DataSheet1_New deformed Heisenberg algebra from the μ-deformed model of dark matter.PDF

Recently, the μ-deformation-based approach to modeling dark matter, which exploits μ-deformed thermodynamics, was extended to the study of galaxy halo density profile and of the rotation curves of a number of (dwarf or low brightness) galaxies. For that goal, μ-deformed analogs of the Lane–Emden equation (LEE) have been proposed, and their solutions describing density profiles obtained. There are two seemingly different versions of μ-deformed LEE which possess the same solution, and so we deal with their equivalence. From the latter property we derive new, rather unusual, μ-deformed Heisenberg algebra (HA) for the position and momentum operators, and present the μ-HA in few possible forms (each one at μ → 0 recovers usual HA). The generalized uncertainty relation linked with the new μ-HA is studied, along with its interesting implications including the appearance of the quadruple of both maximal and minimal lengths and momenta.

近年来，利用μ形变热力学（μ-deformed thermodynamics）的暗物质建模μ形变方法（μ-deformation-based approach）被拓展应用于星系晕密度轮廓研究，以及一批矮星系或低光度星系的旋转曲线分析。为此，研究者提出了兰道-爱姆登方程（Lane–Emden equation, LEE）的μ形变对应形式，并通过求解得到了可描述密度轮廓的解。现有两种形式看似不同的μ形变兰道-爱姆登方程，却拥有完全一致的解，因此二者实为等价形式。基于这一等价性，我们推导得到了一套新颖且颇具特殊性的位置与动量算符μ形变海森堡代数（μ-deformed Heisenberg algebra, μ-HA），并给出了该μ-HA的多种等价表述形式（当μ→0时，所有形式均可退化为常规海森堡代数）。我们还研究了与该新型μ-HA相关的广义不确定关系（generalized uncertainty relation），探讨了其蕴含的若干有趣推论，包括同时涵盖极大与极小长度、极大与极小动量的四重特征。