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Abstract

Consider bounded solenoids in the space or planar approximate models of infinitely long solenoids. It is proved that the solenoid impenetrability, by means of a sequence of Hamiltonians with diverging potentials, converges in the norm resolvent sense to the usual Aharonov–Bohm model with Dirichlet boundary condition. The framework is that of nonrelativistic quantum mechanics. Mathematics Subject Classification (2000)82D99–47B25–81Q10
Impenetrability of Aharonov-Bohm solenoids: proof
of norm resolvent convergence
esar R. de Oliveira
Departamento de Matem´atica – UFSCar, S˜ao Carlos, SP, 13560-970 Brazil
Marciano Pereira
Departamento de Matem´atica e Estat´ıstica – UEPG,
Ponta Grossa, PR, 84030-000 Brazil
October 25, 2010
Abstract
Consider bounded solenoids in the space or planar approximate
models of infinitely long solenoids. It is proved that the solenoid impen-
etrability, by means of a sequence of Hamiltonians with diverging po-
tentials, converges in the norm resolvent sense to the usual Aharonov-
Bohm model with Dirichlet boundary condition. The framework is
that of nonrelativistic quantum mechanics.
Keywords: Aharonov-Bohm Hamiltonian, norm resolvent convergence, bounded
solenoids.
MSC: 82D99; 47B25; 81Q10.
1 Introduction
The Aharonov-Bohm (AB) effect is an authentic quantum phenomenon and
was first considered in [19, 18, 1]. There are many discussions about the
justification of the famous Aharonov-Bohm Hamiltonian and interpreta-
tions (see, for instance, [2, 7, 8, 9, 11, 15, 20, 22, 26, 28, 30] and references
therein); the questions are particularly interesting for the more realistic case
of solenoids Sof radii greater than zero. Sometimes it involves the quan-
tization in multiply connected regions, and the main points to be clarified
Corresponding author. Email: oliveira@pq.cnpq.br
1
are the presence of the vector potential Ain the operator action (occasion-
ally in regions with no magnetic field), and the (natural) choice of Dirichlet
boundary conditions at the solenoid border.
Apparently, Kretzschmar [24] was the first one to propose a model of
the solenoid impenetrability via increasing positive potentials Vnso that
Vn→ ∞ precisely on the interior Sof the solenoid (see also comments
in [7]), but a mathematical rigorous approach was only published many
years later by Magni and Valz-Gris [26], whose main result was a proof a
strong resolvent convergence.
In [13], the present authors have considered an increasing sequence of
cylindrical solenoids of finite length together with the same shielding process
of [26], and it was proved that both limits commute and converge, in the
strong resolvent sense, to the usual AB Hamiltonian HAB given by
dom HAB =H2(S0)∩ H1
0(S0),
HAB =pq
cA2
,p=i,
where S0denotes the exterior region of the solenoid. Note that the Dirich-
let boundary conditions are selected (due to the space H1
0(S0) in the limit
operator domain). cand qare the light speed and particle electric charge,
respectively. Recently [15], other boundary conditions at the cylindrical
solenoid that are compatible with quantum mechanics, in the sense that
they give rise to self-adjoint Hamiltonians, were investigated, and the scat-
tering of the AB Hamiltonian with Robin boundary conditions has been
considered and compared with the traditional Dirichlet case.
The goal of this note is to prove that, in some cases of interest, one has the
norm resolvent convergence instead of just the strong one. This is important
since spectral and eigenfunctions convergences are usually guaranteed only if
norm resolvent convergence holds. The important assumption, necessary to
get such norm convergence, is that the region border S=Sis a bounded
set, since some arguments involving compactness will be evoked. Hence, the
results obtained below apply, in particular, to any finite cylindrical solenoid
as well as to toroidal solenoids in R3[27], and, due to the symmetry along the
vertical coordinate, to approximate models in R2of infinitely long cylindrical
solenoids.
