33 cm. ThePlanck scale is thought to be the smallest length scale that makes sense: beyond thisquantum gravity effects become important and it’s no longer clear that the conceptof spacetime makes sense. The largest length scale we can talk of is the size of thecosmological horizon, roughly 10 60 l p .ObservableUniverse~ 20 billion light yearsEarth 10 10 cmCosmologicalConstantAtoms−810 cmNuclei−1310 cmLHCPlanck Scale−3310cmlengthEnergy−3310eV−3 10eV10 1112−10eV= 1 TeV10 2819eV= 10 GeVFigure 2: Energy and Distance Scales in the UniverseSome useful scales in the universe are shown in the figure. This is a logarithmic plot,with energy increasing to the right and, correspondingly, length increasing to the left.The smallest and largest scales known are shown on the figure, together with otherrelevant energy scales. The standard model of particle physics is expected to hold up– 5 –
Page 5 and 6: 4.7.2 Some Useful Formulae: Inner a
Page 7 and 8: 0. Introduction“There are no real
Page 9: At distances shorter than this, the
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
Page 57 and 58: The interaction picture is a hybrid
Page 59 and 60: 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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