### When China Rules the World

The title of the post can be found on Wikipedia with additional information. But this post is more personal. Indeed China rulez! Few weeks ago I received the following email:

Quantum World-2017

Time: 16th-18th October, 2017

Place: Changsha, Hunan Province, China

On behalf of the organizing committee of CQW-2017, we sent you a letter a few days ago invited you to join us and give a speech at first Annual Conference of Quantum World (CQW-2017), which will be held on 16th-18th October, in Changsha, Hunan Province, China. It seems you have not received that letter yet, so I am writing again to extend our sincere invitation. As we have learnt your valuable contribution to Asymptotic Formula for Quantum Harmonic Oscillator Tunneling Probabilities…, we believe your inspirational speech and participation will highlight this congress a lot!

Under the theme “From E=MC2 to Quantum Industry”, the first Annual Conference of Quantum World (CQW-2017) aims at 200+ oral presentations in Quantum Physics and Mechanics, Quantum Information Science, Quantum Chemistry, Quantum Optics, Quantum Materials, and Quantum System, Quantum Engineering and Application, which cover hot topics with both theoretical and experimental contributions.

The conference venue Changsha, as the capital of Hu’nan province, is a beautiful, creative, historical and cultural city with comfortable climate, unique scenery and convenient transportation. It will give you a special experience on the colligation and integration of the Huxiang Culture with the modern civilization.

Worth mentioning that partial scientific program with speakers’ profile and excellent speech titles has been updated on website, kindly click here to view and give us your valuable advice.

Sincerely yours,

CQW-2017 Organizing Committee

Of course I was surprised, because I do not expect anybody but few experts in the whole world care about my paper. Looks to me like a huge conference industry.

In my blog post Lorentz transformation from an elementary point of view – from blogging to science publishing I wrote about a paper that came as the result of blogging. This is the second paper that I wrote together with prof. Jerzy Szulga, a continuation of our previous paper, A Comment on “On the Rotation Matrix in Minkowski Space-time” by Ozdemir and Erdogdu, http://arxiv.org/abs/1412.5581, Reports on Mathematical Physics, 74(1), 2014, 39-44,
DOI: 10.1016/S0034-4877(14)60056-2. Today I received from my coauthor a message with a copy of another invitation from China:

Dear Dr. ….,

I’m writing to follow-up my last invitation as below, would you please give me a tentative reply? Thank you very much. I apologize for the inconvenience if the letter disturbed you more than once.

It is our great pleasure and privilege to welcome you to join the 8th World Gene Convention-2017, which will take place in Macao, China during November 13-15, 2017. We would like to welcome you to be the chair/speaker in Theme 902: Agriculture, Food and Plant Biotechnology while presenting about A Comment on “On the Rotation Matrix in Minkowski Space-Time” by Ozdemir and Erdogdu…….

If the suggested thematic session is not your current focused core, you may look through the whole sessions and transfer another one that fit your interest (more info about the program is available athttp://www.bitcongress.com/wgc2017/ProgramLayout.asp

Under our SAB members’ contributions and endeavor, BIT’s 7th World Gene Convention-2016 (WGC-2016), successfully held in Shanghai, China during November 13-15, 2016. Totally, there were nearly 300 world-renowned experts, professors, laboratory principals, project leaders and representatives of well-known enterprises attended the WGC-2016. Participants from the international enterprises, academic and research institutions enjoyed the three days scientific program. Depending on the warmly support and good suggestions from all of the participants, we are confident in organizing WGC-2017 which would be better and more successful than WGC-2016.

WGC-2017 features a very strong technical program, mainly focused on: breakthroughs in gene, advances genomics & genetics, new research of DNA and RNA, focus on basic research, the frontier research of life sciences, new biotherapy discovery, emerging areas for medicine applications, robust technology development, and cutting-edge Biotechnology. It aims to provide a platform for all experts from academia, industry and national labs to discuss latest hot researches and achievements. Attendees will hear world-class speakers discussing the challenges and opportunities facing the gene, biotechnology and life sciences field. The business & academic experts who are from home and abroad will give excellent speeches.

