13
Local Non-Abelian (Gauge) Symmetries
. . . The difference between a neutron and a proton is then a purely
arbitrary process. As usually conceived, however, this arbitrariness is
subject to the following limitations: once one chooses what to call a
proton, what a neutron, at one space time point, one is then not free
to make any choices at other space time points.
It seems that this is not consistent with the localized field concept
that underlies the usual physical theories. In the present paper we wish
to explore the possibility of requiring all interactions to be invariant
under independent rotations of the isotopic spin at all space time points
. . . .
—Yang and Mills (1954)
Consider the global SU(2) isospinor transformation (12.32), written here again,
1
2
1
2
ψ
( )′ (x) = exp(iα · τ /2)ψ
( ) (x)
(13.1)
1
2
1
2
for an isospin doublet wavefunction ψ
( ) (x). The dependence of ψ
( ) (x) on
the space-time coordinate x has now been included explicitly, but the parameters α are independent of x, which is why the transformation is called a ‘global’
one. As we have seen in the previous chapter, invariance under this transformation amounts to the assertion that the choice of which two base states
– (n, p), (u, d), . . . – to use is a matter of convention; any such non-Abelian
phase transformation on a chosen pair produces another equally good pair.
However, the choice cannot be made independently at all space-time points,
only globally. To Yang and Mills (1954) (cf the quotation above) this seemed
somehow an unaesthetic limitation of symmetry: ‘Once one chooses what to
call a proton, what a neutron, at one space-time point, one is then not free
to make any choices at other space-time points.’ They even suggested that
this could be viewed as ‘inconsistent with the localised field concept’, and
they therefore ‘explored the possibility’ of replacing this global (space-time
independent) phase transformation by the local (space-time dependent) one
1
2
1
2
ψ
( )′ (x) = exp[igτ · α(x)/2]ψ
( ) (x)
(13.2)
in which the phase parameters α(x) are also now functions of x = (t, x) as
39
Local Non-Abelian (Gauge) Symmetries
. . . The difference between a neutron and a proton is then a purely
arbitrary process. As usually conceived, however, this arbitrariness is
subject to the following limitations: once one chooses what to call a
proton, what a neutron, at one space time point, one is then not free
to make any choices at other space time points.
It seems that this is not consistent with the localized field concept
that underlies the usual physical theories. In the present paper we wish
to explore the possibility of requiring all interactions to be invariant
under independent rotations of the isotopic spin at all space time points
. . . .
—Yang and Mills (1954)
Consider the global SU(2) isospinor transformation (12.32), written here again,
1
2
1
2
ψ
( )′ (x) = exp(iα · τ /2)ψ
( ) (x)
(13.1)
1
2
1
2
for an isospin doublet wavefunction ψ
( ) (x). The dependence of ψ
( ) (x) on
the space-time coordinate x has now been included explicitly, but the parameters α are independent of x, which is why the transformation is called a ‘global’
one. As we have seen in the previous chapter, invariance under this transformation amounts to the assertion that the choice of which two base states
– (n, p), (u, d), . . . – to use is a matter of convention; any such non-Abelian
phase transformation on a chosen pair produces another equally good pair.
However, the choice cannot be made independently at all space-time points,
only globally. To Yang and Mills (1954) (cf the quotation above) this seemed
somehow an unaesthetic limitation of symmetry: ‘Once one chooses what to
call a proton, what a neutron, at one space-time point, one is then not free
to make any choices at other space-time points.’ They even suggested that
this could be viewed as ‘inconsistent with the localised field concept’, and
they therefore ‘explored the possibility’ of replacing this global (space-time
independent) phase transformation by the local (space-time dependent) one
1
2
1
2
ψ
( )′ (x) = exp[igτ · α(x)/2]ψ
( ) (x)
(13.2)
in which the phase parameters α(x) are also now functions of x = (t, x) as
39