Cosmin Crucean: Perturbative QED on de Sitter Universe.

PERTURBATIVE Q.E.D ON DE SITTER UNIVERSE WEST UNIVERSITY OF TIMISOARA FACULTY OF PHYSICS author: Crucean Cosmin 1

THE THEORY OF FREE FIELDS ON DE SITTER UNIVERSE Dirac ﬁeld • De Sitter metric: 1 ds2 = dt2 − e2ωtdx 2 = (dt2 − dx 2), ω > 0, ωtc = −e−ωt (1) (ωtc)2 c ˆ gµν = ηαβ eαeβ ˆ ˆ ˆ µ ν (2) • In the Cartesian gauge the nonvanishing tetradic components are: ˆ 1 ˆ ˆ 1 e0 = −ωtc, eˆ = −δˆωtc, e0 = − , ei = −δj . ˆ 0 i j i j 0 j i (3) ωtc ωtc 2

• Action for Dirac ﬁeld √i ¯ α S [e, ψ] = d x −g [ψγ ˆ Dαψ − (Dαψ)γ αψ] 4 ˆ ˆ ˆ 2 ¯ −mψψ , (4) where Dα = eµ Dµ = ∂α + Γα ˆ α ˆ ˆ ˆ (5) • In this gauge the Dirac operator reads: 3iω 0 ED = iγ 0∂t + ie−ωtγ i∂i + γ 2 0 i 3iω 0 = −iωtc(γ ∂tc + γ ∂i) + γ (6) 2 • the Dirac equation (ED − m)ψ = 0 (7) • The fundamental solutions are: 1 πk/2 (1) πp/ω e Hν− (qe−ωt)ξλ(p ) Up, λ(t, x ) = i 2 (1) eip·x−2ωt (2π)3/2 λe−πk/2Hν+ (qe−ωt)ξλ(p ) (2) πp/ω −λe−πk/2Hν− (qe−ωt)ηλ(p ) Vp, λ(t, x ) = i 1 πk/2 (2) e−ip·x−2ωt , (8) (2π)3/2 2 e Hν+ (qe−ωt)ηλ(p ) 3

• Charge conjugation ¯ Up, λ(x) → Vp, λ(x) = iγ 2γ 0(Up, λ(x))T . (9) • The orthonormalization relations : ¯ d3x(−g)1/2Up, λ(x)γ 0Up , λ (x) = ¯ d3x(−g)1/2Vp, λ(x)γ 0Vp , λ (x) = δλλ δ 3(p − p ) ¯ d3x(−g)1/2Up,λ(x)γ 0Vp ,λ (x) = 0, (10) d3p Up, λ(t, x )Up, λ(t, x ) + Vp, λ(t, x )Vp, λ(t, x ) = e−3ωtδ 3(x − x ). + + (11) λ • The ﬁeld operator ψ(t, x ) = ψ (+)(t, x ) + ψ (−)(t, x ) = d3p Up, λ(t, x )a(p, λ) + Vp, λ(t, x )b+(p, λ) . (12) λ • Canonic cuantization {a(p, λ), a+(p , λ )} = {b(p, λ), b+(p , λ )} = δλλ δ 3(p − p ) , (13) 4

{ψ(t, x ), ψ(t, x )} = e−3ωtγ 0δ 3(x − x ) , (14) SF (t, t , x − x ) = i 0|T [ψ(x)ψ(x )]|0 = θ(t − t )S (+)(t, t , x − x ) −θ(t − t)S (−)(t, t , x − x ) . (15) [ED (x) − m]SF (t, t , x − x ) = −e−3ωtδ 4(x − x ). (16) • Conserved operators Pi =: ψ, P iψ := d3p pi [a+(p, λ)a(p, λ) + b+(p, λ)b(p, λ)] (17) λ W =: ψ, W ψ := d3p pλ[a+(p, λ)a(p, λ) + b+(p, λ)b(p, λ)] . (18) λ iω 3 i + ↔ + ↔ H= d pp a (p, λ) ∂ pi a(p, λ) + b (p, λ) ∂ pi b(p, λ) . (19) 2 λ 5

