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 -

--

'zEna.r@mpr.p )= - 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