Perturbative microstates in string theory
289
String theory provides a unitary, quantum theory of gravity, so infonnation cannot
be lost. Thus, the infonnation loss entailed in Hawking's derivation must be
an artefact of the semi-classical approximation used. However, it is not clear
precisely where or how the approximation does breakdown nor, if it does, how
the infonnation is returned. We shall not pursue this topic further either. Instead,
we turn to the second puzzle thrown up by the Bekenstein-Hawking result and
the one on which string theory has been able to shed some light, namely whether
there is an explanation of black-hole entropy in tenns of microstates.
10.4 Perturbative microstates in string theory
Thermodynamics is only an approximation to a more fundamental description
based on the statistical properties of the microstates of the system. So the
fact that black holes have entropy suggests that this too should be understood
microscopically. Roughly speaking, a thermodynamic system with entropy S is
associated with a number e S of microstates of the system. For the Schwarzchild
black hole, with a horizon area given by (10.8), we see from (10.75) that the
entropy is Sbh = 4rrG N M2 loc. For the moment, the only important feature is
that the associated number of states grows like e M2 • We might have hoped that
this approximates the number of perturbative string states with mass M. However,
this is not the case, as we shall now demonstrate.
It suffices to consider the open bosonic string. This has mode expansion [13]
X/L = xiL + I: p/L T + its L !at: cos(nu)
(10.77)
n#) n
where the oscillator coefficients at: are creation (annihilation) operators for
n < 0 (n > 0). In units where the string length scale Is = I/..!i?i is 1 (T is
the string tension), the mass eigenvalues are given by the eigenvalues of
!M2 = N-I
(10.78)
where
00
" i
i
(10.79)
N = ~a_nan
n=1
is the number operator; the sum over i is over the DT = 24 transverse dimensions
of the bosonic string. We wish to estimate the number of (degenerate) states dn
that have number-eigenvalue n. It is convenient to define a generating function
00
G(w) = trw N = Ldnw n
(10.80)
n=O
289
String theory provides a unitary, quantum theory of gravity, so infonnation cannot
be lost. Thus, the infonnation loss entailed in Hawking's derivation must be
an artefact of the semi-classical approximation used. However, it is not clear
precisely where or how the approximation does breakdown nor, if it does, how
the infonnation is returned. We shall not pursue this topic further either. Instead,
we turn to the second puzzle thrown up by the Bekenstein-Hawking result and
the one on which string theory has been able to shed some light, namely whether
there is an explanation of black-hole entropy in tenns of microstates.
10.4 Perturbative microstates in string theory
Thermodynamics is only an approximation to a more fundamental description
based on the statistical properties of the microstates of the system. So the
fact that black holes have entropy suggests that this too should be understood
microscopically. Roughly speaking, a thermodynamic system with entropy S is
associated with a number e S of microstates of the system. For the Schwarzchild
black hole, with a horizon area given by (10.8), we see from (10.75) that the
entropy is Sbh = 4rrG N M2 loc. For the moment, the only important feature is
that the associated number of states grows like e M2 • We might have hoped that
this approximates the number of perturbative string states with mass M. However,
this is not the case, as we shall now demonstrate.
It suffices to consider the open bosonic string. This has mode expansion [13]
X/L = xiL + I: p/L T + its L !at: cos(nu)
(10.77)
n#) n
where the oscillator coefficients at: are creation (annihilation) operators for
n < 0 (n > 0). In units where the string length scale Is = I/..!i?i is 1 (T is
the string tension), the mass eigenvalues are given by the eigenvalues of
!M2 = N-I
(10.78)
where
00
" i
i
(10.79)
N = ~a_nan
n=1
is the number operator; the sum over i is over the DT = 24 transverse dimensions
of the bosonic string. We wish to estimate the number of (degenerate) states dn
that have number-eigenvalue n. It is convenient to define a generating function
00
G(w) = trw N = Ldnw n
(10.80)
n=O