• [H, P i] = iωP i. (20) Scalar ﬁeld • Action of the scalar ﬁeld ∗ √ √ S[φ, φ ] = 4 d x −g L = d x −g ∂ µφ∗∂µφ − m2φ∗φ , 4 (21) • The Klein-Gordon equation: 1 √ √ ∂µ −g g µν ∂ν φ + m2φ = 0 . (22) −g ∂t2 − e−2ωt∆ + 3ω∂t + m2 φ(x) = 0 . (23) • Solutions of Klein-Gordon equation: 1 π e−3ωt/2 −πk/2 (1) p −ωt ip·x fp (x) = 3/2 e Hik e e , (24) 2 ω (2π) ω 6

9 m k= µ2 − , 4 µ = , m > 3ω/2 (25) ω • The orthonormalization relations: ↔ 3 1/2 ∗ i d x (−g) fp (x) ∂t fp (x) = ↔ 3 1/2 ∗ −i d x (−g) fp (x) ∂t fp (x) = δ 3(p − p ) , (26) ↔ 3 1/2 i d x (−g) fp (x) ∂t fp (x) = 0 , (27) ↔ ∗ i 3 d p fp (t, x ) ∂t fp (t, x ) = e−3ωtδ 3(x − x ) . (28) • Canonic cuantization ∗ φ(x) = φ(+)(x) + φ(−)(x) = d3p fp (x)a(p ) + fp (x)b+(p ) , (29) [a(p ), a†(p )] = [b(p ), b†(p )] = δ 3(p − p ) . (30) • 7

[φ(t, x ), ∂tφ†(t, x )] = ie−3ωtδ 3(x − x ) . (31) • The commutator functions: D(±)(x, x ) = i[φ(±)(x), φ(±)†(x )], (32) ∗ D(+)(x, x ) = i d3p fp (x)fp (x ), ∗ D(−)(x, x ) = −i d3p fp (x)fp (x ). (33) • G(x, x ) is a Green function of the Klein-Gordon equation if: ∂t2 − e−2ωt∆x + 3ω∂t + m2 G(x, x ) = e−3ωtδ 4(x − x ) . (34) DR(t, t , x − x ) = θ(t − t )D(t, t , x − x ) , (35) DA(t, t , x − x ) = − θ(t − t)D(t, t , x − x ) , (36) 8

• The Feynman propagator, DF (t, t , x − x ) = i 0|T [φ(x)φ†(x )] |0 = θ(t − t )D(+)(t, t , x − x ) − θ(t − t)D(−)(t, t , x − x ) , (37) The electromagnetic ﬁeld • The action √ 1 √ S[A] = d4x −g L = − d4x −g Fµν F µν , (38) 4 Fµν = ∂µAν − ∂ν Aµ • Field equation √ ∂ν ( −g g ναg µβ Fαβ ) = 0 , (39) • In the Coulomb gauge A0 = 0, (Ai) ; i = 0 ,and chart {tc, x} ¸ (∂t2c − ∆)Ai = 0 (40) 9

• The ﬁeld operator (+) (−) Ai(x) = Ai (x) + Ai (x) = d3k ei(nk , λ)fk (x)a(k, λ) + [ei(nk , λ)fk (x)]∗a∗(k, λ) λ (41) 1 1 fk (x) = √ e−iktc+ik·x , (42) (2π)3/2 2k • the Hermitian form ↔ ↔ 3 ∗ f, g = i d x f (tc, x ) ∂tc g(tc, x ) , f ∂ g = f (∂g) − g(∂f ) (43) • The orthonormalization relations: ∗ ∗ fk , fk = − fk , fk = δ 3(k − k ) , (44) ∗ fk , fk = 0 , (45) ↔ 3 ∗ i d k fk (tc, x) ∂tc fk (tc, x ) = δ 3(x − x ) , (46) k · e (nk , λ) = 0 , (47) 10

