Download The propagators of the free particle and particle in a box in polymer

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 )
.
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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 ).
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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.