Editing Goddard–Thorn theorem (section)

== Formalism ==
There are two naturally isomorphic functors that are typically used to quantize bosonic strings.  In both cases, one starts with [[positive-energy representation]]s of the Virasoro algebra of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms.  Here, "Virasoro-invariant" means ''L<sub>n</sub>'' is adjoint to ''L''<sub>−''n''</sub> for all integers ''n''.

The first functor historically is "old canonical quantization", and it is given by taking the quotient of the weight 1 primary subspace by the radical of the bilinear form.  Here, "primary subspace" is the set of vectors annihilated by ''L<sub>n</sub>'' for all strictly positive ''n'', and "weight 1" means ''L''<sub>0</sub> acts by identity.  A second, naturally isomorphic functor, is given by degree 1 BRST cohomology.  Older treatments of BRST cohomology often have a shift in the degree due to a change in choice of BRST charge, so one may see degree −1/2 cohomology in papers and texts from before 1995.  A proof that the functors are naturally isomorphic can be found in Section 4.4 of Polchinski's ''String Theory'' text.

The Goddard–Thorn theorem amounts to the assertion that this quantization functor more or less cancels the addition of two free bosons, as conjectured by Lovelace in 1971.  Lovelace's precise claim was that at critical dimension 26, Virasoro-type Ward identities cancel two full sets of oscillators.  Mathematically, this is the following claim:

Let ''V'' be a unitarizable Virasoro representation of central charge 24 with Virasoro-invariant bilinear form, and let {{pi}}{{supsub|1,1|''λ''}} be the irreducible module of the '''R'''<sup>1,1</sup> Heisenberg Lie algebra attached to a nonzero vector ''λ'' in '''R'''<sup>1,1</sup>.  Then the image of ''V''&nbsp;⊗&nbsp;{{pi}}{{supsub|1,1|''λ''}} under quantization is canonically isomorphic to the subspace of ''V'' on which ''L''<sub>0</sub> acts by 1-(''λ'',''λ'').

The no-ghost property follows immediately, since the positive-definite Hermitian structure of ''V'' is transferred to the image under quantization.