PT symmetric field theory

物理系统由一个称为哈密顿量的量来描述。在传统的量子物理学中，使用了两种哈密顿量：(I)厄米哈密顿量，它支配孤立系统的行为；(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化系统，其总能量和几率是守恒的；这样的系统的能级是真实的。非厄米哈密顿人一般从他们的环境中接收能量和/或将能量耗散到他们的环境中，因此他们通常不处于平衡状态，他们的能量和几率不守恒，并且他们的能级是复杂的，因为能级是不稳定的。这一建议涉及一类所谓的PT对称哈密顿量，它具有厄米特哈密顿和非厄米特哈密顿的性质，介于保守系统和耗散系统之间。与非厄米系统一样，PT对称系统不是孤立的，但它们与环境的接触受到限制，从而使从环境中获得的收益和对环境的损失完全平衡。因此，尽管PT对称系统不是孤立的，但它们在平衡状态下的行为类似于厄米系统，它们的能级是真实的。然而，与厄米系统不同的是，PT对称系统可以表现出从能量为实的连续平衡相到能量为复数的破裂非平衡相的转变。厄米系统永远不可能有复杂的能量，因此也不可能有这样的相变。PT相变是在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关键，它涉及有限个自由度。然而，粒子是量子场的激发，量子场定义在所有的空间和时间上。量子场论有无限多个自由度。因此，任何涉及PT对称性的基本理论都需要PT对称场论的表述。此外，即使在厄米量子场论的框架内，计算中也经常出现非厄米PT对称特征。这些特征往往被忽略，要么是因为与从计算中的发散表达式中提取有限数的处方有关的不严格的数学基础，要么是因为物理模型的不完整性。这一建议通过研究以下问题，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，PT对称量子力学系统与厄米量子力学的特征是否有相似之处？2.PT对称场论中表现出相似特征的类型是否有任何限制？3.由于厄米特场论中的分歧而产生的非厄米特征，能否通过PT对称量子场论框架内的程序来处理？4.PT对称场论能否为基础物理模型带来新的可能性，在较低的空间维度中，可以在实验室中实现吗？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.