arXiv:hep-th/9304011 v1 Apr 5 1993

The formulae (10.1) and (10.2), while elegant, do not make manifest the physics of
the models we are discussing. To address this problem, we examine these formulae at
genus zero.
Since κ counts loops, we can regard the expectation value in (10.1) as a
“quantum-mechanical” expectation value, with κ playing the role of ¯h, and obtain the
genus zero approximation to the loop formulae as follows. Using the Campbell–Baker–
Hausdorff formula to separate exponentials of ˆp 2 and u, and then inserting a complete set
of eigenfunctions
we obtain
〈p|x〉 ≡ 1 √
2πκ
e ipx/κ , (10.3)
〈
W (l)
〉h=0 = ∫ ∞
µ
dz 〈 z|e −lˆp2 e −lu |z 〉 = 1
2πκ
=
∫ ∞
µ
∫
1 ∞
2 √ πκl 1/2
∫ ∞
dz dp e −lp2 e −lu
−∞
µ
dz e −lu .
(10.4)
Let us consider this formula first for the case of pure gravity. If we wish to calculate the
expectation value of a loop with the cosmological constant inserted, we take a derivative
with respect to µ to bring the operator down from the action, yielding
〈 ∫ 〉
W (l) e γφ 1
=
2 √ πκl 1/2 e−lu(µ) . (10.5)
In pure gravity, the string equation is the Painlevé I equation (7.17):
u 2 − κ2
3 u′′ = z , (10.6)
and the genus zero equation becomes simply u(z) = z 1/2 . The matrix model result for the
wavefunction of the cosmological constant is thus
l
V
=
〈
W (l)
precisely as expected from the continuum theory (e.g. sec. 4.3).
(10.7)
Exercise. Spectrum of 2D Gravity
Using the KPZ formula (4.6), show that the spectrum of numbers ν in the WdW
equation (4.7) for the case of pure gravity is ν = ν j = j + 1 2 , j ≥ 0.
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