The toroidal case is particularly important, since for the physics com-
munity this setting is considered to present the only trustful experimental
evidence of the AB effect [27, 31, 32, 10]; in case of straight solenoids there
is a leakage of magnetic flux and the experiments are not generally accepted.
Note that the recent experiments reported in [10] show that the experimen-
tal results of [31, 32], with toroidal solenoids, can not be explained by the
action of a force. It is also worth mention the rigorous proofs [4, 5, 6] that
the original Ansatz of Aharonov and Bohm [1], adapted to the toroidal case,
can be approximated by real solutions of the Schr¨odinger equation. Hence,
2
it is the three-dimensional case with some bounded solenoids the physically
important one, and this setting is included in Theorem 1 below.
In the next section the results of this work are precisely stated, whereas
their proofs appear in Section 3. Concluding remarks are discussed in the
last section of this note.
2 Statement of the Results
If λCand Tis a linear operator on a Hilbert space, Rλ(T) = (Tλ1)1
denotes the resolvent operator of Tat λ. Let Sdenote a (nonempty)
bounded open subset of Rd,d= 2,3, Sits boundary, S0its exterior, that is,
S0=Rd\(S ∪ S), and its closure will be indicated by ˜
S=S∪ S, whereas
the closure of its exterior by ˆ
S=S0∪ S.
Let Bdenote a (bounded) magnetic field on Rdand Aa (bounded)
magnetic potential so that B=∇ × A; furthermore, assume that divA
L2
loc, and so Hnbelow is essentially self-adjoint when defined on C
0(Rd) [25].
In R2it is supposed that the only possible nonzero component of Bis the
one perpendicular to the plane, so that, in fact, the magnetic field is reduced
to a scalar function in this case.
As already indicated, think of Sas a finite solenoid of general shape,
and the most interesting situation here is when Bvanishes on S0(but not
necessarily on S), but the results below are not restricted to this setting.
Particularly relevant situations are: (1) Sis a torus in R3, and (2) Sis a
circle in R2.
In order to prevent a quantum particle to penetrate the interior re-
gion S, as in the Aharonov-Bohm effect, consider an increasing sequence
of potential barriers Vn(x) = (x), where χ=χSis the characteristic
function of this set, that is, χ(x) = 1 if x∈ S and χ(x) = 0 otherwise. The
question is: What does happen with the particle Hamiltonian as n→ ∞?
This is a way to model the impenetrability of the region S[24, 26], and so to
find the quantum Hamiltonian describing the particle motion in a magnetic
field and in Rdwith a hole S.
The Hamiltonian of a quantum particle in this situation is given by the
self-adjoint operator, for n0 and d= 2,3,
dom Hn=H2(Rd), Hn=pq
cA2
+Vn=H0+Vn,
where H2denotes a usual Sobolev space in L2(Rd), i.e., the domain of the
free Hamiltonian ∆ (the negative laplacian). Note that the operator H0is
implicitly defined by this relation. Recall that, under the above hypotheses
on A, C
0(Rd) is a core of Hn[25].
The impenetrable limit n→ ∞ was rigorously considered in [26] for a
particular region, however the same arguments hold in a much more general
3
situation. By using the Kato-Robinson theorem [12, 14], it was shown that
Hnconverges to HAB in the strong resolvent sense as n→ ∞, and since
elements of H1
0(S0) vanish at the solenoid border (in the sense of Sobolev
traces), Dirichlet boundary conditions have showed up in this limit. Note,
however, that Hnand HAB act in different Hilbert spaces; since Vn→ ∞
in S, one restricts the resolvent convergence to the subspace of functions
that vanish a.e. in S, and if P0denotes the projection onto the subspace
L2(S0) of L2(Rd), the resolvent convergence will be understood in the sense
Rλ(Hn)P0Rλ(HAB)P0, for λ > 0 (see [12, 26, 29] for details); but in
order to simplify the notation, here it will be simply written Rλ(Hn)
Rλ(HAB). This construction works since Hnis a monotonically increasing
sequence of self-adjoint operators, and so the strong resolvent convergent is
directly related to the convergence of a monotonic family of quadratic forms
[12, 29]. A discussion on the nontrivial relation between resolvents and
quadratic forms for general bounded from below sequences of self-adjoint
operators, with some applications to singular quantum limits, can be found
in [16].
