## Welcome to My Home Page

My name is Sungwook Lee. I am an associate professor of mathematics at the University of Southern Mississippi.

Ever since I was a high school freshman, I have been greatly passionate about physics and my deam of youth was to become a theoretical physicist like Benjamin Lee, a prominant Korean-American theoretical particle physicist who could have won a Nobel prize (would rightly have deserved it if he did) if tragic accident hadn't occurred to him. So I studied both mathematics and physics when I was an undergraduate student. What happened along the way was that the more I studied mathematics the more I was fascinated with the beauty of abstract mathematics so at some point I decided to become a mathematician instead. My passion about physics, however, has never been dissipated, rather it has been getting stronger. Although I am trained as a pure mathematician (a differential geometer), in recent years my research focus has shifted to theoretical physics. I am particularly interested in studying quantum physics, high energy physics and cosmology. Besides physics, I am also interested in studying theoretical computer science and mathematical finance. I like to tackle difficult real life problems (most interesting real life problems are indeed extremely difficult to solve) using mathematics and physics, in particular I am trying to understand (often not so well-defined) complex mathematical structures emerging from biological systems, economics and finance.

### Why am I doing mathematics?

### What fascinate me most in mathematics?

Many things in mathematics fascinate me but for me one thing definitely stands out above everything else. It's the notion of infinite sum. If I were a formalist, I would have been less fascinated with it. From physical point of view, I do not think we fully understand what it is. One question I want to find out an answer to is:
#### What should be physically meaningful definition of infinite sum \(\displaystyle\sum_{n=0}^\infty a_n\) of numbers? Is there a truly divergent series?

#### Does continuum really exist?

#### Riemann hypothesis and quantum physics

### What fascinate me most in physics?

#### What is the shape of a black hole?

#### How can a wormhole be created?

#### Is time travel possible?

#### What are particles?

#### Geometrization of magnetism?

Among known forces in nature (there are four of them: gravitation, electromagnetism, strong force and weak force), magnetism is the only one that we can actually see (meaning visualize). For instance, as we have seen in elementary school science class, iron filings placed in a magnetic field shows magnetic field lines. I wonder if magnetic field actually bends space around its source and magnetic field lines are in fact geodesics of the resulting curved space. If it were true, we would possibly be able to geometrize magnetism as Einstein did gravitation. Even more, in my wild imagination, we may be able to create an artificial black hole or an artificial wormhole using a strong magnetic field.
#### Symmetry and quantum physics

#### A new approach to gauge theory

#### Wick rotation and Euclideanization of quantum field theory

#### Why are antiparticles so rare in the universe?

There is no unique answer as to why one does mathematics. The answer will vary depending on individual mathematicians. A lot of mathematicians are doing mathematics because they are simply fascinated with the mathematical beauty itself. Majority of them are also philosophically aligned with Formalism where mathematics is considered to be a game that is played by a set of rules. For them mathematics does not have to have a connection to physical reality. I used to be one of them when I was particularly fascinated with the simple mathematical beauty presented in algebra, logic and topology. Partly due to my strong interest in physics, my philosophical perspective of mathematics has changed over time. I am now more interested in mathematics inspired by physics and philosophically I am more inclined to Platonism as it is my belief that mathematical reality is also physical reality. This viewpoint may provide an answer to the question once posed by Albert Einstein "How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?"

There are a number of questions arising in mathematics and physics that fascinate me more than others and my research revolves around them hoping to be able to answer some of these questions fully or even partially in the near future.There is no unique way to define what a series \(\displaystyle\sum_{n=0}^\infty a_n\) converges means. The convention is that we define it as the limit \(\displaystyle\lim_{n\to\infty}s_n\) of the \(n\)-th partial sum \(s_n=\displaystyle\sum_{k=0}^n a_k\) as we learned in high school. But this is not the only way we can define the infinite sum. For example, the Grandi's series \(1-1+1-1+1-1+\cdots\) diverges with the conventional summation but it converges to \(\frac{1}{2}\) with Cesàro summation. The sum of all natural numbers \(1+2+3+\cdots\) diverges with the conventional summation but it can be shown to converge to \(-\frac{1}{12}\) using analytic continuation. (Srinivasa Ramanujan was the first one to prove this by assuming that the sum exists but such an assumption is not necessary with analytic continuation.) For more details please read here.

