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Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscilator. A parallel is traced with gravity for stressing the relevance of such discrete interaction models. Fundamental interactions, according to quantum field theory, are realized through the exchange of interaction quanta-packets of matter-energy with defined quantum numbers, viz. momentum-energy, spin, electric charge, etc. Thei are discrete interactions, in contradistinction to the classical continuous picture. The Bohr's correspondence principle, a useful guideline in the early days of quantum mechanics, states that in the limit of very large quantum numbers the classical idea of continuity must result from the quantum discreteness as an effective description. It would be very interesting to see in a clear way how this transition discrete-to-continuous occurs. This is the objective of the present letter with the use of a simple model of discrete classical interaction for studying this transition in the classical simple harmonic oscillator. We should not forget, however, that the harmonic potential, although being an extremely useful tool in all branches of modern physics, is not itself a fundamental interaction, which, as well known, are just the gravitational, the electromagnetic, the weak and the strong interactions; actually it is just an effective description. This may just valorize the importance of studying how it can be understood as an effective * PIVIC-UFES
Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscilator. A parallel is traced with gravity for stressing the relevance of such discrete interaction models. Fundamental interactions, according to quantum field theory, are realized through the exchange of interaction quanta-packets of matter-energy with defined quantum numbers, viz. momentum-energy, spin, electric charge, etc. Thei are discrete interactions, in contradistinction to the classical continuous picture. The Bohr's correspondence principle, a useful guideline in the early days of quantum mechanics, states that in the limit of very large quantum numbers the classical idea of continuity must result from the quantum discreteness as an effective description. It would be very interesting to see in a clear way how this transition discrete-to-continuous occurs. This is the objective of the present letter with the use of a simple model of discrete classical interaction for studying this transition in the classical simple harmonic oscillator. We should not forget, however, that the harmonic potential, although being an extremely useful tool in all branches of modern physics, is not itself a fundamental interaction, which, as well known, are just the gravitational, the electromagnetic, the weak and the strong interactions; actually it is just an effective description. This may just valorize the importance of studying how it can be understood as an effective * PIVIC-UFES
We derive the Euclidean time formulation for the equilibrium canonical ensemble of the type IIA and type IIB superstrings, and the spinð32Þ=Z 2 heterotic string. We compactify on R 8 × T 2 , and twist by the Neveu-Schwarz sector antisymmetric 2-form B-field potential, spontaneously breaking supersymmetry at low temperatures, while preserving the tachyon-free low-energy gravitational field theory limit. We verify that the super partners of the massless dilaton-graviton multiplet obtain a mass which is linear in the temperature. In addition, we show that the free energy for the superstring canonical ensemble at weak coupling is always strongly convergent in the ultraviolet, high-temperature, regime dominated by the highest mass level number states. We derive the precise form of the exponential suppression as a linear power of the mass level, which erases the exponential Hagedorn growth of the degeneracies as the square root of mass level number. Finally, we close a gap in previous research giving an unambiguous derivation of the normalization of the one-loop vacuum energy density of the spinð32Þ=Z 2 perturbative heterotic string theory. Invoking the O(32) type IB-heterotic strong-weak duality, we match the normalization of the one loop vacuum energy densities of the T-dual O(32) type IA open and closed string with that of the spinð32Þ=Z 2 heterotic string on R 9 × S 1 , for values of the compactification radius, R ½Oð32Þ , R IB ≫ α 01=2 , with R IA < α 01=2 . We show that the type IA thermal solitonic winding spectrum is a simple model for finite temperature pure QCD, transitioning above the critical duality phase transformation temperature to the deconfined ensemble of thermally excited IB gluons. 1 There are some typos in both [2] and [3,4] in the expressions for the string free energy, but they do not take away from the pioneering elegance of these early papers. PHYSICAL REVIEW D 90, 126005 (2014) 1550-7998=2014=90 (12)=126005(34) 126005-1
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