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2 ∂x (24) In (24) we have taken k⊥ 2 = ky2 + kz2 , (25) 2 M 2 = k⊥ + m2 . Equation (24) is of the form (12) with the turning points given by (ω + qE0 x) = ±M. (26) It is better to have a convenient form of (24) and (26), making the following change of variables: ρ= ω qE0 x + √ , q E0 M2 λ= , qE0 (27) in (24). We get ∂2φ + (ρ2 − λ)φ = 0, (28) 2 ∂ρ √ so that the turning points are at ρ = ± λ. Before we apply cwkb to (28), let us try to understand the origin of complex paths in the formalism. Let us restrict to (1 + 1) dimension and identify xc = −ω/qE0 , and the 2 acceleration a = qE0 /M (now M = m since k⊥ = 0).

The basics of cwkb is thus discussed in Section 3 very briefly. (For details and the equivalence of the cwkb-pair production with other standard methods, the reader is refered to Refs. ) Before we enter into the discussion of particle production through cwkb, let us try to understand the origin of complex paths for an electromagnetic background. According to Schwinger [19] the imaginary part of the effective Lagrangian is given by (for constant electric field E0 ) ∞ Im Leff = 1 (qE0 )2 (−1)n+1 exp 3 2 2 (2π) n n=1 − πm2 n , qE0 (5) where m is the mass and E0 is the magnitude of the electric field.

For details and the equivalence of the cwkb-pair production with other standard methods, the reader is refered to Refs. ) Before we enter into the discussion of particle production through cwkb, let us try to understand the origin of complex paths for an electromagnetic background. According to Schwinger [19] the imaginary part of the effective Lagrangian is given by (for constant electric field E0 ) ∞ Im Leff = 1 (qE0 )2 (−1)n+1 exp 3 2 2 (2π) n n=1 − πm2 n , qE0 (5) where m is the mass and E0 is the magnitude of the electric field.

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General relativity and gravitation Vol. 32


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