We prove the existence of a countably infinite number of “excited” states for the Lorentzian-signature Taub–Wheeler–DeWitt (WDW) equation when a cosmological constant is present using the Euclidean-signature semi-classical method. We also find a “ground” state solution when both an aligned electromagnetic field and cosmological constant are present; as a result, conjecture that the Euclidean-signature semi-classical method can be used to prove the existence of a countably “infinite” number of “excited” states when the two aforementioned matter sources are present. Afterward, we prove the existence of asymptotic solutions to the vacuum Taub–WDW equation using the “no boundary” and “wormhole” solutions of the Taub Euclidean-signature Hamilton–Jacobi equation and compare their mathematical properties. We then discuss the fascinating qualitative properties of the wave functions we have computed. By utilizing the Euclidean-signature semi-classical method in the above manner, we further show its ability to prove the existence of solutions to Lorentzian-signature equations without having to invoke a Wick rotation. This feature of not needing to apply a Wick rotation makes this method potentially very useful for tackling a variety of problems in bosonic relativistic field theory and quantum gravity.
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