In string theory the Goddard-Thorn theorem (Goddard-Thorn 72, Thorn 87, Thorn 12) establishes the consistency of the usual quantization of the bosonic string sigma-model for target space dimension 26, and of the NSR superstring sigma-model for target space dimension 10.
In the original “old covariant quantization” approach to this problem the proof amounts to demonstrating that the inner product on the would-be Hilbert space of quantum states is indeed positive definite. Since would-be quantum states of negative norm have been known as “ghost states”, the theorem has become widely known as the no-ghost theorem of string theory. More modern derivations proceed via or compare to quantization via the BRST complex of the string sigma-model.
In particular the theorem thus establishes the superstring excitations of vanishing mass, and that these coincide with the quanta of perturbative supergravity in 10 dimensions (type IIA supergravity or type IIB supergravity).
The original articles are
P. Goddard, Charles Thorn, Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys. Lett., B 40, No. 2 (1972), 235-238.
R. C. Brower, Phys. Rev. D 6 (1972) 1655
Charles Thorn, A detailed study of the physical state conditions in covariantly quantized string theories, Nuclear Physics B 286 (1987): 61-77 (doi:10.1016/0550-3213(87)90431-7)
Charles Thorn, Improved Proof of the No-ghost Theorem for Fermion States of the Superstring, Nuclear Physics B Volume 864, Issue 2, 11 November 2012, Pages 285-295 (arXiv:1110.5510)
The proof via DDF operators is also recalled in
See also