Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville Theory

Transcription

1 Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville Theory Justin Rivera Wentworth Institute of Technology May 26, 2016 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

2 Outline Brief introduction to the project We are looking for the space of physically allowable wavefunctions for 2d quantum gravity Solutions to Schrödinger s equation Solving for eigenfunctions of the Hamiltonian gives a basis for wavefunctions given the Hamiltonian is self adjoint Sturm-Liouville Theory and orthogonality of eigenfunctions of the Hamiltonian Conclusion Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

3 The Model Spacetime is M = R S 1 equipped with a 2-metric g x 0 parametrizes time R l(t) is the arc length around the spatial manifold {x M x 0 = t} Arc length is calculated with respect to the spatial part of the metric g Results in a Lagrangian for l as a dynamical variable(nakayama 1994) L = 1 4l(x 0 ) ( l(x 0 ) ) 2 λl(x 0 ) (m )2 l(x 0 ) t x Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

4 Rationale for studying 2-D Our Hamiltonian for 2-D quantum gravity ( H m = Π l lπ l + m l 2) 1 + λl Notice the factor ordering ambiguity in the kinetic term Also our configuration variable l must be positive Both key issues in 4-D General Relativity that can be studied in our 2-D model Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

5 Varied Factor Ordering Our Hamiltonian ( H m = Π l lπ l + m l 2) 1 + λl Notice the factor ordering ambiguity Choose a two-parameter family of orderings where i + j + k = 1 l i Π l l j Π l l k Apply Schrödinger quantization: l ˆl, Π l i d dl, ĤΨ = EΨ 2 l d 2 Ψ dl 2 2 (1 (i k)) dψ dl + ( ) ( 2 (i k) 2 (i + k) 2 ( ) l 1 + m l 4 2) 1 + λl E Ψ = 0. Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

6 Self Adjointness When the Hamiltonian is of the form Ĥ = h2 d 2 2m + V (x) it is dx 2 formally self adjoint i.e. ) (ĤΨ ĤΨ, Φ Φdx = ) Ψ (ĤΦ dx Ψ, ĤΦ With varied factor ordering the operator is not self adjoint with respect to dl and the wave function is not normalizable To impose formal self adjointness we use the measure l k i dl Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

7 Transformation to a Confluent Hypergeometric Equation The equation we need to solve 2 l d 2 Ψ dl 2 2 (1 (i k)) dψ dl + ( ) ( 2 (i k) 2 (i + k) 2 ( ) l 1 + m l 4 2) 1 + λl E Ψ = 0 The general form of a confluent hypergeometric equation is Substitutions are xy + (c x)y ay = 0 z = αl Ψ(l) = e βz z γ y(z) α, β, γ are constants that we choose, a, c are to be determined. Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

8 Transformation The 3 free parameters were α = 2 λ (i k) ± γ = Which determines a and c to be 1 ± a = c = 1 ± β = 1 2 (i + k) 2 + 4(m+ 1 2) 2 2 (i + k) 2 + 4(m+ 1 2) 2 2 h 2 2 (i + k) ( m E 2 λ Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19 h 2 ) 2

9 Transformation ( ) zy (z) + (i + k) 2 + 4(m+ 1 2) 2 z y (z) h 2 1 ± (i + k) 2 + 4(m+ 1 2) 2 2 h 2 + E 2 λ y(z) = 0 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

10 Solutions to Schrödinger s Equation Using the known solutions to the confluent hypergeometric equation we get the following energy eigenfunction and energy eigenvalues Ψ n (l) = Ne λl l γ L (c 1) n (αl) E n = 2 1 ± (i + k) 2 + 4(m+ 1 2) 2 λ n + h 2 2 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

11 Distinct Sets of Energy Eigenfunctions Notice that gamma has a plus-minus (i k) ± γ = The energy eigenfunction involves γ Ψ n (l) = Ne λl (i + k) 2 + 4(m+ 1 2) l γ L (c 1) n (αl) Both the positive and negative branch yield a distinct set of energy eigenfunctions and eigenvalues µ = ± (i + k) ( m ) 2 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