The main goal of this note is to show that, in case of bounded regions S,
the above strong convergence can be substantially improved, that is, it will
be proven that the norm resolvent convergence is in effect.
Theorem 1. Let Sbe a (nonempty) bounded open subset of Rd,d= 2,3,
and Hnand HAB as above. Then, Hnconverges to HAB in the norm resol-
vent sense as n→ ∞.
3 Proofs
3.1 Preliminary Lemmas
In this subsection two technical results that will be used to prove Theorem 1
will be discussed. Such results are based on references [3, 25]. Without loss
set c= 1 = q.
Let Band Cbe bounded operators on L2(Rd) and denote by B
(Rd)
the set of bounded Borel functions that vanish at infinity (with the sup
norm). Write B˙
Cto indicate that || ≤ C|ψ|for all ψL2(Rd),
that is, ||(x)(C|ψ|) (x). The following results will be used ahead, and
references to their proofs are indicated:
(i) If B˙
Cand Cis a compact operator, then Bis also compact (The-
orem 2.2 in [3]).
(ii) If λ > 0, aL2
loc(Rd)dand the function VL1
loc(Rd) is nonnegative,
then
(i∇ − a)2+V+λ1˙
(∆ + λ)1
(Lemma 6 in [25]).
4
(iii) If f, g B
(Rd), the operator f(x)g(p) on L2(Rd) is compact (Sec-
tion 11.4.1 in [14]).
Lemma 1. If fB
(Rd),aL2
loc(Rd)dand VL1
loc(Rd)is nonnegative,
then f(x)Rλ(i∇ − a)2+V(x)is compact, for all λ > 0.
Proof. Let λ > 0 and fB
(Rd). By (ii) above,
Rλ(i∇ − a)2+V˙
Rλ(∆),
that is,
Rλ(i∇ − a)2+Vψ
Rλ(∆)|ψ|,ψL2(Rd).
Hence
f(x)Rλ(i∇ − a)2+Vψ
≤ |f(x)|Rλ(∆)|ψ|,ψL2(Rd),
that is,
f(x)Rλ(i∇ − a)2+V˙
≤ |f(x)|Rλ(∆).
Since the operator on the right hand side is compact by (iii), the lemma
follows by (i).
Lemma 2. Let λ,f,aand Vbe as in Lemma 1. Let H(a)be the strong
resolvent limit of the sequence of operators (i∇ − a)2+nV as n→ ∞.
Then fRλ(H(a)) is a compact operator.
Proof. Note that such strong resolvent limit H(a) is supposed to exist.
By (ii), for all none has
Rλ(i∇ − a)2+nV ˙
Rλ(∆),
that is,
Rλ(i∇ − a)2+nV ψ
Rλ(∆)|ψ|,ψL2(Rd).
Take n→ ∞, and since the strong resolvent limit exists in L2(Rd), for each
ψthere exists a subsequence that converges a.e., so that
|Rλ(H(a)) ψ| ≤ Rλ(∆)|ψ|,ψL2(Rd).
Hence,
|f(x)Rλ(H(a)) ψ|≤|f(x)|Rλ(∆)|ψ|,ψL2(Rd),
and so
f(x)Rλ(H(a)) ˙
≤ |f(x)|Rλ(∆).
The lemma follows by applying first (iii) and then (i).
5
The next simple remarks will be used several times in the arguments
of Subsection 3.2. If a bounded operator Ton a Hilbert space is positive
T0, then Tis self-adjoint and
kTk= sup
kξk=1
hT ξ, ξi= sup
kξk=1
hT1/2ξ, T 1/2ξi
= sup
kξk=1
kT1/2ξk2=kT1/2k2.
And if S, P are bounded operators, then kS P k2=kPSSP k.
