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. . .