e (nk , λ) · e (nk , λ )∗ = δλλ , (48) ∗ k ik j ei(nk , λ) ej (nk , λ) = δij − 2 . (49) k λ • In local frame · Aˆ (x) = e ˆj Aj = i i· d3k wˆ k,λ)(x)a(k, λ) + [wˆ k,λ)(x)]∗a∗(k, λ) . i( i( (50) λ • Conformal transformation gµν (x) = Ω(x)ηµν , (51) g µν (x) = Ω−1(x)η µν , ds2 = Ω ds2, c (52) Aµ = Aµ ; Aµ = Ω−1A µ. (53) • Quantization [a(k, λ), a†(k , λ )] = δλλ δ 3(k − k ) . (54) [Ai(tc, x ), π j (tc, x )] = [Ai(tc, x ), ∂tc Aj (tc, x )] = i δij (x − x ) , tr (55) 11

• The momentum density j √ δL π = −g = ∂tc Aj (56) δ(∂tc Aj ) tr 1 q iq j δij (x ) = d3q δij − 2 eiq·x (57) (2π)3 q • Conserved operators 1 Pl = δij : Ai, (P l A)j := d3k k l a†(k, λ)a(k, λ) , (58) 2 λ 1 H= δij : Ai, (HA)j : (59) 2 1 W = δij : Ai, (W A)j : = d3k k λ a†(k, λ)a(k, λ) (60) 2 λ 12

[H, P i] = iωP i , [H, W] = iωW , [W, P i] = 0 , (61) 3 i (Hfk ) (x) = −iω k ∂ki + f (x) (62) 2 k iω 3 i † ↔ H= d kk a (k, λ) ∂ ki a(k, λ) , (63) 2 • The commutator functions (±) (±) (±) † Dij (x − x ) = i[Ai (x), Aj (x )] (64) (+) 3 ∗ k ik j Dij (x −x) = i d k fk (x)fk (x ) δij − 2 (65) k (+) i d3k k ik j )−ik(tc −tc ) Dij (x −x)= δij − 2 eik·(x−x (66) (2π)3 2k k 13

• at equal times, (+) 1 tr ∂tc Dij (tc − tc, x − x ) = δij (x − x ) . (67) tc =tc 2 • The transversal Green functions tr ∂t2c − ∆x Gij (x − x ) = δ(tc − tc)δij (x − x ) (68) si ∂iGi· (x) = 0. ¸ ·j R Dij (x − x ) = θ(tc − tc)Dij (x − x ) , (69) A Dij (x − x ) = −θ(tc − tc)Dij (x − x ) , (70) • The Feynman propagator , F Dij (x − x ) = i 0|T [Ai(x)Aj (x )] |0 (+) (−) = θ(tc − tc)Dij (x − x ) − θ(tc − tc)Dij (x − x ) , (71) 14

Interaction between Dirac field and electromagnetic field • The action for interacting fields: √ i ¯ α S [e, ψ, A] = 4 d x −g [ψγ ˆ Dαψ − (Dαψ)γ αψ] ˆ ˆ ˆ 2 ¯ 1 ¯ ˆ ˆ −mψψ − Fµν F µν − eψγ αAαψ , (72) 4 • Equations for fields in interaction iγ αDαψ − mψ = eγ αAαψ, ˆ ˆ ˆ ˆ 1 ¯ ˆ ∂µ (−g)F µν = eeν ψγ αψ. α ˆ (73) (−g) 15

Interaction between scalar field and electromagnetic field • The action √ µ † 1 2 † S[φ, A] = d x −g (∂ φ ∂µφ − m φ φ) − Fµν F µν 4 4 −ie[(∂µφ†)φ − (∂µφ)φ†]Aµ + e2φ†φAµAµ . (74) • Equations for fields in interaction 1 √ √ √ ∂µ −g g ∂ν φ − ie −gφAµ + m2φ = ie(∂µφ)Aµ + e2φAµAµ, µν −g 1 ∂µ (−g)F µν = −ie[(∂ ν φ†)φ − (∂ ν φ)φ†] (−g) +2e2φ†φAν . (75) 16

