Antilinearity rather than Hermiticity as a guiding principle for quantum theory
Antilinearity rather than Hermiticity as a guiding principle for quantum theory
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反线性而不是厄米性作为量子理论的指导原则
DOI：
10.1088/1751-8121/aac035
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. Mannheim
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文献类型：
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作者：
P. Mannheim
目前有很多兴趣在哈密顿不是厄米，而是拥有一个反线性PT对称。在这里，我们试图把这样的PT对称理论到尽可能普遍的背景。在简要概述了PT对称性程序之后，我们证明了具有作用于定义明确的希尔伯特空间的反线性对称性是可以强加于量子理论的最一般条件，对于量子理论，可以具有定义明确的时间无关的内积，具有自伴的哈密顿量，并且具有全部为真实的的能量本征值。对于这些性质中的每一个，厄米性只是一个充分条件，而不是必要条件，因此厄米性是哈密顿量既有反线性又有厄米性的特殊情况。除了是能量本征值存在的必要条件外，反线性还允许出现一些物理上有趣的情况，即明显非厄米但自伴的哈密顿量，其能量本征值出现在复共轭对中，或者是约旦块，根本不能对角化。我们表明，这些想法可以扩展到量子场论，与时间无关的内积和复杂的洛伦兹变换下的不变性的存在的双重要求，迫使反线性对称性，唯一的CPT。因此，我们将CPT定理推广到非厄米特哈密顿算子。对于分别是电荷共轭不变的理论，PT对称性随后，与PT对称性程序的物理相关性的情况下，从而被提前。虽然CPT对称性可以在路径积分量子化过程中的每个经典路径的经典水平上定义，但相反，在这样的路径积分中，根本没有参考哈密顿量或其作用的量子希尔伯特空间的厄米性，因为它们是严格量子的只有在路径积分量子化和量子希尔伯特空间被构造之后才能定义的力学概念。因此，CPT对称性超越了厄米性，并在它之上具有首要地位，我们的工作提出了厄米性如何进入量子理论的问题。为此，我们表明，CPT不变的理论是否有一个哈密尔顿是厄米是一个属性的解决方案的理论，而不是哈密尔顿本身。因此，根本不需要假设厄米性。
Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT. We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT-symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT-invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.