Lecture
2
8.324
Relativistic
Quantum
Field
Theory
II
Fall
2010
8.324
Relativistic
Quantum
Field
Theory
II
MIT
OpenCourseWare
Lecture
Notes
Hong
Liu,
Fall
2010
Lecture
2
1
Λ
Figure
1:
An
element
Λ
of
the
manifold
of
the
Lie
group
G
,
and
the
Lie
algebra
g
as
the
tangent
space
of
the
identity
element.
Some
facts
about
Lie
groups
and
Lie
algebras:
1.
Different
Lie
groups
can
have
the
same
Lie
algebra.
The
Lie
algebra
determines
the
Lie
group
up
to
discrete
choices
of
global
structure.
For
example,
SU
(2)
=
S
3
,
SO
(3)
=
S
3
/
2
.
2.
An
invariant
subalgebra
is
a
subset
of
a
Lie
algebra
g
′
�
g
which
is
closed
under
the
action
of
g
.
That
is,
[
g
,
g
′
]
�
g
′
. A
simple
Lie
algebra
is
a
Lie
algebra
which
does
not
contain
invariant
subalgebras
and
which
is
not
Abelian.
The
complex
simple
Lie
algebras
are
completely
classified:
su
(
n
)
,
so
(2
n
),
so
(2
n
+
1),
sp
(
n
)
, E
6
,
7
,
8
, F
4
and
G
2
are
the
only
possibilities.
3.
For
a
compact
Lie
group,
it
is
always
possible
to
choose
a
basis
of
T
a
so
that
f
abc
=
f
a
bc
is
truly
antisymmetric
(there
is
no
distinction
between
upper
and
lower
indices).
All
internal
symmetry
groups
are
compact.
For
example,
SU
(
n
)
(the
set
of
n
×
n
unitary
matrices):
U
=
exp
[
i
Λ
a
T
a
]
, a
= 1
, . . . , n
2
−
1
,
(1)
where
Tr(
T
a
) = 0
,
(
T
a
)
†
=
T
a
,
(2)
that
is,
the
generators
are
hermitian
and
traceless,
and
hence
we
can
choose
[
T
a
, T
b
] =
if
abc
T
c
,
(3)
where
the
f
abc
are
fully
antisymmetric.

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- Fall '10
- Liu
- Quantum Field Theory, Lie group, Lie algebra