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Nonlocal phenomena in quantum mechanics with fractional calculus. (English) Zbl 1451.81059

Summary: In this study, nonlocal phenomena in quantum mechanics are investigated by making use of fractional calculus. in this context, fractional creation and annihilation operators are introduced and quantum mechanical harmonic oscillator has been generalized as an important tool in quantum field theory. Therefore wave functions and energy eigenvalues of harmonic oscillator are obtained with respect to the order of fractional derivative which corresponds to influence of nonlocal effects. In order to investigate nonlocality in quantum field theory, Einstein’s coefficients are taken into consideration in the framework of fractional calculus. For this purpose, all energy modes of photons are considered as fractional quantized harmonic oscillators and thus Einstein’s coefficients are obtained. In the case \(\alpha=1\), where space becomes continuous, results of conventional physical models are recovered.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
81T10 Model quantum field theories
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
26A33 Fractional derivatives and integrals
81V80 Quantum optics
47A25 Spectral sets of linear operators
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