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

 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: Noncommutative Quantum Cosmology

 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: Motivation – Noncommutative (NC) space-time: Noncommutative Quantum Cosmology Understand initial conditions of our universe starting from a full NC framework

  ij e  ij antisymmetric 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 Phase Space Noncommutative Extension of Quantum Mechanics: Noncommutative Quantum Cosmology (1)

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

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

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

 Canonical quantization of the Classical Hamiltonian constraint,  »0  Wheeler De Witt (WDW) Equation:  Solutions for commutative WDW Equation:  K i : modified Bessel functions KS Cosmological Model – Quantum Model: Noncommutative Quantum Cosmology ,  ~ L P ~ 1 (11) (12) (13) Planck unities

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

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

 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: Solutions – Noncommutative WDW Equation: Noncommutative Quantum Cosmology From constraint (19):(23) (24) (25) (26) + (27)

Solutions – Noncommutative WDW Equation: Noncommutative Quantum Cosmology 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

 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. Analysis of Solutions – Noncommutative WDW Equation: Noncommutative Quantum Cosmology

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