Feynman Path Integral Formulation

5.3 Pair Creation in Constant Electric Fields 153
S(A) =−trlogdet(i ̸∂ + e ̸A + m − iε )+ trlogdet(i ̸∂ + m − iε ) , (5.56)
where the zero external field, A = 0, contribution has been subtracted out in order to
avoid spurious divergences. Due to charge conjugation invariance one can re-write
the above traces as expressions involving the square of the Dirac operator,
{ (i ̸∂ + e ̸A)
S(A) = 1 2 2 trlogdet + m 2 }
− iε
−∂ 2 + m 2 . (5.57)
− iε
The pair creation probability at the point x will be denoted by P(x), and is given by
|Z[A]| 2 = e −2S(A) = e −∫ d 4 xP(x) , (5.58)
so it is this P(x) that one needs to extract from the trace in Eq. (5.57) (Schwinger,
1951; Brezin and Itzykson, 1970). After making use of the identity
log x ∫ ∞
y = ds
(
e is(y+iε) − e is(x+iε)) , (5.59)
s
0
and the explicit form for the square of the Dirac operator (which is the Klein-Gordon
operator, plus the spin part σ μν F μν ), one obtains for the probability of pair creation
at x
∫ ∞
ds
P(x) =Retr
0 s e−is(m2 −iε)
(
×〈x| exp ( is[(i∇ μ + eA μ ) 2 + 1 2 eσ μν F μν ] ) − exp ( is(i∇ μ ) 2)) |x〉 >
(5.60)
where now the trace is over spinor indices only.
In general, for arbitrary x-dependent fields A(x), this is still a rather formidable
expression. To simplify things a bit, one can consider a static uniform electric field
along the z axis, for which A 3 = −Etwith E constant. In this gauge the 〈x|(...)|x〉 >
matrix element can be evaluated by inserting a complete set of momentum eigenfunctions,
which reduces the problem to computing the trace of the time evolution
operator for a harmonic oscillator with imaginary frequency ω 0 = 2ieE. Since the
energy levels for such a system are well known, one can easily compute the trace,
which then reduces the problem to the evaluation of a single s integral. This last integral
can then be evaluated by residues, and one finds for the probability of creating
a pair the exact result, for constant uniform field E,
P = 2e2 E 2
(2π) 3 ∞
∑
n=1
(
1
n 2 exp −
)
nπ m2
|eE|
. (5.61)
Since the scale for the external field is set by the electron mass squared (m 2 ), the
effect is generally very small in atoms. It is important to note that the result is