The in out fields • The equation of the interaction fields in the Coulomb gauge ¯ (∂t2c − ∆x)A(x) = eψ(x)γψ(x) (76) • Solution: ˆ A(x) = A(x) − e ¯ d3ydtc DG(x − y)ψ(y)γψ(y), (77) ˆ A(x) = A(R/A)(x) − e ¯ d3ydtc DR/A(x − y)ψ(y)γψ(y), (78) ˆ • the free fields , A(R/A)(x) satisfy: ˆ lim (A(x) − A(R/A)(x)) = 0. (79) t→ ∞ • The fields in/out: √ ¯ z3Ain/out(x) = A(x) + e d3ydtc DR/A(x − y)ψ(y)γψ(y). (80) 17

• The in/out fields, final expressions: √ z3Ain/out(x) = A(x) + d3ydtc DR/A(x − y)(∂t2c − ∆y )A(y), (81) ↔ 3 a(k, λ)in/out = i d xfk (x)e (nk , λ) ∂tc Ain/out(x) (82) • The case of interaction with scalar field ˆ + e d4y √−g G(x, y) √ i ∂µ[√−gφ(y)Aµ(y)] φ(x) = φ(x) −g +i[∂µφ(y)]Aµ(y) + eφ(y)Aµ(y)Aµ(y) , (83) ˆ √ i √ φ(x) = φR/A(x) + e d y −gDR/A(x, y) √ ∂µ[ −gφ(y)Aµ(y)] 4 −g +i[∂µφ(y)]Aµ(y) + eφ(y)Aµ(y)Aµ(y) . (84) 18

• The in/out ﬁelds: √ √ zφin/out(x) = φ(x) − d y −gDR/A(x, y)[EKG(y) + m2]φ(y). 4 (85) • ↔ 3 ∗ a(p )in/out = i d x(−g)1/2fp (x) ∂t φin/out(x) ↔ † 3 1/2 b (p )in/out = i d x(−g) fp (x) ∂t φin/out(x). (86) 19

Perturbation theory • The expression of Green functions 1 ˆ ˆ ˆ G(y1, y2, ..., yn) = 0|T [φ(y1)φ(y2)...φ(yn), S]|0 , (87) 0|S|0 ∞ (x)d4 x (−i)n S = T e−i LI =1+ T [LI (x1)...LI (xn)]d4x1...d4xn, (88) n=1 n! • The interaction Lagrangian √ LI (x) = −ie −g : [(∂µφ†)φ − (∂µφ)φ†]Aµ : (89) (−i)n ˆ ˆ ˆ 0|T [φ(y1)φ(y2)...φ(yn), LI (x1)...LI (xn)]|0 d4x1...d4xn. (90) n! 0|S|0 20

• the amplitude: 3 1 ∗ out, 1(p )|in, 1(p ) = δ (p − p ) − −g(y) −g(z)fp (y)[EKG(y) + m2] z × 0|T [φ(y)φ†(z)]|0 [EKG(z) + m2]fp (z)d4yd4z. (91) • The scattering amplitude in the ﬁrst order of perturbation theory ↔ ∗ Ai→f = −e −g(x) fp (x) ∂µ fp (x) Aµ(x)d4x, (92) Ai→f = −ie ¯ −g(x)Up ,λ (x)γµ A µ ˆ (x)Up,λ (x)d4x, (93) 21

Coulomb scattering for the charged scalar field • External field ˆ 0 Ze −ωt A = e (94) |x| • initial and final states are: φf (x) = fpf (x), φi(x) = fpi (x). (95) • Scattering amplitude −αZ pi ∞ (2) (1) (2) (1) Ai→f = − z Hik (pf z)Hik−1(piz) − Hik (pf z)Hik+1(piz) dz 8π|pf − pi|2 2 0 pf ∞ (2) (1) (2) (1) + z Hik−1(pf z)Hik (piz) − Hik+1(pf z)Hik (piz) dz , (96) 2 0 e−ωt z= . (97) ω 22

