Master's Thesis in Theoretical Physics - Universiteit Utrecht

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Note that this is a matrix equation since Ξ and M 2 are matrices, thus one should keep inmind that all the other terms are multiplied by the 2×2 unit matrix. Comparing this to Eq.(5.53), we see that we have the same field equations for the two-Higgs doublet model as forthe real scalar field. There are only two differences. 1) Instead of one field we now have twoscalar fields. 2) The scalar fields are complex instead of real fields. We now take these twodifferences into account when we quantize the two-Higgs doublet model.We now perform canonical quantization for the complex scalar fields φ a and φ b by imposingthe commutator[ˆφi (⃗x,η), ˆπ j (⃗x ′ ,η) ] = iδ i j δ 3 (⃗x −⃗x ′ ), (i, j = 1,2), (5.114)where the canonical momentum is by definitionWe could also writeˆπ i =∂L = a 2∂∂ η φ ˆφ∗′ i, (i = 1,2). (5.115)i( ) (ˆπ1ˆΠ ˆφ∗′≡ˆπ = a 2 12ˆφ ∗′2)≡ a 2 ˆΦ ∗′ . (5.116)The system of field equations for the complex fields φ 1 and φ 2 that we want to solve is linear(Eq. (5.113)), and since the nonminimal coupling and mass matrices are hermitean, we canexpress the complex fields in terms of annihilation and creation operators asˆφ i (x) = ∑ ∫ d 3 k[l (2π) 3 e ik·x φ k,il (η)â − k,l + e−ik·x ϕ ∗ k,il (η) ˆb]+k,l= ∑ ∫ d 3 k[ eik·xl (2π) 3 φ k,il (η)â − k,l + ϕ∗ k,il (η) ˆb]+−k,l,(i, l = 1,2 ; k = |k|). (5.117)Notice the differences with the expansion for the single real scalar field in Eq. (5.57). Firstof all there are now creation and annihilation operators for the fields φ 1 and φ 2 , since wehave the canonical commutation relation for both fields, see the δ i j in Eq. (5.114). Secondlywe defined two sets of creation and annihilation operators a ± and b ± and two modefunctions φ k and ϕ k . We did this because the fields are complex, i.e. φ ∗ i ≠ φ i. Physically,excitations of the field φ describes particles, whereas excitations of its complex conjugatedescribes antiparticles. The operators a ± create or annihilate a particle, whereas the operatorsb ± create or annihilate an antiparticle.From the expansion (5.117) we can find the conjugate momentumˆπ j (x) = a 2 ˆφ∗′ j= a 2 ∑ ∫ d 3 k[ eik·xm (2π) 3 φ ∗′k, jm (η)â+ k,m + ϕ′ k, jm (η) ˆb]−−k,m,( j, m = 1,2 ; k = |k|). (5.118)The equations for ˆφi and ˆπ j can also be written in matrix form asˆΦ(x) =∫d 3 k(2π) 3 eik·x [ Φ k (η) · Â− k + Ψ∗ k (η) · ˆB + −kˆΠ(x) = a 2 ∫ d 3 k(2π) 3 e−ik·x [ Φ ∗′k (η) · Â+ k + Ψ′ k (η) · ˆB − −k], (5.119)]
whereΦ k =( ) ( ) ( ) ( )φk,aa φ k,abϕk,aa ϕ, Ψ k,abφ k,ba φ k =, Â −k,bb ϕ k,ba ϕk = âk,aˆbk,a, ˆB −k,bb âk = k,bˆb k,bThe creation and annihilation operators a ± and b ± form two independent sets and obey thecommutation relations[ ]â − k,i , â+ = δk ′ , j i j (2π) 3 δ(k − k ′ )[ˆb−k,i , ˆb]+ = δk ′ , j i j (2π) 3 δ(k − k ′ )[â − k,i , ˆb] [ ]+ = ˆb−k ′ , j k,i , â+ k ′ , j= 0. (5.120)Again we have the relations a + = (a − ) † and b + = (b − ) † . Using the expansion of the fields ˆφi inEq. (5.117) and the conjugate momenta ˆπ j in Eq. (5.118) together with these commutationrelations we can write the canonical commutator from Eq. (5.114) as[ˆφi (x,η), ˆπ j (x ′ ,η) ] = a 2 ∑ m∫d 3 k(2π) 3 eik·(x−x′ ) ( φ k,im (η)φ ∗′k, jm (η) − ϕ∗ k,im (η)ϕ′ k, jm (η) )∫ d= a 2 3 k( eik·(x−x′ )(2π) 3 Φ k (η) · Φ †′k (η) − Ψ∗ k(η) · ΨT′k)i (η) j= iδ i j δ 3 (⃗x −⃗x ′ ), (i, j = a, b),where I have imposed the canonical commutation relation in the last line. This implies that()Φ k (η) · Φ †′k (η) − Ψ∗ k(η) · ΨT′k (η) = Φ k(η) · Φ †′Tk (η) − Ψ ′ k (η) · Ψ† k (η) i =a 2 . (5.121)Now we substitute the mode expansion for Φ from Eq. (5.119) into the field equation Eq.