3.2 Proof of Theorem 1
Denote Rn:= R1(Hn), n0, and RAB := R1(HAB), and recall that Rn
converges strongly to RAB [26].
Pick an open subset O ⊂ Sso that dist (O,S)>0, and another open
subset U ⊃ ˜
S\O ⊃ S whose closure Uis compact. Choose a nonnegative
(real-valued) function ϕC
0(Rd) with ϕ= 1 on U.
One may write
RnRAB =χO(RnRAB) + χS\O ϕ(RnRAB )
+χˆ
Sϕ(RnRAB) + χˆ
S(1 ϕ) (RnRAB),
and the proof that follows consists in showing that each term on the right
hand side vanish in the norm of the space of bounded operators. This is the
motivation for the calculations below.
Lemma 3. Let ϕalso denote the multiplication operator by the function ϕ.
Then
0ϕ(RnRAB)ϕϕ(R0RAB )ϕ,
and the operators ϕ(RnRAB)ϕare compact, for all n0. Furthermore,
ϕ(RnRAB)ϕconverges strongly to zero as n→ ∞.
Proof. It is known that RnRAB 0; see the proof of Theorem 10.4.2
in [14]. Thus, by Proposition 9.3.1 in [14], one has
hψ, ϕ (RnRAB )ϕψi=hϕψ, (RnRAB)ϕψi
=D(RnRAB)1/2ϕψ, (RnRAB)1/2ϕψE
=
(RnRAB)1/2ϕψ
0,ψL2(Rd).
To conclude the second inequality in the statement of the lemma, con-
sider the quadratic forms bnassociated with the operators Hn,n0, and
note that b0bn, for all n0. Hence, by Lemma 10.4.4 in [14], one has
RnR0. Thus,
ψ, [ϕ(R0RAB )ϕϕ(RnRAB)ϕ]ψ
=ψ, ϕ (R0Rn)ϕψ=
(R0Rn)1/2ϕψ
0,
6
for all ψL2(Rd).
Now apply Lemma 1 with f=ϕand R1(i∇ − ~
a)2+V=Rn, to
conclude that ϕRnis compact and, so, that ϕRnϕis compact since it is the
composition of a compact operator with a bounded one. Similarly, apply
Lemma 2 with f=ϕand R1(H(~
a)) = RAB to conclude that both ϕRAB
and ϕRABϕare also compact. Since Hnconverges to HAB in the strong
resolvent sense as n→ ∞, for ψL2(Rd) one has
kϕ(RnRAB)ϕψk≤kϕk k(RnRAB) (ϕψ)k → 0,
as n→ ∞, since ϕis a bounded function. Therefore, the sequence of
operators ϕ(RnRAB )ϕconverges strongly to zero as n→ ∞.
The above lemma, combined with Theorems VIII-3.3 and VIII-3.5 in [23],
ensures the norm convergence
kϕ(RnRAB)ϕk → 0, n → ∞,
and so
(RnRAB)1/2ϕ
2
=kϕ(RnRAB)ϕk → 0,
as n→ ∞. Hence, one also gets the following norm convergence
k(RnRAB)ϕk=
(RnRAB)1/2(RnRAB )1/2ϕ
(RnRAB)1/2
(RnRAB)1/2ϕ
(R0RAB)1/2
(RnRAB)1/2ϕ
0,
(1)
as n→ ∞.
By following the proof of Lemma 2.3 in [21], one finds that there exists
a constant κ1 so that
kRnχOk ≤ κ n1/2,n1.
Indeed, it is possible to find nonnegative functions η1, η2C(Rd) so that
η2
1+η2
2= 1, with η1= 1 on Oand supp η1⊂ S . Furthermore, it is possible
to assume that |∇η1|and |∇η2|are bounded functions. Thus,
Hn=η1Hnη1+η2Hnη2− |∇η1|2− |∇η2|2.
