Applying the Euclidean-signature semi-classical method to the quantum Taub models with a cosmological constant and aligned electromagnetic field

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.

Automated summary

By utilizing the Euclidean-signature semi-classical method, this study proves the existence of a countably infinite number of excited states for the Lorentzian-signature Taub-Wheeler-DeWitt equation in the presence of a cosmological constant. Additionally, a ground state solution is found when both an aligned electromagnetic field and cosmological constant are present. The method is also shown to be capable of proving solutions to Lorentzian-signature equations without needing to apply a Wick rotation, making it potentially useful for various problems in bosonic relativistic field theory and quantum gravity.