86: Chapter 86 Two Types of Classrooms

After the second mock exam, once the path on the platform was cleared, Jiang Lin could simply take and complete orders.

He no longer needed to devote all his energy to it, and he could carve out time to go to Jiangda University again.

Having not seen Teacher Lu and his senior fellow apprentices for a long time, he felt quite a bit of missing them.

He rode his bicycle to Building B of the School of Physics and headed straight for the third floor.

The door of B304 was half open.

Meng Che was sitting in front of an oscilloscope, pinching a probe in his hand and adjusting the waveform.

Yin Hang was sitting by the window with a printed paper spread out before him.

Yao Siyu was standing in front of a white board covered with dense derivations, evidently thinking about the next step.

As Jiang Lin walked in, Meng Che was the first to react; he put down the probe, turned around, and wore that signature gossipy expression on his face.

"I have not seen you in a long time. I heard you have been auditing classes at the School of Physics recently?"

"Mhm."

The movement between the two finally caught the attention of Yin Hang and Yao Siyu.

Yin Hang closed the paper, and Yao Siyu turned away from the whiteboard; both of them looked at Jiang Lin with obvious curiosity in their eyes.

"What classes have you been auditing? Tell us about them!" Meng Che pressed further.

Seeing that all three looked quite interested, Jiang Lin thought for a moment and said, "A few undergraduate core courses: Quantum Mechanics, Electrodynamics, Theoretical Mechanics, Statistical Physics, and so on. I have listened to a bit of each."

When he finished saying this, the laboratory fell quiet for a few seconds.

"These four courses belong to the four hardest undergraduate core courses at the Jiangda University School of Physics," Yao Siyu sighed, her tone full of emotional reminiscence.

Obviously, learning these courses back then had not been easy either.

"How does it feel to audit them?" Yin Hang could not help asking either.

"Some chapters are explained in more detail than the textbooks I have read before," Jiang Lin weighed his wording, "but overall, the textbooks I have read are perhaps deeper than what these classes teach."

"What do you mean by deeper?" Meng Che asked.

Although all three knew Jiang Lin's level was very high, they had not really tested just how high it was.

Jiang Lin thought for a moment and explained: "For example, when Professor Zhao taught the chapter on the WKB Approximation, he mainly covered standard one-dimensional barrier penetration and the WKB conditions for bound states, and explained the derivation of connection formulas. But he did not talk about the relationship between the WKB Approximation and the asymptotic expansion of Quantum Field Theory, nor did he talk about the extension of the WKB Approximation in nonlinear systems."

Hearing this, Meng Che and Yin Hang simultaneously wore expressions of as expected.

Only Yao Siyu's thoughts were slightly complicated.

She had read for her undergraduate degree at the Jiangda University School of Physics, was recommended for graduate studies under Professor Lu Zhixing, and had always been ranked first all the way through.

Although she knew that her high school junior fellow apprentice, Jiang Lin, was amazing and capable of helping Professor Lu solve problems, sometimes she still subconsciously thought that the junior apprentice was only outstanding in certain fields.

Hearing the things mentioned by the junior apprentice just now, she realized that the other person's breadth of knowledge might exceed her own.

Just as Yao Siyu was muttering to herself, Professor Lu Zhixing returned.

"Teacher Lu."

"Jiang Lin is here."

Professor Lu Zhixing placed the folder on the desk, picked up the cup of tea on the desk that had been brewed who knows when, and took a sip.

Sitting down in front of his desk piled with papers and printouts, he beckoned to Jiang Lin.

"Jiang Lin, I heard you have been auditing classes at the School of Physics recently. How does it feel? Is it too simple?"

"A little bit."

"Since you find it simple, why do you not audit graduate classes?"

Jiang Lin was slightly stunned.

"I can introduce you to a few professors who teach them."

Professor Lu Zhixing opened the folder and pulled out a printed class schedule from it, on which three courses were circled in red by him.

"Introduction to Quantum Field Theory, Professor Xu Wenqing. Advanced Quantum Mechanics, Professor Sun Changqing. Advanced Electrodynamics, Professor Li Huamin."

He handed the class schedule to Jiang Lin, who took it and looked at the red circles on it.

