Kiefer C. Quantum gravity

THE FREE BOSONIC STRING 83
and
Ĥ 1 Ψ[X µ (σ)] ≡ δΨ
i Xµ′ =0, (3.49)
δX
µ
where the factor ordering has been chosen such that the momenta are on the
right. Note that in contrast to the examples in Section 3.1, one has now to deal
with functional derivatives, defined by the Taylor expansion
∫
Ψ[φ(σ)+η(σ)] = Ψ[φ(σ)] + dσ
δΨ η(σ)+··· . (3.50)
δφ(σ)
The above implementation of the constraints is only possible if there are no
anomalies; see the end of this section and Section 5.3.5. An important property
of the quantized string is that such anomalies in fact occur, preventing the validity
of all quantum equations (3.48) and (3.49). Equations such as (3.48) and (3.49)
will occur at several places later and will be further discussed there, for example
in the context of parametrized field theories (Section 3.3).
Note that this level of quantization corresponds to a ‘first-quantized string’ in
analogy to first quantization of point particles (Section 3.1). The usual ‘second
quantization’ would mean to elevate the wave functions Ψ[X µ (σ)] themselves
into operators (‘string field theory’). It must also be emphasized that ‘first’ and
‘second’ are at best heuristic notions since there is just one quantum theory (cf.
in this context Zeh (2003)).
In the following, we shall briefly discuss the connection with the standard
textbook treatment of the bosonic string; see, for example, Polchinski (1998a).
This will also be a useful preparation for the discussion of string theory in Chapter
9. Usually one starts with the Polyakov action for the bosonic string,
S P = − 1
4πα
∫M
′ d 2 σ √ hh αβ (σ, τ)∂ α X µ ∂ β X µ , (3.51)
where h αβ denotes the intrinsic (not induced) metric on the worldsheet, and
h ≡|deth αβ |. In contrast to the induced metric, it consists of independent degrees
of freedom with respect to which the action can be varied. The action (3.51)
can be interpreted as describing ‘two-dimensional gravity coupled to D massless
scalar fields’. Since the Einstein–Hilbert action is a topological invariant in two
dimensions, there is no pure gravity term present, and only the coupling of the
metric to the X µ remains in (3.51). One can also take into account a cosmological
term, see Chapter 9. In contrast to (3.37), the Polyakov action is much easier to
handle, especially when used in a path integral.
The Polyakov action has many invariances. First, it is invariant with respect
to diffeomorphisms on the worldsheet. Second, and most importantly, it possesses
Weyl invariance, that is, an invariance under the transformations
h αβ (σ, τ) ↦→ e 2ω(σ,τ ) h αβ (σ, τ) (3.52)
with an arbitrary function ω(σ, τ). This is a special feature of two dimensions
where √ hh αβ ↦→ √ hh αβ . It means that distances on the worldsheet have no