Since
hψ, Hnψi=hψ, H0ψi+hψ, nχSψi
=hH1/2
0ψ, H 1/2
0ψi+nZS
|ψ|2nkψk2,ψC
0(S),
7
one has
hψ, η1Hnη1ψi=hη1ψ, Hnη1ψi ≥ nhη1ψ, η1ψi
=nhψ, η2
1ψi ≥ nhψ, χOψi,ψC
0(Rd),
that is,
η1Hnη12
1O,
and one may find a constant κso that HnOκ1, that is, Hn+κ1
O; without loss, one may assume that κ1.
Write Qn:= Rκ(Hn) = (Hn+κ1)1and S=Q1/2
nχO. It then follows
that
hψ, Q1/2
nχOQ1/2
nψi=DQ1/2
nψ, χOQ1/2
nψE
n1DQ1/2
nψ, (Hn+κ1)Q1/2
nψE
=n1Dψ, Q1/2
n(Hn+κ1)Q1/2
nψE=n1hψ, ψi,
for all ψC
0(Rd), that is,
(Hn+κ1)1/2χO(Hn+κ1)1/2n1.
Note that Sis bounded, S=χOQ1/2
nand
SS=Q1/2
nχOχOQ1/2
n=Q1/2
nχOQ1/2
n
is self-adjoint and bounded. Hence,
kQ1/2
nχOk2=kSk2=kSSk=kQ1/2
nχOQ1/2
nk ≤ n1,
and so kQ1/2
nχOk ≤ n1/2. Therefore,
kQnχOk=kQ1/2
nQ1/2
nχOk≤kQ1/2
nkkQ1/2
nχOk
≤ kQ1/2
nχOk ≤ n1/2,
with kQ1/2
nk ≤ 1, since κ1. Thus,
(Hn+κ1)1χO
(Hn+κ1)1/2χO
n1/2.
By the first resolvent identity, one has Rn=Qn+ (κ1)RnQn,so that
RnχO=QnχO+ (κ1)RnQnχO,
and since kRnk ≤ 1,
kRnχOk ≤ kQnχOk+ (κ1)kRnkkQnχOk
n1/2+ (κ1)n1/2=κ n1/2,
8
that is, kRnχOk ≤ κ n1/2. This inequality implies that
(2) k(RnRAB)χOk=kRnχOk → 0,
as n→ ∞, since RABχO0.
Recall that ˆ
S=S ∪ S0and write ρ= (1 ϕ)χˆ
S; then one has ρC.
Compute
(H0+1) [ρ(R0RAB)] = (ρ) (R0RAB )2 (ρ)· ∇ (R0RAB )
+ρ(H0+1)R0ρ(H0+1)RAB
+ 2i(A· ∇ρ) (R0RAB),
but since (H0+1)R0=1and, due to the presence of the projection opera-
tor P0(see just before Theorem 1), (H0+1)RAB (HAB +1)RAB =1, it
then follows that
(H0+1) [ρ(R0RAB)] = (ρ) (R0RAB )2 (ρ)· ∇ (R0RAB )
+ 2i(A· ∇ρ) (R0RAB),
and, since ρ= (1 ϕ)χˆ
S, one has
(H0+1) [ρ(R0RAB)] = (∆ϕ)χˆ
S(R0RAB)
+ 2 (ϕ)χˆ
S· ∇ (R0RAB )
2i(A· ∇ϕ)χˆ
S(R0RAB),
so that, after applying R0on the left, and using Lemmas 1 and 2 again, it
is found that the operator
ρ(R0RAB) = R0χˆ
S(∆ϕ) (R0RAB)
+ 2R0χˆ
S(ϕ)· ∇ (R0RAB )
2iR0χˆ
S(A· ∇ϕ) (R0RAB)
is compact.
The same arguments above imply that ρ(RnRAB) is also compact,
and the inequality ρ(RnRAB)ρρ(R0RAB)ρimplies that
(3)
(RnRAB) (1 ϕ)χˆ
S
0, n → ∞.