"These three are core courses at the graduate level, a tier higher than undergraduate core courses. Professor Sun's Advanced Quantum Mechanics leans towards mathematical structure; Professor Li's Advanced Electrodynamics is famous for its rigor; and Professor Xu's Introduction to Quantum Field Theory is actually not introductory at all—it has great depth. You can start with these three. After listening to them, if you still feel it is too simple, let me know."

"Okay." Jiang Lin nodded, folded the class schedule, and put it into his pocket.

"If you feel it is not simple, let me know too," Professor Lu Zhixing added.

"Thank you, Teacher."

"Go back to your busywork," Professor Lu Zhixing finally said. "If you have any questions, come to B304 anytime."

April 20th, Wednesday.

Today was the first audited class for Advanced Quantum Mechanics, from 2:00 PM to 3:30 PM, located in Classroom A402.

Jiang Lin arrived at one fifty-five.

Classroom A402 was a small classroom, much smaller than the lecture halls used for undergraduate classes, capable of seating only around twenty people.

The desks and chairs were those old-fashioned connected wooden desks, with the tabletops carved full of formulas and doodles by countless generations of students with pencils and ballpoint pens.

The blackboard still retained the handwriting from the previous class; one could tell it was group theory content, filled with the Lie algebra structure of SU(3).

There were already more than a dozen graduate students in the classroom.

Completely different from undergraduate classes, no one was talking, no one was playing on their phones, and most people had notebooks and printed handouts spread out in front of them—some were reading books with their heads down, others were resting their eyes with them closed.

There was a quiet pressure in the air, as if a group of people were all preparing to face a known difficult matter.

When Jiang Lin walked into the classroom, a few people raised their heads to glance at him, but their gazes touched and instantly parted.

No one showed surprise.

Strange students occasionally appeared in the graduate student circle.

They might be auditors from other majors, undergraduates taking courses in advance, or visiting students of a certain professor.

It was not strange for a younger-looking person to sit in the classroom.

Jiang Lin chose a seat in the second row near the aisle.

This position was close enough to the blackboard to see every detail clearly, while not being in the first row.

At two o'clock sharp, the door of the classroom was pushed open.

Professor Sun Changqing walked in.

He was around sixty-some years old, with completely white hair combed very neatly.

He wore an old-fashioned pair of gold-rimmed glasses with very thick lenses, showing severe nearsightedness.

He was wearing a dark blue Zhongshan-style jacket that was slightly washed out but very clean, with the top collar button fastened.

In his left hand, he held an English textbook with frayed edges from being flipped through, and in his right hand, he held an enamel teacup, on the body of which the red characters "Institute of Physics, Chinese Academy of Sciences" were printed, though the words were almost worn away.

He placed the textbook and teacup on the podium without taking attendance, without pleasantries, and without even saying hello to the classmates.

He merely glanced at the students in the classroom, his gaze pausing on Jiang Lin for a brief moment before moving away.

"This class will continue discussing the Dirac Equation."

Professor Sun turned around and wrote the covariant form of the Dirac Equation on the blackboard, then began lecturing.

His lecturing style was very special; unlike some teachers who tried to chew up knowledge and feed it to students, he was like an old scholar discussing problems with colleagues.

Many basic derivations were skipped directly, assuming that those listening already knew them.

He mentioned the commutation relations of the Dirac matrices and moved right past them.

For the classification of spinor representations, he drew a table on the blackboard with chalk and then continued moving forward.

For the form of plane wave solutions, he wrote down two formulas and began discussing their physical significance.

Jiang Lin could completely keep up.

In fact, not only could he keep up, but he could also effortlessly cross-reference his existing cognition one by one in his heart.

Every form of the Dirac Equation, every derivation path, and every physical application precisely corresponded to the contents in his memory.

During the second Wasteland period, he had repeatedly used this equation.

When dealing with relativistic electrons in a plasma environment, the Dirac Equation was a fundamental tool.

When calculating quantum state evolution under strong magnetic field conditions, the exact solutions of the Dirac Equation were what he relied on to survive.

The content Professor Sun lectured on almost completely overlapped with what Jiang Lin already knew by heart.

Until around forty minutes in, Professor Sun wrote a line of heading on the blackboard.

Extension of the Dirac Equation in Curved Spacetime.

Professor Sun's lecturing speed finally slowed down.

He seemed to know that this was a difficult point, the place where graduate students were most likely to fall behind.

