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Noncommutative Quantum Cosmology: Exploring Phase Space Dynamics

This paper presents an in-depth study on noncommutative quantum cosmology, driven by the desire to understand the initial conditions of our universe within a noncommutative framework. It discusses noncommutative space-time motivated by string theory, the phase space extension of quantum mechanics, and applies the Kantowski-Sachs cosmological model. The paper analyzes the noncommutative Wheeler-DeWitt equation, exploring solutions and their implications for the quantum model. Key features such as damping behavior in wave functions are examined, elucidating the impact of noncommutativity on quantum dynamics.

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Noncommutative Quantum Cosmology: Exploring Phase Space Dynamics

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  1. Noncommutative Quantum Cosmology Catarina Bastos 21 Dezembro 2007 C. Bastos, O. Bertolami, N. Dias and J. Prata, “Phase Space Noncommutative Quantum Cosmology” DF/IST-8.2007

  2. Noncommutative Quantum Cosmology: • Motivation – Noncommutative space-time • Phase Space Noncommutative Extension of Quantum Mechanics • Kantowski-Sachs Cosmological Model: • Classical Model • Quantum Model • Solutions: • Noncommutative WDW Equation • Analysis of Solutions • Conclusions Noncommutative Quantum Cosmology

  3. Motivation – Noncommutative (NC) space-time: • String Theory / M-Theory (configuration space NC) • Gravitational Quantum Well: • Measurement of the first two quantum states for ultra cold neutrons • Phase Space NC extension • Feature of quantum gravity : • Significative effects at very high energy scales (?) • Configuration space NC (?) • Phase Space NC (?) • NC Quantum Cosmology: Understand initial conditions of our universe starting from a full NC framework Noncommutative Quantum Cosmology

  4. Phase Space Noncommutative Extension of Quantum Mechanics: (1) • ij e ijantisymmetric real constant (dxd) matrices • Seiberg-Witten map: class of non-canonical linear transformations • Relates standard Heisenberg algebra with noncommutative algebra • States of system: • wave functions of the ordinary Hilbert space • Schrödinger equation: • Modified ,-dependent Hamiltonian • Dynamics of the system Noncommutative Quantum Cosmology

  5. The Cosmological Model – Kantowski Sachs: (2) • ,: scale factors, N: lapse function • ADM Formalism Hamiltonian for KS metric: • P , P: canonical momenta conjugated to ,  • Lapse function (gauge choice): (3) (4) Noncommutative Quantum Cosmology

  6. KS Cosmological Model - Classical Model: (5) • Commutative Algebra: • Equations of motion in the constraint hypersurface, »0: • Solutions for  and : (6) (7) Noncommutative Quantum Cosmology

  7. KS Cosmological Model – Classical Model: (8) • Noncommutative Algebra: • Equations of motion: • Numerical solutions only! • Constant of motion: (9) (10) Noncommutative Quantum Cosmology

  8. KS Cosmological Model – Quantum Model: Planck unities , ~ LP ~ 1 (11) • Canonical quantization of the Classical Hamiltonian constraint, »0 • Wheeler De Witt (WDW) Equation: • Solutions for commutative WDW Equation: • Ki: modified Bessel functions (12) (13) Noncommutative Quantum Cosmology

  9. KS Cosmological Model – Quantum Model: (14) • Noncommutative Algebra: • Non-unitary linear transformation, SW map: • Relation between dimensionless parameters,  and : Invertible only if (15) (16) (17) Noncommutative Quantum Cosmology

  10. KS Cosmological Model – Quantum Model: • Noncommutative WDW Equation: • Exhibits explicit dependence on noncommutative parameters, • No analitical solution! • Noncommutative version of constant of motion (10): (18) (19) Noncommutative Quantum Cosmology

  11. Solutions – Noncommutative WDW Equation: From constraint (19): (23) (24) • Solutions of Eq. (18) are simultaneously eigenstates of Hamiltonian and constraint (23). • If a(c,c) is an eigenstate of operator (23) with eigenvalue aÎ: • Eq. (25) into (18) yields: (25) + (26) (27) Noncommutative Quantum Cosmology

  12. Solutions – Noncommutative WDW Equation: P(0)=0 , P(0)=0.4 , (0)=1.65 , (0)=10 (a) ==0 , a=0.4 (b) =5 , =0 , a=0.4 (c) =0 , =0.1 , a=0.565 (d) =5 , =0.1 , a=0.799 Noncommutative Quantum Cosmology

  13. Analysis of Solutions – Noncommutative WDW Equation: • For typical=5, wave function with damping: 0.05<<0.12 • The wave function blows up for c>0.12 • For >, varying  affects numerical values of (z) but its qualitative features remain unchanged • The range for possible values of  where the damping occurs is slightly different • The lower limit for  seems to be 0.05 for all possible values of  • For >, the damping behaviour of the wave function is more difficult to observe, only for certain values of  that the wave function does not blow up ([1,2]) • For large z, the qualitative behaviour of the wave function is analogous to the one depicted in Figures. Noncommutative Quantum Cosmology

  14. Conclusions: • Classical constraint allow us to solve numerically the NCWDW equation • Quantum Model is affected by the introduction of noncommutativity in momenta Introduces a damping behaviour for the wave function which is more peaked for small values of  Natural Selection of States Noncommutative Quantum Cosmology

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