División de Ciencias e Ingenierías Campus León Universidad de Guanajuato. O. Obregón. M(atrix) Theory. (Supersymmetric Quantum Cosmology)

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1 División de Ciencias e Ingenierías Campus León Universidad de Guanajuato O. Obregón M(atrix) Theory (Supersymmetric Quantum Cosmology)

2 Summary We use the M(atrix) model arising from the quantization of the eleven dimensional membrane in the Light Cone Gauge (LQG). We use a Dirac-Like Gamma matrix representation for the fermionic degrees of freedom. The relevant operators are the supercharges and the SU(N) constraint operators. They generate multicomponent wave eigenfunction solutions. We restrict to the SU() group and to and 4 dimensions. Key Results Normalized solutions exist (Restricted models). The method can be extended to any SU(N) group and to any dimensions. The 4-dimensional solution resembles a solution found independently in SUSY quantum cosmology. Supersymmetric generalization of M(atrix) cosmology. (Freedman Gibbons).

3 M(atrix) theory and supergravity M-Theory l d-supergravity l Five theories of strings l Supermembranes l M(atrix) theory l Mechanics of D0 particles The Degrees of freedom are NxN matrices and solutions are relevant not only for large N (Susskind).

4 Physics of Matrix theories Matrix theory D0-Brane Mechanics (Susskind, Fischler, Shenker, Banks) Black hole thermodynamics Matrix cosmology Physics of Supermembranes M(atrix) Theory Supermembranes (Nicolai, Halpern, De Witt) Supersymmetric quantum mechanics Supersymmetric Quantum Cosmology (our work)

5 M(atrix) Theory Hamiltonian Bosonic and Fermionic Hamiltonians H B = m a m a H F = + g4 4 f abc m b n c f ade m d n e ig~ f abc a ( m ) m b c Supercharges Q =( m a ) m a + igf abc ( mn a ) m b n c G a = f abc ( m b m c i~ b c )

6 Supercharge algebra { Q,Q } = H +g( m ) m a G a The eigenfunction has to satisfy Q i =0 G a i =0 Bosonic Degrees of freedom (usual Rep.) m a m a [ m a, n b ]=i~ ab mn Fermionic Degrees of freedom (Matrices) { a, b } = ab

7 SU() M(atrix) equations In the SU() dimensional model we have: supercharges 3 G-Operators 6 Bosonic and 6 fermionic DOF Q = g ( 3 3 ) g ( 3 3 ) g 3( ) G = i~( ) With the choice of Dirac-Like gamma matrices for fermionic DOF (8x8 matrices) Q i =0 G a i =0 Matrix Differential equation for the 8 components of the wave function

8 Matrix Equations We can restrict to particular configurations on the bosonic variables (bosonic potentials) [3f ( ~ ) 3] +[i 3 +3f ( ~ )] +[ 3f 3 ( ~ )] 4 [i + ] 8 =0, [3f ( ~ ) i 3 ] +[ 3 3f ( ~ )] +[ 3f 3 ( ~ )] 3 [i + ] 7 =0, [ 3f 3 ( ~ )] +[3f ( ~ ) 3] 3 +[i 3 + 3f ( ~ )] 4 +[i + ] 6 =0, [ 3f 3 ( ~ )] +[ i 3 3f ( ~ )] 3 +[ 3 3f ( ~ )] 4 +[i + ] 5 =0, [ i ] 4 +[3f ( ~ ) 3] 5 +[i 3 +3f ( ~ )] 6 +[ 3f 3 ( ~ )] 8 =0, [ i ] 3 [i 3 + 3f ( ~ )] 5 +[ 3 3f ( ~ )] 6 +[ 3f 3 ( ~ )] 7 =0, [i ] +[ 3f 3 ( ~ )] 6 +[3f ( ~ ) 3] 7 +[i 3 + 3f ( ~ )] 8 =0, [i ] +[ 3f 3 ( ~ )] 5 [i 3 +3f ( ~ )] 7 +[ 3 3f ( ~ )] 8 =0, f ( ~ )=( ), Potentials f ( ~ )=( 3 3 ), f 3 ( ~ )=( 3 3 ).

9 Form of wave eigenfunctions Particular configuration = = 3 = x = = 3 = y i =(,, 3, 4, 5, 6, 7, 8 ) G a i =0 i =(,,i,i, i, i,, ) Q i =0 x = r cos, y = r sin Solution = ±ar apple e iapple ± br apple e iapple, = ar apple e iapple br apple e iapple.

10 Extended model (4-dimensions) bosonic and fermionic DOF 0 Q, Q... Q,64 Q,... Q, Q 64,... Q 64,64 (64x64) matrix operators C A C A = C A Particular configuration -> 4 = x, f( ~ )=(c dx) = 4 3 = c, 3 = d. Solution with pair of coupled components 37 = c i exp h 3 i c dx x + c i exp h i 3 c dx x.

11 Solution to the extended model Particular values to constants in the last solution 4 = R(t), = 4 3 = r Mc 3, 3 = p applec 3 3G The solution becomes 37 h = c exp i h + c exp i Mc ~ R p applec 3 G~ R i Mc ~ R p applec 3 G~ R i. This is the Solution of a particular Supersymmetric Quantum Cosmology model as We Will see.

12 Supersymmetric Quantum Cosmology (O. Obregón, V. Tkach, J. Rosales, J. Torres, Phys. Lett. A 340 5) S = Cosmological FRW model with Matter (Dust). Z " c NG R dr dt + N applec4 G R NMc # dt Superspace (Supertime) Cosmological Action In terms of super fields. S = Z apple c G N RD RD R + c3p apple G R McR d d dt N(t,, ) =N(t)+i (t)+i (t)+v (t) R(t,, ) =R(t)+i (t)+i (t)+b(t)

13 Susy Quantum Cosmology Constrarints Explicit Constraints for the Supersymmetric Cosmological model H (R) =0, S (R), S (R) =0, F (R) =0 H = G c R R S = S = applec 4 R G M G R + c M p apple ig / c p appler / cr / R G / + MG/ R / ig / cr / R + cp appler / MG / G / R / c p apple R MG cr A Dirac-like Matrix representation was also used = p ~, = p ~ +, ± = ( ± ),

14 Solutions of SUSY QC and M(atrix) Theory Supersymmetric Quantum Cosmology h = C exp = C h exp Mc ~ R p i applec 3 G~ R Mc ~ R p applec 3 G~ R i M(atrix) Theory exact Solution 37 = c i + c i h exp h exp Mc ~ R p applec 3 G~ R i Mc ~ R p applec 3 G~ R i. Same Normalizable solutions Different Hilbert spaces

15 Conclusions Exact Normalizable solutions exist (Restricted models). Finite N Physically Relevant Solutions. Supersymmetric extension to Matrix Cosmology (Freedman- Gibbons). The model can be extended to any SU(N) group and Any Dimension. Perspectives Solution to some other(s) Supersymmetric cosmological model(s); Anisotropic models (Bianchi models). Our procedure allows to construct an algorithm for the search of solutions without any restriction (Any Group any dimension) and for large N. Connection with Super String Cosmology.

16 ThAnK YoU AlL!