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In the present work, we quantize three Friedmann-Robertson-Walker models in the presence of a negative cosmological constant and radiation. The models differ from each other by the constant curvature of the spatial sections, which may be positive, negative or zero.They give rise to Wheeler-DeWitt equations for the scale factor which have the form of the Schrödinger equation for the quartic anharmonic oscillator. We find their eigenvalues and eigenfunctions by using a method first developed by Chhajlany and Malnev. After that, we use the eigenfunctions in order to construct wave packets for each case and evaluate the time-dependent expected value of the scale factors. We find for all of them that the expected values of the scale factors oscillate between maximum and minimum values. Since the expectation values of the scale factors never vanish, we conclude that these models do not have singularities.

The modeling of the early universe is done through the quantization of a Friedmann-Robertson-Walker model with positive curvature. The material content consists of two fluids: radiation and Chaplygin gas. The quantization of these models is made by following the Wheeler and DeWitt's prescriptions. Using the Schutz formalism, the time notion is recovered and the Wheeler-DeWitt equation transforms into a time dependent Schrödinger equation, which rules the dynamics of the early universe, under the action of an effective potential V ef . Using a finite differences method and the Crank-Nicholson scheme, in a code implemented in the program OCTAVE, we solve the corresponding time dependent Schrödinger equation and obtain the time evolution of a initial wave packet. This wave packet satisfies appropriate boundary conditions. The calculation of the tunneling probabilities shows that the universe may emerge from the Planck era to an inflationary phase. It also shows that, the tunneling probability is a function of the mean energy of the initial wave packet and of two parameters related to the Chaplygin gas. We also show a comparison between these results and those obtained by the WKB approximation.

Abstract. The thermodynamics of the spin-S anisotropic quantum XXZ chain with arbitrary value of S and unitary norm, in the high-temperature regime, is reported. The single-ion anisotropy term and the interaction with an external magnetic field in the z-direction are taken into account. We obtain, for arbitrary value of S, the β-expansion of the Helmholtz free energy of the model up to order β 6 and show that it actually depends on 1 S(S+1). Its classical limit is obtained by simply taking S → ∞. At h = 0 and D = 0, our high temperature expansion of the classical model coincides with Joyce's exact solution [11]. We study, in the high temperature region, some thermodynamic quantities such as the specific heat and the magnetic susceptibility as functions of spin and verify for which values of S those thermodynamic functions behave classically. Their finite temperature behavior is inferred from interpolation of their highand low-temperature behavior, and shown to be in good agreement with numerical results. The finite temperature behavior is shown for higher values of spin.PACS. 05.30.Ch Quantum ensemble theory -75.10.Jm Quantized spin models -75.10.Hk Classical spin models

We consider the XYZ chain model of arbitrary spin S in the high-temperature region, with external magnetic field and single-ion anisotropy term. Our high-temperature expansion of the Helmholtz free energy is analytic in the parameters of the model for S, which may range from 1 / 2 to the classical limit of infinite spin ͑S → ϱ͒. Our expansion is carried out up to order ͑J␤͒ 5 . Our results agree with numerical results of the specific heat per site for S =1/2, obtained by the Bethe ansatz, with h = 0 and D = 0. Finally, we show that the magnetic susceptibility and magnetization of the quantum model can be well approximated by their classical analog in this region of temperature.