Chapter 228: The Fare Energy Levels

Three days later, in the Grand Auditorium of the Royal Academy of Sciences.

The venue was packed to capacity.

Everyone’s face bore a similar mixture of confusion and anticipation.

They were like travelers lost in a mathematical forest, desperately in need of a map.

And Lia Farrien was the one who had promised to provide them with that map.

Masters Albin and Lorentz sat in the front row, their expressions solemn, ready at any moment to challenge any statement they deemed to be “heresy.”

Henry and Victor, the two protagonists of the former wave-particle debate, also sat together below the stage today, maintaining a polite but distant gap between them.

Heisenberg sat in an inconspicuous corner, his hat brim pulled low as he tried to hide himself in the shadows.

His heart was both nervous and excited; he felt like a prisoner awaiting judgment, yet also like a believer about to witness a divine revelation.

Ninth-Circle Archmage Horace and President Samuel sat on the rostrum on the second floor; their presence alone represented the highest level of prestige for this public lecture.

Klein accompanied Lia, shielding her from any unnecessary disturbances.

He poured a cup of warm water for Lia and placed it beside the podium.

At exactly nine in the morning, Lia stepped onto the podium.

The entire auditorium fell silent instantly.

Lia scanned the room, her gaze calm and clear.

“Ladies and gentlemen, esteemed mages.”

Her voice, carried through an amplification array, reached everyone’s ears clearly.

“I know that Mister Heisenberg’s paper has brought much confusion to everyone.”

“Physical quantities as matrices, multiplication that does not satisfy the commutative law—these concepts subvert the mathematical understanding we have built since childhood.”

“Today, I stand here not to overthrow anything, nor to prove anything.”

“I simply wish to provide everyone with a way to understand Mister Heisenberg’s thoughts—a metaphor.”

She picked up a piece of white chalk and wrote the title of the lecture on the massive blackboard: “On the Wave Interpretation of Heisenberg’s Matrices.”

“Before we get to the main topic, I would like to talk to everyone about the public carriages that can be seen everywhere in our capital.”

This opening remark left everyone stunned.

They had prepared themselves for a profound and esoteric storm of theory, but they hadn’t expected to hear about public carriages.

Lia ignored the reaction from the crowd and continued, “As everyone knows, riding a public carriage requires paying a fare.”

“Suppose there is a route that starts from Station A and passes through Stations B, C, and D in sequence, eventually reaching the terminus, Station E.”

“For convenience, we have established a simple set of tiered fare standards.”

She drew a straight line on the blackboard and marked five points.

“From Station A to Station B, the fare is two copper coins.”

“From Station B to Station C, the distance is shorter, so it is one copper coin.”

“From Station C to Station D, it is two copper coins.”

“From Station D to Station E, the distance is the furthest, so it is four copper coins.”

“Calculating the total fare then becomes very simple. For example, if a passenger boards at Station B and gets off at Station E, the fare they need to pay is one plus two plus four, which equals seven copper coins.”

The mages below the stage looked at each other; this was practically an arithmetic exercise for apprentices before they even entered the academy.

‘What exactly is Professor Lia trying to say?’

“Now, our transportation department needs to produce a fare schedule to be posted inside the carriages for passenger reference.”

“Lady Elena, who is also the wife of Archmage Horace, provided a very intuitive proposal.”

Lia’s tone carried a hint of playfulness, bringing a slight smile to Archmage Horace on the rostrum.

“Lady Elena suggested that we could set a ‘coordinate’ for each station. Let’s assume the coordinate for Station A is 0.”

“Then, based on the tiered fares, the coordinate for Station B would be 2, Station C would be 3, Station D would be 5, and Station E would be 9.”

“In this way, the fare from any starting point to any destination is simply the absolute value of the difference between the coordinates of the two stations.”

“This ‘coordinate’ could be referred to as a ‘fare energy level’.”

This explanation was very clear and followed the logic mages were accustomed to.

The world is continuous, and everything can be marked within a coordinate system.

“However, Mister Heisenberg raised an objection.”

Lia’s tone shifted as her gaze fell upon the young man in the corner.

Heisenberg’s whole body jolted, and he subconsciously sat up straighter.

