5
12.2. The flavour symmetry SU(2) f
this further in chapter 16. In weak interactions, a third realization appears:
the gauge quanta acquire mass via (it is believed) a second instance of spontaneous symmetry breaking, as will be explained in Part VII. In fact a further
application of this idea is required in the electroweak theory, because of the
chiral nature of the gauge symmetry in this case: the quark and lepton masses
also must be ‘spontaneously generated’.
12.2 The flavour symmetry SU(2) f
12.2.1 The nucleon isospin doublet and the group SU(2)
The transformations initially considered in connection with the gauge principle
in section 2.5 were just global phase transformations on a single wavefunction
ψ
′ = e
iα ψ.
(12.1)
The generalization to non-Abelian invariances comes when we take the simple step – but one with many ramifications – of considering more than one
wavefunction, or state, at a time. Quite generally in quantum mechanics, we
know that whenever we have a set of states which are degenerate in energy (or
mass) there is no unique way of specifying the states: any linear combination
of some initially chosen set of states will do just as well, provided the normalization conditions on the states are still satisfied. Consider, for example, the
simplest case of just two such states – to be specific, the neutron and proton
(figure 12.1). This single near coincidence of the masses was enough to suggest
to Heisenberg (1932) that, as far as the strong nuclear forces were concerned
(electromagnetism being negligible by comparison), the two states could be
regarded as truly degenerate, so that any arbitrary linear combination of neutron and proton wavefunctions would be entirely equivalent, as far as this
force was concerned, for a single ‘neutron’ or single ‘proton’ wavefunction.
This hypothesis became known as ‘charge independence of nuclear forces’.
Thus redefinitions of neutron and proton wavefunctions could be allowed, of
the form
ψ p → ψ p
′ = αψ p + βψ n
(12.2)
ψ n → ψ n
′ = γψ p + δψ n
(12.3)
for complex coefficients α, β, γ, and δ. In particular, since ψ p and ψ n are
degenerate, we have
Hψ p = Eψ p ,
H ψ n = Eψ n
(12.4)
from which it follows that
Hψ
′
= H(αψ p + βψ n ) = αHψ p + βHψ n
(12.5)
p
= E(αψ p + βψ n ) = Eψ
′
(12.6)
p
12.2. The flavour symmetry SU(2) f
this further in chapter 16. In weak interactions, a third realization appears:
the gauge quanta acquire mass via (it is believed) a second instance of spontaneous symmetry breaking, as will be explained in Part VII. In fact a further
application of this idea is required in the electroweak theory, because of the
chiral nature of the gauge symmetry in this case: the quark and lepton masses
also must be ‘spontaneously generated’.
12.2 The flavour symmetry SU(2) f
12.2.1 The nucleon isospin doublet and the group SU(2)
The transformations initially considered in connection with the gauge principle
in section 2.5 were just global phase transformations on a single wavefunction
ψ
′ = e
iα ψ.
(12.1)
The generalization to non-Abelian invariances comes when we take the simple step – but one with many ramifications – of considering more than one
wavefunction, or state, at a time. Quite generally in quantum mechanics, we
know that whenever we have a set of states which are degenerate in energy (or
mass) there is no unique way of specifying the states: any linear combination
of some initially chosen set of states will do just as well, provided the normalization conditions on the states are still satisfied. Consider, for example, the
simplest case of just two such states – to be specific, the neutron and proton
(figure 12.1). This single near coincidence of the masses was enough to suggest
to Heisenberg (1932) that, as far as the strong nuclear forces were concerned
(electromagnetism being negligible by comparison), the two states could be
regarded as truly degenerate, so that any arbitrary linear combination of neutron and proton wavefunctions would be entirely equivalent, as far as this
force was concerned, for a single ‘neutron’ or single ‘proton’ wavefunction.
This hypothesis became known as ‘charge independence of nuclear forces’.
Thus redefinitions of neutron and proton wavefunctions could be allowed, of
the form
ψ p → ψ p
′ = αψ p + βψ n
(12.2)
ψ n → ψ n
′ = γψ p + δψ n
(12.3)
for complex coefficients α, β, γ, and δ. In particular, since ψ p and ψ n are
degenerate, we have
Hψ p = Eψ p ,
H ψ n = Eψ n
(12.4)
from which it follows that
Hψ
′
= H(αψ p + βψ n ) = αHψ p + βHψ n
(12.5)
p
= E(αψ p + βψ n ) = Eψ
′
(12.6)
p