String Theory Demystified

CHAPTER 3 Symmetries and Worldsheet Currents 53
The trace is easy to calculate:
αβ
αβ µ ν 1 αβ
h T = h ∂ X ∂ X η − h h αβ
α β µν
αβ
2
ρσ µ ν
h ∂ X ∂ X η
ρ σ µ νν
αβ
= h ∂
µ ν
X ∂ X η
ρσ µ
−h ∂ X ∂
ν
X η
αβ
(because h h
α
β
µν
ρ
( )
σ
µν
= δ = 2)
α
αβ α
αβ µ ν
αβ µ ν
= h ∂ X ∂ X η −h ∂ X ∂ X ηη (relabel dummy indices, ρ →α, σ → β)
α β µν
α β µν
= 0
So we’ve established our fi rst fact to fi le away about the energy-momentum tensor
for the string. It’s traceless:
T α
α = 0 (3.5)
In a moment we will learn the physical reason why the energy-momentum tensor is
traceless.
Symmetries of the Polyakov Action
In this section we will list some of the symmetries of the Polyakov action (and
hence of the bosonic string in Minkowski space-time). There are three symmetry
groups of the Polyakov action. These include
• Poincaré transformations
• Reparameterizations of the worldsheet coordinates
• Weyl transformations
The concept of symmetries is so important we will take a momentary digression
to discuss the topic before describing the symmetries of the Polyakov action.
Symmetries in physics can be global symmetries or local symmetries. These are
defi ned as follows:
• A global symmetry is one that holds at all points in space-time. The
parameters of the transformation will not depend on space-time.
• A local symmetry is one that acts differently at different space-time points.
In this case, the parameters of the transformation will be functions of the
space-time coordinates.
You should recall from your studies of classical mechanics and quantum fi eld
theory that a symmetry in physics leads to a conservation law. The formal statement