String Theory Demystified

CHAPTER 5 Conformal Field Theory Part I
There are two important properties of conformal fi eld theories. These can be
summed up by saying that a conformal fi eld theory is scale invariant. This manifests
itself in two ways:
• Conformal fi eld theories have no length scale.
• Conformal fi eld theories have no mass scale.
To see why this is important, we can consider the quantum fi eld theory of a scalar
fi eld. Let φ( x) be a scalar fi eld in d space-time dimensions. Consider a rescaling of
the coordinates which we write as a scale transformation:
x′ = λ x
91
(5.3)
Under a scale transformation, a scalar fi eld φ( x) transforms using the classical
scaling dimension ∆= ( d −2)/
2as follows:
d
− +
2 1
−∆
φ( x) → φ′ ( λx) = λ φ( x) = λ φ(
x)
For a fi eld theory to be scale invariant, we require that the action be invariant
under this transformation. This is, in fact, true when we consider a free, massless
scalar fi eld. The action in this case is
S d x
d
= ∫ ∂µφ∂ φ
µ
Under the transformation x′ = λ x,
it is clear that dx′ = λ dx,
that is,
d
d d
d x′ = d( λx ) d( λx ) …d( λx ) = ( λ⋅λ… λ)
d( x ) d( x ) …d( x ) dx = λ
0 1 d−1
0 1
Now recall that ∂ µ is shorthand for ∂/( ∂x
)
µ . So we pick up a copy of λ under a
scale transformation:
So we have
∂ ∂ 1 ∂
→ =
∂x ∂(
λx ) λ ∂x
µ µ µ
∂ ∂ → ⎛ ⎞
⎝
⎜
⎠
⎟ ∂ ′ ⎛ ⎞
⎝
⎜
⎠
⎟ ∂ ′ = ⎛
µ 1 1 µ 1 ⎞
µ φ φ
µ ( φ ) ( φ )
⎝
⎜ 2
λ λ λ ⎠
⎟ ∂ ⎛
d d
− + ⎞ µ ⎛ − + ⎞
2
µ
⎝
⎜ λ φ
⎠
⎟ ∂
⎝
⎜ λ φ
⎠
⎟
1
2 1
d−
1
= ⎛ 1 µ µ
⎞ − d+ 2
−d
⎝
⎜ 2 ⎟ λ ∂µ φ∂ φ = λ ∂µ φ∂ φ
λ ⎠