Relativistic free-motion time-of-arrival operator for massive spin-0 particles with positive energy

Instantaneous tunneling of relativistic massive spin-0 particles

“…The resulting value coincides with the Cauchy principal value for j = 0 and to the Hadamard finite part for non-negative integer values of j. Reference [30] has already evaluated the case when j = k = 0, and we can use exactly the same contour, shown in fig. 1, to generalize their method for non-negative integers {j, k}.…”

“…The operator corresponding to the TKF TF (η, ζ) coincides with the Rigged Hilbert space extension of Razavi's relativistic free TOA operator studied in ref. [30], where it was shown that the associated physical quantities are consistent with special relativity. The TOA-operator equation ( 7) was derived under the assumption that the interaction potential is analytic, but since the TKF equation ( 11) is in integral form, then eqs.…”

“…The earliest was proposed by Razavi [28] which is a relativistic version of the Aharonov-Bohm TOA operator [29], and studied in detail by ref. [30]. We will thus first propose a formalism on the construction of relativistic TOA operators via quantization using refs.…”

“…We will thus first propose a formalism on the construction of relativistic TOA operators via quantization using refs. [30][31][32][33][34], and then construct the corresponding barrier traversal time operator for the first time here.…”

“…for all real a, b [30] which can also be obtained using the calculus of residues. Combining the results yields…”