arXiv:hep-th/9304011 v1 Apr 5 1993

If we isolate one Euclidean coordinate, call it φ, as a special coordinate, then we can
√p
write the Euclidean on-shell wavefunctions as e ipx e ± 2 +m 2φ . As long as there are no
tachyons in the theory, these wavefunctions have exponential growth and are not normalizable.
Conversely, the existence of Euclidean on-shell normalizable wavefunctions is a
signal of tachyons in the theory.
In order to describe off-shell physics, we introduce the normalizable states which
diagonalize the Wheeler–DeWitt operator: e ipx+iEφ , with eigenvalue E 2 + p 2 + m 2 . For
example, in a mixed position-space/momentum-space representation where we Fourier
transform with respect to all other coordinates, we may describe the propagator as
∫ ∞
G(ϕ 1 , p; ϕ 2 , −p) = dE e−iEϕ 1
e iEϕ 2
−∞ E 2 + ⃗p 2 + m 2
1
√⃗p
= θ(φ 1 − φ 2 ) √
⃗p 2 + m 2 e− 2 +m 2 |ϕ 1 −ϕ 2 | + [1 ↔ 2] .
(5.6)
Exercise. Back to the wall
What happens if φ is restricted to be semi-infinite? Put a boundary condition that
the wavefunctions vanish at φ = log µ and calculate the analog of (5.6).
Interactions and Topology-Change
One-dimensional quantum gravity from the target space viewpoint provides a useful insight
into the origin of the violation of the Wheeler–DeWitt constraint in topology-changing
processes. In this case, a topology-changing process corresponds to one 0-dimensional space
splitting into two as in
p 1
.
p
2
p = p + p
3 1 2
(5.7)
The “violation of the Wheeler–DeWitt constraint” is simply the familiar fact that if p 2 1 =
p 2 2 = −m 2 are on-shell momenta then in general p 2 3 = (p 1 +p 2 ) 2 ≠ −m 2 will not be on-shell.
This above basic phenomenon can also be realized as the result of a contact term
arising from a singularity at the boundary of “moduli space.” Consider the wavefunction
of a particle that interacts with an external potential V so that the wavefunction becomes
˜ψ(τ) =
∫ τ
−∞
dτ ′ e −H(τ−τ ′) V ψ , (5.8)
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