Starting from the principle of general covariance, he step-by-step introduced the concept of the vielbein, explaining that in curved spacetime, the Dirac matrices were no longer constant matrices, but fields depending on spacetime coordinates.

He wrote out the expression for the spin connection, derived the form of the covariant derivative in curved spacetime, and gave the complete Dirac Equation in curved spacetime.

Jiang Lin began to take notes.

This was his first time actively taking notes in a graduate class.

During the previous forty minutes, he had only occasionally written down a few keywords and listened most of the time.

The vielbein, spin connection, and Lorentz connection decomposition involved in the curved spacetime Dirac Equation were fields he had hardly touched upon when studying plasma Physics in the Wasteland.

In the Wasteland, he mainly focused on applied Physics problems in flat spacetime; although he knew the basic concepts of curved spacetime, he had never systematically deduced them.

And Professor Sun Changqing explained it in great detail.

He used a geometric intuitive method to explain the meaning of the vielbein.

At every point in curved spacetime, you need to establish a local flat coordinate system; the vielbein is the bridge connecting this local coordinate system and the global coordinate system.

The Dirac spinor transforms under the local coordinate system, but its covariant derivative needs to incorporate the effects of spacetime curvature—this effect is the spin connection.

He wrote in his notebook:

[Vielbein — Transforming global coordinates into local Lorentz transformations]

[Crucial: What the Dirac matrices satisfy in curved spacetime is the Clifford algebra of curved spacetime, not that of flat spacetime].

When writing them line by line, he occasionally drew a simple diagram to aid understanding.

The graduate student sitting next to him, a bespectacled boy who looked about twenty-four or twenty-five, inadvertently glanced at Jiang Lin's notes and then froze for a moment.

Jiang Lin's notes did not copy the blackboard at all; instead, he used his own language to reorganize the framework of the derivation, and the marginal annotations were filled with understanding and supplements to the derivation steps.

"Is this not what the professor often calls cognitive transformation?" the bespectacled boy muttered.

On the podium, Professor Sun Changqing finished writing the last formula for the curved spacetime extension, put down the chalk, and patted the dust off his hands.

"This section is relatively difficult."

This was the first sentence he said throughout the entire class that was not Physics content.

"Go back and deduce it yourselves; understanding it and being able to deduce it are two different things."

Having said that, he glanced at his watch, picked up the enamel teacup, and took a sip.

The tea was obviously completely cold, but he did not care.

Then he announced the end of class.

Some people gathered around to ask questions.

Jiang Lin sat in his seat, read his notes from beginning to end again, and marked several places with a red pen.

These were places where he felt he needed to further consult literature.

In the evening, he added a small section to his notebook.

[Curved Spacetime Dirac Equation — Vielbein Method — The key difference from flat spacetime lies in the spin connection term — This term degenerates into the spinor-gravity interaction term in the weak field limit].

After writing these, he opened a General Relativity textbook on his computer and began checking several key concepts mentioned in Professor Sun's class.

This was the first time during his sixth return that he truly learned something new in class.

The next day, the class Jiang Lin wanted to audit was Advanced Electrodynamics.

According to Teacher Lu's introduction before class, Professor Li Huamin was one of the strictest professors in the Jiangda University School of Physics. His research direction was classical Electrodynamics and radiation theory, and he had published several papers in PRL in this field.

He was notoriously strict in class, frequently arguing on the spot with graduate students over the details of a formula. It was said that a student had once been questioned by him to the point of standing on the podium for ten minutes without being able to say a word.

But he deeply appreciated students with depth.

At nine o'clock sharp, Li Huamin pushed the door open and entered.

He was around fifty years old, with a slender build, a very straight back, and a fast walking pace.

He wore a dark brown suit jacket, with the shirt buttons inside fastened to the very top one, without a tie.

He wore rimless glasses; his eyes behind the lenses were not large, but his gaze was very sharp, sweeping across the classroom like making some kind of rapid evaluation.

He placed a stack of handouts and a textbook on the podium, and spoke directly without looking at anyone.

"Today we will discuss higher-order expansions in radiation theory."

Without any pleasantries, without an opening remarks, and not even mentioning where the last class left off.

He assumed everyone was prepared and that everyone had previewed today's content.

Li Huamin's pace was faster than Professor Sun Changqing's, and the information density was greater.

He did not explain basic concepts.

