Quantum Field Theory

More documents

Recommendations

Info

We have analogous results for the negative frequency solutions, which we may write as( √p )· σ ηs with vr †v s (⃗p) =− √ (⃗p) · v s (⃗p) = 2p 0 δ rs(4.124)p · ¯σ η s and ¯v r (⃗p) · v s (⃗p) = −2mδ rsWe can also compute the inner product between u and v. We haveū r (⃗p) · v s (⃗p) = ( ξ r † √ p · σ , ξ r †√ p · ¯σ ) γ 0 ( √p · σ ηs− √ p · ¯σ η s )= ξ r†√ (p · ¯σ)(p · σ)η s − ξ r †√ (p · ¯σ)(p · σ)η s = 0 (4.125)and similarly, ¯v r (⃗p) · u s (⃗p) = 0. However, when we come to u † · v, it is a slightlydifferent combination that has nice properties (and this same combination appearswhen we quantize the theory). We look at u r † (⃗p) · v s (−⃗p), with the 3-momentum inthe spinor v taking the opposite sign. Defining the 4-momentum (p ′ ) µ = (p 0 , −⃗p), wehaveu r † (⃗p) · v s (−⃗p) = ( ξ r † √ p · σ , ξ r †√ p · ¯σ ) ( √ )p′ · σ η s− √ p ′ · ¯σ η s= ξ r†√ (p · ¯σ)(p ′ · σ)η s − ξ r †√ (p · ¯σ)(p ′ · σ)η s (4.126)Now the terms under the square-root are given by (p·¯σ)(p ′·σ) = (p 0 +p i σ i )(p 0 −p i σ i ) =p 2 o − ⃗p 2 = m 2 . The same expression holds for (p ′ · ¯σ)(p · σ), and the two terms cancel.We learnOuter Productsu r † (⃗p) · v s (−⃗p) = v r † (⃗p) · u s (−⃗p) = 0 (4.127)There’s one last spinor identity that we need before we turn to the quantum theory. It is:Claim:2∑s=1u s (⃗p) ū s (⃗p) = /p + m (4.128)where the two spinors are not now contracted, but instead placed back to back to givea 4 × 4 matrix. Also,2∑s=1v s (⃗p) ¯v s (⃗p) = /p − m (4.129)– 104 –
Proof:2∑s=1u s (⃗p) ū s (⃗p) =2∑s=1( √p · σ ξs√ p · ¯σ ξs)(ξs † √ p · ¯σ , ξ s †√ p · σ ) (4.130)But ∑ s ξs ξ s † = 1, the 2 × 2 unit matrix, which then gives us( )2∑m p · σu s (⃗p)ū s (⃗p) =p · ¯σ ms=1which is the desired result. A similar proof works for ∑ s vs (⃗p)¯v s (⃗p).(4.131)□– 105 –
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 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
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: HelicityThe helicity operator is th
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 and 154: momenta, we have the amplitudes giv
Page 155: 6.7 AfterwordIn this course, we hav
show all

Quantum Field Theory

Create successful ePaper yourself

Delete template?

Save as template?