004 - s02 - the lorentz group: suppose a light flashes at xi , and then in a Galilee-boosted frame xi ; then, the
constancy of the speed of light between these two frames tells us, c c s 2 x x c2t 2 xi xi s2 x x c2t2 xixi; c 1 x x t 2 xi xi ;
(1.1)
First, let the 0 coordinate be the time-coordinate, and second let c 1 ; we then compactify the appearance of any four-vector, 2 0 0 i i s x x x x x x x g x ; (1.2) g g g g00 g11 g 22 g33 g ii ; th
Problem: Show that x x x x g g , in which x x .
The linear transform is x x 0 x0 i xi . This implies, x x g x x g
g x x Problem: Let L be the matrix of
x
x g
x x g x x
g x x g
(1.3)
g ;
-values, and g, x be similarly defined. Show that the invariance of g
under Lorentz transform requires det L 1.
1 proper (1.4) 1 i m p r o p e r
[invariance] s 2 xT gx g LT gL det g det(LT gL) (det L) 2 det g det L 1
Note that L g is permissible by the definition of (1.4)--we are not talking not talking about the familiar boost-transform matrix just yet, but rather some general properties L must have. Problem: Show that 00 1 . 1 0 g
0
0 ( 0 ) ( 0 ) 0 0
0
2
i
2
0
0 0 1 ortho-chronous 1 0 0 1 non-ortho-chronous
(1.5)
The two possibilities each in (1.4) and (1.5) constitute four possible types of Lorentz transforms. i ij j Examples (all satisfy the above criteria): (a) rotations ( x a x ), (b) boosts x x , (c) time-inversion ( x0 , xi ) ( x 0 , xi ) , (d) full-inversion ( x0 , xi ) ( x 0 , xi ) x ; in matrix form, these appear as,
1 0 tatio on-matrix trix ; aT a1 La ; a a 3 3 rotati 0 a cosh sinh Lb 0 0
sinh 0 0 cosh 0 0 ; 0 1 0 0 0 1 1 0 0 1 Lc 0 0 0 0
00 cosh 1; cosh
1 1
2
; sinh
1 0 0 0 0 1 0 0 0 0 ; L 1; d 0 0 1 0 1 0 0 1 0 0 0 1 0
(1.6)
; 2 1
(1.7)
0
(1.8)
1
Problem : An infinitesimal Lorentz transformation and its inverse can be written as, ; x ( g ) x x ( g ) x
(1.9)
(a) show, from the definition of the inverse, that . Inverting both sides of the definition of the inverse: The prime denotes inversion. Taking the inverse of both
sides of x ( g ) x from (1.9), and using the definition of the inverse provided, we use the obvious
Minkowski-metric inverse g
1 x (( x ) )
2
g , and write ,
1
x [(g )x ] (g )x (g )x
g x x g x g (g )g x ;
(1.10)
Noting x x , we massage (1.10) into obtaining a term that “acts like” the identity-operator ,
x x x g x g ( g ) g x [( g ) g ][( g ) g ]x ;
(1.11)
By (1.11), we identify [( g ) g ][( g ) g ] . The implications of this statement are, 3
( g ) g ( g ) g ( ( g ) g ( g )) g
( g g ) g g ( )
(1.12)
b) show from the preservation of the norm (e.g., x x x x ) that . x x x x ( g
) x g ( g ) x ( g )( g ) g x x
x x [ g g O2 ( )]x x x x g g
x x x x g x x x x x x x x x x
(1.13)
Problem: Take the following as given: [ L 1 2 , L 3 4 ] i ( g
[S
14
12
3
g L g L g L ); L i( x x ) L ;
3 4
g S g S g S ); ] i( g S
L2
,S
1 3
2
1 4
4
2 3
2 3
1 4
1 3
2
2
4
4
1 3
2 3
1 4
2 4
(1.14)
1 3
The (1.14) is the Lie algebra of the generator of rotations: x 12 i L x x . Using (1.14),
M S L ; S † S ;
(1.15)
Consider spatial-indices only: (1 , 2 , 3 , 4 ) (i, j, k , ) (1, 2,3) . Then, direct calculation says, (1.16)
1
See, also: EM 04 - 570 - pr 09 - infintesimal generators of lorentz transforms.
2
In the 2nd to last step:
x ( x ) ( g [ x ]) ( g [( g ) x ]) g ( g ) g x , by
definition of the inverse. 3
Using:
g g ; ; g 0; g ; .
Introduce the following new operators: J i 12 ijk M jk ; Ki M 0i ; Ni 12 ( Ji i K i );
(1.17)
Derive the following commutation relations from (1.14) through (1.17), [ J i , J j ] i ijk J k ; [ Ki , K j ] i ijk J k ; [ J i , K j ] i ijk K k ; [ Ni , N j ] 0; [ Ni , N j ] i ijk N k ; [N i , N j ] i ijk N k ; †
†
†
†
(1.18)
