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 so-called Faddeev–Popov ghosts, or 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 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.