For all those years I have thought the divergence issues (renormalization and regularization) in particle physics have something to do with the way quantum field theory is formulated. For example, in quantum field theory elementary particles are treated as point particles and such a treatment of elementary particles may contribute to the divergence issues in particle physics. However, it could also have something to do with the way infinite summation is defined. I speculate that one day physics may tell us what a correct infinite summation is by resolving all the divergence issues arising in particle physics.

For being a differential geometer, I am quite familiar with smooth manifolds. Most physical theories are built upon smooth manifolds in particular on spacetime continuum. But quantum mechanics made me wonder if space and time are really continua:Max Planck was able to derive so-called Planck's law of black-body radiation by assuming that energy is quantized and this led to the dawn of quantum mechanics. As far as I know there is no observed physical phenomenon that may justify that space and time should also be quantized, albeit quantized space, quantized time or quantized spacetime are no strangers in theoretical physics especially in quantum gravity. Notably Hideki Yukawa's elementary domain theory was an attempt to quantize spacetime in an effort to resolve divergence issues in quantum field theory. What if space and time are really quantized with their respective quanta, say the Planck length \(\ell_P\) and the Planck time \(t_P\)? While there is a possibility that quantized spacetime may eliminate infinities in quantum field theory, it may pose an intriguing fundamental question. Are we using wrong mathematics to study quantum physics? If spacetime is quantized, shouldn't we reformulate quantum physics using quantum calculus (calculus without limits) rather than infinitesimal calculus? On the other hand, quantized spacetime may have an interesting ramification in mathematics. If physical space is not a continuum but a quantized discrete space, it may also imply that irrational numbers do not exist because as seen from Dedekind's cuts the existence of irrational numbers results the existence of real continuum. In fact, no one has ever really seen \(\sqrt{2}\). All we have seen are floating point approximations of \(\sqrt{2}\). If physical space is really quantized, irrational numbers like \(\sqrt{2}\) would be merely an illusion that exists only in our mind.

The Riemann zeta function is the analytic continuation of the Dirichlet's series \[\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}.\] The Riemann hypothesis conjectures that the Riemann zeta function has zeros only at the negative integers (trivial zeros) and complex numbers with real part \(\frac{1}{2}\). An interesting question may be if the imaginary parts of non-trivial zeros of the Riemann zeta function are eigenvalues (energies) of a Hamiltonian \(\hat H\). This is called the Hilbert-Pólya conjecture. Such a Hamiltonian, if it exists, is assumed to be Hermitian but it may not necessarily be the case. Another class of Hamiltonians called \(\mathscr{PT}\)-symmetric Hamiltonians or more generally \(\mathscr{P}\)-Hermitian Hamiltonians may be complex Hamiltonians but they still have all real eigenvalues. So there is a much broader class of Hamiltonians than people previously thought that can be examined for the Hilbert-Pólya conjecture.

There are so many things that fascinate me in physics but only to name a few that I understand to a degree and I have some of my own ideas about in no particular order:

Birkhoff's theorem states that any spherically symmetric vaccum Einstein field equations \(R_{ij}=0\) must be static and asymptotically flat. (The solutions of vacuum Einstein field equations describe empty space outside of a massive body.) This means that the Schwarzchild solution is, up to diffeomorphisms, the only spherically symmetric solution of vacuum Einstein field equation. One may naturally wonder if a static and asymptotically flat solution of vacuum Einstein field equations must be spherically symmetric. One can easily show that there is no static asymptotically flat solution with cylindrical symmetry. However, as shown by Kip Thorn there can be a rotational dust solution of vaccum Einstein field equations with cylindrical symmetry. I do not know if there can be a static asymptotically flat toroidal (a doughnut shape) black hole. I read somewhere that Stephen Hawking proved in his book with G.F.R. Ellis "The Large Scale Structure of Space-Time" that any static and asymptotically flat solution of vaccum Einstein field equations must be spherically symmetric. But I also read his proof is incomplete according to some. I have not had a chance to check these claims by myself yet though.

