概率时间
Probabilistic time
摘要：
时间的概念在经典统计系综中作为一种有序结构出现。概率分布$p_τ(t)$在给定的时间$t$通过整合过去和未来获得。我们讨论所有的时间概率分布，实现了一个单一的时间演化所描述的旋转的真实波函数$q_τ(t)=\pm \sqrt{p_τ(t)}$。我们建立了一个映射到量子物理和薛定谔方程。合适的经典观测量映射到量子算子。算子乘积的非交换性可追溯到局部时间子系统的不完全统计量。我们对经典统计的研究是基于两个级别的可观测值，取值为1或0。然后，波函数可以映射到格拉斯曼代数的元素。费米子的量子场论自然产生于我们对概率时间的表述。
英文摘要：
The concept of time emerges as an ordering structure in a classical statistical ensemble. Probability distributions $p_τ(t)$ at a given time $t$ obtain by integrating out the past and future. We discuss all-time probability distributions that realize a unitary time evolution as described by rotations of the real wave function $q_τ(t)=\pm \sqrt{p_τ(t)}$. We establish a map to quantum physics and the Schrödinger equation. Suitable classical observables are mapped to quantum operators. The non-commutativity of the operator product is traced back to the incomplete statistics of the local-time subsystem. Our investigation of classical statistics is based on two-level observables that take the values one or zero. Then the wave functions can be mapped to elements of a Grassmann algebra. Quantum field theories for fermions arise naturally from our formulation of probabilistic time.
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