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MOT QM PQM FPP PPB SP The propagators of the free particle and particle in a box in polymer quantum mechanics. Ernesto Flores González Departamento de Fı́sica, Universidad Autónoma Metropolitana-Iztapalapa, Adviser: Hugo A. Morales Técotl Collaboration: Carlos M. Reyes and Juan D. Reyes. Mexi-Lazos 2012 November 9th MOT QM PQM FPP PPB SP Motivation General: Polymer quantum mechanics is a “toy model”which gives a friendly introduction to technical and conceptual issues of Loop Quantum Gravity by simple mechanics systems. 1 Polymer → Schrödinger: ψpoly → ψsch and Epoly → Esch . A. Ashtekar, S. Fairhurst and J. L. Willis, Classical Quantum Gravity 20 (2003). A. Corichi, T. Vukasinac and J. A. Zapata, Phys. Rev.D76, 044016 (2007). 2 Free Scalar Field Propagator: G.M. Hossain, V. Husain, and S.S. Seahra, Phys. Rev.D82, 124032 (2010). MOT QM PQM FPP PPB SP Schrödinger Representation Canonical Quantization: {q, p} = 1 7→ [b q , pb] = i~b 1. Dynamics Kinematics 2 q̂, p̂ 7→ Hsch = L (R, dµ) b p̂) |ψi = E |ψi 7→ Hsch H(q̂, Weyl Algebra U(λ)V (µ) = e −iλµ V (λ)U(µ), U(λ1 )U(λ2 ) = U(λ1 + λ2 ), V (µ1 )V (µ2 ) = V (µ1 + µ2 ). Stone-Von Neumann Theorem: b If U(λ) and Vb (µ) are weakly continuous in λ and µ then the representation is unitarily equivalent to the Schrödinger representation and ∃ q̂ and p̂ self-adjoint such that b U(λ) = e iλx̂ , i Vb (µ) = e ~ p̂µ . MOT QM PQM FPP PPB SP Kinematics (PQM). xj → |xj i , hxi |xj i = δi,j , |ψi = n X ai |xi i , i=1 E .V . = |ψi = n X j=1 aj |xj i , ∀ γ = {xj } ⊂ R ψ(x) = N X j=1 aj δxj (x), MOT QM PQM FPP PPB SP Polymer Hilbert Space Hpoly = ∞ X j=−∞ aj δxj (x) : ∞ X j=−∞ |aj |2 < ∞ , Hpoly = L2 (R, dµc ). Representation of the Weyl Algebra. Û(λ) |xj i = e iλxj |xj i & V̂ (µ) |xj i = |xj − µi ; but in this representation Vb is not weakly continuous in µ: b lı́m hxi | V̂ (µ) |xj i = 6 hxi | V̂ (0) |xj i ⇒ @ p̂, U(λ) O.K. ⇒ ∃ x̂. µ7→0 POLYMER REPRESENTATION IS UNITARILY INEQUIVALENT TO THE SCHRÖDINGER ONE. MOT QM PQM FPP PPB SP Dynamics (PQM) How is Hb introduced in Hpoly if @ p̂? p2 ∼ i −i i ~2 h 2 − e ~ µ0 p − e ~ µ0 p 2 µ0 7→ i ~2 h pb2 = 2 2 − Vb (µ0 ) − Vb (−µ0 ) . µ0 The eigenvalue equation of the Hamiltonian becomes a difference equation: 2mµ20 (j−1) + ψ = 2 − ψx(j+1) (E − V(x + jµ )) ψx(j) . 0 0 x0 0 0 ~2 The free particle A Corichi et al i Eµ0 ,n ψp0 (xj ) = e − ~ xj p , ~2 = mµ 1 − cos µ~0 p . 