3
Let's revisit the tie algebra of the Lorentz group,
but Lnow with Einstein index rotation .
M = It EX → AT -
- Tut E WT
ymtym-
- I → % ,Hey
"
No -- ri or Ann : yer
-
- Mnr-
-
transposes co - tract"
SE to NW"
contraction"
NE to Sw"
Plug in : yr ,I r 's te w 's )y " I oh Ew ? ) -
- or ;
oft E yr ,
why"in t Ey . ,r 's y
" who TOLE ) = %\
C- 7h, y" wtpt C- 7u×y'
' ow no = O
E Mu ,
"w 's tytowmo ) -
- o
w" n
t Wnt = O
=3 w is an antisymmetric 2- index tensor w 16 independent
components ( note order of indices ! This matters
A general infinitesimal Lorentz transformation Can be written
X = iz Wmu M"
= i ( Wo,
Mo'
t worm"
two ,
MM - w, ,
M"
t w, ,M'
't w
, ,M"
)
If we take
Mio= -
moi= Ki and M
"= - M = Eiikjk
,we can write
O Won Won "3
= xqg ,a
fat matrix with× = i
( Wo , Ow, ,
- Wo"
tow}6 independent components
Wow
wog - Wiz Wrs
Alternative parametrization? X = it 'T tip ' I ( Bi -
- Woi,
Ei -
- Eisner )
Covariant rotation : ( Mnf) : -
- i ( y" Oh - guar;)
generator flabel matrix
index 0 I O O
ex . Hoy : -
-
ily" oh - n' ar ;)
-
- i (
go.gg/---kx+ I ifIo
,p= ,T
- I if a =L,
M -
- O
Now can compute commutator ! 4
cnn.mg ; .
. in- T : cnn.gr.
- ing :c my ;L
=- Gnr : -
marker; - you :) .. Gear: - go
.
only - v ; - purr :)-
- - -
=-
yn-
ye-
of t go-
que of t ( 3 similar )
=- in "
i ( go-
o ; - y"
r : ) t 13 similar I
= - i yup Mom) : t 13 similar )
= 7 [ M"
,M
" ) -
-- i ( y M
"-
y- - mm , - yr
- mm - quemno
)-
Poincare' transformations inmatrix fo -n
:
Xm - xnt I can be implanted as a matrix with one extra entry :
i
:÷÷÷H¥÷H¥÷÷÷÷,
so a general Lorentz t translation Can be written as
"" " fat
.O
"" minimal .
t.no#I--te''
it.
Infinitesimal translation is still a vector,
let 's call it F :
Po -
oil.Oo ,
P'
-
-
if-9! )
,
etc.
O O
C O instead of 1 because p-
is
[ Pm,
P"
) = O ( HW ) infinitesimal : x - x' ti is Hee Ma - I -
. Iter
One last commutation relation to compute : SLcnn.rs
:too . tooO O
O O
A p-
transforms like a-
-
( , Ofa
. )A - vector
, as it should
0
So this is proportional to some P ( Lorentz part is o ) :
if yur ;-
guar;Xpy,
But 6,
tu ng we defined P,
CPT i OF,
so
[ M"
,PT ,
-
- - yur ; org try"
or ; of
= iynrfir ; ) - in-
Cir : )
[ mm,
PT =
ifynopu- gu
-
pm)
We now have the Complete commutation relations fo - the Lie
algebra of the Poincare group'
.
[ MY
MN) -
-- i ( y
me m"
-
y- - mm + yr
- mm - quemno
)
:÷÷÷*r
Casimiro6L
Now that we have the algebra,
What can we do with it ?
If we find an object that Commutes with all gene - autos,
a
theorem from math tells usit must be proportional to the
identity operator on anyirreducible representation
: this is called a Casimir operator.
Irreducible ⇐ 7 can't write as block - diagonal Like
R, Ototal
Here 's one such operator ? p'
=p-
pm
Proof . [ Pn,
Po ) = 0 since all P 's co - - - te
CP"
,m
" ] .
- pncpn,
m" ] t Cpr ,
n' 7pm
( using CAB,
C) .
- ACB, C) tCA
,
CT B )
=p- ( - ish poetic :P
'
) tf - in"
pot iynop e) pm
= - i pep - tip - pr - if pot i pope- -
p 's commute,
so each term cancels
= O
⇒ P'
is a constant times he identity.
Lets call he constant m
-^
.
we will identify it with the physical squared mass of a particle .
The Poincare'
algebra has a second Casini -, but it's a bit trickier
.
Let's define#tzEnMP( Pauli - Kubinski pseudo vector )
Emp ,is the totally antisymmetric tensor with Gaz ,
= - I
>LFirst
,observe that
Wnpn = -
'
zfnupom" Po pm = O since pop - is symmetric but
C-rupo is antisymmetric in on
, n
Let's apply a Lorentz transformation Such that P-
= ( n, 0,0
,o ) .
Then W ; =
- I Eino Mik Po = m Ji where Ji is the Lorentz generator
For rotators
Furthermore,
Won pm -
- o = > Wo Po - trip =3 Wo = TIPI a O,
So Wn -
- ( o,
MT )
W' = Wnw = - m
' J . J ← re laced to total spin J-
Note : this only works if m> O ! ! Will come back to n
-
- O.
Claim : w"
commie , with all pm and m- ✓
To show this,
first compute [ Wn,
PV ) and ( Wn,
M"
]
Then few, f) = WTwr.PT t Cw ;'m ]Wn
,etc .
[ we,
PT -
--
'[email protected] )= - then.me/mmlpYiPvJtfmYpypr)= - ttnanrfi ) ( gape - you pay pv
But PB Pr is symmetric , so E symbol kills it : C Wn,
Pu ) = O
Can also show ( Wn,
M" ) -
- i ( of w'
- oh Wo ) ( Hhs
and here [ WT m"
7=0