The document summarizes key aspects of quantum field theory on de Sitter spacetime, including solutions to the Dirac, scalar, electromagnetic, and other field equations. It presents:
1) Fundamental solutions for the Dirac equation and orthonormalization relations for Dirac spinor modes.
2) Solutions to the Klein-Gordon equation for a scalar field and corresponding orthonormalization relations.
3) Quantization of electromagnetic vector potentials in the Coulomb gauge and orthonormalization relations for photon modes.
Cosmin Crucean: Perturbative QED on de Sitter Universe.
1. PERTURBATIVE Q.E.D
ON DE SITTER UNIVERSE
WEST UNIVERSITY OF TIMISOARA
FACULTY OF PHYSICS
author: Crucean Cosmin
1
2. 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
3. • 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
4. • 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
5. {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
6. •
[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
7. 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
8. [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
9. • 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
10. • 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
11. 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
12. • 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
13. [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
14. • 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
15. 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
16. 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
17. 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
18. • 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
19. • 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
20. 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
21. • 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
22. 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