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We describe in detail the quantum tunneling of massive particles from Kerr black hole by using complex trajectories, which are solutions to the Hamilton's equations of motion with imaginary proper time.
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In performing the discretization we use the resolution of identity as ∫ | x ⟩ ⟨ x | d 4 x \int\ket{x}\bra{x}d^{4}x so that transition amplitude between initial and final points on the trajectory is given as a scalar density. Accordingly, the basis vector | x ⟩ \ket{x} transforms as vector density of weight 1 2 \frac{1}{2} . Alternatively, using: ∫ | x ⟩ ⟨ x | − g d 4 x \int\ket{x}\bra{x}\sqrt{-g}\,d^{4}x with the normalization ⟨ x ⟩ p = exp ( − i p μ x μ ) / ( 2 π ( − g ) 1 / 4 ) \braket{x}{p}=\text{exp}\left(-ip_{\mu}x^{\mu}\right)/(2\pi(-g)^{1/4}) results in the cancellation of the − g \sqrt{-g} factor in the volume element. The net effect of using the the latter choice is that the overall prefactor multiplying the momentum integrals in ( 49
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Cesim K. Dumlu, “Multidimensional quantum tunneling in the Schwinger effect,” Phys. Rev. D 93
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