## My Talks

### 2020

- Numbers and Quantum Physics, 9/25/2020, Theoretical Physics Seminar
### 2019

- The Curvature, the Einstein Equations, and the Black Hole, 10/11/2019, 10/18/2019 & 10/??/2019, Physics Seminar
My lecture note with the same title, which is an extended version of the above lectures, is available here.

- What is Linux?, 6/3/2019, 2019 NSF-HBCU Summer School on Computational Modeling of Disordered Solids at the University of Southern Mississippi. Here is the list of basic Linux commands that we discussed in the afternoon lab session.
### 2018

- What is Riemann Hypothesis?, 10/16/2018, Physics Seminar
### 2017

- What is a Quantum Computer?, 5/24/2017, TKX Seminar, School of Computing
### 2016

- on \(\mathscr{P}\)-Hermitian Quantum Mechanics, 5/06/2016, Department of Physics and Astronomy Weekly Physics Seminar
### 2014

- Doing Quantum Physics with Split-Complex Numbers, 2/21/2014, Department of Mathematics Weekly Colloquium (For this time, it is a Mathematics-Physics Joint Colloquium)
### 2013

- Time Travel and Parallel Universes, 9/30/2013, Science Café, University Libraries, University of Southern Mississippi
- Surfaces of Revolution in Hyperbolic 3-Space, 4/26/2013, Department of Mathematics Weekly Colloquium
- Non-Relativistic Quantum Mechanics as a Gauge Theory, 03/01/2013, 2013 LA/MS Section Meeting of the Mathematical Association of America

Abstract: Ever since a novel by H. G. Wells "The Time Machine" was published in 1895, the idea of time travel has been one of the most popular themes in Sci-Fi genre. Even for physicists, time travel is an intriguing notion and they have been arguing without a consensus or a settlement whether it is physically possible for someone to travel forward or backward in time.

In my talk, I will first discuss possibilities of time travel that have been studied by physicists and then I will discuss my own take on time travel, a possibility of the existence of parallel universes (these are not the same parallel universes appeared in string theory) as a consequence if time travel is permitted, and the emergence of a new physics.

Abstract: I discuss how to construct surfaces of revolution with constant mean curvature \(H=c\) in hyperbolic 3-space \(\mathbb{H}^3(-c^2)\) of constant sectional curvature \(-c^2\). It is intriguing to see that while the hyperbolic 3-space flattens to Euclidean 3-space \(\mathbb{E}^3\) as \(c\to 0\), those surfaces approach catenoid, the minimal surface of revolution in \(\mathbb{E}^3\). I also discuss how to construct minimal surface of revolution in \(\mathbb{H}^3(-c^2)\). This work was done with Kinsey Zarske as her undergraduate research project.

Abstract: I propose a new approach to a gauge theoretic treatment of quantum mechanics. In this model, state functions are th elifts of ordinary \(\mathbb C^n\)-valued state functions to the holomorphic tangent bundle \(T^+(\mathbb C^n)\), where we regard \(\mathbb C^n\) as an \(n\)-dimensional Hermitian manifold. When an external field is introduced, the unique Hermitian connection on the holomorphic tangent bundle \(T ^+(\mathbb C^n)\) gives rise to the energy and momentum operators for a particle influenced by the presence of the external field. This obtains the Schrödinger equation that describes the motion of a particle under the influence of an external field.

In this talk, I discuss the lifted quantum mechanics model for the abelian case i.e. when the external field is electromagnetic field.

Abstract: Quantum mechanics is a remarkably intriguing subject of physics and even after more than a century from its birth physicists still do not fully understand it. In fact, we need an interpretation to understand quantum mechanics. While there is a leading interpretation, the so-called København interpretation, there are indeed many different interpretations of quantum mechanics and there is no consensus among physicists as to what quantum mechanics is.