(5.113) to get(∂ 2 η + k2 + a 2[ R(Ξ − 1 6 ) + M2]) (aΦ k (η)) = 0(∂ 2 η + k2 + a 2[ R(Ξ − 1 6 ) + M2]) (aΨ ∗ k(η)) = 0, (5.122)and similarly for the complex conjugates of the fields Φ k and Ψ ∗ . We see that we obtainkthe same equations as for the real scalar field, Eq. (5.62). Thus, again we can solve thisequation in quasi de Sitter space and find the propagator. This is exactly what we will doin the next section.5.3.1 Propagator for the two-Higgs doublet modelIn quasi de Sitter space with the scale factor (2.23), the general solutions of Eq. (5.122) areΦ k = Φ (1)kΨ ∗ k= Φ (1)k· α + Φ(2) k · β· γ + Φ(2) · δ, (5.123)kwhere α,β,γ,δ are 2 × 2 matrices and the functions Φ (1) and Φ(2)k kare√− πη (−kη)Φ (1)k (η) = 1 aΦ (2)k (η) = 1 a√− πη4 H(1) ν4 H(2) ν(−kη). (5.124)
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INFLATION AND BARYOGENESIS THROUGH
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AbstractIn this thesis we consider
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6 The fermion sector 716.1 Fermion
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and particle physics. The Einstein
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2.2 An expanding universeWe are now
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Here, H 0 = H(t = 0) and we have ch
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Since the spatial background is hom
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into neutral hydrogen at a temperat
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horizon during inflation isl phys (
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VΦSlowrollregime0 MpΦFigure 3.1:
Page 24 and 25: wavelengths that are larger than th
Page 26 and 27: In the Standard Model, the only sca
Page 28 and 29: esearch of density perturbations in
Page 30 and 31: Finally, we can also vary the actio
Page 32 and 33: such that we get for Eqs. (4.18)Fro
Page 34 and 35: Φt252015105N8060402000 200 400 600
Page 36 and 37: 4.2 The conformal transformationThe
Page 38 and 39: Eq. (4.44) is invariant under a con
Page 40 and 41: Note that we have succeeded in remo
Page 42 and 43: Figure 4.6: WMAP measurements of th
Page 44 and 45: Note that the matrices Ω and ˜Ξ
Page 46 and 47: and we can writeξ ++ =ξ −− =
Page 48 and 49: value of ξ. With the explicit expr
Page 50 and 51: 0.600.800.550.75Φ1t0.50Φ2t0.700.4
Page 53 and 54: Chapter 5Quantum field theory in an
Page 55 and 56: with equation of motion0 = ¨φk +
Page 57 and 58: Eq. (5.18). If we now act with the
Page 59 and 60: (5.28) with α = 1 and β = 0, and
Page 61 and 62: 5.2 Quantization of the nonminimall
Page 63 and 64: If we now substitute the expansion
Page 65 and 66: the adiabatic regime, whereas the t
Page 67 and 68: the lowest energy state at a later
Page 69 and 70: wherey ≡ y ++ (x, x ′ ) = −(|
Page 71 and 72: Now we plug all these expansions in
Page 73: of the field and expand the action
Page 77 and 78: This again exactly the same express
Page 79 and 80: Chapter 6The fermion sectorIn the p
Page 81 and 82: In this representation the projecti
Page 83 and 84: These relations allow us to rewrite
Page 85 and 86: 6.2.1 Fermion propagator in curved
Page 88 and 89: We now again rescale the field χL/
Page 90 and 91: φψψFigure 6.1: Feynman diagram f
Page 92 and 93: Note that the fact that the scalar
Page 94 and 95: The next step is to get rid of the
Page 96 and 97: such that we can findχ(x ′ ) = e
Page 98 and 99: where in all the equations z = k∆
Page 100 and 101: We can now compare this equation to
Page 103 and 104: Chapter 7Summary and outlookCosmolo
Page 105 and 106: esulting action as the action of a
Page 107: In future work we hope to study som
Page 111: Appendix AConventions• Throughout
Page 114 and 115: Finally, we use the Bianchi identit
Page 116 and 117: Note that we could also have calcul
Page 118: [24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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