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| 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 space]]s. It is named after [[Peter Goddard (physicist)|Peter Goddard]] and [[Charles Thorn]].
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| 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 [[Faddeev–Popov ghost|vectors of negative norm]] for ''r'' ≠ 0. The name "no-ghost theorem" is also a word play on the phrase [[no-go theorem]].
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| == Formalism ==
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| Suppose that ''V'' is a [[vector space]] with a [[nondegenerate bilinear form]] (·,·).
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| Further suppose that ''V'' is acted on by the [[Virasoro algebra]] in such a way that the [[adjoint]]{{disambiguation needed|date=December 2013}} of the operator ''L<sub>i</sub>'' is ''L<sub>-i</sub>'', that the [[central element]] of the Virasoro algebra acts as multiplication by 24, that any vector of ''V'' is the sum of [[eigenvector]]s of ''L''<sub>0</sub> with non-negative integral [[eigenvalue]]s, and that all [[eigenspace]]s of ''L''<sub>0</sub> are finite-dimensional.
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| Let ''V<sup>i</sup>'' be the subspace of ''V'' on which ''L''<sub>0</sub> has eigenvalue ''i''. Assume that ''V'' is acted on by a [[group (mathematics)|group]] ''G'' which preserves all of its structure.
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| Now let <math>V_{II_{1,1}}</math> be the [[vertex algebra]] of the [[Double covering group|double cover]] <math>\hat{I}I_{1,1}</math> of the two-dimensional even [[unimodular lattice|unimodular]] [[Lorentzian lattice]] <math>II_{1,1}</math> (so that <math>V_{II_{1,1}}</math> is <math>II_{1,1}</math>-graded, has a bilinear form (·,·) and is acted on by the Virasoro algebra).
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| Furthermore, let ''P''<sup>1</sup> be the subspace of the vertex algebra <math>V\otimes V_{II_{1,1}}</math> of vectors ''v'' with ''L''<sub>0</sub>(''v'') = ''v'', ''L''<sub>i</sub>(''v'') = 0 for ''i'' > 0, and let <math>P^1_r</math> be the subspace of ''P''<sup>1</sup> of degree ''r'' ∈ <math>II_{1,1}</math>. (All these spaces inherit an action of ''G'' from the action of ''G'' on ''V'' and the trivial action of ''G'' on <math>V_{II_{1,1}}</math> and '''R'''<sup>2</sup>).
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| Then, the [[quotient space|quotient]] of <math>P^1_r</math> by the [[nullspace]] of its bilinear form is naturally [[isomorphic]] (as a [[G-module|''G''-module]] with an invariant bilinear form) to <math>V^{1-(r,r)/2}</math> if ''r'' ≠ 0, and to <math>V^1 \oplus \mathbb{R}^2</math> if ''r'' = 0.
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| ==Applications==
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| The theorem can be used to construct some [[generalized Kac–Moody algebra]]s, in particular the [[monster Lie algebra]].
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| == References ==
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| * P. Goddard and C. B. Thorn, ''[http://preprints.cern.ch/cgi-bin/setlink?base=preprint&categ=CM-P&id=CM-P00058839 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.
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| {{DEFAULTSORT:Goddard-Thorn theorem}}
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| [[Category:Abstract algebra]]
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| [[Category:String theory]]
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| [[Category:Theorems in mathematical physics]]
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As they call me Gabrielle. Vermont also has always been my livelihood place and I enjoy everything that I have to have here. As a complete girl what I totally like is going - karaoke but I haven't so much made a dime as well as. I am the cashier and I'm ordering pretty good financially. See what's new on my own, personal website here: http://circuspartypanama.com
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