arXiv:hep-th/9304011 v1 Apr 5 1993

may take the double scaling limit of the fermion operator by first defining an operator ˆΨ N
with a smooth N → ∞ limit,
1 ( λ
ˆΨ N (λ, x) ≡ ˆΨ √ , Nx ) , (11.29)
(2N)
1/4 2N
where we have substituted x = X/N. 46 We now rescale
a ɛ (ε i ) →
a ɛ (ν)
( 1
π log √ 2N ) 1/2 , (11.30)
so that in the N → ∞ limit we have
∫
ˆΨ N (λ, x) → ˆψ(λ, x) =
dν e iνx a ɛ (ν) ψ ɛ (ν, λ) , (11.31)
where ψ ɛ (ν, x) are normalized as in appendix A (eqns. (A.1, A.2)). The vacuum of the
double scaled field theory is defined by
a ɛ (ν)|µ〉 = 0 ν < µ
(11.32)
a † ɛ (ν)|µ〉 = 0 ν > µ ,
where the Fermi level µ is as in (11.19).
11.4. Macroscopic Loops at c = 1
The discussion of macroscopic loop operators given in chapters 8 and 9 above continues
to hold at c = 1 with some minor modifications [90]. We now wish to compute the
continuum limit of the macroscopic loop operator
∫
∫
W lattice (L, q) = dX e iqX tr e LΦ(X) → dλ dx e iqx ψ † ψ(λ, x) e λl = W cont (l, q) . (11.33)
In particular, the boundary condition on the loop is of Dirichlet type, x(σ) = x.
A subtlety that arises is that the eigenvalue density is concentrated on both sides of
the quadratic maximum, or, in double-scaled coordinates, on (−∞, −2 √ µ ] ∪ [2 √ µ, ∞).
This means there are two (perturbatively) disjoint “worlds” and we cannot simultaneously
46 Note that in this chapter we have used X to denote the lattice target space coordinate and
in what follows we use x to denote the continuum target space coordinate, in minor conflict with
the notation of chapt. 5 in which X denoted the continuum target space coordinate.
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