从时间的认识论处理发展方程
Evolution equations from an epistemic treatment of time
摘要：
相对论上，时间$t$是一个可观测的，就像位置$r$一样。在量子理论中，$t$是一个参数，与可观测的$r$相反。这种差异表明，存在着一种更为复杂的时间形式化，它包含了两种观点。本文提出了这样一种形式化。演化是按照连续时间$n\in \mathbf{\mathbb{N}}$来描述的，每次事件发生时都会更新。序列时间$n$与关系时间$t$是分开的，后者描述了时空中事件之间的距离。有一个与每个$n$相关联的时空，其中$t$表示时间$n$上关于时间关系的知识。波函数的演变描述的参数$σ$之间插入顺序时间$n$。对于自由物体，我们得到一个Stueckelberg方程$\frac{d}{dσ}Ψ(r_{4},σ)=\frac{ic^{2}\hbar}{2\langle ε\rangle}\BoxΨ(r_{4},σ)$，其中$r_{4}=(r,ict)$ .在此，$σ$描述了在时间$n$的实验开始与时间$n+m$的观察之间经过的时间$m$。假设参数化是自然的，意味着$\frac{d}{dσ}\langle t\rangle=1$，其中$\langle t\rangle$是定义$n$和$n+m$的事件之间的预期时间距离。平方静止能量$ε_{0}^{2}$与描述"稳态“$Ψ(r_{4},σ)=ψ(r_{4},\tildeσ)e^{i\tildeσσ}$的本征值$\tildeσ$成比例。狄拉克方程作为定态方程的“平方根”，其条件是$\tildeσ>0$，这是由$n$的定向性质得出的。因此，形式主义意味着所有可观测的物体都有非零的静止质量，包括基本费米子。$n$的引入释放了$t$，因此它可以被视为具有不确定性$Δt$的可观测量。
英文摘要：
Relativistically, time $t$ is an observable just like position $r$. In quantum theory, $t$ is a parameter, in contrast to the observable $r$. This discrepancy suggests that there exists a more elaborate formalization of time, which encapsulates both perspectives. Such a formalization is proposed in this paper. The evolution is described in terms of sequential time $n\in \mathbf{\mathbb{N}}$, which is updated each time an event occurs. Sequential time $n$ is separated from relational time $t$, which describes distances between events in space-time. There is a space-time associated with each $n$, in which $t$ represents the knowledge at time $n$ about temporal relations. The evolution of the wave function is described in terms of the parameter $σ$ that interpolates between sequential times $n$. For a free object we obtain a Stueckelberg equation $\frac{d}{dσ}Ψ(r_{4},σ)=\frac{ic^{2}\hbar}{2\langle ε\rangle}\BoxΨ(r_{4},σ)$, where $r_{4}=(r,ict)$. Here $σ$ describes the time $m$ passed between the start of the experiment at time $n$ and the observation at time $n+m$. The parametrization is assumed to be natural, meaning that $\frac{d}{dσ}\langle t\rangle=1$, where $\langle t\rangle$ is the expected temporal distance between the events that define $n$ and $n+m$. The squared rest energy $ε_{0}^{2}$ is proportional to the eigenvalue $\tildeσ$ that describes a 'stationary state' $Ψ(r_{4},σ)=ψ(r_{4},\tildeσ)e^{i\tildeσσ}$. The Dirac equation follows as a `square root' of the stationary state equation from the condition that $\tildeσ>0$, which follows from the directed nature of $n$. The formalism thus implies that all observable objects have non-zero rest mass, including elementary fermions. The introduction of $n$ releases $t$, so that it can be treated as an observable with uncertainty $Δt$.
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