Introduction to Conformal Field Theory: With Applications to String ...

2.3 The Energy–Momentum Tensor 19
2.3 The Energy–Momentum Tensor
Usually, a Field Theory is defined in terms of a Lagrangian action from which one
can derive various objects and properties of the theory. In particular, the energy–
momentum tensor can be deduced from the variation of the action with respect to
the metric and so it encodes the behaviour of the theory under infinitesimal transformations
gμν ↦→ gμν + δgμν with δgμν ≪ 1.
Since the algebra of infinitesimal conformal transformations in two dimensions
is infinite dimensional, there are strong constraints on a conformal field theory. In
particular, it turns out to be possible to study such a theory without knowing the
explicit form of the action. The only information needed is the behaviour under
conformal transformations which is encoded in the energy–momentum tensor.
Implication of Conformal Invariance
In order to study the energy–momentum tensor for CFTs, let us recall Noether’s
theorem which states that for every continuous symmetry in a Field Theory, there is
a current jμ which is conserved, i.e. �μ jμ = 0. Since we are interested in theories
with a conformal symmetry x μ ↦→ x μ + ɛμ (x), we have a conserved current which
can be written as
jμ = Tμν ɛ ν , (2.19)
where the tensor Tμν is symmetric and is called the energy–momentum tensor. Since
this current is preserved, we obtain for the special case ɛ μ = const. that
0 = � μ jμ = � μ � Tμν ɛ ν� = � � μ � ν
Tμν ɛ
⇒ � μ Tμν = 0 . (2.20)
For more general transformations ɛμ (x), the conservation of the current (2.19) implies
the following relation:
0 = � μ jμ = � � μ � � ν μ ν
Tμν ɛ + Tμν � ɛ �
= 0 + 1
2 Tμν
� μ ν ν μ
� ɛ + � ɛ � = 1
2 Tμνη μν� � · ɛ � 2 1
=
d d Tμ μ� � · ɛ � ,
where we used Eq. (2.3) and the fact that Tμν is symmetric. Since this equation has
to be true for arbitrary infinitesimal transformations ɛ(z), we conclude
In a conformal field theory, the energy–momentum tensor Tμν is traceless,
that is, Tμ μ = 0.