arXiv:hep-th/9304011 v1 Apr 5 1993

egarded as space, the time coordinate t is a negative signature c = 1 field obtained by
analytically continuing X. The tachyon background
〈
T (φ, t)
〉
= µ e
√
2φ
(13.1)
acts as a repulsive wall for incoming bosons and the dilaton background leads to a spatiallyvarying
coupling
κ eff (φ) = κ 0 e 1 2 Qφ . (13.2)
Because the S-matrix of massless bosons in two-dimensions is a subtle object, we
begin with some precise mathematical definitions of what we are talking about. We begin
with the string definition. As explained in sec. 5.4, the vertex operators are V ± ω
(5.22). Using (11.61), we write
given by
Def 1: The connected string scattering matrix elements are asymptotic expansions in κ
given by
S ST
c
( k∑
ω i →
i=1
l∑
i=1
ω ′ i
)
= A n (Vω − 1
, . . . Vω − k
, V + ω
, . . . V +
1
′ ω
) . (13.3)
′
l
Mathematically it is easier to use a Euclidean signature boson X via the analytic
continuation |q| → −iω:
V + ω → V q q > 0
V − ω → V q q < 0 .
(13.4)
We’ll refer to the S-matrix elements calculated with V q as the “Euclidean S-matrix.”
According to the matrix model hypothesis, these amplitudes may be calculated via
the c = 1 matrix model according to the discussion of sec. 11.7. If one is interested
in the S-matrix and not in the macroscopic loop amplitudes (which contain much more
information), then it is most efficient to calculate the collective field S-matrix which we
describe next. 54
In collective field theory we define the S-matrix according to the coordinate-space
version of the LSZ prescription, that is, we isolate the piece of the large spacetime asymptotics
of time-ordered Green’s functions which is proportional to the product of on-shell
incoming and outgoing wavefunctions.
54 Indeed, defining the S-matrix directly via asymptotics in τ-space [132], as presented below,
was an important technical advance over the original method [90] of calculating loop amplitudes
and then shrinking the loops.
152