arXiv:hep-th/9304011 v1 Apr 5 1993

Dirichlet boundary conditions imply the free propagator for the fermions χ ± :
∫ ∞
0
dE cos Eτ 1 cos Eτ 2
E 2 + p 2 . (12.12)
The change of variables to (12.11) can be pursued more carefully and perturbation theory
calculations can be performed in this formalism. See [76] and [16] for details.
Remark: The one-loop free energy in this 1+1 dimensional relativistic field theory
can be calculated [76] and goes as log µ, which is hence interpreted as the volume of φ-space.
Further attempts to interpret this result may be found in [15] (see also sec. 11.6).
12.4. τ-space and φ-space
Collective field theory is a theory of a massless boson that represents fluctuations in
the eigenvalue density. On the other hand, the massless boson of string theory is a scalar
field T (φ, t). In this section we discuss the nontrivial relation between these two bosonic
fields [91]. 50
As we have seen, the macroscopic loop and tachyon field are essentially the same. In
turn, W and ρ are related by a Laplace-like transform,
W (l, x) =
∫ ∞
2 √ µ
dλ e −lλ ρ = 2√ µ
K 1 (2 √ ∫ ∞
µl) + dτ e −2l√ µ cosh τ ∂ τ ζ , (12.13)
l
0
where in the second equation we have shifted the field ρ by its genus zero one-point function
ρ = √ λ 2 − 4µ + ∂ λ χ, and changed variables to λ = 2 √ µ cosh τ.
Using this transformation on fields, we can understand the relation between the tree
level propagators of the W -theory, (11.46, 11.47) and those of the collective field theory
(12.12). The key relation is provided by the kernel e −2l√ µ cosh τ , which satisfies the
differential equation:
H φ e −2l√ µ cosh τ ≡
(− ∂2
∂φ 2 + 4µl2 )
e −2l√ µ cosh τ = − ∂2
∂τ 2 e−2l√ µ cosh τ . (12.14)
50 Many authors continue to identify τ with the Liouville coordinate φ. While both spaces
share many qualitative features, they cannot be the same. The matrix model coordinate λ has
no (obvious) geometric meaning in the discrete worldsheet sum. Rather, it is the loop operator
W (L) that has a geometric meaning and is related to the worldsheet metric, and therefore to the
Liouville field φ.
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