希尔伯特空间或Gelfand三重态-时间对称或时间非对称量子力学
Hilbert Space or Gelfand Triplet - Time Symmetric or Time Asymmetric Quantum Mechanics
摘要：
内在微物理不可逆性是在指数衰减状态中观察到的时间不对称性。它由量子物理系统的哈密顿量$\QTR{it}{H}$生成的半群来描述，而不是由描述由于外部储存器或测量装置的影响而引起的不可逆性的Liouvillian $\QTR{it}{L}$生成的半群来描述。由$\QTR{it}{H}$生成的半群时间演化在Hilbert空间（HS）理论中是不可能的，HS理论只允许时间对称的边界条件和酉群时间演化。这导致了HS理论中衰变概率的问题。为了克服这些和其他问题（不存在的狄拉克kets）所造成的勒贝格积分的HS，一个扩展的HS的凝胶'fand三重，其中不仅包含狄拉克kets，但也广义本征向量的自伴$\QTR{it}{H}$与复杂的本征值（$E_R-iΓ/2$）和Breit-Wigner能量分布。这些伽莫夫态$ψ^G$具有时间非对称指数演化。我们可以从量子力学$\QTR{cal}{P}(t)=Tr(|ψ^G> < ψ^G|Λ)$的基本公式中推导出伽莫夫态衰变为$Λ$所描述的衰变产物的几率，在HS量子力学中该几率为零。从这个结果可以导出衰减率$\QTR{group}{\dot c}(t)$和共振散射和衰减现象学中使用的$\QTR{group}{\dot c}(0)$、$Γ$和寿命$τ_R$之间的所有标准关系。在玻恩近似中，我们得到狄拉克的黄金法则。
英文摘要：
Intrinsic microphysical irreversibility is the time asymmetry observed in exponentially decaying states. It is described by the semigroup generated by the Hamiltonian $\QTR{it}{H}$ of the quantum physical system, not by the semigroup generated by a Liouvillian $\QTR{it}{L}$ which describes the irreversibility due to the influence of an external reservoir or measurement apparatus. The semigroup time evolution generated by $\QTR{it}{H}$ is impossible in the Hilbert Space (HS) theory, which allows only time symmetric boundary conditions and an unitary group time evolution. This leads to problems with decay probabilities in the HS theory. To overcome these and other problems (non-existence of Dirac kets) caused by the Lebesgue integrals of the HS, one extends the HS to a Gel'fand triplet, which contains not only Dirac kets, but also generalized eigenvectors of the self-adjoint $\QTR{it}{H}$ with complex eigenvalues ($E_R-iΓ/2$) and a Breit-Wigner energy distribution. These Gamow states $ψ^G$ have a time asymmetric exponential evolution. One can derive the decay probability of the Gamow state into the decay products described by $Λ$ from the basic formula of quantum mechanics $\QTR{cal}{P}(t)=Tr(|ψ^G> < ψ^G|Λ)$, which in HS quantum mechanics is identically zero. From this result one derives the decay rate $\QTR{group}{\dot c}(t)$ and all the standard relations between $\QTR{group}{\dot c}(0)$, $Γ$ and the lifetime $τ_R$ used in the phenomenology of resonance scattering and decay. In the Born approximation one obtains Dirac's Golden Rule.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
