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Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra



A. Farmany
 
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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 position-angular momentum Heisenberg uncertainty principle.

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  How to cite this article:

A. Farmany , 2011. Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra. Journal of Applied Sciences, 11: 411-413.

DOI: 10.3923/jas.2011.411.413

URL: https://scialert.net/abstract/?doi=jas.2011.411.413
 
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 energy-momentum 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):

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(1)

where Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra and gμv is the space-time metric.

The canonical momenta are obtained from the Hamiltonian analysis as:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(2)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(3)

where, four primary constraints are:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(4)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(5)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(6)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(7)

Note that ∏μVμ = 0 and λ is the angle variables. We consider the second class constraint set (Vμ, χv) (Ghosh, 1994). The inverse of Poisson bracket is:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(8)

where Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra and Tμv is the torsion term.

In the coordinates system the generic Dirac bracket is defined by:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(9)

From Eq. 9 we can write (Farmany, 2005),

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(10)

where, Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra and Sμv is an anti-symmetric 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 Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra and Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra 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:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(11)

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 (Franke-Arnold et al., 2004). But we should consider the effect of the non-commutativity on the localization process. Let us reconsider it. In this framework, there is a particle in a non-commutative space whose coordinates satisfies the Heisenberg algebra.

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(12a)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(12b)

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(12c)

From Eq. 12a-c, we can set up the Jacobi identity as:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(13)

As a result of Eq. 13, an important consequence of the non-commutative coordinates is that neither the position operator does satisfy the usual low:

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(14)

and nor the angular momentum operator satisfy the standard Eq. 3 algebra

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(15)

In fact we can write,

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(16)

for position operator and

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(17)

for angular momentum operator, respectively. Comparing Eq. 14 with Eq. 16 we find a new term as Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra in Eq. 16. From Eq. 16 we have,

Image for - Zitterbewegung Anyons and Deformed Position-angular Momentum Uncertainty Principle Algebra
(18)

that shows a deformed algebra of position-angular 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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