My research interests fall into three categories: PT-symmetric quantum mechanics, quantum field theory, and mathematical physics.

1. PT-symmetric quantum mechanics

One of the well accepted axioms in quantum mechanics is that the Hamiltonian of a non-dissipative system must be represented by a Hermitian matrix. Hermiticity guarantees all energy eigenvalues to be real. However, Hermiticity is only a sufficient condition to have a real spectrum. It is not necessary. Many Hamiltonians with an entirely real spectrum appear to be non-Hermitian. Some of these Hamiltonians were shown to exhibit PT symmetry. The most striking feature of a PT-symmetric quantum system is that the PT symmetry may be spontaneously broken. This phenomena is called PT phase transition. PT phase transition has not only been predicted in theoretical studies, but also been found in various experiments.

There are many open questions in this quickly expanding field. In the long term, I am trying to answer the following questions.
How to classify all the non-Hermitian PT-symmetric matrices?
What kind of Jodan block structures may form in a PT-symmetric Hamiltonian?
Besides PT phase transition, what other distinction is there between a PT-symmetric system and a Hermitian system?
When PT symmetry is not broken, Hamiltonian may be self-adjoint with respect to a dynamical metric operator. What is the physical consequences of a dynamical metric operator?
How to formulate time-dependent PT-symmetric quantum mechanics?
And many other analogues of quantum mechanical problems in a PT-symmetric system.
By studying PT-symmetric quantum mechanics, we will gain deeper understanding on quantum phenomena in general. Although my research is mainly in the theoretical side of PT-symmetric quantum mechanics, it is highly possible for me to collaborate with experimentalists on the realizations and examinations of PT theories. In particular, I may collaborate with colleagues in the area of quantum optics.

2. Quantum field theory

The standard model is the best theory so far for mankind to understand the fundamental physical world. It is written in the language of quantum field theory. The recently discovered Higgs boson is the last missing piece of the standard model. However, the story is not complete yet. In conventional quantum field theory, the model to described the self-interaction of Higgs particle is called the phi-4 theory. There is a critical flaw for the phi-4 theory: It is not asymptotic free, which make the theoretical prediction difficult in very high energy. PT symmetry may help us to overcome this problem. There is evidence shown that the PT-symmetric version of the phi-4 theory is asymptotic free. The PT-symmetric quantum field theory is far from complete. I am working on various issues in this field. For example, I am studying how to perform path integrals consistently in a PT-symmetric quantum field theory. I am also studying how to compute physical quantities correctly in such a theory.

With my high energy background, I am always interested in open questions in conventional quantum field theory and cosmology. One question I am working on is about the fluctuations in various cosmological models on inflation.

3. Mathematical physics

Every physics student knows that the harmonic oscillator is extremely important in quantum mechanics. The reason is that the corresponding eigenvalue problem is soluble. The harmonic oscillator solution has many applications in perturbation theory and computation physics. Several years ago, together with my PhD advisor, Carl M. Bender, we discovered a new class of exactly soluble eigenvalue problems. We found a new series of polynomials associated with the problem. Now I want to study more details of these polynomials. With these new soluble models, I expect to find applications in perturbation theory and computation physics.

I am also interested in understanding the perturbation theory better and the application for Pade approximation to a perturbative series.