^{1, 2}

^{3}

^{2}

^{1}

^{2}

^{3}

We consider noncommutative two-dimensional quantum harmonic oscillators and extend them to the case of twisted algebra. We obtained modified raising and lowering operators. Also we study statistical mechanics and thermodynamics and calculated partition function which yields the free energy of the system.

It is believed from quantum gravity theories that space time must change its nature at distances comparable to the Planck scale. It is well known also that Einstein’s theory of gravity, that is, general relativity, cannot be consistently quantized with the rules developed in the framework of quantum field theory. Using perturbation theory around a flat metric, the ultraviolet divergences require infinitely many counterterms. Quantum gravity has an uncertainty principle which prevents one from measuring positions to better accuracies than the Planck length [

The quantum system thus described is much simpler and more rigid than its classical analogue. One thus obtains a nonnegligible payoff for abandoning the commutativity of classical mechanics. Though less intuitive, quantum mechanics is more directly accessible by virtue of its simplicity and its contact with spectroscopy.

In the recent work [

In this work we would like to consider two-dimensional quantum harmonic oscillators with the twisted conformal algebra [

The system of quantum harmonic oscillators [

In the ordinary conformal algebra, we have the following operators:

In order to rewrite the above operators and conformal algebra in noncommutative space we introduce the following noncommutative relation between coordinates [

In that case we have the following extended expressions:

Noncommutative version of the Hamiltonian (

Effect of lowering operators on the ground states is similar to the ordinary noncommutative version;

These will be useful in the context of factorization method [

In this section we briefly study statistical mechanics and thermodynamics of the noncommutative quantum harmonic oscillators in two dimensions with twisted relation (

By using the results of [

By using the partition function one can obtain Helmholtz free energy as follows:

(a) Free energy in terms of

In this work we considered a system of quantum harmonic oscillators in noncommutative space with twisted conformal transformation and studied statistical mechanics and thermodynamics. We generalized recent work [

We found that higher order noncommutative parameters modified ladder operators and the Hamiltonian. Then, we obtained partition function, entropy, and Helmholtz free energy. In that case one can obtain other thermodynamic quantities such as entropy. We found that the free energy is an increasing function of noncommutative parameter while the entropy is a decreasing function of noncommutative parameter. Also we found that the effect of noncommutativity is at finite temperature.

The authors declare that there is no conflict of interests regarding the publication of this paper.