Just as he was pondering another difficult point, the phone suddenly rang.
Seeing Daniel's name displayed on the screen, Robert Stephen eventually answered the call, showing respect.
"Hi, Robert, did you look at the problem I sent you in the email?"
"Hmm? I haven't checked my emails today."
"Oh, if you're still researching Super Helical Space Algebra, I suggest you take a look now. I sent you a set of problems on Super Helical Space Algebra that the Institute for Advanced Study specially designed, you can try to solve them."
"Don't mention it, just remember to buy me a drink next time you're in Princeton. By the way, if you can't solve them, you can contact Edward and ask him for the answers... Sixty percent of the problems in this set were proposed by him, but he doesn't plan to publish the answers yet."
"Got it, thanks again."
After hanging up the phone, Robert immediately opened his email.
There's no doubt that practicing problems is one of the fastest ways to enter a new mathematical field.
Unfortunately, when it comes to Super Helical Space Algebra, a brand new direction, devising problems requires a deep understanding of the related theories first.
So, as things currently stand, even wanting to practice problems is difficult.
Soon, the relevant file was downloaded.
Opening the file, Robert Stephen first browsed through all the problems quickly.
There were a total of six problems, but it was clear they were of high quality.
Robert Stephen then focused his energy on the first problem:
"Consider a one-dimensional Super Helical Space Algebra model, with Hamiltonian H=−t∑(over N, under j)=1 (cfjcj+1↑+cfjcj+1↓+h.c.)+U∑j=1Nnj↑nj↓−μ∑j=1N(nj↑+nj↓)"
"where cjσ and †cjσ† are the annihilation and creation operators for electrons at position j. σ=↑,↓ denotes spin, njσ=cjσ†cjσ is the electron number operator. t is the electron transition strength, U is the Hubbard interaction strength, μ is the chemical potential."
"a, Prove the commutation relation [H,cjσ]=−t(cj−1σ+cj+1σ)+U(nj,−σ−njσ)cjσ.
"b, Consider the mean-field approximation of the system, and assume ⟨cjσ†clσ′⟩=δj,lδσ,σ′⟨cjσ†cjσ⟩, where ⟨cjσ†cjσ⟩ is the average electron number at spin σ and position j. Write the Hamiltonian Hmf under the mean-field approximation."
It has to be said that this problem was very well-crafted.
Having researched superspiral space algebra for two months, Robert could see that this question tested a basic understanding of the superspiral space algebra model. It has to be said, in targeting new algebra research, Princeton has once again moved to the forefront of the field.
Quickly, Robert became engrossed in the problem.
It must be said, being able to solve problems while researching a brand new field of mathematics is also a form of happiness.
After scribbling and tweaking for three hours, Robert finally completed the problem-solving process, matching the second problem's answer nj↑nj↓≈nj(nj−1).
A full sense of accomplishment.
Excited, Robert took a photo of his solution process and sent it straight to Edward Witten, casually asking, "Is my solution correct?"
After sending the email, Robert glanced at the time; it was already one o'clock in the morning.
He had no expectation that Edward Witten would reply to his email at this time.
Feeling a hint of drowsiness, Robert was planning to clean up and go to sleep. Just as he put away all the manuscripts on his desk, the speaker suddenly emitted the sound of an email notification. Instinctively, he opened his mailbox and saw... Edward was also up.
"Congratulations, Professor Stephen, although there are some minor flaws in the proof of the first question, on the whole, it is correct. I'd also like to know, what do you think of these problems? Aside from the six example problems in the first part, there are six more problems in the second part, and I am considering releasing this entire question bank to the public."
After pondering for a moment, Robert began drafting an email.
"The problems are very meaningful, Professor Witten, and they are very helpful to me, helping me to organize some basic concepts of this new algebra direction. Due to the often lacking symmetry in this special space, the commutation problem can only be considered under very special circumstances, which leads to an extremely abstract mathematical framework."
"Your problems help materialize some abstract theories, which is significant for everyone's understanding of superspiral space algebra. If I may be so honored, I would very much like to join your team!"
After clicking the reply button, Robert Stephen suddenly felt wide awake.
One could only say that the dedication of mathematicians is hard for the average person to imagine.
Luckily, Edward Witten soon replied to him.
"Thank you for your feedback, and you're very welcome to join us. Unfortunately, we have missed out on something. This has resulted in our slow progress in understanding and researching the foundational theories of superspiral space algebra. Tonight, I will put all the problems into the shared question bank at the Institute for Advanced Study in Princeton and update them periodically.
"Of course, if you have any good problems, you can also send them to me or to Professor William. After cross-checking, they will also be added to the corresponding question bank. I must admit, this is indeed a very interesting research direction. Qiao's research is astonishing."
After reading this reply, Robert Stephen just felt dispirited.
It was that Qiao Ze again.