Multipole expansion, Liénard–Wiechert potentials, near-field and far-field decomposition of radiation fields...

These were things he thought graduate students should already know by heart.

He started directly from the higher-order corrections of the radiation field tensor, writing tensor expression after tensor expression on the blackboard, the sound of chalk hitting the board being fast and dense.

Jiang Lin followed very closely.

Electrodynamics was one of the fields he used most in the Wasteland; electromagnetic field calculations in plasma environments were his old specialty.

He was familiar with most of what Li Huamin taught.

But he noticed that Li Huamin's handling methods in certain places were very special, different from the standard derivations in other textbooks.

For example, when dealing with the nonlinear radiation corrections of an accelerating charge, Li Huamin used a direct expansion method in covariant form rather than the component expansion method used in usual textbooks, which made the derivation process much more concise while demanding higher proficiency in tensor operations.

About an hour later, Li Huamin stopped and wrote a problem on the blackboard.

[Problem: Consider the radiation of an accelerating charge in a uniform magnetic field, and prove that the polarization state of its radiation exhibits new components under higher-order approximation.]

After finishing writing, he turned around and pushed up his glasses.

"Ten minutes," he said.

A sound of turning pages echoed in the classroom.

The graduate students lowered their heads to start solving the problem; some frowned, some bit the ends of their pens, and some whispered discussions with the people next to them.

Radiation of an accelerating charge in a uniform magnetic field was a standard Electrodynamics exercise, and most people could produce a basic solution.

The principal component of the radiation field corresponded to the direction of the charge's acceleration, and the polarization state was determined by the geometric relationship between the acceleration vector and the observation direction.

But Professor Li asked about the new components appearing under higher-order approximation.

Jiang Lin glanced at the problem and thought of the key point in about two seconds.

Standard methods for handling this problem would expand the radiation field into various order time derivatives of acceleration, and then take the lowest-order non-zero term to obtain the principal component.

But Professor Li asked about the situation under higher-order approximation.

This meant that higher-order expansion terms of the radiation field needed to be considered.

Under normal circumstances, because these higher-order terms contained the time derivatives of acceleration, they were very small in the non-relativistic limit and were directly ignored.

But under certain special conditions, such as when the charge acceleration and the magnetic field direction had a specific coupling relationship, a component orthogonal to the polarization direction of the principal component would split out from the higher-order terms.

According to Jiang Lin's judgment, this so-called new component corresponded to a term appearing in a higher-order expansion of the radiation field tensor. This term was related to covariance, was non-trivial, and was ignored in most standard textbook exercise solutions.

This was a subtle detail.

Professor Li was clearly testing whether the graduate students could notice this detail.

Jiang Lin picked up his pen and began to calculate.

He did not use the component expansion method.

That was too slow and prone to missing terms.

He directly used the covariant form of the radiation field tensor to couple the four-dimensional acceleration of the charge with the magnetic field tensor, and then performed an eigenvalue decomposition on the polarization tensor of the radiation field.

The derivation process involved four-dimensional curl operations and the dual form of the electromagnetic field tensor, requiring careful handling of the tensor indices at every step.

It took him about six minutes to write out the complete result.

In the polarization tensor of the radiation field, besides the principal component corresponding to the direction of acceleration, a secondary component orthogonal to both the direction of the magnetic field and the direction of acceleration appeared, with an intensity proportional to the product of the magnetic field strength and the magnitude of the acceleration.

After writing the final equals sign, Jiang Lin put down his pen and checked it once.

There were no problems.

He raised his hand.

Li Huamin noticed.

He walked down from the podium at a quick pace, came over to Jiang Lin, picked up the draft paper, and began to read.

His way of reading was different from others.

Instead of reading the derivation process from start to finish, he first looked at the final conclusion, then worked backwards to check the rationality of the key steps.

This was the reading style of experienced scholars; they did not need to follow the derivation step by step, and could judge whether the entire derivation held up by only looking at the key nodes.

He looked at it for two minutes and then looked up.

"You have one more term in this component than my standard answer."

Jiang Lin was slightly stunned.

He was very confident in what he had calculated.

On the Wasteland, he had performed countless electromagnetic field calculations far more complex than this, involving nonlinear radiation corrections in high-density plasmas, synchrotron radiation polarization evolution under strong magnetic fields, and even processing quantum corrections of radiation fields under certain extreme conditions.