While we understand pretty well how a certain type of black holes (stellar black holes to be exact but this is not the only type of black holes) can be created by gravitational collapse. A wormhole and a white hole are, on the other hand, merely mathematical solutions of Einstein field equations at the moment and no physical mechanism whatsoever is known as to why and how such things can exist. My wild speculation is that a wormhole and a white hole may be created as a consequence of a black hole. Matter being sucked into a black hole would not just disappear but has to go somewhere. From geometry point of view, since spacetime has no edges, the trajectories of particles fell into the event horizon of a black hole must continue unless particles hit the black hole singularity in which case we don't know what's really going to physically happen to those particles. (Mathematically we know what will happen to those particles due to Penrose's celebrated singularity theorem. It also guarantees that black hole singularity exists. While I do not doubt math in his theorem, I am not so convinced that *physically* it will be absolutely the case.) This hints that there may be an exit for these particles which bypassed the singularity (if it exists), a white hole and the black hole and the white hole are connected by a tubelike region, a wormhole. Is there any way we can observe a wormhole? I cannot think of a direct way to observe a wormhole but there may be an indirect way to observe a wormhole. If we can detect an emission of electromagnetic waves (radio waves and visible light) as powerful as a quasar jet but not coming from any known source such as a quasar, a supernova or a hypernova, it is possible that it is coming from a white hole. If the source is a white hole, it indirectly shows the existence of a wormhole.

My take on this question is that you cannot travel forward in time but you *can* travel backward in time. I claim that you cannot travel forward in time because it would violate causality. Traveling backward in time would also violate causality if one assumes that time is one-dimensional. But if time can be multi-dimensional, the well-known grandfather paradox can be avoided. I personally believe that nature has a built-in chronology protection mechanism, namely nature does not allow any violation of causality nor any tampering with already recorded history while she allows timewarps and time machines. This is a stonger version of Novikov's consistency conjecture. If any tampering with history (including someone traveling backward in time) occurs, a new timeline will be created to prevent history from being rewritten. As a consequence a new reality will be created from that moment a tampering occurred. Is there any means for us to travel backward in time? The answer is affirmative. According to Einstein's general relativity, gravitational field bends space-time around it i.e. it not only bends space but also time. In order to prevent violation of causality, timewarp due to a presence of a black hole will occur in past direction and this will lead to a white hole opening up at an event in the past. Some wormholes may be stable and large enough so that we can travel through. Even if a wormhole is not traversable, we may be able to send a signal to the past through a wormhole. (A message from the future?) If we observe an emission of powerful electromagnetic waves from a white hole, it is likely coming from the future (of a reality different from that of ours). For more details please read my blog article Time Travel, Parallel Universes (Alternate Realities) and the Greys.

This is not a simple question. Since Huygens and Newton, we have known that light exhibits the nature of both particle and that of wave. For material particles, in his 1924 Ph.D. thesis de Broglie assigned wave properties to particles and hypothesized that what is true for photons must be true for any type of particles. de Broglie's hypothesis was first confimed by Davisson-Germer experiment and also later by double-slit experiment. So what kind of entity can possibly exhibit two entirely different features, a particle and a wave? In quantum mechanics, partcles are modeled by localized wave packets. (A wave packet can be considered as, simply put, a bunch of waves that are superposed.) A wave packet actually behaves like a particle as well. Another possible candidate is a particle being a vibrating string. If a vibrating string is very tiny, it may be observed as a particle. Also we know wave equation \(\Box\psi=0\), where \(\Box=-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial x^2}\) is the d'Alembertian, has two different types of solutions: one is Fourier's picture and the other is d'Alembert's picture. Fourier's picture describes a vibrating string and d'Alembert's picture describes a propagating wave. It can be easily shown using trigonometric identities that a vibrating string can be viewed as a propagating wave. So vibrating string model of particle may elucidate wave-particle duality. For this and some other reasons I favor string model of particles while I am not in favor of the mainstream string theory.

