Poincare transformation

 

Transformation

Quantum Labels

Generators

Algebra

translation

P

P1 , P2 , P3 

[Tx , Ty]=0

rotation

Spin  S, Sz

J1 , J2 , J3 

[Ji , Jj]=iεijk Jk

[Ji , Pj]=iεijk Pk

boost

Not used

K1 , K2 , K3 

[Ki , Kj]=-iεijk Jk

[Ji , Kj]=iεijk Kk

[Ji , Pj]=iδijk H

 

Time evolution

E

H

[Ji , H]=0

[Pi , H]=0

[Hi , H]=0

[Ki , H]=i Pi

Parity

“improper transformation”, P

 

Jè+ J, Kè- K, Pè-P, HèH

Time Reversal

“improper transformation”, T Antiunitary operator

Jè-J, Kè+K, Pè-P, HèH

Charge Conjugation

“particle-antiparticle”, C not included as Poincare Space-time

 

 

for particles with mass in rest frame.

 

Note: Don’t include special relativity but assume Galilean relativity then mass drops out and there is no reason not to mix states of different mass.  This is referred to as a “projective representation”. One can introduce a mass operator and add it to the group so that mass becomes a reasonable state label.  One application for Homotopy classes is to prove general properties about representations (basis states) based on groups such as the Poincare group.

 

Neutral kaons