论文标题
在经典和量子波函数的运算符起源上
On the Operator Origins of Classical and Quantum Wave Functions
论文作者
论文摘要
我们研究了经典的Koopman-Von neumann Wave函数$ψ_{kvn} $的操作符代数起源以及量子机械一个$ψ_{qm} $。我们介绍了基于非共同泊松，符号和非共同差异结构的操作机械师（OM）的形式主义。 OM充当量子前代数，随后是与现实世界和量子力学相关的代数结构。特别是，$ψ_{kvn} $和$ψ_{qm} $都是此量子前形式主义的后果。不需要先验的希尔伯特空间。 OM承认在没有调用状态的情况下，运营商期望值的代数概念。一个相位空间捆绑$ {\ cal e} $从此遵循。 $ψ_{kvn} $和$ψ_{qm} $显示为$ {\ cal e} $中的部分。 $ψ_{kvn} $和$ψ_{qm} $之间的差异来自量化图，该量化图被解释为$ {\ cal e} $的部分的“扭曲”。我们还表明，Schrödinger方程是从Koopman-Von Neumann方程获得的。这表明Schrödinger方程和量子波函数都不是基本结构。相反，它们都源自前量子操作员代数。最后，我们评论这些操作员之间的纠缠如何暗示空间的出现。这种形式主义对现场理论的可能扩展。
We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $ψ_{KvN}$ as well as the quantum mechanical one $ψ_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $ψ_{KvN}$ and $ψ_{QM}$ are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${\cal E}$ follows from this. $ψ_{KvN}$ and $ψ_{QM}$ are shown to be sections in ${\cal E}$. The difference between $ψ_{KvN}$ and $ψ_{QM}$ originates from a quantization map interpreted as "twisting" of sections over ${\cal E}$. We also show that the Schrödinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.
