×

Discrete phase space and continuous time relativistic quantum mechanics. I: Planck oscillators and closed string-like circular orbits. (English) Zbl 1514.81158

Summary: The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length \(l\) is investigated. Fundamental physical constants such as \(\hbar\), \(c\), and \(l\) are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles \(S_n^1\) of radii \(2E_n=\sqrt{2n + 1}\) within the discrete \((1+1)\)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities \(S_n^1\times\mathbb{R}\subset \mathbb{R}^2\) and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion \(S^{\#}\) matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein-Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group \(\mathcal{I}[O(3, 1)]\).

MSC:

81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
39A12 Discrete version of topics in analysis
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
70M20 Orbital mechanics
81R60 Noncommutative geometry in quantum theory
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
81U20 \(S\)-matrix theory, etc. in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Das, A., Nuovo Cimento18, 482 (1960). · Zbl 0094.42703
[2] Greiner, W., Relativistic Quantum Mechanics, 3rd edn. (Springer-Verlag, 2000). · Zbl 0998.81503
[3] Weyl, H., Z. Phys.46, 1 (1927). · JFM 53.0848.02
[4] Wigner, E., Phys. Rev.40, 749 (1932). · Zbl 0004.38201
[5] Heisenberg, W., Z. Phys.33, 879 (1925). · JFM 51.0728.07
[6] Das, A. and Smozynski, P., Found. Phys. Lett.7, 21 (1994).
[7] Das, A. and Smozynski, P., Found. Phys. Lett.7, 127 (1994).
[8] Das, A., Can. J. Phys.88, 73 (2010).
[9] Das, A., Can. J. Phys.88, 93 (2010).
[10] Das, A., Can. J. Phys.88, 111 (2010).
[11] Das, A. and DeBenedictis, A., Sci. Voyage1, 45 (2015).
[12] Das, A., Chatterjee, R. and Yu, T., Mod. Phys. Lett. A35, 24 (2020). · Zbl 1443.81048
[13] Das, A. and Haldar, S., Phys. Sci. Int. J.17, 1 (2018).
[14] Zwiebach, B., A First Course in String Theory, 2nd edn. (Cambridge Univ. Press, 2009). · Zbl 1185.81005
[15] Polchinski, J., String Theory, Vol. 1 (Cambridge Univ. Press, 1998). · Zbl 1006.81521
[16] Einstein, A., The Meaning of Relativity (Methuen & Co. Ltd., 1951). · Zbl 0063.01229
[17] Synge, J. L., Relativity: The General Theory (North Holland Publishing Co., 1960). · Zbl 0090.18504
[18] Lanczos, C., The Variational Principles of Mechanics (Univ. of Toronto Press, 1970). · Zbl 0257.70001
[19] Lorch, E. R., Spectral Theory (Oxford Univ. Press, 1962). · Zbl 0105.09204
[20] Schiff, L. I., Quantum Mechanics (McGraw Hill, 1968).
[21] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products (Academic Press, 1980). · Zbl 0521.33001
[22] Jordan, C., Calculus of Finite Differences (Chelsea Publishing, 1965). · Zbl 0154.33901
[23] Pettofrezzo, A.J., Introductory Numerical Analysis (D.C. Heath & Co., 1967). · Zbl 0154.16701
[24] Wigner, E., Ann. Math.40, 149 (1939). · Zbl 0020.29601
[25] Weinberg, S., The Quantum Theory of Fields I (Cambridge Univ. Press, 1995). · Zbl 0959.81002
[26] Das, A., Tensors: The Mathematics of Relativity Theory and Continuum Mechanics (Springer-Verlag, 2007). · Zbl 1138.53001
[27] Hamilton, W. R., Collected Mathematical Papers II (Cambridge Univ. Press, 1940). · JFM 66.0984.01
[28] Clark, C., The Theoretical Side of Calculus (Wadsworth Publ. Co., 1972). · Zbl 0249.26001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.