Poincare transformation
Transformation 
Quantum Labels 
Generators_{} 
Algebra 

translation 
P 
P_{1} , P_{2} , P_{3} _{} 
[T_{x} , T_{y}]=0 

rotation 
Spin S, S_{z} 
J_{1} , J_{2} , J_{3} 
[J_{i} , J_{j}]=iε_{ijk} J_{k}_{} [J_{i} , P_{j}]=iε_{ijk} P_{k}_{} 

boost 
Not used 
K_{1} , 
[K_{i} , K_{j}]=iε_{ijk} J_{k}_{} [J_{i} , K_{j}]=iε_{ijk} K_{k}_{} [J_{i} , P_{j}]=iδ_{ijk} H_{} 

Time evolution 
E 
H 
[J_{i} , H]=0 [P_{i} , H]=0 [H_{i} , H]=0 [K_{i} , H]=i P_{i} 

Parity 
“improper transformation”, P 
Jè+ J, Kè K, PèP, HèH 

Time Reversal 
“improper transformation”, T Antiunitary operator 
JèJ, Kè+K, PèP, HèH 

Charge Conjugation 
“particleantiparticle”, C not included as Poincare Spacetime 


_{}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