In addition to the dynamic scientific program, you will benefit from the wonderful experience in Macao, China. Macao is an international free port. It’s famous for light industry, tourism, and hotel. Macao is also one of the most developed and richest regions in the world, this is a city of amazing and fascinating cultural wealth. The unique blend of European and Oriental cultures existing here creates a pleasurable and laid back atmosphere in a truly beautiful city. We hope you will enjoy your stay in this beautiful city with all its feature, beauty, architecture and hospitality!

We expect your precious comments or suggestions; also your reference to other speakers will be highly appreciated. We look forward to receiving your replies on the following questions:

1. What is the title of your speech?

2. Do you have any suggestions about our program?

We look forward to see you in Macao in 2017 for this influential event.

Sincerely yours,

Ms. Teresa Xiao

Organizing Committee of WGC-2017

It is only Chinese people can have such a broad and brave imagination and vision of the future. To connect Lorentz and spin groups of matrices with genes and agriculture – it is a real feat.

### Pauli, rotations and quaternions

[latexpage]
In Nobody understands quantum mechanics, but spin is fun we met three Pauli matrices $\sigma_1,\sigma_2,\sigma_3.$. They are kinda OK, but they are not quite fitting our purpose. Therefore I will replace them with another set of matrices, and I will call these matrices $s_1,s_2,s_3.$ They are defined the same way as Pauli matrices, except that $s_2$ has a different sign:

$$s_1=\begin{bmatrix}0&1\\1&0\end{bmatrix},\quad s_2=\begin{bmatrix}0&i\\-i&0\end{bmatrix},\quad s_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.$$
We have
$$(s_i)^*=s_i,\quad (i=1,2,3),$$
$$(s_1)^2=(s_2)^2=(s_3)^2=I,$$
$$s_1s_2=-is_3,\quad s_2s_3=-is_1,\quad s_3s_1=-is_2.\label{eq:pro}$$

The Pauli’s matrices $\sigma_i$ would have in Eq. (\ref{eq:pro}) plus signs on the right. We want minus signs there. Why? Soon we will see why. But, perhaps, I should mention it now, that when Pauli is around many things go not quite right. That is the famous “Pauli effect“. From Wikipedia article Pauli Effect:

The Pauli effect is a term referring to the apparently mysterious, anecdotal failure of technical equipment in the presence of Austrian theoretical physicist Wolfgang Pauli. The term was coined using his name after numerous instances in which demonstrations involving equipment suffered technical problems only when he was present.

One example:

In 1934, Pauli saw a failure of his car during a honeymoon tour with his second wife as proof of a real Pauli effect since it occurred without an obvious external cause.

You can find more examples in the Wikipedia article. The car breakdown during a honeymoon with Pauli’s second wife is somehow related to the change of the sign of the second matrix above. And now it is easy to relate our second set of matrices to quaternions. It was Sir William Hamilton who introduced quaternions in 1843. There were three imaginary units $\mathbf{i},\mathbf{j},\mathbf{k}$ satisfying:
$$\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1,$$
$$\mathbf{i}\mathbf{j}=\mathbf{k},\quad \mathbf{j}\mathbf{k}=\mathbf{i},\quad \mathbf{k}\mathbf{i}=\mathbf{j}.$$

