Toward a geometrical foundation for physics
Toward a geometrical foundation for physics
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迈向物理学的几何基础
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发表时间：
2015
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通讯作者：
S. Eitan
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作者：
S. Eitan
我们提出的猜想，物理学的基本概念，包括这样的基本概念，如自旋，电荷（S），质量和时空，出现从一个数学理论与2-流形的度量的波动。这涉及到看一组2-流形，其度量依赖于全局参数$$t$$t，其中每个流形也与一组六个参数相关联，这些参数的值沿着$$t$$t连续变化，这取决于所有其他流形的度量和相关参数。更直观、更具体地说，我们所做的是定义在由全局参数$$t$$t参数化的$$R^3$$R3中的仿射平面上产生振荡的对象，其中与每个对象相关联的平面根据其他仿射平面上的类似对象产生的振荡而连续沿着$$t$$t变化。我们研究了一些典型的结构时出现的振荡的振幅趋于零，并表明，在任何值的$$t$$t的每个这样的结构的状态可以与一个真实的参数$0\leqslane\beta < 1$$0 β<1，并与其他类似的结构存在一种特殊类型的相互作用，这导致重新调整$$\beta $$β的值。我们继续认为，通过赤平投影，$t$$t的任何值的$R^3$R3都对应于$$R^4$$R4中的超球，其中$t$$t被标识为同心超球中的$$R^4$$R4的叶理中的超球半径的对数。这就导致我们将某些由我们所研究的构造产生的曲线解释为对应于某些S^2S2曲线。我们使用最后一个对应关系的两边来导出自由狄拉克方程作为抽象点粒子的经典运动方程，其中我们的典型结构和它们的相互作用被理解为与电子与电磁辐射场相互作用有关，时空中任何点的涌现描述通常涉及无限多个离散值。因此，狄拉克方程是从我们的特殊曲线族的弗伦内方程推导出来的，其物理意义由推导所需的物理和几何实体的识别来确定。我们提出，我们理论中无穷小振荡的起源与欧几里得平面的公理化和真实的数的构造有关。我们简要地讨论了这些想法的相关性强相互作用和重力。
We present the conjecture that the fundamental notions of physics, including such basic concepts as spin, charge(s), mass and spacetime, emerge from a mathematical theory having to do with fluctuations in the metrics of 2-manifolds. This involves looking at a set of 2-manifolds whose metrics depend on a global parameter $$t$$t, where each manifold is also associated with a set of six parameters whose values change continuously along $$t$$t depending on the metrics and associated parameters of all the other manifolds. More intuitively, and specifically, what we do is define objects that generate oscillations on affine planes in $$R^3$$R3, parameterized by the global parameter $$t$$t, where the plane associated with each object changes continuously along $$t$$t depending on the oscillations that are generated by similar objects on other affine planes. We study some typical structures that arise when the oscillations’ amplitudes tend to zero, and show that the state of each such structure at any value of $$t$$t can be associated with a real parameter $$0 \leqslant \beta < 1$$0⩽β<1, and that a special type of interaction with other similar structures exists which results in readjusting the value of $$\beta $$β. We go on to think of $$R^3$$R3 at any value of $$t$$t as corresponding to a hypersphere in $$R^4$$R4 via the stereographic projection, where $$t$$t is identified as the logarithm of hypersphere radius in the foliation of $$R^4$$R4 in concentric hyperspheres. This leads to interpreting some curves arising from the structures we study as corresponding to certain $$S^2$$S2 curves. We use both sides of that last correspondence to derive the free Dirac equation as the classical equation of motion of an abstract point particle, where our typical structures and their interactions are understood in connection with electrons interacting with an em radiation field, and the emergent description of any point in spacetime typically involves an infinite number of discrete values of $$t$$t. The Dirac equation is thus derived from the Frenet equations of our special family of curves, and the physical meaning is determined by the identification of physical and geometrical entities required by the derivation. We propose that the origin of the infinitesimal oscillations in our theory has to do with the axiomatization of the Euclidean plane and the construction of the real numbers. We briefly discuss the relevance of these ideas to the strong interaction and to gravity.