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Chapter 3: Symmetries Symmetry is very important in physics and especially particle physics. Symmetries are connected to conservation laws (rotational invarianceangular momentum conservation; translational invariancemomentum conservation) Transformations can be continuous or discrete e.g. translations, rotations, Lorentz transformations are continuous Spatial reflection through the origin (parity) is discrete. Associated conservation laws are additive (cont symmetries) Associated conservation laws are multiplicative (discrete symmetries) Two types of symmetry breaking: Explicit breaking (when the interactions do not respect the symmetry. The corresponding quantum numbers are not conserved in the interaction.) Spontaneous breaking (when the interaction does respect the symmetry; mathematically the Lagrangian is invariant under the corresponding group but its states are not). Next example: Parity or mirror reflection is a discrete transformation (x, y, z) ® (-x,-y,-z) P = +1 "even parity" P= -1 "odd parity" y = cos x Þ Py = cos(-x) = cos(x) = +y Þ P = +1 y = sin x Þ Py = sin(-x) = - sin(x) = -y Þ P = -1 y = cos x + sin x Þ Py = cos x - sin(x) Þ No definite parity Question: What does the parity do to the following quantities Spatial position r Time t Momenta p Angular momentum r x p Intrinsic spin Parity of hydrogenic wave functions Y(r,q , f ) = c (r)Yl m (q , f ) (2l +1)(l - m)! m =c (r) Pl (cosq )eimf 4p (l + m)! Let’s focus on the angular part imf e im(p +f ) ®e imp imf =e e m imf = (-1) e Pl m (cosq ) = Pl m (cos(p - q )) = (-1)l+m Pl m (cosq ) The overall angular wave function Y_lm is the product of these two pieces. m l+m (-1) (-1) = (-1) (-1) l 2m Question: What is the parity of the hydrogenic wave functions ? Answer: Parity = (-1)l ; we will use this result in the future. Parity is a multiplicative quantum number. Parity is found to be conserved in electromagnetic and strong interactions. Question: What about the weak interaction ? Elementary particles have intrinsic parity Pp = -1 PN = +1 (nucleon: n or p) Pp = -1 PN = +1 (nucleon: n or p) p+ p ®p+ + p + n Is this a strong, weak or electromagnetic interaction ? What is the product of intrinsic parities in the initial and final states ? Ans: +1 and -1. How can parity be conserved ? Ans: orbital angular momentum between a pair of particles in the final state.