• Scattering amplitude in ﬁnal form −αZ Ai→f = B , 2 k (98) 8π|pf − pi| (pf + pi) 1 pf pf Bk = √ 2iδ(pf − pi) + θ(pi − pf ) h∗k − gk pf pi pi pi pi 1 ∗ pi pi + θ(pf − pi) gk − hk . (99) pf pf pf 3 1 2 1 3 2 e−πk χik 2 F1 2 , 2 + ik; 1 + ik; χ 2 F1 2 , 2 + ik; 1 + ik; χ gk (χ) = − sinh2(πk) 1 B( 2 − ik; 1 + ik) B(− 1 − ik; 1 + ik) 2 3 1 2 1 3 2 eπk χ−ik 2 F1 2 , 2 − ik; 1 − ik; χ 2 F1 2 , 2 − ik; 1 − ik; χ + − sinh2(πk) 1 B( 2 + ik; 1 − ik) B(− 1 + ik; 1 − ik) 2 3 1 2 1 3 2 χik 2 F1 2 , 2 + ik; 1 + ik; χ 2 F1 2 , 2 + ik; 1 + ik; χ + − sinh2(πk) 1 B(− 2 ; 1 + ik) 1 B( 2 ; 1 + ik) 3 1 2 1 3 2 χ−ik 2 F1 2 , 2 − ik; 1 − ik; χ 2 F1 2 , 2 − ik; 1 − ik; χ + − , sinh2(πk) 1 B(− 2 ; 1 − ik) 1 B( 2 ; 1 − ik) (100) 23

3 3 1 1 e−πk χ−ik 2 F1 , 2 − ik; 2 − ik; χ2 2 F1 2 , 2 − ik; −ik; χ 2 hk (χ) = 2 · χ2 − sinh2(πk) B(− 1 + ik; 2 − ik) 2 B(−ik; 1 + ik) 2 3 3 2 1 1 2 eπk χik 2 F1 2 , 2 + ik; 2 + ik; χ 2 F1 2 , 2 + ik; ik; χ + · χ2 − sinh2(πk) 1 B(− 2 − ik; 2 + ik) B( 1 − ik; ik) 2 1 1 2 3 3 2 χik 2 F1 2 , 2 + ik; ik; χ 2 F1 2 , 2 + ik; 2 + ik; χ + − · χ2 sinh2(πk) B( 1 ; ik) 2 B(− 1 ; 2 + ik) 2 1 1 2 3 3 2 χ−ik 2 F1 2 , 2 − ik; −ik; χ 2 F1 2 , 2 − ik; 2 − ik; χ + − · χ2 , sinh2(πk) 1 B( 2 ; −ik) 1 B(− 2 ; 2 − ik) (101) pf pi • the notations are χ = pi or pf . • The Minkowski limit (pf + pi) B∞ = √ 2iδ(pf − pi). (102) pf pi • Minkowski case (Ef + Ei) BM = 4iδ(Ef − Ei), (103) Ef Ei 24

• Ration of the two amplitudes δ(pf − pi) 1 = . (104) 2δ(Ef − Ei) 2 • The incident ﬂux ↔ ∗ Jµ = −i fp (x) ∂µ fp (x) , (105) ∗ ↔ πpie−3ωt (2) pi −ωt (1) pi −ωt j(t) = −i fpi (x) ∂i fpi (x) = 3 Hik e Hik e . (106) 2ω(2π) ω ω 25

Coulomb scattering with fermions • The scattering amplitude √ ¯ ˆ Ai→f = −ie −g Upf ,λf (x)γ0A0(x)Upi,λi (x)d4x. (107) √ ∞ pi pf Ai→f = −iαZ ξλf (pf )ξλi (pi) eπk + (2) (1) dzzHν+ (pf z)Hν− (piz) 8π|pf − pi|2 0 ∞ +sgn(λf λi)e−πk (2) (1) dzzHν− (pf z)Hν+ (piz) (108) 0 26

