The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can $curve$ the classical spacetime. It is shown that the gravitational field produced by $smearing$ a point-mass $M_o$ at $r=0$ throughout all of space (in a spherically symmetric fashion) can be interpreted as the gravitational field generated by a self-gravitating anisotropic fluid droplet of mass density $4\pi M_o{r}^2{\phi }^{\ast }\left(r\right)\phi \left(r\right)$ and which is sourced by the $probability$ $cloud$ (associated with a spinless point-particle of mass $M_o$) $permeating$ a 3-spatial domain region ${\mathcal{D}}_3=\int 4\pi r^{2}dr$ at any time $t$. Classically one may smear the point mass in any way we wish leading to arbitrary density configurations $\rho \left(r\right)$. However, Quantum Mechanically this is $not$ the case because the radial mass configuration $M\left(r\right)$ must obey a key third order nonlinear differential equation (nonlinear extension of the Klein–Gordon equation) displayed in this work and which is the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We conclude by extending our proposal to the Lagrange–Finsler and Hamilton–Cartan geometry of (co) tangent spaces and involving the relativistic version of Bohm’s Quantum Potential. By further postulating that the quasi-probability Wigner distribution $W(x,p)$ $curves$ phase spaces, and by encompassing the Finsler-like geometry of the cotangent-bundle with phase space quantum mechanics, one can naturally incorporate the $noncommutative$ and non-local Moyal star product (there are also non-associative star products as well). To conclude, Phase Space is the arena where to implement the space–time–matter unification program. It is our belief this is the right platform where the quantization $of$ spacetime and the quantization $in$ spacetime will coalesce.

这项工作中提出的量子力学的几何化是基于以下假设：量子概率密度可以 $Cü\mathrm{[R}vË$古典时空。结果表明，由$s米Ë\mathrm{一种}\mathrm{[R}\mathrm{一世}ñG$ 点质量 ${\mathrm{中号}}_{Ø}$ 在 $\mathrm{[R}=0$ 在整个空间中（以球对称的方式）可以解释为由质量密度自重各向异性的液滴产生的引力场 $4\pi {\mathrm{中号}}_{Ø}{\mathrm{[R}}^2{\phi }^{\ast }（\mathrm{[R}）\phi （\mathrm{[R}）$ 并由 $p\mathrm{[R}Øb\mathrm{一种}b\mathrm{一世}升\mathrm{一世}Ťÿ$ $C升Øüd$ （与质量的无旋转点粒子相关 ${\mathrm{中号}}_{Ø}$） $pË\mathrm{[R}米Ë\mathrm{一种}Ť\mathrm{一世}ñG$ 3空间域区域 ${\mathcal{d}}_3=\int 4\pi {\mathrm{[R}}^{2}d\mathrm{[R}$ 随时 $Ť$。传统上，我们可能希望以任何方式涂抹点质量，从而导致任意密度配置$\rho （\mathrm{[R}）$。但是，从量子力学的角度来看，这是$ñØŤ$ 这种情况是因为径向质量配置 $\mathrm{中号}（\mathrm{[R}）$必须遵守这项工作中显示的关键三阶非线性微分方程（Klein-Gordon方程的非线性扩展），它是牛顿-薛定er方程的静态球对称相对论类似物。最后，我们将提议扩展到（共）切空间的Lagrange-Finsler和Hamilton-Cartan几何，并涉及相对论形式的Bohm量子势。通过进一步假设拟概率Wigner分布$\mathrm{w\; ^}（X，p）$ $Cü\mathrm{[R}vËs$ 相空间，并通过相空间量子力学包含余切束的Finsler型几何结构，自然可以将 $ñØñCØ米米üŤ\mathrm{一种}Ť\mathrm{一世}vË$和非本地Moyal星级产品（也有非关联星级产品）。总而言之，相空间是实施时空问题统一计划的场所。我们相信这是量化的正确平台$ØF$ 时空与量化 $\mathrm{一世}ñ$ 时空将合并。