Richard Feynman was skeptical about string theory and he said something like (perhpas not in his exact words) "if string theory can calculate the mass of an electron, I will believe it." In spite of its many wonderful mathematical achievements, string theory hasn't been able to produce the mass of an electron yet. In string model of particles, unlike quark model, there is no hierarchy meaning all particles are equal and no particle is more fundamental than others. This is called *nuclear democracy*. If my recollection is right, the term was coined by Berkely particle physicists in 1970's. There are two types of strings in terms of topology or in terms of boundary conditions: open and closed strings. What follows is not the way open and closed strings are characterized in the mainstream string theory. I speculate that open strings at rest would correspond to massless particles as open strings at rest are indistinguishable so they would not have distinct rest masses. On the other hand, closed strings at rest can be distinguished by their radii. I speculate that closed strings would correspond to massive particles and there would be an explicit relationship bewteen the radius of a closed string at rest and its rest mass, say the smaller the radius is the heavier the string (particle) is. (In mainstream string theory closed strings are corresponded to gravitons, quanta of gravitation.)

In quantum mechanics symmetry plays an important role. For example, the conservation of angular momentum is obtained by the rotational invariance i.e. \(\mathrm{SO}(3)\) symmetry of quantum mechanical system. Hamiltonian being Hermitian (i.e. self-adjoint) is also closely related to \(\mathrm{SO}(3)\) or its universal cover \(\mathrm{SU}(2)\) symmetry. \(\mathrm{SO}(3)\) symmetry is the symmetry of \(\mathbb{R}^3\), so considering such symmetry would make sense for non-relativistic quantum mechanics as particles are stationary or moving much slower than the speed of light. However, massive partcles are actually moving close to the speed of light. So in reality (i.e. relativistically) one may consider that a single free particle is in \(\mathbb{R}^{2+1}\) rather than in \(\mathbb{R}^3\) whose symmetry is \(\mathrm{SO}^+(2,1)\). If we accept \(\mathrm{SO}^+(2,1)\) symmetry instead of \(\mathrm{SO}(3)\) symmetry, we can obatin a new quantum theory which is called \(\mathscr{P}\)-Hermitian quantum mechanics, where \(\mathscr{P}\) stands for the parity operator. If a Hamiltonian \(\hat H\) is of the form \(\hat H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\hat V(x)\) (one-dimensional case), \(\hat H\) is \(\mathscr{P}\)-Hermitian if and only if it satisfies \begin{equation}\overline{\hat H(-x)}=\hat H(x).\end{equation} Physicists call \(\hat H\) satisfying (1) a \(\mathscr{PT}\)-symmetric Hamiltonian but in mathematics a function \(f(x)\) satisfying \(\overline{f(-x)}=f(x)\) is called a Hermitian function. It can be easily shown that if \(f(x)\) is a Hermitian function, the real part of \(f(x)\) is an even function and the imaginary part of \(f(x)\) is an odd function. So the examples of \(\mathscr{P}\)-Hermitian potential \(\hat V(x)\) would include \(x^2+ix^3\), \(\cos x+i\sin x\), \(ix^3\), \(ix^5\), etc. One may think that complex Hamiltonians are unphysical but \(\mathscr{P}\)-Hermitian Hamiltonians have all real eigenvalues. \(\mathscr{P}\)-Hermitian quantum mechanics may very well be a viable alternative theory of quantum mechanics with a caveat that \(|\psi|^2\) is no longer interpreted as a probability (because it can be negative) rather it is regarded as a quantity that is required to be preserved under time evolution.