“Mister Heisenberg believes there is a fundamental problem with this line of thinking.”

“And that is: an ordinary passenger can never, and will never, be able to directly ‘observe’ what the ‘absolute coordinate’ of a certain station is.”

“For example, if I am a passenger traveling from Station C to Station D, I can only observe that I paid two coins.”

“I have no way of knowing through any means that the coordinate of Station C is 3 or the coordinate of Station D is 5.”

“The only thing a passenger can directly observe and verify is the fact that it costs two copper coins to go from Station C to Station D.”

“This is the most real, the most solid physical reality.”

“Our theory must be built upon these facts that can be observed and verified, rather than upon those so-called coordinates that we subjectively set but cannot directly perceive.”

These words were like a bolt of lightning, clearing the fog in many people’s hearts.

They began to understand why Heisenberg wanted to abandon the concept of an “orbit.”

Because an electron’s orbit, like the “absolute coordinate” of a station, cannot be directly observed!

“Based on this idea, Mister Heisenberg gave up on using one-dimensional coordinates to describe fare rules; instead, he provided a two-dimensional table.”

Lia picked up a piece of yellow chalk and drew a five-by-five grid on the blackboard.

The horizontal and vertical axes of the table were labeled A, B, C, D, and E, respectively.

“The horizontal axis of this table represents the starting station, and the vertical axis represents the destination station.”

“Every number in this table represents the fare from the corresponding start to the corresponding destination.”

She quickly filled the table.

For instance, in the first row, third column, she wrote the number 3, representing the fare from Station A to Station C.

In the third row, fifth column, she wrote the number 7, representing the fare from Station C to Station E.

“Now, every number in this table is a real, observable quantity that can be personally verified by any passenger by taking a trip.”

“As long as you actually travel from Station A to Station C, you can confirm the correctness of this number.”

“This is the prototype of a matrix. It abandons the description of unobservable ‘internal states’ and only concerns itself with ‘inputs’ and ‘outputs’ that can be experimentally verified.”

In the corner, Heisenberg clenched his fists tightly.

‘Professor Lia… she completely understands my ideas! She explained them even more clearly and thoroughly than I could!’

The mages below also erupted into an irrepressible murmur of astonishment.

‘So that’s how it is!’

‘So a matrix isn’t a monster fabricated out of thin air, but a tool that inevitably arose from the pursuit of physical reality!’

Master Albin’s face turned somewhat unsightly. He wanted to object, but found he had no opening to do so.

Lia’s example was too simple and the logic too clear for him to find any flaw to attack.

“Alright, now let us return to the atomic world.”

Lia erased the carriage route from the blackboard but left the two-dimensional table.

“We now have two physical quantities: one called ‘p’, representing the momentum of an electron; and another called ‘q’, representing its position.”

“In the classical world, they are just ordinary numbers. Multiplying them together—p times q or q times p—yields the same result.”

“But now, Mister Heisenberg tells us that in the microscopic world, both ‘p’ and ‘q’ should be represented by such ‘tables,’ which is to say, matrices.”

“This brings about an unprecedented difficulty.”

“How should we multiply two ‘tables’ together? What exactly does it mean to multiply ‘p’ by ‘q’?”

“Or rather, what is the physical meaning of the act of multiplying two tables?”

Lia’s question caused the hall, which had just become a bit clearer, to fall back into deep thought.

Yes, they understood multiplying numbers.

But a table multiplied by a table? What was the meaning behind that?

“To make it easier for everyone to understand, I would like to leave you with a mathematical exercise at the apprentice level.”

Lia wrote two two-by-two matrices on the blackboard.

“Please calculate Matrix I times Matrix II, as well as Matrix II times Matrix I.”

“In the next class, I will reveal the answer and explain the physical significance of matrix multiplication.”

Just then, the chimes for the end of the lesson rang out melodiously.

The lecture came to a temporary close.

Yet, not a single mage in the entire auditorium left.

They put their heads together, debating the problem on the blackboard and the meaning of matrices with great intensity.

The doors to a brand-new mathematical world were slowly creaking open before them.

📚 Also on NHV Novels

SSS-Class Soulmate

FantasyHealinghunter

Read Now →
Not interested