We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter T and equations of motion converted into difference equations. These equations are solved and the physical limit T → 0 then taken. In principle, this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projective geometry. Quantization via Schwinger’s action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter.

中文翻译：

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超空间费米子、莫比乌斯变换、柯林空间、费米子加倍、暗物质

我们开发了一种经典和量子力学的方法，其中连续时间通过无穷小的参数 T 和转换为差分方程的运动方程来扩展。求解这些方程，然后采用物理极限 T → 0。原则上，该策略应恢复原始连续时间微分方程的所有标准解。我们发现这对玻色子变量有效，而对于费米子，会出现其他解。对于玻色子和费米子，运动的差分方程可以与射影几何中的莫比乌斯变换相关。通过施温格作用原理的量化恢复了玻色子的标准粒子-反粒子模式，但在费米子的情况下，希尔伯特空间必须被克林空间取代。