130
5 General-Relativistic Matrix Kinetic Theory
where
s C lm :=
l 2 − m 2
l 2 − s 2
l 2
4l 2 − 1
,
(5.11.42)
we then obtain the variation
δ η T lm ( ˆ
ω) := ˜ ˆ
T lm ( ˆ
ω) − T lm ( ˆ
ω)
≡
d ˜
s Y
∗
lm ( ˜
ϑ) ˜ ˆ
T ( ˜
ϑ, ˆ
ω) −
d s Y
∗
lm (ϑ) ˆ
T (ϑ, ˆ
ω)
(5.11.43)
= −η
d
s C l+1,m s Y
∗
l+1,m (ϑ)
ˆ
ω∂ ˆ
ω + l + 2
−
sm
l(l + 1)
s Y
∗
lm (ϑ)
ˆ
ω∂ ˆ
ω + 1
+ s C lm s Y
∗
l−1,m (ϑ)
ˆ
ω∂ ˆ
ω − l + 1
ˆ
T (ϑ, ˆ
ω) + O(η
2 )
≡ −η
s C l+1,m
ˆ
ω∂ ˆ
ω + l + 2
ˆ
T l+1,m ( ˆ
ω) −
sm
l(l + 1)
ˆ
ω∂ ˆ
ω + 1
ˆ
T lm ( ˆ
ω)
+ s C lm
ˆ
ω∂ ˆ
ω − l + 1
ˆ
T l−1,m ( ˆ
ω)
+ O(η
2 ) .
Finally, it is straightforward to compute the variation of the total brightness
ˆ
T lm :=
∞
0
d ˆ
ω ˆ
ω
3 ˆ
T lm ( ˆ
ω) ,
(5.11.44)
by integrating by parts the ∼ ∂ ˆ
ω terms
δ η ˆ
T lm = −η
2 C l+1,m (l − 2) ˆ
T l+1,m +
3sm
l(l + 1)
ˆ
T lm − 2 C lm (l + 3) ˆ
T l−1,m
+ O(η 2 ) ,
(5.11.45)
which is in agreement with [23].
References
1. L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (CRC Press, Boca Raton, USA, 1989)
2. J. Berges, Introduction to nonequilibrium quantum field theory. AIP Conf. Proc. 739, 3 (2004).
https://doi.org/10.1063/1.1843591. arXiv: hep-ph/0409233
3. O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich et al., Transporttheoretical description of nuclear reactions. Phys. Rept. 512, 1 (2012). https://doi.org/10.1016/
j.physrep.2011.12.001. arXiv: 1106.1344
4. R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That (Addison-Wesley, Redwood City, USA, 1989)
5. A.D. Dolgov, Neutrinos in the early universe. Sov. J. Nucl. Phys. 33, 700 (1981)
6. R. Barbieri, A. Dolgov, Neutrino oscillations in the early universe. Nucl. Phys. B 349, 743
(1991). https://doi.org/10.1016/0550-3213(91)90396-F
7. G. Sigl, G. Raffelt, General kinetic description of relativistic mixed neutrinos. Nucl. Phys. B
406, 423 (1993). https://doi.org/10.1016/0550-3213(93)90175-O
5 General-Relativistic Matrix Kinetic Theory
where
s C lm :=
l 2 − m 2
l 2 − s 2
l 2
4l 2 − 1
,
(5.11.42)
we then obtain the variation
δ η T lm ( ˆ
ω) := ˜ ˆ
T lm ( ˆ
ω) − T lm ( ˆ
ω)
≡
d ˜
s Y
∗
lm ( ˜
ϑ) ˜ ˆ
T ( ˜
ϑ, ˆ
ω) −
d s Y
∗
lm (ϑ) ˆ
T (ϑ, ˆ
ω)
(5.11.43)
= −η
d
s C l+1,m s Y
∗
l+1,m (ϑ)
ˆ
ω∂ ˆ
ω + l + 2
−
sm
l(l + 1)
s Y
∗
lm (ϑ)
ˆ
ω∂ ˆ
ω + 1
+ s C lm s Y
∗
l−1,m (ϑ)
ˆ
ω∂ ˆ
ω − l + 1
ˆ
T (ϑ, ˆ
ω) + O(η
2 )
≡ −η
s C l+1,m
ˆ
ω∂ ˆ
ω + l + 2
ˆ
T l+1,m ( ˆ
ω) −
sm
l(l + 1)
ˆ
ω∂ ˆ
ω + 1
ˆ
T lm ( ˆ
ω)
+ s C lm
ˆ
ω∂ ˆ
ω − l + 1
ˆ
T l−1,m ( ˆ
ω)
+ O(η
2 ) .
Finally, it is straightforward to compute the variation of the total brightness
ˆ
T lm :=
∞
0
d ˆ
ω ˆ
ω
3 ˆ
T lm ( ˆ
ω) ,
(5.11.44)
by integrating by parts the ∼ ∂ ˆ
ω terms
δ η ˆ
T lm = −η
2 C l+1,m (l − 2) ˆ
T l+1,m +
3sm
l(l + 1)
ˆ
T lm − 2 C lm (l + 3) ˆ
T l−1,m
+ O(η 2 ) ,
(5.11.45)
which is in agreement with [23].
References
1. L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (CRC Press, Boca Raton, USA, 1989)
2. J. Berges, Introduction to nonequilibrium quantum field theory. AIP Conf. Proc. 739, 3 (2004).
https://doi.org/10.1063/1.1843591. arXiv: hep-ph/0409233
3. O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich et al., Transporttheoretical description of nuclear reactions. Phys. Rept. 512, 1 (2012). https://doi.org/10.1016/
j.physrep.2011.12.001. arXiv: 1106.1344
4. R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That (Addison-Wesley, Redwood City, USA, 1989)
5. A.D. Dolgov, Neutrinos in the early universe. Sov. J. Nucl. Phys. 33, 700 (1981)
6. R. Barbieri, A. Dolgov, Neutrino oscillations in the early universe. Nucl. Phys. B 349, 743
(1991). https://doi.org/10.1016/0550-3213(91)90396-F
7. G. Sigl, G. Raffelt, General kinetic description of relativistic mixed neutrinos. Nucl. Phys. B
406, 423 (1993). https://doi.org/10.1016/0550-3213(93)90175-O