Now, one can write
RnRAB =χO(RnRAB) + χS\O ϕ(RnRAB )
+χˆ
Sϕ(RnRAB) + χˆ
S(1 ϕ) (RnRAB),
and by equations (1), (2) and (3), it follows that
kRnRABk → 0,
as n→ ∞, that is, Hnconverges to HAB in the norm resolvent sense. This
finishes the proof of Theorem 1.
9
4 Final Remarks
An important point in the proof of the norm resolvent convergence here
was the compactness of the closure of the region S, since this allowed the
choice of the nonnegative function ϕC
0(Rd) with ϕ= 1 on the compact
set U ⊃ ˜
S. In some sense, such function ϕC
0(Rd) is responsible for the
compactness of the auxiliary operators in the proof of Theorem 1.
The magnetic field may vanish outside ˜
Sor not. Note that Theorem 1
also applies to regions like (assume that a, L,  > 0, and <a)
S=(x, y, z)R3:L < z < L, a <x2+y2< a,
which is a “fat” cylinder of radius aand finite length L. This may be a way
to take into account the difference between the particle running into the
finite cylindrical solenoid at a point with z=±L, from entering through
the lateral border of that cylinder.
Think of an infinitely long cylindrical solenoid in R3(i.e., the standard
Aharonov-Bohm setting). A common situation is to take into account the
symmetry along the vertical coordinate by considering approximations by
two-dimensional models [1, 28]. Note that the norm resolvent convergence
holds for such approximate models in R2, but the proof presented here does
not work for the original model in R3, since in this case the solenoid border
is not included in a compact subset.
Similarly, for fixed intensity n, if one considers a sequence of solenoids
whose lengths go to infinity in R3(e.g., as discussed in [13]), the norm
resolvent convergence is not expected to hold (furthermore, the convergence
of the corresponding sequence of vector potentials is not uniform in general).
In this way, the commutation diagram at the end of reference [13] should
hardly be improved to accommodate norm resolvent for all convergences
therein presented.
Finally, note that the result of Theorem 1 is easily adapted to subsets of
Rd, for any d2; the restriction to d= 2,3, was only due to the physical
interpretations related to the magnetic Aharonov-Bohm effect.
Acknowledgments
The authors acknowledge partial support from CNPq (Brazil).
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13
... In Quantum Mechanics, addressing the singularities arising from the AB magnetic potential is most effectively achieved by imposing a vanishing condition on the eigenfunction at these singularities. Notably, researchers [22] [23] [24] [25], in dealing with the initial Aharonov-Bohm Hamiltonian, employed the natural shielding method and opted for the Dirichlet boundary condition, wherein wave functions vanish at the solenoid. In a recent development, [26] proposed a modification of the AB Hamiltonian that is essentially self-adjoint, signifying ...
... By applying a natural shielding method to the initial AB Hamiltonian [18,20,27,29], the Dirichlet boundary condition (i.e., wavefunctions vanish at the solenoid) is selected. In [21] we have recently proposed a modification of the AB Hamiltonian that is essentially self-adjoint, that is, a model for which there is exactly one selfadjoint extension; the physical interpretation is the absence of contact of the particle with the solenoid in this case. ...
Preprint
We propose a simple situation in which the magnetic Aharonov-Bohm potential influences the values of the deficiency indices of the initial Schr\"odinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov-Bohm effect.
... By applying a natural shielding method to the initial AB Hamiltonian [18,20,27,29], the Dirichlet boundary condition (i.e., wavefunctions vanish at the solenoid) is selected. In [21] we have recently proposed a modification of the AB Hamiltonian that is essentially self-adjoint, that is, a model for which there is exactly one selfadjoint extension; the physical interpretation is the absence of contact of the particle with the solenoid in this case. ...
Article
Full-text available
We propose a simple situation in which the magnetic Aharonov–Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov–Bohm effect.