12 Questions about the Eigenfunctions Are the eigenfunctions in L 2? Ψ n (l) = Ne λl l γ L (c 1) n (αl) Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

13 Questions about the Eigenfunctions Are the eigenfunctions in L 2? Ψ n (l) = Ne λl l γ L (c 1) n (αl) Always in L 2 when positive branch of µ is taken When negative brach of µ we need µ < 1 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

14 Questions about the Eigenfunctions Are the eigenfunctions in L 2? Ψ n (l) = Ne λl l γ L (c 1) n (αl) Always in L 2 when positive branch of µ is taken When negative brach of µ we need µ < 1 Are the sets of eigenfunctions over complete in L 2? Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

15 Questions about the Eigenfunctions Are the eigenfunctions in L 2? Ψ n (l) = Ne λl l γ L (c 1) n (αl) Always in L 2 when positive branch of µ is taken When negative brach of µ we need µ < 1 Are the sets of eigenfunctions over complete in L 2? Is each set of eigenfunctions mutually orthogonal? Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

16 Questions about the Eigenfunctions Are the eigenfunctions in L 2? Ψ n (l) = Ne λl l γ L (c 1) n (αl) Always in L 2 when positive branch of µ is taken When negative brach of µ we need µ < 1 Are the sets of eigenfunctions over complete in L 2? Is each set of eigenfunctions mutually orthogonal? Are linear combinations of each set of eigenfunctions dense in L 2? Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

17 Sturm-Liouville Theory Given a second order differential equation of the form P(x)y + Q(x)y + R(x)y = 0 we can put it in Sturm-Liouville form [ µ(x)p(x)y (x)] + µ(x)r(x)y(x) = 0 If we began with an eigenvalue problem, the S.L. form gives us the S.L. eigenvalue problem [ ] [ ] p(x)y (x) + κr(x) q(x) y(x) = 0 If y 1 and y 2 are eigenfunctions with distinct eigenvalues and appropriate boundary conditions then the solutions are orthogonal in L 2 ((a, b), r(x)dx) Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

18 Sturm-Liouville Theory Our Schrödinger s equation with varied factor ordering is put into Sturm-Liouville form and we read off the eigenvalue problem [ l (1 (i k)) Ψ ] + [ ( ( E λ (i k) 2 (i + k) 2 )) ] 2 lk i 2 l1 (i k) l (1+i k) = 0 4 q(l) = λ ( (i k) 2 (i + k) 2 ) 2 l1 (i k) l (1+i k) 4 p(l) = l 1 (i k) r(l) = l k i κ = E 2 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

19 Boundary Conditions The boundary condition lim b a 0 [N 1 N 2 e 2 λl Take positive branch of µ P 1 P 2l 2γ+1 (i k) N 1 N 2 e 2 λl P 2 P 1l 2γ+1 (i k)] b = 0 a lim a 0 [N 1 N 2 e 2 λl Take negative branch of µ lim a 0 [N 1 N 2 e 2 λl P 1 P 2l 1+µ N 1 N 2 e 2 λl P 2 P 1l 1+µ ] a P 1 P 2l 1 µ N 1 N 2 e 2 λl P 2 P 1l 1 µ] a Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

20 Orthogonality The eigenfunctions given by the positive brach of µ are orthonormal in L 2 ((0, ), l k i dl) The eigenfunctions given by the negative brach of µ are orthonormal in L 2 ((0, ), l k i dl) when µ < 1 It is necessary to impose a boundary condition to get just one set of sturm-liouville eigenfunctions, which shows that a boundary condition is necessary information to characterize space of physically allowed wavefunctions Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

21 Conclusion Applied varied factor ordering to the Hamiltonian Solved Schrödinger s equation through a transformation to a confluent hypergeometric equation S.L. gives us proof of orthogonality of our eigenfunctions Factor ordering matters Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

22 Future work Try to find different transformations that will lead to different eigenfunctions Apply knowledge of factor ordering to 4-D General Relativity Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19

23 Thank you for your time. Questions? Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville May 26, Theory / 19