Feynman Path Integral Formulation

44 1 Continuum FormulationSimilarly one can expand the constraint T ab in Fourier modes; it is more convenientto write these constraints as Ẋ 2 R ≡ T −− = 0 and Ẋ 2 L ≡ T ++ = 0. One defines∫ πL m ≡ dσ T −− = 1 20+∞∑n=−∞α m−n · α n , (1.207)and similarly for ˜L m in terms of Ẋ 2 L and therefore ˜α μ n . A little algebra then gives theclassical Poisson bracket{L m ,L n } = i(m − n)L m+n , (1.208)and an analogous expression for ˜L m . A simple interpretation for the occurrence ofthe above algebra in the closed string case is that it is obeyed by the generators D nof the infinitesimal “diffeomorphisms” on the unit circle S 1D n = ie inθ ddθ . (1.209)In a quantum mechanical treatment for the operators α μ n and ˜α μ n one has to be carefulabout ordering ambiguities, which were not taken into account when deriving theclassical result of Eq. (1.208). These do not affect the above result unless m+n = 0,in which case a new term can arise, the so-called central extension of the Virasoroalgebra. In particular one needs to be careful to restrict the physical Hilbert spacethrough the conditionsand the normalization requirementL m |φ〉 = 0 (m > 0)(L 0 − a)|φ〉 = 0 , (1.210)〈0|[L 2 ,L −2 ]|0〉 = d 2 , (1.211)where a is an arbitrary parameter (it will turn out to be a = 1). Thus by a carefultreatment of the operator ordering problem and a suitable physically motivatedchoice of the oscillator ground state one finds that the quantum-mechanical versionof Eq. (1.208) is[L m ,L n ]=(m − n)L m+n + 112 d (m3 − m)δ m+n,0 . (1.212)The origin of the central term proportional to δ m+n is a requirement that the operatorL 0 be normal ordered so as to obtain a finite matrix element.Of great interest is of course the ground state of the bosonic string. From the formof the string Hamiltonian the mass M of the closed string excitations (for α ′ = 1 2 )isgiven by