His tensor operations almost never had errors.

This was the muscle memory honed by the Wasteland World.

But Professor Li said he had one extra term.

This was somewhat intriguing.

Li Huamin continued to look at the draft paper.

His gaze swept back and forth across Jiang Lin's derivation steps twice, his lips moving slightly as if recalculating in his mind.

Then his brow furrowed, and he mused, "No, you didn't calculate too many terms."

His voice wasn't loud, but the classroom was so quiet that everyone heard it.

"My standard answer missed a term."

Several graduate students raised their heads and looked over this way.

A female student sitting next to Jiang Lin even showed an expression of surprise and confusion.

Li Huamin looked at Jiang Lin, his gaze completely different from before.

Previously it was scrutiny, and now it was a gaze carrying an evaluative undertone: "How did you calculate this term?"

Jiang Lin thought about how to explain it.

What he actually used was a high-order expansion in covariant form, directly processing the four-dimensional form of the electromagnetic field tensor and then extracting the polarization structure of the radiation field from it.

After a brief moment of thought, he simplified his answer and said, "There is a term in the high-order expansion of the radiation field tensor, which corresponds to the coupling between the charge acceleration and the magnetic field direction, rather than simply the charge acceleration. When performing component expansion using standard methods, this term is easily missed because it requires considering the mixed tensor structure of both the electric and magnetic fields simultaneously."

After listening, Li Huamin smiled.

This was the first time Li Huamin smiled throughout the entire class, and it was a heartfelt, gratified smile.

Then he briefly explained, "On this problem, they usually can only calculate the principal component. Being empirical, I forgot to add this term while preparing the lesson."

Finishing speaking, he turned and walked back to the podium, picked up a piece of chalk, and said to the class, "Class, hold on a moment."

Everyone raised their heads.

Li Huamin pointed in Jiang Lin's direction: "There are actually two components to this problem. My previous standard answer missed one, and this student has calculated both complete components. I will now write the correct answer on the blackboard, and you should make a note of it."

He began to rewrite the complete answer on the blackboard.

The first term was the standard principal component, the result calculated by most graduate students.

The second term was the extra component calculated by Jiang Lin.

Li Huamin framed it separately with red chalk and drew two horizontal lines beneath it.

As he wrote, he explained the physical origin of this component.

The polarization rotation caused by the coupling of acceleration and the magnetic field was very small in the non-relativistic limit, but would be significantly enhanced under conditions of high-energy charges or strong magnetic fields.

All the graduate students raised their heads; some picked up their phones to record video, while others took notes quickly.

When the bell rang for the end of class, Li Huamin had just finished writing the last formula.

The graduate students began to pack their things, and Jiang Lin also closed his notebook.

But before he could stand up, Li Huamin had already walked up to him.

"Your ability to capture details like this is very rare; don't let it be buried."

Jiang Lin was slightly stunned.

These words were not a compliment; they were heavier than a compliment.

A compliment meant you did well, whereas what Li Huamin said meant not letting it go to waste.

The premise of this sentence was that he already acknowledged that you possessed something uncommon, and he was worried that this thing would be wasted.

Jiang Lin stood up and found that his voice was lighter than expected: "Thank you, Teacher."

Li Huamin nodded and turned to leave.

Jiang Lin stood beside his seat and lowered his head to glance at his draft paper.

That sheet of paper was densely filled with formulas, and in the bottom right corner of the paper, he had drawn a small circle containing an abbreviated note.

[Covariant form automatically captures cross-coupling — component expansion easily missed].

He folded the draft paper, tucked it into his notebook, and walked out of A401.

The sunlight outside was very bright, shining on the gray exterior wall of the Physics building and reflecting a warm hue.

Jiang Lin got on his bicycle, and as the breeze blew over, he realized that his back was slightly damp.

It was not nervousness, but a physical reaction after high concentration.

On the Wasteland, he was frequently in this state, but in the Real World, he hadn't experienced it for a long time.

April 22, Friday.

Introduction to Quantum Field Theory.

Professor Xu Wenqing, who taught the course, was one of the most prestigious professors in the School of Physics at Jiangda University. Around sixty years old, he was a senior scholar in the field of quantum field theory in China, with research directions primarily focusing on the renormalization group and quantum chromodynamics.

