Abstract 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.
Abstract We consider quantum field theoretic many-body Green’s function approach to solve the Coulomb many-body problem. The earlier beyond Born-Oppenheimer Green’s function theories are absolute in nature and are based on the non-reduced Hamiltonian. Motivated by the issues following this approach we have developed a reduced Green’s function theory which is relational in nature. The central differences between these approaches trace back to the relational-absolute debate, which has continued for over two thousand years since the time of Aristotle and still persists today. We highlight that these aspects of the theories are connected to several areas of modern physics, including relational quantum mechanics, quantum reference frames, superselection rules, the separation of different types of motion and spontaneous symmetry breaking. The starting point of any such exact theory must be absolute, given that the global conservation laws hold. However, for the observables to be meaningful, they must be defined with respect to the relative space. Approximations, such as the one introduced by Born and Oppenheimer, can break global symmetries and make certain parts of the absolute description well-defined. We highlight that spontaneous symmetry breaking is not necessary to explain the existence of solids, but entities such as phonons can be naturally explained within the relational theory.
Our current understanding of quantum chaos in many-body quantum systems hinges on the random matrix theory (RMT) behavior of eigenstates and their energy level statistics. Although RMT has been remarkably successful in describing "coarse" features of many-body quantum Hamiltonians in chaotic regimes, such as the Wigner-Dyson level spacing statistics or the volume-law behavior of eigenstate entanglement entropy, it remains a challenge to describe their "finer" features, particularly those arising from spatial locality. Here, we show that we can accurately describe the statistical behavior of eigenstate ensembles in many-body Hamiltonians by using pure random states with physical constraints that capture the essential features of the Hamiltonian, specifically spatial locality and symmetries. We demonstrate our approach on local spin Hamiltonians with a scalar U(1) charge. By constructing ensembles of constrained random states that account for two commuting scalar charges playing the role of energy and magnetization, we describe the patterns of entanglement of midspectrum eigenstates beyond their average volume-law behavior, including O(1) corrections and fluctuations, analytically and numerically. When defining the correspondence between quantum chaotic eigenstates in many-body Hamiltonians and RMT ensembles, our work highlights the important role played by spatial locality in describing universal features beyond the volume-law behavior.
Generators of continuous-time quantum Markovian evolution, known as Lindblad operators, play a key role in the theory of open quantum systems. The question of how to define and characterize random Lindblad operators has attracted considerable attention in recent years. Purely dissipative random Lindbladians have been extensively studied using toolboxes of free probability and non-Hermitian random matrix theories, which helped to uncover a characteristic lemon-like spectral support. In this work, we focus on random Lindbladians with a Hamiltonian component. We analyze three ensembles of increasing complexity, designed to illustrate the joint effects of dissipative and Hamiltonian dynamics. While the first two allow for an exact analytic treatment with standard probabilistic and scalar-valued free-probability methods, the most general case requires the more general framework of operator-valued free probability. We explain how the spectral support changes with the relative Hamiltonian strength and the rank of the dissipative component and characterize the effect of Hamiltonian-induced spectral squeezing.