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

• Parity
• Charge conjugation