2 0 The particle in a box G Chacón et al q ψn0 (j) = N2 sin nπj , N 2 ~ 1 − cos nπ . Eµ0 ,n = mµ 2 N 0 MOT QM PQM FPP PPB SP Free particle propagator (PQM). Propagator (SAKURAI) i b k(xj , t; xr , t0 ) := hxj | exp − Hpoly (t − t0 ) |xr i , ~ xj − xr ~(t − t0 ) = i l Jl (z) e −iz , z = , l =j −r = . µ0 mµ20 Properties ψp (xj , t) = ∞ X i k(xj , t; xr , t0 )ψp (xr , t0 ) = exp − E (t − t0 ) ψp (xj , t0 ). ~ r =−∞ “Composition”:k(xs , t2 ; xp , t0 ) = lı́mt−→t0 k(xj , t; xr , t0 ) = i r −j J P∞ j=∞ r −j (0) k(xs , t2 ; xj , t1 )k(xj , t1 ; xp , t0 ). o.k. = δxj ,xr . o.k. ∂ bpoly . K (xs , t2 ; xp , t0 ) is Green’s function of the operator: i~ ∂t −H MOT QM PQM FPP PPB SP , , Continuum approximation: k(xj , t; xr , t0 )poly −→ K (x, t; x , t )Sch 1 We have z = λ 2πµ0 l, if λ ∼ 10−10 m, µ0 ∼ lp ∼ 10−35 m → z l. Therefore Jl (z) is the dominant factor in k(xi , xj ; t, t0 )poly , moreover we can use its asymptotic expansion. k(xi ,xj ;t,t0 )poly µ0 2 µ0 dµc → dµSchr ⇒ 3 But it is insufficient to take only the limit µ0 → 0, we need also to consider := k(xj , t; xr , t0 )Shr . the limit l → ∞, keeping xj − xr = µ0 l fixed. This yields k(xj , t; xr , t0 ) = l→∞ µ0 →0 µ0 ⇒ lı́m lı́m r m exp 2iπ~(t − t0 ) im(xj − xr )2 2~(t − t0 ) . MOT QM PQM FPP PPB SP The method of images. In agreement with the method of images we obtain kci (xj , t; xr + 2kN, t0 ) = The potential for one point charge between two infinite plane conductors can be written in the form Φ = −q ∞ X n=−∞ kpi (xj , t; xr + 2kN, t0 ) −kpi (xj , t; −xr + 2kN, t0 ), ∞ n X −iz = e i j−r −2kN Jj−r −2kN (z) k=−∞ {Gf (~ r , [2nd + y ]x̂) − Gf (~ r , [2nd + (−y )]x̂)} o − i j+r −2kN Jj+r −2kN (z) , MOT QM PQM FPP PPB SP Relationship between kci (xj , t; xr + 2kN, t0 ) and kce (xj , t; xr , t0 ). N−1 2 X nπj nπr −iz [1−cos( nπ N )] . = sin sin e (1) N n=1 N N P∞ If we introduce the Jacobi-Anger expansion e iz cos θ = k=−∞ i k Jk (Z )e ikθ in (1) it turns kce (xj , t; xr , t0 ) out kce (xj , t; xr , t0 ) = N−1 ∞ e −iz X X k nπ(j − r − k) nπ(j + r − k) i Jk (z) cos − cos . N n=1 N N k=−∞ We can verify N−1 X n=1 cos nπp N = −1 + ∞ X m−∞ Nδp,2mN 1 p + [1 − (−1) ]δp,m , 2 and then obtain kce (xj , t; xr , t0 ) = kci (xj , t; xr + 2kN, t0 ). MOT QM PQM FPP PPB SP Summary and perspectives 1 We have calculated the propagators of the free particle and the particle in a box in the polymer representation and we have checked some consistency properties they satisfy. 2 We have put in contact both propagators with their continuum analogs. 3 Future: To do the analysis of the continuum limit of the propagators through the Wilson’s renormalization group: A. Corichi, T. Vukasinac and J. A. Zapata, Classical Quantum Gravity 24 (2007). 4 Future: To calculate the propagator for the simple harmonic oscillator.