标题： 特征时间算符作为量子钟 显示英文标题

标题： Characteristic time operators as quantum clocks

摘要： 我们考虑在[E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)]中引入的特征时间算子$\mathsf{T}$，它是有界且自伴的。对于满足某些增长条件的半有界离散哈密顿量$\mathsf{H}$，$\mathsf{T}$在希尔伯特空间的一个稠密子空间中满足规范关系$[\mathsf{T},\mathsf{H}]|\psi\rangle=i\hbar|\psi\rangle$对$|\psi\rangle$。虽然$\mathsf{T}$不具有协变性，但我们证明它在称为时间不变集$\mathscr{T}$的总测度为零的时间集合中仍然满足规范关系。 在每个时间$t$附近的$\mathscr{T}$中，$\mathsf{T}$仍然与$\mathsf{H}$拥有规范共轭关系，其期望值给出了参数时间。 在其附近$\mathscr{T}$的二维投影满足时间-能量不确定关系，并且与泡利矩阵$\sigma_y$成正比。 因此，可以通过测量一个相容的可观测量来构建一个量子钟，在$\mathscr{T}$附近指示时间。 显示英文摘要

摘要： We consider the characteristic time operator $\mathsf{T}$ introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $\mathsf{H}$ with some growth condition, $\mathsf{T}$ satisfies the canonical relation $[\mathsf{T},\mathsf{H}]|\psi\rangle=i\hbar|\psi\rangle$ for $|\psi\rangle$ in a dense subspace of the Hilbert space. While $\mathsf{T}$ is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set $\mathscr{T}$. In the neighborhood of each time $t$ in $\mathscr{T}$, $\mathsf{T}$ is still canonically conjugate to $\mathsf{H}$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $\mathscr{T}$, and is proportional to the Pauli matrix $\sigma_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathscr{T}$ by measuring a compatible observable.