... The justification of the Dirichlet boundary condition is usually based on a shielding method in the definition of the Hamiltonian which was considered for the first time in Ref. 16. In the mathematical framework, Magni and Valz-Gris 19 proved convergence of this limiting process in the strong resolvent sense; the proof of convergence in the norm resolvent sense was published in Ref. 10. In the other direction, de Oliveira and Pereira 11 have classified all self-adjoint extensions ofḢ κ in the Sobolev space H 2 and then compared the scattering of three different self-adjoint extensions, which are associated with Dirichlet, Neumann, and Robin boundary conditions (the Dirichlet case was considered in Ref. 23 and has motivated other studies). ...
Article
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).
Article
Full-text available
A magnetic potential is included in the so-called (quantum) dynamical confinement to open sets of Rd; gauge transformations are also discussed. Then, the results are applied to the Aharonov–Bohm model in the plane (the solenoid is a disk of radius greater than zero) in order to get some examples of operators confining the electron outside the solenoid.
Book
This Element offers an opinionated and selective introduction to philosophical issues concerning idealizations in physics, including the concept of and reasons for introducing idealization, abstraction, and approximation, possible taxonomy and justification, and application to issues of mathematical Platonism, scientific realism, and scientific understanding.
Article
By appealing to resources found in the scientific understanding literature, I identify in what senses idealizations afford understanding in the context of the (magnetic) Aharonov-Bohm effect. Three types of concepts of understanding are discussed: understanding-what, which has to do with understanding a phenomenon; understanding-with, which has to do with understanding a scientific theory; and understanding-why some phenomenon occurs. Consequently, I outline an account of understanding-with that is suggested by the historical controversy surrounding the Aharonov-Bohm effect.
Article
In this article we argue that idealizations and limiting cases in models play an exploratory role in science. Four senses of exploration are presented: exploration of the structure and representational capacities of theory; proof-of-principle demonstrations; potential explanations; and exploring the suitability of target systems. We illustrate our claims through three case studies, including the Aharonov-Bohm effect, the emergence of anyons and fractional quantum statistics, and the Hubbard model of the Mott phase transitions. We end by reflecting on how our case studies and claims compare to accounts of idealization in the philosophy of science literature such as Michael Weisberg’s three-fold taxonomy.
Article
Full-text available
Two approaches to understanding the idealizations that arise in the Aharonov–Bohm (AB) effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations.
Article
Full-text available
The paper interprets and proves the charge superselection rule within the framework of local relativistic field theory as the statement that the charge operator commutes with all quasilocal observables. Once the basic formalism expressing the property of locality of the observables has been accepted, the proof is an elementary application of Gauss law relating the electric charge in a region to the flux of electric field through the boundary of the region. Most of the paper is devoted to the evidence that the indefinite metric formalism and its accompanying definitions of gauge, gauge transformation, and gauge invariance are internally coherent and consistent with the evidence from free field theory and the renormalized perturbation theory of coupled fields. The paper closes with speculations on analogous explanations of the baryon and lepton superselection rules within the framework of gauge models of strong and weak interactions.
Chapter
The more celebrated of the two effects described by Aharonov and Bohm (1959) is that the behaviour of a quantum charged particle is modified by magnetic flux φ threading a region from which the particle is excluded. They solved Schrödinger’s equation for the scattering of a plane wave of particles by a thin infinite impenetrable cylinder containing a solenoid generating Ф(considered as concentrated onto a single flux line), and showed, as have later analyses for cylinders that are not thin, or for different geometries (e.g. tori) that observable properties do depend on Ф. More precisely, they depend on the quantum flux parameter αqΦ/h \alpha \, \equiv \,q\,\Phi /h (1) where q is the charge, the dependence on α being periodic with period unity. A thorough review of the theory of the Aharonov–Bohm (AB) predictions, and experiments carried out to test them, has been written by Olariu and Popescu (1985).
Book
Groups and semigroups of linear operators, their generalizations and applications
Article
The ΓΓ-convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e. the vanishing of the cross-section diameter) of the Laplace operator with Dinchlet boundary conditions; a procedure to obtain the effective Schrödinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.