
Mini Review


Zitterbewegung Anyons and Deformed Positionangular Momentum Uncertainty Principle Algebra 

A. Farmany



ABSTRACT

In this study, existence of a generalized uncertainty principle in the anyon Zitterbewegung model is explored. It is shown that the localization of anyon with high energy probes, receives a deformed algebra for positionangular momentum Heisenberg uncertainty principle.





Received: August 07, 2010;
Accepted: November 22, 2010;
Published: January 22, 2011


INTRODUCTION
In the particle physics, by using high energy probes, it could be possible
to measure the location of a particle precisely and minimize the localization
uncertainty. Since gravity couples to the energymomentum tensor, the high energy
probe would generate the large fluctuations on the metric. Because the situation
of this probe is quantum mechanically, the metric fluctuations are indeterminable
components. Thus, there exist a limitation to precise the localization; this
is named generalized uncertainty principle. The heuristic derivations of generalized
uncertainty principle are made in the black hole physics, microphysics, quantum
mechanics and other area of physics (Witten, 1996; Yoneya,
2000; Kempf and Managano, 1997; Adler
et al., 2001; Maggiore, 1994; Adler
and Santiago, 1999; Farmany and Dehghani, 2010;
Farmany, 2010; Farmany et al.,
2007, 2008). In this letter we obtain a generalized
uncertainty principle in the Zitterbewegung model of anyons. According to Chou
et al. (1993), constructing a model for anyons as charged particles,
is not economical way and a sensible model of anyons may be constructed as a
model that focuses on anyons as spinning particle. Recently, this model received
a much attention (Chou et al., 1993; Plyushchay,
1990a, b; Jackiw and Nair, 1991,
1994; Wilczek, 1990; Jackiw
and Pi, 1990; Farmany, 2005; Ghosh,
1994; Banerjee et al., 1996; Farmany,
2011). This new model of anyons, exhibits the classical analogue of the
quantum phenomenon Zitterbewegung. Also the model is derived from existing spinning
particle model and retains the essential features of anyon in the non relativistic
regime. In this letter, the existence of a generalized uncertainty principle
in the Zitterbewegung model is explored.
Let we begin with the Lagrangian of an anyon (Ghosh, 1994):
where
and g_{μv} is the spacetime metric.
The canonical momenta are obtained from the Hamiltonian analysis as: where, four primary constraints are:
Note that ∏_{μ}V^{μ} = 0 and λ is the angle
variables. We consider the second class constraint set (V^{μ},
χ^{v}) (Ghosh, 1994). The inverse of Poisson
bracket is:
where
and T_{μv} is the torsion term.
In the coordinates system the generic Dirac bracket is defined by:
From Eq. 9 we can write (Farmany, 2005),
where,
and S^{μv} is an antisymmetric matrix. According to quantum mechanics,
the momentum and the position of a particle could not be measured precisely.
This is due to the fact that operators
and
are not commute. In fact in the quantum mechanics, the act of measurement interferes
with the system and modifies it, since there is a large uncertainty in the final
velocity (or coordinate) of a particle as:
So, an attempt to localize a particle with a minimum uncertainty in Δx,
the momentum uncertainty Δp will be increase. According to Eq.
11 for an anyon to be observed by means of a photon with momentum p, the
usual Heisenberg arguments leads to a position uncertainty given by Eq.
11 (FrankeArnold et al., 2004). But we should
consider the effect of the noncommutativity on the localization process. Let
us reconsider it. In this framework, there is a particle in a noncommutative
space whose coordinates satisfies the Heisenberg algebra.
From Eq. 12ac, we can set up the Jacobi
identity as:
As a result of Eq. 13, an important consequence of the noncommutative coordinates is that neither the position operator does satisfy the usual low: and nor the angular momentum operator satisfy the standard Eq. 3 algebra In fact we can write, for position operator and
for angular momentum operator, respectively. Comparing Eq. 14
with Eq. 16 we find a new term as
in Eq. 16. From Eq. 16 we have,
that shows a deformed algebra of positionangular momentum uncertainty. CONCLUSION It is interesting that generalized uncertainty principle may be derived in the quantum field theory, string theory, black hole physics, quantum mechanics and other area of physics. In this essay, the existence of a generalized uncertainty principle in the anyon Zitterbewegung model is explored.

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