An interesting aspect of quantum mechanics is that unlike other branches of physics, it is built upon complex numbers. How were complex numbers brought into building quantum mechanics? The answer is unlikely found in quantum mechanics textbooks. In this talk, we will attempt to answer this question. The marriage between complex numbers and quantum mechanics, however, does not appear to be an entirely happy one. The main conflict comes with the path integral which calculates particle amplitudes and the remedy physicists came up with, while brilliant, is outrageously weird from both mathematical and physical points of view. We will discuss this also.

It turns out quantum mechanics may be built upon an entirely different number system called split-complex numbers. Interestingly the forementioned conflict with the path integral disappears in this version of quantum mechanics. There are many intriguing ramifications of this version of quantum mechanics. We will discuss some of those. This idea is currently being developed and at the moment we have more questions than answers.

Abstract: I gave a talk on some basic concepts of quantum computing to an audience of mostly school of computing folks who are participating in TKX project, a DOD funded project of School of Computing.

Abstract: The standard quantum mechanics is built upon complex numbers with the standard positive definite Hermitian product. The complex numbers can be also equipped with an indefinite Hermitian product. Can we build quantum mechanics with the indefinite Hermitian product? If so, how different would it be from the theory of quantum mechanics that we know?

It turns out that we can indeed build a theory of quantum mechanics with the indefinite Hermitian product. It is called \(\mathscr{P}\)-Hermitian quantum mechanics. In this talk, I begin my discussion with the 2-state \(\mathscr{P}\)-Hermitian quantum mechanical system and extend it to the general continuum case. I show that so-called \(\mathscr{PT}\)-symmetric quantum mechanics that has been studied by physicists for more than a decade is in fact \(\mathscr{P}\)-Hermitian quantum mechanics. I also show that contrary to the belief of \(\mathscr{PT}\)-symmetric quantum physicists, \(\mathscr{P}\)-Hermitian quantum mechanics (hence \(\mathscr{PT}\)-symmetric quantum mechanics) is not a generalization of the standard quantum mechanics but an alternative theory of quantum mechanics. \(\mathscr{P}\)-Hermitian quantum mechanics exhibits distinctive features. It admits a whole new class of Hamiltonians that cannot be considered in the standard quantum mechanics. The symmetry of \(\mathscr{P}\)-Hermitian quantum mechanics is also very different from that of the standard quantum mechanics. I will also address some serious misconceptions of the current \(\mathscr{PT}\)-symmetric quantum mechanics and offer resolutions that may lead to a viable alternative theory of quantum mechanics.

Abstract: Earlier investigations on the nature of light show that, light must be described by electromagnetic waves or by particles (wave-particle duality). de Broglie hypothesised that what is true for photons should be valid for any particle. We may assign a particle with mass \(m\), propagating uniformly with velocity \(v\) through field-free space, an energy \(E\) and momentum \(\mathbf{p}\). In the wave picture, the same particle may be described by a frequency \(\omega\) and a wave vector \(\mathbf{k}\). We require that these quantities satisfy the equations \[E=\hbar\omega,\ \mathbf{p}=\hbar\mathbf{k}.\] In fact, these equations are satisfied by the photon and that a photon is described by the plane wave \[\psi(\mathbf{r},t)=A\exp[i(\mathbf{k}\cdot\mathbf{r}-\omega t)].\] Following de Broglie, to every free particle, a plane wave shown above is assigned. So, this is how complex numbers entered in the formulation of quantum mechanics.

In this talk, I show that complex numbers are really for light (photons) and assert that split-complex numbers might have been the right choice to describe other particles. I show that quantum mechanics can be completely rebuilt based upon split-complex numbers. The new quantum mechanics exhibits distinct features. Anti-particles show up naturally in this new quantum mechanics. Remarkably, the path integral can be calculated in Minkowski spacetime without turning it into Euclidean path integral via Wick rotation. This new quantum mechanics may also offer an explanation on baryon asymmetry, i.e. an explantion as to why there aren't as many anti-particles as particles in the universe.