He had a massive influence in the domestic field theory community. Back then, after completing his postdoc at Cornell University, he returned to China and single-handedly established the field theory research direction at Jiangda University.

According to Teacher Lu, the characteristic of Professor Xu's lectures was an extremely fast pace and extremely high concept density, posing considerable difficulty even for graduate students.

"You might find this course challenging."

According to the syllabus, the first few classes of this course covered canonical quantization, free field theory, and perturbative expansion, all of which he was very familiar with.

On the Wasteland, he had used field theory methods to handle collective excitations in plasma, calculated nonlinear response functions using perturbation theory, and performed Feynman diagram calculations many times.

But today's topic was different.

Professor Xu looked slightly younger than Professor Sun; his hair was graying yet still retained some black, cut very short.

He wore rimless glasses, a light gray shirt with the sleeves rolled up to the middle of his forearms, revealing lean and sturdy forearms.

He held a stainless steel thermos cup in his hand.

Walking up to the podium, he first unscrewed the cup, took a sip, glanced at his lecture notes, and then said leisurely, "Today we will discuss the basic concepts of Quantum Mechanics and the renormalization group."

His tone was very flat, as if he were talking about something that didn't need much emphasis.

But Jiang Lin noticed that several graduate students in the classroom straightened their postures simultaneously.

A boy sitting in front-left of him took a deep breath and opened the textbook in front of him that was as thick as a brick.

《An Introduction to Quantum Field Theory》 by Peskin & Schroeder.

This book had a universally recognized evaluation among graduate students in the Physics department.

The first half was a field theory textbook, and the second half was something that made one doubt life.

And the renormalization group was right in the second half.

Professor Xu directly cut into the concept of the Wilsonian renormalization group, discussing momentum space truncation, the integration of high momentum modes, and the evolution equation of the effective action.

He only explained each concept once and immediately moved on to the next one.

Not giving time for thought, or rather, he defaulted to thinking being something students should do after class, using classroom time solely to transmit information.

Jiang Lin kept up completely in the first class.

He was very clear on the core idea of the Wilson renormalization group.

Introduce a cutoff Λ in momentum space, integrate out the momentum modes higher than the cutoff to obtain a low-energy effective theory, and then observe the evolution of the coupling constant by changing the cutoff.

He had encountered this train of thought when studying the effective field theory of plasma on the Wasteland; although he didn't delve deeply into it at the time, he was familiar with the basic framework.

Before entering the second class, Xu Wenqing began to talk about the β function and fixed-point manifolds.

The β function described the evolution of the coupling constant with energy scale.

When the cutoff dropped from Λ to Λ / b, the rate of change of the coupling constant was the β function.

Physically, the zeros of the β function corresponded to the fixed points of the theory where the coupling constant did not change with scale, and the system was in a scale-invariant state.

Jiang Lin could understand these concepts.

But next, Xu Wenqing talked about the Callan-Symanzik equation, the fundamental equation describing the scaling behavior of correlation functions in momentum space, involving the calculation of renormalization constants and anomalous dimensions.

Then came non-perturbative renormalization.

Instead of perturbative expansion, it directly dealt with the global properties of the renormalization group flow.

Down to critical exponents: near the phase transition point, the relationship between the exponents of the power-law behavior of various physical quantities varying with reduced temperature and the eigenvalues of the renormalization group fixed points.

Jiang Lin felt as if he had returned to the time when he first encountered Quantum Mechanics, beginning to struggle.

It wasn't that he couldn't understand at all.

He could understand the general direction.

The β function described the scale evolution of the coupling constant, fixed points corresponded to scale invariance in critical phenomena, and the renormalization group had profound applications in both condensed matter Physics and particle Physics.

These physical pictures were clear to him.

But for the specific derivation steps, Xu Wenqing used some techniques he had never systematically learned.

For instance, the construction of effective field theory under specific truncation schemes.

Professor Xu wrote a truncation function form unfamiliar to Jiang Lin on the blackboard, and then derived the evolution equation of the effective action starting from this truncation.

For example, the Feynman diagram technique of multi-loop expansion.

When deriving the two-loop correction of the β function, Professor Xu rapidly drew a series of Feynman diagrams and used dimensional regularization to handle divergences; the entire process was smooth like flowing water, but the computational techniques involved were completely new to Jiang Lin.

