Research Article
Heisenberg Algebra in the Bargmann-Fock Space with
Natural Cutoffs
Maryam Roushan
1
and Kourosh Nozari
2
1
Department of Physics, Faculty of Basic Sciences, Islamic Azad University, Shahreza Branch, Shahreza, Iran
2
Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
Correspondence should be addressed to Maryam Roushan; m57.roshan@gmail.com
Received 4 September 2013; Revised 9 November 2013; Accepted 9 November 2013; Published 2 February 2014
Academic Editor: Elias C. Vagenas
Copyright © 2014 M. Roushan and K. Nozari. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited. e publication of this article was funded by SCOAP
3
.
We construct a Heisenberg algebra in Bargmann-Fock space in the presence of natural cutos encoded as minimal length, minimal
momentum, and maximal momentum through a generalized uncertainty principle.
1. Introduction: The Generalized Uncertainty
Principle and Fuzzy Spacetime
According to the equivalence principal in general relativity,
gravitational eld is coupled to everything. is means that
photons in Heisenberg gedankenexperiment are actually
coupled with electrons gravitationally and this leads to
modication of the standard uncertainty principle. It has
been characterized that gravity in very small length scales
causesseriouschangeinthestructureofspacetime.Itcauses
minimal uncertainty in positions of atomic and subatomic
particles [1–15]. In fact, there is absolutely smallest uncer-
tainty in position measurement of any quantum mechanical
system and this feature leads nontrivially to the existence
of a minimal measurable length in the order of Planck
length. Existence of this natural cuto requires deformation
of the standard Heisenberg uncertainty principle to the
so-called generalized uncertainty principle (GUP) (see, for
instance, [13, 14, 16–20]). In one dimension of position and
momentum operators, the deformed Heisenberg algebra can
be represented as
[
,
]
=1+
2
. (1)
In general, for two symmetric operators and ,wehave
≥
|[
,
]|
.
(2)
So the generalized uncertainty principle can be deduced
as
≥
2
1+
(
)
2
.
(3)
While in ordinary quantum mechanics canbemade
arbitrarily small by letting grows correspondingly, this
is no longer the case if (3) holds. If for decreasing ,
increases, the new term ()
2
on the right hand side of (3)
will eventually grow faster than the le hand side. Hence
can no longer be made arbitrarily small [16, 18]. To obtain this
minimal uncertainty, we saturate inequality in (3)andsolve
the resulting equation for ,
=
±
(
)
2
−
2
.
(4)
e reality of solutions requires positivity of the term in
square root, leading to
(
)
0
=
.
(5)
is being the smallest uncertainty in position mea-
surement leads nontrivially to the existence of a minimal
measurable length. In fact, a key characteristic of quantum
theory is the emergence of uncertainties, and one might
expect that the distance observable would also be aected by
Hindawi Publishing Corporation
Advances in High Energy Physics
Volume 2014, Article ID 353192, 6 pages
http://dx.doi.org/10.1155/2014/353192
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