[论文解读] PT symmetry as a necessary and sufficient condition for unitary time evolution

Philip D. Mannheim|arXiv (Cornell University)|Dec 14, 2009

Quantum Mechanics and Non-Hermitian Physics被引用 8

一句话总结

本文证明了在非厄米量子系统中，PT对称性——等价于久期方程的实性——是实现幺正时间演化的必要且充分条件。研究证明，对于任意PT对称的哈密顿量，存在一个与时间无关的算符V，使得VHV⁻¹ = H†，从而通过V依赖的内积保持规范不变，确保即使本征值为复共轭对或实数且完全时，时间演化仍保持幺正性。

ABSTRACT

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper we provide conditions that are both necessary and sufficient. We show that $PT$ symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any $PT$-symmetric Hamiltonian $H$ there always exists an operator $V$ that relates $H$ to its Hermitian adjoint according to $VHV^{-1}=H^{\\dagger}$. When the energy spectrum of $H$ is complete, Hilbert space norms $<\\psi_1|V|\\psi_2>$ constructed with this $V$ are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete the operator $V$ is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with $PT$-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast non-Hermitian but $PT$-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. We show that through $PT$ symmetry the fourth-order derivative Pais-Uhlenbeck theory can be stabilized against transitions to states negative frequency.

研究动机与目标

• 确定非厄米哈密顿量实现幺正时间演化的普遍条件。
• 证明PT对称性（或久期方程的实性）是实现幺正性的必要且充分条件，超越了传统上仅充分但非必要的厄米性条件。
• 构造一个与时间无关的算符V，使其将H与它的共轭H†联系起来，从而为所有能级本征态提供一个保持范数的内积。
• 将幺正性推广至谱不完备的情形以及约当块哈密顿量的情形。
• 证明PT对称系统满足因果性，并可通过具有实欧几里得延续的路径积分实现一致的量子化。

提出的方法

• 定义一个V依赖的内积⟨Rj(t)|V|Ri(t)⟩，以替代非厄米系统中的标准狄拉克范数。
• 利用薛定谔方程及其共轭，推导V内积的时间演化，导出时间不变性的条件VHV⁻¹ = H†。
• 证明此类V的存在性既是实现幺正时间演化的必要条件也是充分条件，无论本征值是实数还是复共轭对。
• 证明当本征值为实数且谱完全时，V为正定厄米算符，从而可通过其平方根实现到厄米哈密顿量的幺正变换。
• 为两能级模型显式构造P、T和C算符，验证PT对称性与CPT对称性，并利用它们构建一个满足C² = I的归一化C算符。
• 证明PT对称理论的路径积分具有实的欧几里得延续，从而即使在复测度下也能实现一致的量子化。

实验结果

研究问题

• RQ1 PT对称性是否是非厄米量子系统中实现幺位时间演化的必要且充分条件？
• RQ2 能否为任意PT对称哈密顿量构造一个与时间无关的算符V，使得VHV⁻¹ = H†，且该条件是否能确保范数保持不变？
• RQ3 幺正性如何推广至具有复共轭本征值对或谱不完备的哈密顿量？
• RQ4 尽管哈密顿量非厄米，PT对称理论是否仍能保持因果性并实现一致的路径积分量子化？
• RQ5 当V未归一时，C算符在构造正定内积中起什么作用？

主要发现

• PT对称性在非厄米系统中既是实现幺正时间演化的必要条件也是充分条件，推广了标准的厄米性条件。
• 对于任意PT对称哈密顿量H，存在一个与时间无关的算符V，使得VHV⁻¹ = H†，从而通过⟨ψ₁|V|ψ₂⟩保持范数的时间不变性。
• 当能谱完全且为实数时，V为正定厄米算符，其平方根可实现到厄米哈密顿量的幺正变换。
• 即使本征值为复共轭对，V依赖的内积仍保持时间不变，从而保证幺正性。
• 算符V不唯一；通过归一化自由度可构造一个满足C² = I的归一化C算符，满足CPT对称性。
• PT对称理论的路径积分具有实的欧几里得延续，从而即使在复测度下也能实现一致的量子化。