• Final form of the amplitude p2  1 1 −i αZ 1  pf eπk 2F1 −ik 2 ,1 − ik; − ik; 2 f p2 + i Ai→f = ξ (pf )ξλi (pi) δ(pf − pi) + θ(pi − pf ) i 4π |pf − pi|2 λf pi pi cosh(πk) B( 1 , 1 − ik) 2 2 p2  3 3 f e−πk 2F1 2 , 1 + ik; 2 + ik; p2  1+ik 1−ik pf i 1 pi −i + θ(pf − pi) i pi cosh(πk) B(− 1 , 3 + ik) 2 2 pf pf p2 p2  3 3 1 1 e−πk 2F1 2 , 1 − ik; 2 − ik; p2 ieπk 2F1 2 , 1 + ik; 2 + ik; p2  i ik i f pi f × 1 3 − 1 1 cosh(πk) B(− 2 , 2 − ik) pf cosh(πk) B( 2 , 2 + ik) ik 1 pf e−πk +sgn(λf λi)δ(pf − pi) + sgn(λf λi)θ(pi − pf ) i pi pi cosh(πk) p2 p2  1 1 3 3 2 F1 2 ,1 + ik; + ik; 2 f p2 pf 1−ik eπk 2F1 − ik; − ik;2 ,1 2 f p2 i i × −i  B( 2 , 1 1 2 + ik) pi cosh(πk) B(− 1 , 3 − ik) 2 2 2  pi 3 3 1  pi 1+ik e πk 2 F1 2 ,1 + ik; 2 + ik; p2 f +sgn(λf λi)θ(pf − pi) i pf pf cosh(πk) B(− 1 , 3 + ik) 2 2 p2  1 1 pi −ik e−πk 2F1 2 ,1 − ik; − ik; 2 i p2 f −i  . (109) pf cosh(πk) B( 1 , 1 2 2 − ik) 27

• The incident ﬂux eωt ¯ πpie−3ωt (2) pi −ωt (1) pi −ωt j= Upi, λi (x)(piγ)Upi, λi (x) = 3 4ω Hν− e Hν− e | pi | (2π) ω ω (2) pi −ωt (1) pi −ωt +Hν+ e Hν+ e (110) ω ω √ ¯ ˆ Ai→f = −ie −g Upf ,λf (x)γ0A0(x)Vpi,λi (x)d4x (111) 28

References ˘ [1] I.I.Cotaescu, Phys. Rev. D 65, 084008 (2002) ˘ [2] I.I.Cotaescu, R.Racoceanu and C.Crucean, Mod.Phys.Lett A 21, 1313 (2006) [3] C.Crucean, Mod.Phys.Lett A 22, 2573 (2007) [4] C.Crucean and R.Racoceanu, Int.J.Mod.Phys. A 23, 1075 (2008) ˘ [5] I.I.Cotaescu and C.Crucean, Int.J.Mod.Phys. A 23, 1351 (2008) ˘ [6] I.I.Cotaescu, C.Crucean, A.Pop, Int.J.Mod.Phys. A 23 (2008) ˘ [7] I. I. Cotaescu and C. Crucean, Int.J.Mod.Phys. A 23 (2008) [8] C.Crucean, Mod.Phys.Lett A 25, (2010) ˘ [9] I.I.Cotaescu and C.Crucean, qr-qc/0806.2515 [10] C.Crucean, R.Racoceanu and A.Pop, Phys.Lett.B 665, 409 (2008) [11] C.Crucean, math-ph/0912.3659, (2009) 29

Cosmin Crucean: Perturbative QED on de Sitter Universe.

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Cosmin Crucean: Perturbative QED on de Sitter Universe. — Presentation Transcript

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