wang
(Wang)
#1
TheN×N matrixU =A+iB is then automatically unitary and belongs toU(N). We
conclude thatU(N) was indeed the largest symmetry group of theN-dimensional harmonic
oscillator, confirming our earlier study from a different point of view.
9.7 Selection rules
In interpreting the strengths of various physical transitions, it isoften striking how different
the rates for various processes turn out to be. The most important ones are that unless
energy and momentum are conserved in a process, the corresponding transition amplitude
must vanish. Such a condition is often referred to as aselection rule. In the case of energy
momentum conservation, the selection rule is exact and results from the exact symmetry
of the invariance of the laws of nature under translations in time andin space. Another
exact selection rule is the conservation of total angular momentum, associated with the
invariance of the laws of Nature under space rotations. Yet another exact selection rule
is the conservation of electric charge, which is the result of an exact gauge invariance of
electro-dynamics.
The conservation of energy, momentum, angular momentum and electric charge are by
now so well-established selection rules that they their validity is used to detect new forms
of matter. One of the oldest such examples was the discovery of the neutrino by Pauli. By
the 1930’s the neutron was known to decay into a proton and an electron. If these were
the only decay products, however, then angular momentum could not be conserved during
the process, since the spin of the neutron, proton and electron are all ̄h/2, and a ̄h/2 unit
of angular momentum is missing in the balance. An integer unit ̄hcould be carried off by
orbital angular momentum, but a half integer unit cannot. This lack of balance led Pauli to
conjecture the existence of a new particle, the neutrino, with spinh/ ̄ 2.
Some selection rules are only approximate and result not in the vanishing of certain
transition probabilities but in their suppression instead.
In particle physics, a number of such approximate selection rules are associated with
symmetries of the strong and electro-magnetic interactions which, however, fail to be sym-
metries of the weak interactions. For example, the lightest strongly interacting particles are
the threepionsπ^0 andπ±. Their quark composition, mass and life-times are given by
π^0 = ( ̄uu−dd ̄)/
√
2 mπ 0 = 135MeV τπ 0 = 10−^17 s
π+=du ̄ mπ+= 140MeV τπ+= 10−^8 s
π−= ̄ud mπ−= 140MeV τπ−= 10−^8 s (9.56)
The reason for the vast difference in life-times is that strong and electro-magnetic interactions
preserve individual quark number, while the weak interactions do not. Theπ^0 can decay via
