Sungwook Lee

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.