Quantum Gravity

282 STRING THEORYH = 1 2∞∑n=−∞(α −n α n +˜α −n ˜α n ) (9.10)for the closed string, and α −n α n is a shorthand for η µν α µ −n αν n , etc. In (3.60), weintroduced the quantities L m for the open string; one has in particular L 0 = H.One can analogously define for the closed stringL m = 1 ∫ 2ππα ′ dσ e −imσ T −− = 1 ∞∑α m−n α n , (9.11)02n=−∞˜L m = 1 ∫ 2ππα ′ dσ e imσ T ++ = 1 ∞∑˜α m−n ˜α n , (9.12)20n=−∞from which one obtains H = L 0 + ˜L 0 . As we have seen in Section 3.2, the L m(and ˜L m ) vanish as constraints.Recalling that α µ 0 = √ 2α ′ p µ for the open string and, therefore,α 2 0 ≡ αµ 0 αν 0 η µν =2α ′ p µ p µ ≡−2α ′ M 2 ,one obtains from 0 = L 0 = H for the mass M of the open string in dependenceof the oscillatory string modes, the expressionM 2 = 1 α ′∞ ∑n=1and from H = L 0 + ˜L 0 for the mass of the closed string,M 2 = 2 α ′∞ ∑n=1α −n α n , (9.13)(α −n α n +˜α −n ˜α n ) . (9.14)The variables x µ , p µ , α µ n ,and˜αµ n obey Poisson-bracket relations which followfrom the fundamental Poisson brackets between the X µ and their canonicalmomenta P µ (Section 3.2). Upon quantization, one obtains (setting =1)[x µ ,p ν ]=iη µν , (9.15)[α µ m,α ν n]=mδ m,−n η µν , (9.16)[˜α µ m , ˜αν n ]=mδ m,−nη µν , (9.17)[α µ m, ˜α ν n]=0. (9.18)The Minkowski metric η µν appears because of Lorentz invariance. It can causenegative probabilities which must be carefully avoided in the quantum theory.The task is then to construct a Fock space out of the vacuum state |0,p µ 〉,which is the ground state of a single string with momentum p µ , not the no-stringstate. The above algebra of the oscillatory modes can be written after rescaling