Kiefer C. Quantum gravity

86 PARAMETRIZED AND RELATIONAL SYSTEMS
term, one cannot implement the constraints L m ≈ 0 in the quantum theory as
restrictions on wave functions, that is, one cannot have ˆL m |ψ〉 =0forallm.
Instead, one can choose
ˆL n |ψ〉 =0,n>0 , ˆL0 |ψ〉 = a|ψ〉 . (3.63)
It turns out that Weyl invariance can only be preserved at the quantum level
for a =1andD = 26, see for example, Polchinski (1998a) and Chapter 9. This
is achieved by the presence of Faddeev–Popov ghost degrees of freedom whose
central charge cancels against the central charge c of the fields X µ precisely for
D = 26. It is most elegantly treated by ‘BRST quantization’ (see Section 9.2),
leading to an equation of the form Q B |Ψ tot 〉 =0,whereQ B is the BRST charge.
This weaker condition replaces the direct quantum implementation of the constraints.
Going back to the classical theory, one can also define the quantities
˜L n = 1 ∫ ( √ )
dσ e inσ (X′ )
2
2 H ⊥ + H 1 . (3.64)
Using the Poisson-bracket relations between the constraints H ⊥ and H 1 (see in
particular Section 3.3), one can show that
{ ˜L m , ˜L n } = −i(m − n)˜L m+n , (3.65)
which coincides with the Virasoro algebra (3.61). In fact, for the gauge fixing
considered here—leading to (3.58)—one has ˜L n = L n . The result (3.62) then
shows that the naive implementation of the constraints (3.48) and (3.49) may
be inconsistent. This is a general problem in the quantization of constrained
systems and will be discussed further in Section 5.3. Kuchař and Torre (1991)
have treated the bosonic string as a model for quantum gravity. They have shown
that a covariant (covariant with respect to diffeomorphisms of the worldsheet)
quantization is possible, that is, there exists a quantization procedure in which
the algebra of constraints contains no anomalous terms. This is achieved by
extracting internal time variables (‘embeddings’) which are not represented as
operators. 5 A potential problem is the dependence of the theory on the choice
of embedding. This is in fact a general problem; see Section 5.2. Kuchař and
Torre make use of the fact that string theory is an ‘already-parametrized theory’,
which brings us to a detailed discussion of parametrized field theories in the next
section.
3.3 Parametrized field theories
This example is a generalization of the parametrized non-relativistic particle
discussed in Section 3.1.1. As it will be field theoretic by nature, it has similarities
5 The anomaly is still present in a subgroup of the conformal group, but it does not disturb
the Dirac quantization of the constraints.