
Mini Review


Noncommutative Geometry, Quantum Field Theory and Secondclass Constraints 

A. Farmany



ABSTRACT

We review the noncommutative geometry in the secondclass constraints. We show the noncommutativity in the spacetime coordinates based on quantum field theory and matrix theory compactifications has correspondence to the noncommutativity in the spacetime coordinates based on the secondclass constraints.





Received: July 09, 2010;
Accepted: August 13, 2010;
Published: November 10, 2010


INTRODUCTION
Even in the early days of quantum mechanics and quantum field theory, continuous
spacetime and Lorentz symmetry was considered inappropriate to describe the
smallscale structure of the universe. It was also argued that one should introduce
a fundamental length scale limiting the precision of position measurements.
Snyder was the first to formulate these ideas mathematically. He introduced
noncommutative coordinates. Therefore, a position uncertainty arises naturally.
The success of the renormalization program made people forget about these ideas
for some time. However, when the quantization of gravity was considered thoroughly,
it became clear that the usual concepts of spacetime are inadequate and the
spacetime has to be quantized or noncommutative (Connes,
1994; Witten, 1996). As a result, there is a deep
conceptual difference between quantum field theory and gravity: In the former,
space and time are considered as parameters, in the latter as dynamical entities.
To resolve the problem, quantum mechanics in noncommutative space (NCQM) was
developed. If NCQM is a realistic physics, all the low energy quantum phenomena
should be reformulated in it. In literature, NCQM have been studied in detail
(Chaichian et al., 2001; Gamboa
et al., 2001; Hatzinikitas and Smyrnakis, 2002;
Ho and Kao, 2002; Nair and Polychronakos,
2001; Zhang, 2004; Farmany,
2005, 2010; Farmany et al.,
2009ac). In this letter, we focus on the noncommutative
geometry in the secondclass constraints.
NONCOMMUTATIVE GEOMETRY
According to Dirac's procedure for dealing with constrained systems, Dirac
brackets must replace those constraints, which do not commute with all other
(the secondclass constraints), must be solved explicitly or their Poisson brackets,
but the first and secondclass constraints are mixed up. Convincing ways have
been found to disentangle them in a covariant manner by Ghosh
(1994).
Let us begin with the Lagrangian of an anyon in the background gravity: Where:
and g_{μv} is the spacetime metric. In the nonAbelian gauge interactions the canonical momenta are: In our framework, from the four primary constraints we can write:
where, λ = M/J Using analytical methods, from four primary constraints
we can construct further constraint set (Chou et al.,
1993). We focus on the secondclass constraint set (V^{μ},
χ^{μ}) (Dirac, 1964) The Poisson bracket
matrix of the constraint set (V^{μ}, χ^{μ}) is:
In addition, the inverse Poisson bracket is defined by:
Where:
and T_{μv} is the torsion term and N, M and F are three dimensional matrixes. According to Dirac's procedure for dealing with constrained systems, those constraints that do not commute with all other ones the secondclass constraint must be solved explicitly or their Poisson brackets must be replaced by Dirac brackets. Let x^{μ} and x^{v} be the coordinates. Using Eq. 8 we can write the generic Dirac bracket for coordinates as:
Equation 10 can be simplified to:
where, S^{μv} is a matrix and
have dimension of x^{2}. Equation 11 shows the generic
noncommutativity in the geometry of the spacetime.
CONCLUSION The noncommutative geometry in the secondclass constraints is studied. It is shown that the non commutativity in the spacetime coordinates based on quantum field theory and matrix theory compactifications is correspondence to the noncommutativity in the spacetime coordinates based on the secondclass constraints.

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