In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector spaces. It is named after Peter Goddard and Charles Thorn.
The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the vector space inner product is positive definite. Thus, there were no vectors of negative norm for r ≠ 0. The name "no-ghost theorem" is also a word play on the phrase no-go theorem.
Further suppose that V is acted on by the Virasoro algebra in such a way that the adjoint[disambiguation needed] of the operator Li is L-i, that the central element of the Virasoro algebra acts as multiplication by 24, that any vector of V is the sum of eigenvectors of L0 with non-negative integral eigenvalues, and that all eigenspaces of L0 are finite-dimensional.
Let Vi be the subspace of V on which L0 has eigenvalue i. Assume that V is acted on by a group G which preserves all of its structure.
Furthermore, let P1 be the subspace of the vertex algebra of vectors v with L0(v) = v, Li(v) = 0 for i > 0, and let be the subspace of P1 of degree r ∈ . (All these spaces inherit an action of G from the action of G on V and the trivial action of G on and R2).
- P. Goddard and C. B. 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.