8Lie groups in nature

III Symmetries, Fields and Particles 8.1 Spacetime symmetry
In special relativity, we have a metric
ds
2
= dt
2
+ dx
2
+ dy
2
+ dz
2
.
The group of (orientation-preserving and) metric-preserving symmetries gives
us the Lorentz group
SO
(3
,
1). However, in certain cases, we can get something
more (or perhaps less) interesting. Sometimes it makes sense to substitute
τ
=
it
,
so that
ds
2
= dτ
2
+ dx
2
+ dy
2
+ dz
2
.
This technique is known as Wick rotation. If we put it this way, we now have a
It happens that when we complexify this, this doesn’t really matter. The
resulting Lie algebra is
L
C
(SO(3, 1)) = L
C
(SO(4)) = D
2
.
Its Dynkin diagram is
This is not simple, and we have
D
2
= A
1
A
1
.
In other words, we have
so(4) = su(2) su(2).
At the Lie group level,
SO
(4) does not decompose as
SU
(2)
× SU
SU(2) × SU(2) is a double cover of SO(4), and we have
SO(4) =
SU(2) × SU(2)
Z
2
.
Now in general, our fields in physics transform when we change coordinates.
There is the boring case of a scalar, which never transforms, and the less boring
case of a vector, which transforms like a vector. More excitingly, we have objects
known as spinors. The weird thing is that spinors do have an
so
(3
,
1) action,
but this does not lift to an action of
SO
(3
,
1). Instead, we need to do something
funny with double covers, which we shall not go into.
In general, spinors decompose into “left-handed” and “right-handed” com-
ponents, known as Weyl fermions, and these correspond to two different rep-
resentations of
A
1
. We can summarize the things we have in the following
table:
Field so(3, 1) A
1
A
1
Scalar 1 (ρ
0
, ρ
0
)
Dirac Fermion LH RH (ρ
1
, ρ
0
) (ρ
0
, ρ
1
)
Vector 4 (ρ
1
, ρ
1
)
However, the Lorentz group
SO
(3
,
1) is not all the symmetries we can do to
Minkowski spacetime. These are just those symmetries that fix the origin. If we
add in the translations, then we obtain what is known as the Poincar´e group.
The Poincar´e algebra is not simple, as the translations form a non-trivial ideal.