[polar-decomposition-of-Lorentz-group] polar decomposition to rotation and boost (ref-2, Vol.1, p.165)
let where
(ไฝฟ็จ ๅ polar decomposition of positive definite symmetric matrix)
where
is boost, map to
, have diagonal form where
with
[Euler-angle-Lorentz-group] Question
- rotation
ไฝฟ็จ ่ฝด็ๆ่ฝฌๆฅ็ๆ
in ,
in
- boost
ไฝฟ็จ ่ฝด็ boost
in
in
[Lorentz-group-Lie-bracket] with boost and rotation decomposition and Lie-bracket
ๅ ถไธญ ๆจกไปฟ cross product. Example
ๅ ถไธญ ๆจกไปฟ dot product. Example ==> or
ๅไธบๆจกไปฟ cross product
have form where (ref-2, Vol.1, p.180)
[Lorentz-group-orbit-isotropy]
or act on
| orbit type | isotropy-group type |
| or | |
| or | |
| or |
[isotropy-on-lightcone] Prop ไฝ็จไบ lightcone ็ฑปไผผไบ (ๆฐๅฅฝๆฏ Euclidean ไปฟๅฐ็พค)
Proof ไฝฟ็จ spinor ๆๆฏ
Prop ไฝ็จๅจ lightcone (ไธๆฏ projective-lightcone), isotropy ็ฑปไผผไบ
ๆฏ ๆปกๅฐไฝ็จ, orbit ๆฐ , ๆไปฅ่ฎก็ฎ isotropy ๅช้่ฆ่่ ไธ็น
ไฝฟ็จๅ ้ฅๅฐๅฝฑไธ็็น , ่ฎก็ฎ isotropy , where
- ๆฏๅ ้ฅ isotropy
- ๆฏๅฏน ็ไผธ็ผฉ
- ๆฏๅฏน ็็ฉบ้ดๆ่ฝฌ. ่ฝ็ปๅบๆดไธชๅ ้ฅๆช้ข
==> isotropy where
็ฑปไผผไบ isotropy-on-projective-lightcone ็่ฎก็ฎ, ๆญคๅคๅฐ็ฑปไผผไบ
[isotropy-on-lightcone-intuition] isotropy-group of orbit lightcone ็็ด่ง. ๆ นๆฎ
ๅๅผ่ฎจ่ฎบไธค็งๆ ๅต
-
. is rotation in
-
let with boost and rotation decomposition (not the in )
็บฟๆงๅๆๅฐๆฐ็ๅบ
where
- is rotation in
- is boost in
- and ๆฏ lightcone coordinate ็ฑปไผผ็ฉ, ไฟๆ
ๆ่ ๅไธบ
where ๅฐไผๆนๅ , ๅๅบๅฎ
ไบ็ปด Lie algebra is commutative, ๅฏนๅบ ไธญ็ or
ๅจๅ ้ฅไธ (ๅพ)
- ไธ่ฌ ็ๆๅๆฒๅ orbit
- ไธ่ฌ ็ๆๆคญๅๅ orbit
- ๅฏไปฅๆจๅนฟๅฐไธ่ฌ็ lightcone ็ปๅ e.g. , ๅฐ็ๆๆ็ฉๅ orbit
ไฝ็จ็ๅ ทไฝ่ฎก็ฎ
let . metric will be
in