Hamilton, when he invented quaternions, he thought of them as of abstract objects obeying simple algebra rules. But now we can realize them as complex matrices. To this end it is enough to define
$$\mathbf{i}=is_1=\begin{bmatrix}0&i\\i&0\end{bmatrix}, \quad \mathbf{j}=is_2=\begin{bmatrix}0&-1\\1&0\end{bmatrix},\quad \mathbf{k}=is_3=\begin{bmatrix}i&0\\0&-i\end{bmatrix},\label{eq:qmat}$$
and notice that the algebraic relations defining the multiplication table of quaternions are automatically satisfied! If we would have used Pauli matrices $\sigma,$ we would have to use $-i$ in the above equations – which is not so nice. You would have asked: why minus rather than plus?
Some care is needed, however. One should distinguish the bold $\mathbf{i}$ – the first unit quaternion, from the imaginary complex number $i.$ One should also note that the squares of the three imaginary quaternionic units, in their matrix realization, are $-I,$ that is “minus identity matrix”, instead of just number $-1,$ as in Hamilton’s definition.
In the matrix realization we also have:
$$\mathbf{i}^*=-\mathbf{i},\quad \mathbf{j}^*=-\mathbf{j},\quad \mathbf{k}^*=-\mathbf{k},$$
where the star ${}^*$ denotes Hermitian conjugation.
A general quaternion $q$ is a sum:
$q= W+X\mathbf{i}+Y\mathbf{j}+Z\mathbf{k}.$
In matrix realization it is represented by the matrix
$$Q=W I+X\mathbf{i}+Y\mathbf{j}+Z\mathbf{k}=\begin{bmatrix}W+iZ&iX-Y\\iX+Y&W-iZ}\label{eq:Qmat}\end{bmatrix}.$$
The conjugated quaternion, $q^*= W-X\mathbf{i}-Y\mathbf{j}-Z\mathbf{k}$ is represented by Hermitian conjugated matrix. Notice that $qq^*=q^* q=W^2+X^2+Y^2+Z^2.$ Unit quaternions are quaternions that have the property that $qq^*=1.$ They are represented by matrices $Q$ such that $QQ^*=I,$ that is by unitary matrices. Moreover we can check that then
$\det Q=\det \begin{bmatrix}W+iZ&iX-Y\\iX+Y&W-iZ}\end{bmatrix}=W^2+X^2+Y^2+Z^2=1.$
Therefore matrices representing unit quaternions are unitary of determinant one. Such matrices form a group, the special unitary group $\mathrm{SU}(2).$ Yet, to dot the i’s and cross the t’s, we still need to show that every $2\times 2$ unitary matrix of determinant one is of the form (\ref{eq:Qmat}). Let therefore $U$ be such a matrix
$U=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$
Since $\det U=ad-bc=1,$ we can easily check that
$\begin{bmatrix}a&b\\c&d\end{bmatrix} \begin{bmatrix}d&-b\\-c&a\end{bmatrix}=I.$
Therefore
$U^{-1}=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.$
On the other hand $U$ is unitary, that means $U^*=U^{-1}.$ But
$U^*=\begin{bmatrix}\bar{a}&\bar{c}\\\bar{b}&\bar{d}\end{bmatrix}.$
Therefore we must have
$\begin{bmatrix}d&-b\\-c&a\end{bmatrix}=\begin{bmatrix}\bar{a}&\bar{c}\\\bar{b}&\bar{d}\end{bmatrix}.$
That is $d=\bar{a},b=-\bar{c}.$ Writing $a=W+iZ, c=Y+iX$ we get the form (\ref{eq:Qmat}).

We are interested in rotations in our 3D space. Quantum mechanics suggests using matrices from the group $\mathrm{SU}(2).$ We now know that they are unit quaternions in disguise. But, on the other hand, perhaps quaternions are matrices from $\mathrm{SU}(2)$ in disguise? When Hamilton was inventing quaternions he did not know that he was inventing something that will be useful for rotations. He simply wanted to invent “the algebra of space”. Maxwell first tried to apply quaternions in electrodynamics (for $div$ and $curl$ operators), but that was pretty soon abandoned. Mathematicians at that time were complaining that (see Addendum to the Life of Sir William Rowan Hamilton, LL.D, D.C.L. On Sir W. R. Hamilton’s Irish Descent. On the Calculus of Quaternions. Robert Perceval Graves Dublin: Hodges Figgis, and Co., 1891. ):

As a whole the method is pronounced by most mathematicians to be neither easy nor attractive, the interpretation being hazy or metaphysical and seldom clear and precise.’

