arXiv:hep-th/9304011 v1 Apr 5 1993

It is extremely interesting to note that — even for pure gravity — the third quantized
universe propagator is not the naive minisuperspace Wheeler–DeWitt propagator (5.20).
In particular the ultraviolet behavior of the propagator (in E) is completely different from
the naive propagator (5.20). For example, the 1/E 2 behavior in the ultraviolet becomes
e −πE behavior. Our understanding of why this is so is incomplete. (Part of the story is
explained in the next section.)
In general, we can decompose amplitudes as
〈
W (l1 ) · · · W (l n ) 〉 ∫ ∏
=
i
dE i ψ Ei
(l i ) A(E 1 , . . . , E n ) , (10.22)
which (in the case of the four-point amplitude) we depict as
l 1
l 2
∫ ∏
=
i
l 4 l 3
By shrinking l i to zero we get an expansion in terms of local operators. Alternatively, and
equivalently, by doing the integral over the E i we pick up residues corresponding to the
Euclidean on-shell states. A similar picture emerges for all the multicritical points. The
following exercise carries this out in detail for the Ising model.
d
Exercise. The Ising Model
The Ising model has a Z 2 symmetry flipping up spins for down, which, in the
matrix model formulation described in sec. 7.5 is exchange of U ↔ V . Letting W ± (l)
denote the Z 2 odd/even loop operators show that
〈
W± (l 1 )W ± (l 2 ) 〉 ∑
= ± (j + 1/3) I j+1/3 (2 √ µl 1 ) K j+1/3 (2 √ µl 2 )
j,±
∑
∓ (j + 2/3) I j+2/3 (2 √ µl 1 ) K j+2/3 (2 √ µl 2 ) .
j,∓
(10.23)
By summing the infinite series, show that
G(E, ±) = 2π(e πE ± 1 + e −πE sinh πE
)
sinh 3πE . (10.24)
Remark: We have shown that the wavefunctions 〈 σW (l) 〉 satisfy a linear WdW equation.
On the other hand, from the Schwinger–Dyson equations (8.18) and their continuum
analogs, we see that 〈 W (l) 〉 itself satisfies a nonlinear equation. A precise understanding
of the relation of these has never been given (except in special cases [120]). This is an
interesting problem for the future.
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