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Quantum Cosmology in Krylov Space: Complexity and Entropy
Meysam Motaharfar1, Maxwell R Siebersma1, Parampreet Singh1,2
1Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA.
Entropy (Basel, Switzerland)
|July 28, 2026
Summary
We explored quantum cosmology complexity using Krylov space. Loop quantum cosmology (LQC) shows finite complexity at the bounce, unlike Wheeler-DeWitt (WDW) cosmology which diverges at singularities.
Area of Science:
- Cosmology
- Quantum Mechanics
- Theoretical Physics
Background:
- Quantum cosmology models describe the universe's quantum nature.
- Krylov complexity measures quantum system evolution.
- Comparing Wheeler-DeWitt (WDW) and Loop Quantum Cosmology (LQC) is crucial for understanding quantum gravity.
Purpose of the Study:
- To compute Krylov state and operator complexity in WDW and LQC frameworks.
- To investigate the behavior of complexity and entropy in these models.
- To establish a connection between Krylov complexity and canonical quantum cosmology.
Main Methods:
- Analytical construction of Krylov basis using the Lanczos algorithm.
- Evaluation of Krylov state and operator complexity.
- Calculation of Krylov entropy for WDW and LQC.
Main Results:
- Krylov complexity grows quadratically with the scalar field clock in sharply peaked regimes for both WDW and LQC.
- Operator complexity is twice the state complexity in these regimes.
- LQC complexity and entropy remain finite at the bounce, while WDW complexity and entropy diverge at singularities.
Conclusions:
- This study provides the first computation of Krylov complexity for a system with a totally constrained Hamiltonian and no external time.
- It offers a framework for calculating purely quantum-mechanical entropy in quantum cosmology.
- It establishes a bridge between Krylov complexity and canonical quantum cosmology, paving the way to study polymerized quantum geometry effects on complexity and entropy.
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