Feynman Path Integral Formulation

5.3 Pair Creation in Constant Electric Fields 153S(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 toavoid spurious divergences. Due to charge conjugation invariance one can re-writethe 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)− iεThe 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 identitylog x ∫ ∞y = ds(e is(y+iε) − e is(x+iε)) , (5.59)s0and the explicit form for the square of the Dirac operator (which is the Klein-Gordonoperator, plus the spin part σ μν F μν ), one obtains for the probability of pair creationat x∫ ∞dsP(x) =Retr0 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 formidableexpression. To simplify things a bit, one can consider a static uniform electric fieldalong 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 evolutionoperator for a harmonic oscillator with imaginary frequency ω 0 = 2ieE. Since theenergy 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 integralcan then be evaluated by residues, and one finds for the probability of creatinga pair the exact result, for constant uniform field E,P = 2e2 E 2(2π) 3 ∞∑n=1(1n 2 exp −)nπ m2|eE|. (5.61)Since the scale for the external field is set by the electron mass squared (m 2 ), theeffect is generally very small in atoms. It is important to note that the result is