Nowadays quaternions are best understood in the framework of Clifford algebras. Unit quaternions are just one example of what are called “spin groups”. We do not need such a general framework here. We will be quite happy with just quaternions and the group $\mathrm{SU}(2).$ But we still have to relate the matrices $U$ from $\mathrm{SU}(2)$ to rotations, and to learn how to use them. Perhaps I will mention it already here, at this point, that the idea is to consider the rotations as points of a certain space. We want to learn about geometry of this space, and we want to investigate different curves in this space. Some of these curves describe rotations and flips of an asymmetric spinning top. Can they be interpreted as “free fall” in this space under some force of gravitation, when gravitation is related to inertia? Perhaps this way we will be able to get a glimpse into otherwise mysterious connection of gravity to quantum mechanics? These questions may sound like somewhat hazy and metaphysical. Therefore let us turn to a good old algebra.

With every vector $\vec{v}$ in our 3D space we can associate Hermitian matrix $\vec{v}\cdot\vec{s}$ of trace zero:
$$\vec{v}\cdot\vec{s}=v_1s_1+v_2s_2+v_3s_3=\begin{bmatrix}v_3&v_1+iv_2\\v_1-iv_2&-v_3\end{bmatrix}.$$
In fact, as it is very easy to see, every Hermitian matrix of trace zero is of this form. Notice that
$$-\det \vec{v}\cdot\vec{s}=(v_1)^2+(v_2)^2+(v_3)^2=||v||^2.$$
Let $U$ be a unitary matrix. We act with unitary matrices on vectors using the “jaw operation” – we take the matrix representation $\vec{v}\cdot\vec{s}$ of the vector in the jaws:
$\vec{v}\cdot\vec{s}\mapsto U\vec{v}\cdot\vec{s}\, U^*.$
Now, since $\vec{v}\cdot\vec{s}$ is Hermitian and $U$ is unitary, $U\vec{v}\cdot\vec{s}\, U^*$ is also Hermitian. Since $\vec{v}\cdot\vec{s}$ is of trace zero and $U$ is unitary, $U\vec{v}\cdot\vec{s}\, U^*$ is also of trace zero. Therefore there exists vector $\tilde{\vec{v}} =(\tilde{v}_1,\tilde{v}_2,\tilde{v}_3)$ such that:
$\tilde{ \vec{v}}\cdot\vec{s}=U\vec{v}\cdot\vec{s}\,U^*.$
If $U\in\mathrm{SU}(2)$, then $\det{U}=\det(U^*)=1.$ Therefore we must have $||\tilde{\vec{v}}||=||\vec{v}||.$ In other words the transformation (which is linear) $\vec{v}\mapsto \tilde{\vec{v}}$ is an isometry – it preserves the length of all vectors. And it is known that every isometry (that maps $0$ into $0$, which is the case) is a rotation (see Wikipedia, Isometry). Therefore there exists a unique rotation matrix R such that $\tilde{\vec{v}}=R\vec{v}.$ This way we have a map from $\mathrm{SU}(2)$ to $O(3)$ – the group of orthogonal matrices. We do not know yet if $\det(R)=1,$ which is the case, but it needs a proof, some proof….

### Nobody understands quantum mechanics, but spin is fun

[latexpage]
Let me start with a quote from Richard Feynman, “The Character of Physical Law“, The M.I.T. Press 1985, p. 129:

… There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve.
On the other hand, I think I can safely say that nobody understands quantum mechanics.

While we do not understand quantum mechanic, and I am here in a complete agreement with Feynman, we (that is I) find it useful, and we find that we can learn something from it (Quantum Mechanics). We can, rightly or wrongly, get from Her (Quantum Mechanics) some glimpses of “future physics”. In fact many physicists and mathematicians are trying to knock more or less to the same door, though the door still remain closed. The door has a plate on it, and the inscription on the plate reads: Spin, complex, dimensions.

I am going to open this door, just a little, and to stick my head in, to get a glimpse of what is inside.

