arXiv:hep-th/9304011 v1 Apr 5 1993

5.1. Particles in D Dimensions: QFT as 1D Euclidean Quantum Gravity.
In chapters 2 and 4, we have discussed 2D Euclidean quantum gravity. In this section,
we apply the same techniques to 1D Euclidean Quantum Gravity. While the theory is
trivial as a theory of quantum gravity, it has an important and obvious reinterpretation in
terms of target space Euclidean quantum field theory.
Path Integral Approach
An example which will illuminate our later considerations is that of a particle moving
through Euclidean spacetime. This may be thought of as 1D quantum gravity since the
system is described by the action
S = 1 2
We consider the path integral
∫
dτ √ g(τ)
(g ττ ( dX µ ) ) 2
− m
2
. (5.1)
dτ
A(X I , X f ) =
∫ dg dX
Diff eS , (5.2)
with boundary conditions X µ i , Xµ f on Xµ . We can fix the gauge by transforming the
einbein to a constant, f ∗ e = s, where s is the single coordinate invariant quantity (i.e.
modulus), namely the length. The path integral becomes
A(X i , X f ) =
∝
∫ ∞
0
∫ ∞
since the determinant is proportional to s.
Canonical Approach
0
ds (
det ′ (−s −2 ∂ 2
s 1/2 t ) ) (1−D)/2
e
−(∆X) 2 /2s−m 2 s/2
∫
ds
/2s−m 2 s/2 d D p
∝
s D/2 e−(∆X)2 (2π) D
e ip∆X (5.3)
p 2 + m , 2
Turning to the canonical approach, the action (5.1) has a gauge invariance:
δX = ɛ(τ)X ′ δe(τ) = ɛ ′ (τ)e(τ) + ɛ(τ)e ′ (τ) . (5.4)
We can fix the gauge by putting e = 1 at the price of imposing a constraint. The Wheeler–
DeWitt operator, which generates τ diffeomorphisms, is simply H = p 2 + m 2 , where p µ (τ)
is the field canonically conjugate to x µ (τ). The Wheeler–DeWitt equation is the Euclidean
Klein-Gordon equation:
Hψ(x) =
(− ∂2
∂x 2 + m2 )
ψ = 0 . (5.5)
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