In gauge theory, a field \(\psi: \mathbb{R}^{3+1}\longrightarrow\mathbb{C}^n\) interacting with an external gauge potential (physicists call it a minimal-coupling) is a section of a complex vector bundle over Minkowski spacetime \(\mathbb{R}^{3+1}\) associated with a principal \(G\)-bundle over \(\mathbb{R}^{3+1}\). The base space \(\mathbb{R}^{3+1}\) contains information only on the event of a particle and I belive \(\mathbb{C}^n\) must contain much more information on a particle described by the field \(\psi\) as the image of \(\psi\) belongs to \(\mathbb{C}^n\). So shouldn't we study gauge theory using bundles over \(\mathbb{C}^n\) instead? In gauge theory with the holomorphic tangent bundle \(T^+(\mathbb{C}^n)\) over \(\mathbb{C}^n\), gauge groups (structure groups) are naturally determined (in usual gauge theory gauge groups are chosen). Also the unique Hermitian connection on \(T^+(\mathbb{C}^n)\) naturally gives rise to the minimally-coupled gauge invariant derivative. From initial examination, it appears that all the physical information about minimally-coupled fields is encoded in the holomorphic tangent bundle \(T^+(\mathbb{C}^n)\).

In quantum theory, the amplitude of a particle to propagate from a point \(q_I\) to a point \(q_F\) in time \(T\) is given by \[\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{\frac{i}{\hbar}\int_0^TdtL(\dot q,q)}\] where \(L(\dot q,q)\) is the Lagrangian \[L(\dot q,q)=\frac{m}{2}\dot q^2-V(q)\] and \(Dq(t)\) is the Feynman measure given by \[\int Dq(t):=\lim_{N\to\infty}\left(\frac{-im\hbar}{2\pi\delta t}\right)^{\frac{N}{2}}\left(\prod_{k=1}^{N-1}\int dq_k\right)\] with \(\delta t=\frac{T}{N}\). This path integral, while it makes perfect sense physically, does not converge due to the oscillatory factor appeared as the integrand. What Physicists do about this problem is to take the Wick rotation \(t\mapsto it\) which turns Minkowski spacetime to Euclidean spacetime. Accordingly, the path integral turns into Euclidean path integral \begin{equation}\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{-\frac{1}{\hbar}\int_0^TdtL(\dot q,q)}.\end{equation} The integrand becomes a decaying exponential whose maximum value occurs at the minimum of the Euclidean action. Most physicists appear to be satisfied with this resolution, however to me it is troublesome that the path integral cannot be calculated in actual spacetime and that it must be calculated in Euclidean spacetime which is not physical spacetime. Besides, most Euclidean solutions are approximations and there is no guarantee that these solutions will be stable when they are brought to Minkowski spacetime. Furthermore, analytic continuation via Wick rotation works when the spacetime is flat. So Euclideanization will have a problem when the spacetime is curved i.e. gravitation is considered. This issue with the path integral originated from the fact that quantum mechanics was built on complex numbers. So the question is can we build quantum mechanics on a number system other than complex numbers so that we no longer require Euclideanization in quantum theory? The answer is affirmative. There is no absolute reason (physical or otherwise) for a plane wave function to be complex-valued. Quantum mechanics can be build by using plane waves that are split-complex-valued. This alternative version of quantum mechanics, which I call split-Hermitian quantum mechanics, has distinctive features. \(|\psi|^2\) can be negative but not like \(\mathscr{P}\)-Hermitian quantum mechanics case the sign of \(|\psi|^2\) can be interpreted as a unit charge. As a result, the notion of antipartcles is naturally introduced. The path integral can be made non-oscillatory without changing the signature of spacetime. In fact it would take exactly the same form as (2) but it is still defined in \(\mathbb{R}^{3+1}\), not in \(\mathbb{R}^4\). Therefore we can possibly study quantum theory without a need of Euclideanization.

Physicists speculate that in the beginning of the universe the equal amounts of particles and antiparticles were created. However, antiparticles are so rare in the universe. (Such imbalance between matter and antimatter is called baryon asymmetry in physics.) This is actually a good thing for us, otherwise the universe as we know could not have existed. The question still remains though. Why antiparticles are so rare? Split-Hermtian quantum mechanics hints that in the beginning two universes (sounds an oxymoron, perhaps I should call them biverse?) with different signatures (the one that uses east coast convention, the other uses west coast convention) were created. The one with east coast convention (- + + +) made mostly of matter (our universe) and the other with west coast convention (+ - - -) made mostly of antimatter. (The classification of matter and antimatter is relative. I am sure that the folks in our counterpart universe wouldn't call their stuff antimatter.)