7.4 Representation of a Group on a Field
193
The space
M
n
can be the Euclidean space
E
n
if the metric tensor defined on it
is
δ
i j
(in this case we shall be mainly interested in our three-dimensional Euclid-
ean space
E
3
), or, for
n
=
4, the Minkowski space
M
4
of special relativity if the
metric is
η
μν
. It is useful at this point to recall the notations used in
Chap.4
for
describing Cartesian coordinates in the various spaces: The collection of generic
Cartesian coordinates on
M
n
is denoted by
r
=
(
x
i
)
while our familiar Carte-
sian rectangular coordinates are also denoted by
x
=
(
x
,
y
,
z
)
and the space-
time coordinates of an event in Minkowski space are also collectively denoted by
x
=
(
x
μ
)
=
(
ct
,
x
)
. Let us introduce a second
p
-dimensional vector space
V
p
and
let us we consider a map
α
:
M
n
→
V
p
which associates with each point
P
∈
M
n
,
labeled by Cartesian coordinates
x
i
,
i
=
1
, . . .,
n
, a vector in
V
p
, of components
α
(
x
i
), α
=
1
, . . .,
p
,
α
:
∀
x
i
∈
M
n
→
α
(
x
i
)
∈
V
p
.
(7.42)
This function is called a
field
, defined on
M
n
, with values in
V
p
. The index
α
is called
the
internal index
since it labels the
internal components
α
of the field, which are
degrees of freedom not directly related to its space-time propagation. An example is
the index
α
=
1
,
2 labeling the physical polarizations of a photon.
V
p
is consequently
called the
internal space
. If, as
V
p
, we take the space
V
n
k
+
l
of type-
(
k
,
l
)
tensors, the
corresponding field
i
1
...
i
k
j
1
...
j
l
(
x
i
)
is called a
tensor field
. We have already intro-
duced the notion of tensor fields in
Chap.4
,
Sect.4.3
, and illustrated their transforma-
tionpropertiesunderachangeintheCartesiancoordinates(affinetransformations)on
M
n
. There we discussed, as an example, the case of a tensor field
T
i j
k
(
x
i
)
which has
values in the
n
3
-dimensional vector space
V
p
=
V
n
3
of type-
(
2
,
1
)
tensors. Its trans-
formation law is given by (
4.73
), its indices transforming under the homogeneous part
D
=
(
D
i
j
)
(element of GL
(
n
)
) of the affine transformation, according to their posi-
tions. Thinking of
T
i j
k
as the
p
=
n
3
components of a vector in
V
p
, their are subject
to the
linear
action of the matrix
(
D
⊗
D
⊗
D
−
T
)
i js
lmk
defining the representation
the GL
(
n
)
transformation on (2,1) tensors. This transformation property is general-
ized in a straightforward way to generic type-
(k,l)
tensor fields. If we wish to restrict
to transformations preserving the Euclidean or Lorentzian metrics on
E
3
or
M
4
, as
we shall mostly do in the following, we need to restrict the homogeneous part of the
affine transformation to O
(
3
)
or to O
(
1
,
3
)
, respectively.