For instance, the renormalization of non-local operators; when dealing with the anomalous dimensions of composite operators, Professor Xu introduced some operator product expansion techniques that Jiang Lin had never seen before.

These were precisely the things Jiang Lin had actively chosen to abandon during the sixth Wasteland.

The specific contents of statistical field theory and the renormalization group.

At that time, he had more urgent problems to solve in the direction of plasma Physics; although he needed to use the framework of field theory, he didn't have time to delve deeply into it.

As a result, he learned the basic framework of field theory and perturbative calculation techniques, but skipped the details of the renormalization group and multi-loop expansion.

Now, this gap was precisely hit by Professor Xu's class.

A boomerang indeed!

Jiang Lin lowered his head and began taking notes.

Whenever he heard a concept he hadn't systematically learned, he wrote it down in his notebook and marked a question mark next to it.

[Momentum shell integration — choice of truncation scheme — Wilson vs wavefront truncation?]

[Two-loop β function — dimensional regularization — need to supplement multi-loop Feynman diagram calculation techniques]

[Callan-Symanzik equation — anomalous dimensions — relationship with renormalization constant Z]

[Non-local operator renormalization — operator product expansion — to check Wilson's original papers]

At four o'clock sharp, Xu Wenqing closed his lecture notes and said, "Today's content takes time to digest, don't rush, take your time reading it."

Then he carried his thermos cup and left the classroom.

The graduate students began packing their things, some whispering in discussion with an obvious lingering fear in their tone.

Jiang Lin heard two boys talking behind him; one of them said, "Teacher Xu's class is always like this. It's normal not to understand it the first time, and if you can understand it by the third time, you're amazing."

Another replied, "I didn't even take complete notes."

When walking out of the Physics building, the cherry blossoms in front of the building were already withering, covering the ground in white and pink.

Jiang Lin stood at the entrance of the building for a while, watching those petals being blown up by the wind and falling back down.

When returning home in the evening, his mother was cooking in the kitchen.

"You're back."

"Mom, do you need any help?"

"No. Did you go audit a class again today?"

"Yeah."

"Are you tired?"

Jiang Lin thought for a moment: "I'm a bit tired today."

His mother was slightly stunned.

She hadn't heard her son say he was tired for a long time.

At night.

Jiang Lin sat in front of his desk, opened his computer, and created a new document.

[Quantum Field Theory Supplementary Plan — Seventh or Later Wasteland].

[Quantum Field Theory Supplementary Plan]

[Time: Execution starts from the seventh Wasteland, depth determined by the specific duration of the Wasteland]

[Objective: Fill the gap in systematic training on statistical field theory and the renormalization group direction that was voluntarily abandoned during the sixth Wasteland]

[Textbook Selection: Phase 1: 《An Introduction to Quantum Field Theory》 by Peskin & Schroeder — standard textbook from late undergraduate to early graduate school, key chapters 1-10, covering canonical quantization, perturbation theory, Feynman diagrams, and quantum Electrodynamics single-loop corrections. Chapters 11-13 are introductions to the renormalization group and critical phenomena, serving as the main targets to conquer.]

[Phase 2: Three volumes of 《The Quantum Theory of Fields》 by Weinberg — advanced graduate textbooks. Volume 1 establishes the strict mathematical foundation of field theory, Volume 2 delves into the renormalization group and non-Abelian gauge theories, and Volume 3 discusses supersymmetry. Read the sections on renormalization group flow and effective field theory in Volume 2 with emphasis.]

[Phase 3: Two volumes of 《String Theory》 by Polchinski — high-level graduate textbooks. Although string theory is the main thread, the discussion of the renormalization group and conformal field theory in Volume 1 is extremely thorough, providing another perspective for understanding the Wilsonian renormalization group.]

[Phase 4: Wilson's original papers — the initial ideas of the renormalization group. Two Physical Review B papers and one Physics Reports review published by Kenneth Wilson between 1971 and 1974 are the core work that earned him the 1982 Nobel Prize in Physics. The physical intuition and derivation paths in these original papers cannot be replaced by any secondary textbooks.]

[Priority: Renormalization group (derivation and application of the β function and the Callan-Symanzik equation, global structure of Wilson manifolds and fixed points), Feynman diagram techniques for multi-loop expansion, non-perturbative renormalization, scaling laws of critical phenomena and phase transitions]