We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.

1.
W.
Thirring
,
Quantum Mathematical Physics, Atoms, Molecules and Large Systems
(
Springer
,
New York
,
2001
), pp.
85
88
.
2.
M. L.
Boas
,
Mathematical Methods in the Physical Sciences
(
Wiley
,
New York
,
1966
), pp.
260
264
.
3.

This circumstance occurs also in classical electrodynamics. Outside an infinite solenoid with slowly varying current the electric field E is irrotational, but its circulation around the solenoid does not vanish: a potential ϕ(r) cannot be defined in all the space outside the solenoid.

4.

Any single-valued realization of r0rf(x)dx is necessarily discontinuous and the resulting U operator does not implement the required transformation pipi+fi(x).

5.
Y.
Aharonov
and
D.
Bohm
, “
Significance of the electromagnetic potentials in the quantum theory
,”
Phys. Rev.
115
,
485
491
(
1959
).
6.
D. J.
Griffiths
,
Introduction to Quantum Mechanics
(
Prentice Hall
,
Englewood Cliffs, NJ
,
1995
), pp.
343
349
.
7.
Several hundred papers have been written on the Aharonov-Bohm effect. An excellent review is
M.
Peshkin
and
A.
Tonomura
,
The Aharonov-Bohm Effect
(
Springer
,
Berlin
,
1989
).
8.
A similar interpretation was put forward by
E. A.
Carlen
and
M. I.
Loffredo
, “
The correspondence between stochastic mechanics and quantum mechanics on multiply connected configuration spaces
,”
Phys. Lett. A
141
,
9
13
(
1989
).
9.
A similar situation occurs for a particle confined to a segment. There the momentum operator has several nonequivalent self-adjoint extensions. See, for instance,
P.
Garbaczewski
and
W.
Karwowski
, “
Impenetrable barriers and canonical quantization
,”
Am. J. Phys.
72
,
924
932
(
2004
).
10.
L.
Bracci
and
L. E.
Picasso
, “
On the Weyl algebras for systems with semi-bounded and bounded configuration space
,” preprint IFUP-TH/2006-4, to be published in
J. Math. Phys.
11.
W.
Thirring
,
Quantum Mathematical Physics, Atoms, Molecules and Large Systems
(
Springer
,
New York
,
2001
), p.
77
.
12.
Deficiency indices are a pair of non-negative integers associated with a Hermitian operator that determine whether the operator is self-adjoint (vanishing deficiency indices), admits of self-adjoint extensions (nonvanishing but equal indices), or does not admit of self-adjoint extensions. A very readable account of their role is given in
G.
Bonneau
,
J.
Faraut
, and
G.
Valent
, “
Self-adjoint extensions of operators and the teaching of quantum mechanics
,”
Am. J. Phys.
69
,
322
331
(
2001
).
13.
M.
Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness
(
Academic
,
New York
,
1975
), pp.
144
145
.
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