Inside there is certainly spin, and it is complex. So we go through the door, and enter the first office behind. There are three little Pauli matrices there. They are asking us to play with them.

We can’t refuse.
Pauli matrices are $2\times 2$ complex matrices. Here they are, happy, ready to play with, on the yellow background

$$\sigma_1=\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\quad \sigma_2=\begin{bmatrix}0&-i\\ i&0\end{bmatrix},\quad \sigma_3=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}.$$

Two of them are, in fact, real. One is imaginary. But all three of them are Hermitian:
$(\sigma_i)^*=\sigma_i,$
Where ${}^*$ denotes the Hermitian conjugate, that is “transpose and complex conjugate”. Suppose now that $\vec{k}=(k_1,k_2,k_3)$ is a (real) vector in our three-dimensional space. Then we can build the matrix $$\sigma(\vec{k})=k_1\sigma_1+k_2\sigma_2+k_3\sigma_3=\begin{bmatrix}k_3&k_1-ik_2\\k_1+ik_2&-k_3\end{bmatrix}.$$
The matrix $\sigma(\vec{k})$ is again Hermitian. We do not like it. We like to have anti-Hermitian matrix that we will exponentiate, much like in the previous post we were exponentiating antisymmetric matric. Since we are behind the door to a complex space, we can use imaginary $i$. There are a lot of them in every drawer. So, we choose one and exponentiate: $U=\exp(i\sigma(\vec{k})).$
The matrix $U$ is unitary: $UU^*=U^*U=I.$ This follows from a general formula for the exponential:
$\exp(A)^*=\exp(A^*).$
Thus $U^*=\exp(i\sigma(\vec{k}))^*=\exp(-i\sigma(\vec{k}))=U^{-1}.$
There is also a complex version of the Rodrigues rotation formula that we have seen in the previous post Spin – we know that we do not know. It is a fun to derive it, so why not?

Let $\vect{n}$ be a unit vector (that is of norm 1), that is $\vec{n}\cdot\vec{n}=n_1^2+n_2^2+n_3^2=1.$ We can calculate $\sigma(\vec{n})^2$ and, to our surprise, we find that:
$\sigma(\vec{n})^2 = I.$
Since $i^2=-1,$ it follows that
$(i\sigma(\vec{n}))^2 =-I,$
$(i\sigma(\vec{n}))^3 = -(i\sigma(\vec{n})).$
Now we put $\theta$ in front:
$(\theta i\sigma(\vec{n}))^2 =-\theta^2,$
$(\theta\,i\sigma(\vec{n}))^3 =-i\theta^3 \sigma(\vec{n}).$
Calculating the exponential $\exp(\theta i\sigma(\vec{n}))=I+(\theta i\sigma(\vec{n}))+(1/2!)(\theta i\sigma(\vec{n}))^2+…$ we can collect all even powers and all odd powers to get:
$\exp(\theta i\sigma(\vec{n}))=(1-i\theta^2/2!+i\theta^4/4!-…)I+(\theta-\theta^3/3!+\theta^5/5!-…) i\sigma(\vec{n})$
Or
$\exp(\theta i\sigma(\vec{n}))=\cos\theta\, I+ i\sin\theta\, \sigma(\vec{n}).$
So far so good. Complex proves to be simpler. Nature loves it! For real matrices we have three terms in the Rodrigues rotation formula. For complex matrices we have only two! So, we should really be happy. Except that we do not know yet where are these “rotations” now? That will be in the next post. But anticipating, we rewrite the formula above as
$$\exp(\frac{\theta}{2} i\sigma(\vec{n}))=\cos\frac{\theta}{2}\, I+ i\sin\frac{\theta}{2}\, \sigma(\vec{n}).$$
In the middle of writing this post I went for a bicycle ride. To spin my wheels. I came back and my left monitor was dead. I t just died while waiting for me. It was Samsung SyncMaster 913N. While it is true that it was already 12 years old, but i will miss him! R.I.P.

New monitor will come on Monday. In the meantime I will have only one window.