Relation Between Schrödinger and Polymer Quantum Mechanics Alejandro Corichi

36 Pages·2007·0.16 MB·English

Relation Between Schr¨odinger
and Polymer Quantum Mechanics
Alejandro Corichi
UNAM
[email protected]
Work in collaboration with: T. Vukaˇsinac and J.A. Zapata
1
MOTIVATION
• What is the relation between the polymer
and Schr¨odinger representations?
• Can we consider a continuum limit?
• What is then the continuum limit?
• What if it exists?
• What if there isn’t any?
• SHO vs LQC, are they similar?
2
PLAN OF THE TALK
1. Motivation
2. From Schr¨odinger to Polymer
3. Polymer Representation
4. Continuum Limit
5. Simple Harmonic Oscillator
6. Free Particle
7. Polymer Quantum Cosmology
3
2. Preliminaries
From loop quantum gravity, we have learned that the only way to
construct diffeomorphism invariant theories is to start with expo-
H
nentiated objects, like holonomies: h (A) = P exp( A).
e
Is there a price? Yes! The quantum theory becomes discontinuous.
This means that for a system with pˆ and qˆ as fundamental coor-
dinates, one of them becomes ill-defined.
2
But if the Hamiltonian is of the form H = p + V (q), we can
not define it on the Kinematical Hilbert space H. Can we do
something about it?
Not directly!
4
But we can attempt the next best thing:
Regularize the operator!
But, how? General strategy: approximate the non-existing oper-
ator by a different (finite) operator that does exist.
In general, there will be a regulator that does not go away. If we
don’t have diffeo invariance (as in LQG a la Thiemann), how can
we get rid of the regulator?
The objective of this talk is to explore this issue. Namely, can we
take this regulator away and thus arrive at the ‘continuum limit’?
Is there a generic result? Case by case?
5
From Schro¨dinger to Polymer
Our starting point will be Γ = R2 with coordinates (q, p) thereon.
Here the starting point is the algebra generated by the exponentiated versions
of qˆ and pˆ that are denoted by,
i(αqˆ)/~ i(β pˆ)/~
U(α) = e ; V (β) = e
The CCR now become
(−iαβ)/~
U(α) · V (β) = e V (β) · U(α) (1)
and the rest of the product is,
U(α ) · U(α ) = U(α + α ) ; V (β ) · V (β ) = V (β + β )
1 2 1 2 1 2 1 2
The Weyl algebra W is generated by taking finite linear combinations of the
generators U(α ) and V (β ). Quantization means finding an unitary represen-
i i
tation of the Weyl algebra W on a Hilbert space H0.
A ‘Fock type’ representation involves a complex structure J, a linear mapping
from Γ to itself such that J2 = −1. In 2 dimensions, all the freedom is in the
6
~
choice of a parameter d with dimensions of length. For k = p/ , we have
2 2
J : (q, k) 7→ (−d k, q/d )
d
This object together with the symplectic structure: Ω((q, p); (q0, p0)) = q p0 −
p q0 define an inner product on Γ: g (· ; ·) = Ω(· ; J ·) such that:
d d
1 d2
0 0 0 0
g ((q, p); (q , p )) = q q + p p
d ~
d2 2
The representation of the Weyl algebra is then, when acting of functions φ(q):
ˆ iαq/~
U(α) · φ(q) := (e φ)(q)
and,
β
ˆ (q−β/2)
V (β) · φ(q) := e 2 φ(q − β)
d
C
One defines an algebraic state (a positive linear functional) ω : W → ,
d
that must coincide with the vacuum expectation value in the Hilbert space:
ω (a) = haˆi , for all a ∈ W. In our case this the specification of J induces
d 0
7
such a unique state ω that yields,
d
2 2 2
1d α 1β
ˆ − ˆ −
hU(α)i = e 4 ~2 and hV (β)i = e 4d2
vac vac
Wave functions belong to the space L2(R, dµ ), where the measure that dictates
d
the inner product in this representation is given by,
1 2
q
−
dµ = √ e 2 dq
d
d
d π
In this representation, the vacuum is given by the identity function φ (q) = 1
0
that is, just as any other plane wave, normalized.
There is an isometric isomorphism K that maps the d-representation in H
d
to the standard Schr¨odinger representation in H by:
schr
2
q
−
e 2
2d
2 R
ψ(q) = K · φ(q) = φ(q) ∈ H = L ( , dq)
schr
d1/2π1/4
All d-representations are unitarily equivalent. Note also that the vacuum now
8
becomes
1 2
q
−
ψ (q) = e 2 ,
2d
0
d1/2π1/4
The 1/d 7→ 0 limit.
The measure behaves as:
1 2 1
q
−
dµ = √ e 2 dq 7→ √ dq
d
d
d π d π
The inner product between two such states is given by
Z
2 2
−iαq iλq −(λ−α) d
hφ , φ i = dµ e ~ e ~ = e 4~2
α λ d d
It is immediate to see that in the 1/d 7→ 0 limit the inner product becomes,
hφ , φ i 7→ δ (2)
α λ d α,λ
9
with δ being Kronecker’s delta. We see then that the plane waves φ (q)
α,λ α
become an orthonormal basis for the new Hilbert space.
ˆ
Let us see the action of the operator V (β) on the basis φ (q):
α
2
β αβ
Vˆ (β) · φ (q) = e− 2−i ~ e(β/d2+iα/~)q
2d
α
which in the limit 1/d 7→ 0 goes to,
αβ
ˆ i
V (β) · φ (q) 7→ e ~ φ (q)
α α
~
that is continuous on β. Thus, in the limit, the operator pˆ = −i ∂ is well
q
defined. Also, note that in this limit the operator pˆ has φ (q) as its eigenstate
α
with eigenvalue given by α:
pˆ · φ (q) 7→ α φ (q)
α α
10