The exact Wheeler-DeWitt equation for the scale-factor minisuperspace model

Summary

The Wheeler-DeWitt equation for the scale-factor minisuperspace model is derived, resolving operator-ordering ambiguities. The path-integral formulation yields distinct equations due to measure choices. Hermiticity conditions determine the Hilbert-space measure, and probability amplitudes are defined.

Highlights

• The Wheeler-DeWitt equation is derived for the scale-factor minisuperspace model.
• Operator-ordering ambiguities are resolved.
• Path-integral formulation yields distinct equations due to measure choices.
• Hermiticity conditions determine the Hilbert-space measure.
• Probability amplitudes are defined using the Hermitian inner product.
• The model's results establish the universality of quantum prescriptions.
• The approach is limited to locally Minkowskian target spaces.

Key Insights

• The Wheeler-DeWitt equation for the scale-factor minisuperspace model is derived, resolving operator-ordering ambiguities and yielding distinct equations due to path-integral measure choices.
• The Hermiticity condition determines the Hilbert-space measure, ensuring the quantum Hamiltonian is Hermitian and allowing for the definition of probability amplitudes.
• The model's results establish the universality of quantum prescriptions, demonstrating that different choices of path-integral measures lead to equivalent quantum theories.
• The approach is limited to locally Minkowskian target spaces, as the method relies on the existence of a field redefinition yielding a constant metric.
• The results are consistent with previous works, such as Ref. [18], and provide a non-trivial consistency check of the assumption that the discretized path-integral measure yields a time-reparametrization invariant measure.
• The Wheeler-DeWitt equation can be put in an alternative form, involving the wavefunction Ψ and a universal potential term, which is independent of the choice of path-integral measure.
• The approach can be generalized to more involved systems, but the method's applicability is limited to locally Minkowskian target spaces, and further research is needed to explore more general cases.

Mindmap

Citation

Kaimakkamis, E., Partouche, H., Sil, K., & Toumbas, N. (2024). The exact Wheeler-DeWitt equation for the scale-factor minisuperspace model (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18532