Feynman Path Integral Formulation

44 1 Continuum Formulation
Similarly one can expand the constraint T ab in Fourier modes; it is more convenient
to write these constraints as Ẋ 2 R ≡ T −− = 0 and Ẋ 2 L ≡ T ++ = 0. One defines
∫ π
L m ≡ dσ T −− = 1 2
0
+∞
∑
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 the
classical 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 of
the above algebra in the closed string case is that it is obeyed by the generators D n
of the infinitesimal “diffeomorphisms” on the unit circle S 1
D n = ie inθ d
dθ . (1.209)
In a quantum mechanical treatment for the operators α μ n and ˜α μ n one has to be careful
about ordering ambiguities, which were not taken into account when deriving the
classical 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 Virasoro
algebra. In particular one needs to be careful to restrict the physical Hilbert space
through the conditions
and the normalization requirement
L 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 careful
treatment of the operator ordering problem and a suitable physically motivated
choice of the oscillator ground state one finds that the quantum-mechanical version
of Eq. (1.208) is
[L m ,L n ]=(m − n)L m+n + 1
12 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 operator
L 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 form
of the string Hamiltonian the mass M of the closed string excitations (for α ′ = 1 2 )is
given by