PT Symmetry in Quantum Field Theory

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. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-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 PT symmetry also possesses a physical symmetry that we call C. Using C, we show 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 exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

PT-Symmetric Quantum Field Theories and the Langevin Equation

Many non-Hermitian but PT-symmetric theories are known to have a real, positive spectrum, and for quantum-mechanical versions of these theories, there exists a consistent probabilistic interpretation. Since the action is complex for these theories, Monte Carlo methods do not apply. In this paper a field-theoretic method for numerical simulations of PT-symmetric Hamiltonians is presented. The method is the complex Langevin equation, which has been used previously to study complex Hamiltonians in statistical physics and in Minkowski space. We compute the equal-time one-point and two-point Green's functions in zero and one dimension, where comparisons to known results can be made. The method should also be applicable in four-dimensional space-time. This approach may grant insight into the formulation of a probabilistic interpretation for path integrals in PT-symmetric quantum field theories.     

$\mathcal{PT}$-Symmetric Periodic Optical Potentials

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (PT\mathcal{PT}) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn’t led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of PT\mathcal{PT}-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to PT\mathcal{PT}-symmetric Optics and paved the way for the first experimental observation of PT\mathcal{PT}-symmetry breaking in any physical system. In this paper, we present recent results regarding PT\mathcal{PT}-symmetric Optics.     

Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional PT-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.     

PT-symmetric Quantum Mechanics: A Precise and Consistent Formulation

The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour.     

Classical Motion of Complex 2-D Non-Hermitian Hamiltonian Systems

Classical motion of complex 2-D non-Hermitian Hamiltonian systems is investigated to identify periodic, unbounded and chaotic trajectories. The caustic curves, the Lyapunov exponents, and spectral analysis have been used to identify periodic and chaotic trajectories. Using classical trajectories, we were able to predict quantum transition frequaencies of pseudo-Hermitian non-PT symmetric systems accurately. This indicates that there exists a correspondence between classical mechanics and quantum mechanics for certain non-Hermitian Hamiltonians as in the case of real Hermitians.     

Calculating the C Operator in PT-symmetric Quantum Mechanics

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.     

Exotic fermions in Kadyshevsky’s theory and the possibility to detect them

As it is known, the principal research interest of V.G. Kadyshevsky was the development of a geometric approach to quantum field theory with a constraint imposed on the mass spectrum of elementary particles. Non-Hermitian operators arising in this case seemed to be a major obstacle to the development of a consistent theory. These issues have been resolved recently, and the introduction of the pseudo-Hermitian algebraic approach to the construction of quantum theory was a major advance in this physical research. The central point of such theories is the construction of PT-symmetric non-Hermitian Hamiltonians with real eigenvalues. It is important to note that both purely theoretical and experimental studies (e.g., in non-Hermitian optics) are found among the many published papers on this subject. Therefore, we believe that the development of pseudo-Hermitian relativistic quantum theory with a maximal mass may provide favorable opportunities to discuss the possible experimental verification of theoretical results obtained in this field. Kadyshevsky himself regarded the hypothesis of existence of new particles, which he called exotic fermions, as an important prediction of his theory. The possibility of discovery of exotic neutrinos in precision experiments on the determination of the neutrino mass is discussed in the present study.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

多模简并多光子Tavis-Cummings模型中的量子纠缠特性

利用冯·纽曼约化熵理论研究了多模相干态光场与两等同二能级原子简并多光子相互作用系统量子纠缠演化特性,得到了多模光场量子纠缠的解析表达式,并给出了双模光场与两原子相互作用时量子纠缠的数值计算结果.结果表明:量子纠缠随着光子简并度的增大而增强;随着初始平均光子数的增加,量子纠缠演化的周期性变得越来越明显;当场与原子远离共振时,量子纠缠随着频率失谐量的增大而减弱;当失谐量足够大时,场与原子几乎总是处于纠缠状态.这些结论对于纠缠态或纯态的制备及获取光学系统中的量子信息研究中有一定参考价值.     

Symmetry entanglement in polarization biphoton optics

We study kinematic and dynamic ways of forming entangled states of quantum light fields due to their local and global polarization SU(2) symmetries. The kinematic entanglement is shown to be associated with particular polarization bases in the spaces of quantum states of multi-mode radiation, which are generated by the global SU(2) — symmetry. Dynamic entanglement is due to SU(2) symmetries of the Hamiltonians of the matter-radiation interaction. We also define some entanglement measures, which are related to characteristics of light depolarization. Applications of results obtained in biphoton optics are briefly discussed.     

Introduction to 𝒫𝒯-symmetric quantum theory

In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a Hamiltonian, one writes H?=?H ^?, where the symbol ? denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate. In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space?–?time reflection symmetry (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian 𝒫𝒯-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian 𝒫𝒯-symmetric quantum theories.     

How to test for diagonalizability: the discretized PT-invariant square-well potential

Given a non-Hermitian matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note explains how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upon discretizing the configuration space, the system is described by a matrix of dimension three which turns out not to be diagonalizable for a critical strength of the interaction. The systems develops a three-fold degenerate eigenvalue, and two of the three eigenfunctions disappear at this exceptional point, giving a difference between the algebraic and geometric multiplicity of the eigenvalue equal to two. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Product-State Approximations to Quantum States

We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.     

PT-symmetrized supersymmetric quantum mechanics

Supersymmetry between bosons and fermions is modelled withinPT-symmetric quantum mechanics. A non-Hermitian alternative to the Witten’s supersymmetric quantum mechanics is obtained. Work supported by the grant Nr. A 1048004 of GA AS CR. Presented at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems

Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or -symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.     

Experiments in PT-Symmetric Quantum Mechanics

Extended quantum mechanics using non-Hermitian (pseudo-Hermitian) Hamiltonians H = H is briefly reviewed. A few related mathematical experiments concerning supersymmetric regularizations, solvable simulations and large-N expansion techniques are summarized. We suggest that they could initiate a deeper study of nonlocalized structures (quasi-particles) and/or of their unstable and many-particle generalizations. Using the Klein-Gordon example for illustration, we show how the PT symmetry of its Feshbach-Villars Hamiltonian H ^FV might clarify experimental aspects of relativistic quantum mechanics.     

Maximal entanglement of bipartite spin states in the context of quantum algebra

In the framework of the U _q (su(2)) quantum algebra, we investigate the entanglement properties of two-spin systems, of arbitrary spins j ₁ and j ₂, defined in an entanglement of deformed spin coherent states of each of the spins. We derive the amount of entanglement and we give conditions under which bipartite entangled states become maximally entangled. Using these conditions, we construct a large class of Bell states for any choices of the parameters that specify the spin coherent states.     

The C Operator in iϕ 3 Field Theory

The C operator defines a dynamically-determined positive-definite metric in PT-symmetric theories. We show how the operator formalism for the perturbative calculation of C can be extended from quantum mechanics to quantum field theory with a cubic self interaction.