The ? operator in ??-symmetric quantum theories
The ? operator in ??-symmetric quantum theories
复制标题
??对称量子理论中的?算子
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Moritz Reuter
中科院分区：
文献类型：
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作者：
C. Bender;J. Brod;A. Refig;Moritz Reuter
哈密顿量H指定了量子理论的能级和时间演化。量子力学的一个公理是H是厄米的，因为厄米性保证能谱是真实的，并且时间演化是幺正的（概率保持）。本文研究了另一种方法来构建量子理论，其中厄米性的传统要求（组合转置和复共轭）被时空反射的物理上更透明的条件所取代？对称性它表明，如果？？如果哈密顿量H的对称性没有破缺，则H的谱是真实的。例如：？对称非厄米量子力学哈密顿量是H = p2 + ix 3和H = p2 − x4。关键的问题是，非对称的哈密顿量是否能指定物理上可接受的量子理论，其中态的范数是正的，时间演化是么正的。问题的答案是，一个具有连续？？对称性还具有由称为？的线性算子表示的物理对称性。吸毒？说明了如何构造一个其相关范数为正定的内积。其结果是一类新的完全自洽的复量子理论。可观测量被定义，概率是正的，动力学受酉时间演化的支配。在回顾了？？-对称量子力学，新的结果在这里提出的？在具有多个自由度的量子力学理论中，微扰计算算符。
The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of spacetime reflection ?? symmetry. It is shown that if the ?? symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of ??-symmetric non-Hermitian quantum mechanical Hamiltonians are H = p2 + ix3 and H = p2 − x4. The crucial question is whether -symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken ?? symmetry also possesses a physical symmetry represented by a linear operator called ?. Using ? it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of ??-symmetric quantum mechanics, new results are presented here in which the ? operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.