Feynman Path Integral Formulation

1.10 String Theory 45
and for the open string
M 2 = −8a + 8
∞
∑
n=1
α −n · α n , (1.213)
M 2 = −2a + 2
∞
∑
n=1
α −n · α n , (1.214)
with a a constant to be determined from consistency conditions (absence of ghosts).
Thus the mass squared for the for the ground state of closed strings is four times that
for the open strings. In the first case on can show that the vector particle is massless,
but the scalar ground state is a tachyon with m 2 < 0. Further analysis reveals though
that the absence of ghosts (or negative norm states), which in the case of the bosonic
string are associated with the timelike mode of X μ (σ,τ), implies that either d = 26
and a = 1, or d ≤ 25 and a ≤ 1. But the theory can be shown to be truly Lorentz
invariant only in d = 26, which implies that for the bosonic string quantization and
regularization are only consistent at d = 26.
Once the commutator algebra has been specified (as we have seen it is strongly
restricted by the Lorentz and Weyl invariance of the theory) one can start enumerating
the lowest excitations. The structure of the spectrum for closed strings is
|0〉 →tachyon
α † μ
1
˜α † ν
1
|0〉 →massless tensor , (1.215)
while for open strings one finds the following spectrum
|0〉 →tachyon
α † μ
1
|0〉 →massless vector
α † μ
2
|0〉 →massive vector
α † μ
1
α † ν
1
|0〉 →massive tensor , (1.216)
with the mass squared increasing linearly with spin (linear Regge trajectories).
Thus open bosonic string theory contains a massless spin two particle, described
by a traceless symmetric tensor, whose low energy limit should be identified with
the action for general relativity plus higher order corrections. In the original string
theory framework this was regarded as a major disappointment, as no such particle
appeared in the known hadron spectrum (Scherk and Schwartz, 1974).
There is one big problem that remains with the bosonic string discussed so far,
namely that the ground state corresponds to a tachyon, a particle of mass m 2 < 0,
which suggests some sort of fundamental instability of the theory. It is possible that
the tachyon is just an artifact due to the expansion around the wrong string theory
vacuum, but this remains a largely unsolved questions since it is not easy to treat
the bosonic string non-perturbatively. But one possible approach to this problem is
offered by the covariant Euclidean Feynman path integral.