436
M. Group Theory
multiplication is not commutative in general, it happens to be so for these
particular matrices. In fact, the way these four matrices multiply together
is (as the reader can verify) exactly the same as the way the four numbers
(1, i, −1, −i) (in that order) do. Further, the correspondence between the
elements of the two groups is ‘one to one’: that is, if we label the two sets
′
′
of group elements by (e, a, b, c) and (e , a
′ , b , c
′ ), we have the correspondences
′
′
e ↔ e , a ↔ a
′ , b ↔ b , c ↔ c
′ . Two groups with the same multiplication
structure, and with a one-to-one correspondence between their elements, are
said to be isomorphic. If they have the same multiplication structure but the
correspondence is not one-to-one, they are homomorphic.
M.2 Lie groups
We are interested in continuous groups – that is, groups whose elements are
labelled by a number of continuously variable real parameters α 1 , α 2 , . . . , α r :
g(α 1 , α 2 , . . . , α r ) ≡ g(α). In particular, we are concerned with various kinds of
‘coordinate transformations’ (not necessarily space-time ones, but including
also ‘internal’ transformations such as those of SU(3)). For example, rotations
in three dimensions form a group, whose elements are specified by three real
parameters (e.g. two for defining the axis of the rotation, and one for the angle
of rotation about that axis). Lorentz transformations also form a group, this
time with six real parameters (three for 3-D rotations, three for pure velocity
transformations). The matrices of SU(3) are specified by the values of eight
real parameters. By convention, parametrizations are arranged in such a
way that g(0) is the identity element of the group. For a continuous group,
condition (i) takes the form
g(α)g(β) = g(γ(α, β)),
(M.5)
where the parameters γ are continuous functions of the parameters α and β.
A more restrictive condition is that γ should be an analytic function of α and
β; if this is the case, the group is a Lie group.
The analyticity condition implies that if we are given the form of the
group elements in the neighbourhood of any one element, we can ‘move out’
from that neighbourhood to other nearby elements, using the mathematical
procedure known as ‘analytic continuation’ (essentially, using a power series
expansion); by repeating the process, we should be able to reach all group
elements which are ‘continuously connected’ to the original element. The
simplest group element to consider is the identity, which we shall now denote
by I. Lie proved that the properties of the elements of a Lie group which can
be reached continuously from the identity I are determined from elements
lying in the neighbourhood of I.
M. Group Theory
multiplication is not commutative in general, it happens to be so for these
particular matrices. In fact, the way these four matrices multiply together
is (as the reader can verify) exactly the same as the way the four numbers
(1, i, −1, −i) (in that order) do. Further, the correspondence between the
elements of the two groups is ‘one to one’: that is, if we label the two sets
′
′
of group elements by (e, a, b, c) and (e , a
′ , b , c
′ ), we have the correspondences
′
′
e ↔ e , a ↔ a
′ , b ↔ b , c ↔ c
′ . Two groups with the same multiplication
structure, and with a one-to-one correspondence between their elements, are
said to be isomorphic. If they have the same multiplication structure but the
correspondence is not one-to-one, they are homomorphic.
M.2 Lie groups
We are interested in continuous groups – that is, groups whose elements are
labelled by a number of continuously variable real parameters α 1 , α 2 , . . . , α r :
g(α 1 , α 2 , . . . , α r ) ≡ g(α). In particular, we are concerned with various kinds of
‘coordinate transformations’ (not necessarily space-time ones, but including
also ‘internal’ transformations such as those of SU(3)). For example, rotations
in three dimensions form a group, whose elements are specified by three real
parameters (e.g. two for defining the axis of the rotation, and one for the angle
of rotation about that axis). Lorentz transformations also form a group, this
time with six real parameters (three for 3-D rotations, three for pure velocity
transformations). The matrices of SU(3) are specified by the values of eight
real parameters. By convention, parametrizations are arranged in such a
way that g(0) is the identity element of the group. For a continuous group,
condition (i) takes the form
g(α)g(β) = g(γ(α, β)),
(M.5)
where the parameters γ are continuous functions of the parameters α and β.
A more restrictive condition is that γ should be an analytic function of α and
β; if this is the case, the group is a Lie group.
The analyticity condition implies that if we are given the form of the
group elements in the neighbourhood of any one element, we can ‘move out’
from that neighbourhood to other nearby elements, using the mathematical
procedure known as ‘analytic continuation’ (essentially, using a power series
expansion); by repeating the process, we should be able to reach all group
elements which are ‘continuously connected’ to the original element. The
simplest group element to consider is the identity, which we shall now denote
by I. Lie proved that the properties of the elements of a Lie group which can
be reached continuously from the identity I are determined from elements
lying in the neighbourhood of I.