Feynman Path Integral Formulation

152 5 Semiclassical Gravity
the case d = 0. In this case one has
Q[a] ≃
( a
a 0
) c(1+α)
. (5.52)
It is unclear if this simple model is sophisticated enough to capture the essence
of the more complete calculation, but it does suggest that the behavior at small a
could be strongly influenced by the detailed nature of the short distance cutoff, the
form of the effective action as small distances, and the choice of measure over the h
variables.
5.3 Pair Creation in Constant Electric Fields
It is known that if an atom is subject to a sufficiently strong external uniform electric
field it is possible for the field to create pairs. To be specific, if the atomic potential
is approximated by a function with a minimum −V 0 and going to zero at infinity,
then an external electric field, with potential eEx, will give an ionization probability
for a bound electron
∫ V0 /|eE|
P ≃ exp − dx √ 2m(V 0 −|eE|x) =exp
0
(
− 4 )
√ V 0 2mV0
3 |eE|
, (5.53)
using the WKBformula, with V 0 /|eE| the distance between the two classical turning
points. If one thinks of a negative energy electron as trapped in a potential of depth
V 0 ≈−2m, then one obtains for the probability of pair production in a strong electric
field the semiclassical estimate
)
P ≃ exp
(− 16m2 . (5.54)
3|eE|
Clearly in this last instance one is dealing with a relativistic process involving strong
fields, so a fully relativistic treatment would seem more appropriate.
To this end one considers the external field-dependent Feynman amplitude Z[A]
describing the vacuum to vacuum amplitude in the presence of an external electric
field, expressed as a path integral over fermion fields ψ and ¯ψ in a fixed classical
background A,
∫
∫ ]
Z[A] = [dψ][d ¯ψ] exp[
d 4 x ¯ψ(i ̸∂ + e ̸A + m − iε)ψ
= det[i ̸∂ + e ̸A + m − iε] . (5.55)
Using the formal identity detM(A) =exptrlogM(A), the corresponding effective
action S(A) is then given by minus the exponent of the above expression