Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac equaton t s not clear what s the meanng of the Drac matrces. It turns out that they are related to representatons of Lorentz group. The Lorentz group s a collecton of lnear transformatons of space-tme coordnates whch leaves the proper tme x x x (x o ) ( x) x x g x nvarant. Ths requres the transformaton matrx to satsfy the pseudo-orthogonalty relaton,. Generators g g It s useful to nvestgate the group structure by studyng ther n ntesmal elements near the dentty, the generators. For n ntesmal transformaton, we wrte As before, the pseudo-orthogonalty relaton mples g + wth j" j (g + ) (g + )g g " + " We see that " " : So there are 6 ndependent group parameters. Example: oost along x axs cosh snh snh cosh A Ths mples that " ; " () " ; " Smlarly, " ; and " 3 correspond to boosts n y and z drectons respectvely. A
Example: rotaton about z axs Then cos sn sn cos A " ; " " ; " Smlarly, " 3 ;and " 3 correspond to rotatons about x and y axs respectvely. onsder f(x ), an arbtrary functon of x. Under the n ntesmal Lorentz transformaton, the change n f s A f(x ) f(x ) f(x + " x ) f(x ) + " x f + f(x ) + " [x x ]f(x) + Introduce operators M to represent ths change, then f(x ) f(x) " M f(x) + M (x x ) () generators M are called the generators of Lorentz group operatng on functons of space-tme coordnates. Note that for ; ; ; 3 these are just the angular momentum operator L j (x j x j ). Usng the generators gven n Eq() t s straghtforward to work out commutators of these generators, [M ; M ] (g M g M g M + g M ) De ne M j jk J k ; M o K where J k s correspond to the usual rotatons and K the Lorentz boost operators. We can solve for J to get The commutator of J s are, [J ; J j ] J jkm jk " kl" jmn[m kl; M mn] ( ) ( ) " kl " jmn (g lm M kn g km M ln g ln M km + g kn M lm ) ( ) ( ) [ kl " jln M kn + kl " jkn M ln + kl " jml M km kl " jmk M lm ] Usng dentty we get abc " alm ( bl cm bm cl ) [J ; J j ] jk J k (3)
Thus we can dentfy J as the angular momentum operator. Smlarly, we can derve [K ; K j ] jk J k [J ; K j ] jk K k (4) Eqs(3,4) are called the Lorentz algebra. To smplfy the Lorentz algebra, we de ne the combnatons A (J + K ) ; (J K ) Then t s straghtforward to derve the followng commutaton relatons, [A ; A j ] jk A k ; [ ; j ] jk k ; [A ; j ] Ths means that the algebra of Lorentz generators factorzes nto ndependent SU() algebra. The representatons are just the tensor products of the representaton of SU() algebra. We label the rreducble representaton by (j ; j ) whch transforms as (j + )-dm representaton under A algebra and (j + )- dm representaton under algebra.. Smple representatons (a) ( ; ) representaton a Ths -component object has the followng propertes, A a ( ) ab b ) a ) ombnng these realtons we get (J + K ) a ( ) ab b (J K ) a J ( ); K ( ) (b) (; ) representaton a Smlarly, we can get A a ) (J + K ) a a ( ) ab ) (J K ) a ( ) ab b J ( ); K ( ) If we de ne a 4-component by puttng togather these representatons,
Then the acton of the Lorentz generators are J ; K are related to the 4-component Drac eld we studed before, but wth d erent representaton for the matrces. Ths can be seen as follows. onsder Drac matrces n the followng form _ where (; ) ; _ (; ) More explctly, o It s straghtforward to check that n ths case. 5 3 Ths means that n 4-component eld ; s rght-handed and s left-handed. In ths representaton, usually called the Weyl representaton, t s easy to check that (5) j j j j jk k k In the Lorentz transformaton of Drac eld, (x ) S expf 4 " g expf 4 ( " + j " j )g Wrte " ; " j " jk k j j " jk k jl l l " ) 4 ( " + j " j ) + More precsely, (x ) S expf 4 " g exp " + # (6)
If we wrte the Lorentz transformatons n terms of generators, then n terms of the generators J ; K L exp L exp( M " ) J ( ) + K We then see from Eq(6) that for ths ; J ; K are of the form, J ; K These are the same as those n Eq(5). Thsdemonstrate that the wavefuncton whch sats es Drac equaton s just the representaton ; ; under the Lorentz group. Futhermore, the rghthanded components transform as ; represenaton whle left-handed components transform as ; representaton. - Alternatve choce s to use R and the complex conjugate R ( sometme dotted ndce are used for ths bass) nstead of R and L : Snce we get for the complex conjuate J R ( ) R; K R ( ) R J R ( ) R; K R ( ) R It s probably more clearer to use some other notaton for R ; Then J ( ); K ( ) A ( J + K) ; ( J K) ( ) and ndeed belongs to the rrep (; ):