PT symmetric field theory

物理系统是用一个称为哈密尔顿量的量来描述的。在传统的量子物理学中，有两种哈密顿量被使用：（i）厄米哈密顿量，它控制孤立系统的行为;（ii）非厄米哈密顿量，它被用来描述与环境接触的系统的行为。厄米哈密顿量描述了处于平衡状态的理想系统，其总能量和概率是守恒的;这样的系统的能级是真实的。一般来说，非厄米哈密顿从环境中接收能量或将能量耗散到环境中，因此它们通常不处于平衡状态，它们的能量和概率不守恒，并且由于能级不稳定，它们的能级是复杂的。这一提议涉及一类所谓的PT-对称哈密顿算子，它们具有Hermitian和non-Hermitian哈密顿算子的性质，是介于保守和耗散系统之间的一类系统，与非Hermitian系统一样，PT-对称系统不是孤立的，但它们与环境的联系是受约束的，因此从环境中获得的和对环境的损失是完全平衡的。因此，虽然它们不是孤立的，但平衡态的PT对称系统类似于厄米系统，它们的能级是真实的。然而，与厄米特系统不同，PT对称系统可以从能量为真实的的未破平衡相过渡到能量为复的破非平衡相。厄米系统永远不可能有复杂的能量，因此不可能有这样的相变。PT相变是实验中观察到的一个特征信号。量子力学是描述粒子物理的基本方法，涉及有限数量的自由度。然而，粒子是量子场的激发，量子场定义在所有空间和时间上。量子场论有无限多个自由度。因此，PT对称场论的公式化必须描述任何涉及PT对称性的基本理论。此外，即使在厄米量子场论的框架内，非厄米PT对称特征也经常出现在计算中。这些特征往往被忽视，要么是基于与从计算中的发散表达式中提取有限数的处方相关的不严格的数学，要么是物理模型的不完整性。本文通过研究以下问题，直接研究了PT对称性在基本量子场论中的作用和性质：1。在量子场论中是否有类似的特征，区分PT对称量子力学系统和厄米量子力学？2.对于表现出类似特征的PT对称场论的类型有什么限制吗？3.非厄米特的特点，这是由于厄米特场理论的分歧，可以处理的程序范围内的PT对称量子场论？4. PT对称场论能否为基础物理模型带来新的可能性，而这些模型在低空间维度下可以在实验室中实现？这些都是该项目旨在回答的问题。

Physical systems are described by a quantity called the Hamiltonian. In conventional quantum physics two kinds of Hamiltonians are used,(i) Hermitian Hamiltonians, which govern the behaviour of isolatedsystems, and (ii) non-Hermitian Hamiltonians, which have been used to describe the behaviour of systems in contact with the environment.Hermitian Hamiltonians describe idealised systems in equilibrium whose total energy and probability are conserved; the energy levels of such systems are real. Non-Hermitian Hamiltonians in general receive energy from and/or dissipate energyinto their environment, so they are not typically in equilibrium, their energy andprobability are not conserved, and their energy levels are complex, due to the levels being unstable. This proposal concerns a category of so-called PT-symmetricHamiltonians, which share properties of both Hermitian and non-HermitianHamiltonians, being intermediate between conservative and dissipative systems.Like non-Hermitian systems,PT-symmetric systems are not isolated, but their contactwith the environment is constrained so that gain from the environmentand loss to the environment are exactly balanced. Thus, while theyare not isolated, PT-symmetric systems in equilibrium behavelike Hermitian systems and their energy levels are real. However,unlike Hermitian systems, PT-symmetric systems can exhibita transition from an unbroken equilibrium phase, where the energiesare real, to a broken nonequilibrium phase where the energies arecomplex. Hermitian systems can never have complex energies and thuscannot have such a phase transition. The PT phase transition is a characteristicsignature that has been observed in experiments.Quantum mechanics is essential for describing the physics of particles and involves a finite number of degrees of freedom. However, particles are excitations of quantum fields, which are defined over all space and time. Quantum field theorieshave infinitely many degrees of freedom. Consequently the formulation of PT-symmetric field theory is required to describe any fundamental theory involving PT symmetry. Moreover, even within the framework of Hermitian quantum field theories, non-Hermitian PT symmetric features often emerge in calculations. These features tend to be dismissed, either on the basis of nonrigorous mathematics related to prescriptions introduced to extract finite numbers from divergent expressions in calculations, or incompleteness of the physical model. This proposal investigates directly the role and properties of PT symmetry in fundamental quantum field theories by investigating the following questions:1. Are there analogues in quantum field theory of the features that distinguish PT-symmetric quantum mechanical systems from Hermitian quantum mechanics? 2.Are there any restrictions on the type of PT-symmetric field theories that show analogous features? 3. Can non-Hermitian features, which arise due to divergences in Hermitian field theories, be dealt with by procedures within the framework of PT-symmetric quantum field theory?4. Can PT-symmetric field theories lead to new possibilities for models of fundamental physics, which, in low number of spatial dimension, may be realised in the laboratory?These are the questions that the project aims to answer.