Uncertainty Principle in Loop Quantum Cosmology by Moyal Formalism

Abstract

In this paper we derive the uncertainty principle for the Loop Quantum Cosmology homogeneous and isotropic FLWR model with the holonomy-flux algebra. The uncertainty principle is between the variables $c$, with the meaning of connection and $\mu$ having the meaning of the physical cell volume to the power $2/3$, i.e $v^{2/3}$ or a plaquette area. Since both $\mu$ and $c$ are not operators, but rather the random variables, the Robertson uncertainty principle derivation that works for hermitian operators, can not be used. Instead we use the Wigner-Moyal-Groenewold phase space formalism. The Wigner-Moyal-Groenewold formalism was originally applied to the Heisenberg algebra of the Quantum Mechanics. One can derive from it both the canonical and path integral QM as well as the uncertainty principle. In this paper we apply it to the holonomy-flux algebra in case of the homogeneous and isotropic space. Another result is the expression for the Wigner function on the space of the cylindrical wave functions defined on $R_b$ in $c$ variables rather than in dual space $\mu$ variables.