8Lie groups in nature
III Symmetries, Fields and Particles
8.2 Possible extensions
Now we might wonder how we can expand the symmetry group of our theory
to get something richer. In very special cases, we will get what is known as
conformal symmetry, but in general, we don’t. To expand the symmetry further,
we must consider supersymmetry.
Conformal field theory
If all our fields are massless, then our theory gains conformal invariance. Before
we look into conformal invariance, we first have a look at a weaker notion of
scale invariance.
A massless free field φ : R
3,1
→ V has a Lagrangian that looks like
L = −
1
2
(∂
µ
φ, ∂
µ
φ).
This is invariant under scaling, namely the following simultaneous transformations
parameterized by λ:
x 7→ x
0
= λ
−1
x
φ(x) 7→ λ
∆
φ(x
0
),
where ∆ = dim V is the dimension of the field.
Often in field theory, whenever we have scale invariance, we also get conformal
invariance. In general, a conformal transformation is one that preserves all angles.
So this, for example, includes some mixture and scaling, and is more general
than Lorentz invariance and scale invariance.
In the case of 4 (spacetime) dimensions, it turns out the conformal group
is
SO
(4
,
2), and has dimension 15. In general, members of this group can be
written as
0 D
−D 0
P
µ
+ K
µ
P
ν
+ K
ν
M
µν
.
Here the
K
µ
are the special conformal generators, and the
P
µ
are translations.
We will not go deep into this.
Conformal symmetry gives rise to conformal field theory. Theoretically, this
is very important, since they provide “end points” of renormalization group flow.
This will be studied more in depth in the Advanced Quantum Field Theory
course.
Supersymmetry
Can we add even more symmetries? It turns out there is a no-go theorem that
says we can’t.
Theorem
(Coleman-Mondula)
.
In an interactive quantum field theory (sat-
isfying a few sensible conditions), the largest possible symmetry group is the
Poincar´e group times some internal symmetry that commutes with the Poincar´e
group.
But this is not the end of the world. The theorem assumes that we are working
with traditional Lie groups and Lie algebras. The idea of supersymmetry is to
prefix everything we talk about with “super-”, so that this theorem no longer
applies.
To being with, supersymmetry replaces Lie algebras with graded Lie algebras,
or Lie superalgebras. This is a graded vector space, which we can write as
g = g
0
⊕ g
1
,
where the elements in
g
0
as said to have grade 0, and elements in
g
1
have grade
1. We write
|X|
= 0
,
1 for
X ∈ g
0
, g
1
respectively. These will correspond to
bosonic and fermionic operators respectively.
Now the Lie bracket respects a graded anti-commutation relation
[X, Y ] = −(−1)
|X||Y |
[Y, X].
So for Bosonic generators, they anti-commute as usual, but with we have
two fermionic things, they commute. We can similarly formulate a super-
Jacobi identity. These can be used to develop field theories that have these
“supersymmetries”.
Unfortunately, there is not much experimental evidence for supersymmetry,
and unlike string theory, we are already reaching the scales we expect to see
supersymmetry in the LHC. . .