42: Chapter 42 The So-Called Truth

When Jiang Lin saw Professor Lu Zhixing's reply, it was early in the morning when he woke up.

Outside the window, the morning clamor of the old residential area was drilling in through the gaps of the aluminum-alloy windows.

Everything gave him the illusion of being in a different world, as if a lifetime had passed.

He rubbed his swollen eye sockets and reached out to touch the mobile phone he had left on the bedside table before going to sleep.

Ever since he sent out the private message, to make it convenient to see the reply at the first moment, he had used the red envelope money he saved up over the New Year to scour the second-hand market for a smartphone in a passable condition.

The screen lit up, and a small red dot hung in the upper right corner of the Douyin private message icon.

Jiang Lin's heartbeat still instinctively skipped a beat.

There was no help for it; he was far too eager for communication.

In his forty years in the Wasteland, he had scraped for food on the red earth like a primitive man, yet his head was filled with calculus and Joseph-Louis Lagrange's quantities.

It was like being in a dark tunnel stretching for forty years, with only one person tunneling forward.

He needed a synchronized dialogue, needed someone to tell him whether the direction he was groping in this dark tunnel was right or wrong.

Impatiently clicking on the icon, he directly saw the message from Quantum Mechanics Is Not That Mystic.

There were two.

Jiang Lin immediately pulled out a chair and sat down in front of the desk.

His gaze inevitably swept across the upper right corner of his retina, where that eerie blue countdown was still ruthlessly ticking away.

[25:23:41:08]

There were still twenty-five days left.

He took a deep breath and began to read the private message with undivided attention.

He read for about ten minutes.

But during these ten minutes, he was like someone who had been thirsty in the desert for a long time and encountered a pool of clear spring water, not daring to gulp it down in big mouthfuls, sipping it bit by bit, repeatedly chewing every word and phrase in his brain for fear of missing any flavor.

After reading it, he took out paper and pen and wrote down three lines of characters.

[Bell's experiment: What is ruled out are local hidden variable models that satisfy specific assumptions, not proving that all hidden variables are impossible.]

[Interpretation boundary: Experiments rule out certain paths, but do not choose an interpretation for people.]

[Uncertainty: It is not an external height-limit bar, but the self-structure of the state space.]

After writing the third line, he stopped for a very long time.

Professor Lu Zhixing's two letters were not exaggerated in word count, but they casually re-sliced the boundaries of the three problems that had entangled his mind through countless long Wasteland nights.

What did Bell's experiment rule out?

How the question of which slit to take in the double-slit experiment is handled under different interpretations.

Whether the uncertainty relation is ultimately an external restriction or something belonging to the quantum state structure itself.

Especially the last one, which was the most piercing.

Because it hit dead center, poking right into the deepest, most hidden fissure in Jiang Lin's old understanding.

He had indeed previously thought of the uncertainty relation as a kind of constraint line.

And told himself that he must never cross this line.

In his original subconscious, Planck's constant in the formula was this line.

And the particle's position and momentum were like two independent entities squeezing each other.

If you occupy more on this side, the other side must make way.

Therefore, the more accurately the position was measured, the more inaccurate the momentum became.

For a very long time, Jiang Lin had firmly believed in this simple and crude mutual compromise.

The cruel survival rules of the Wasteland had long ago taught him one thing.

When facing rules that could not be resisted, do not sentimentally ask why; just obey and utilize them.

And one had to admit that this image of forcibly classicizing and entitizing the microscopic world was indeed extremely useful during the initial learning stage.

It could help him intuitively understand the formulas, help him correctly solve those complex Physics problems by the faint firelight in that dilapidated Stone House in the Wasteland.

And it could help him avoid some low-level logical errors when establishing preliminary quantum models.

But it was too shallow.

Even in essence, this was just a classic macroscopic residue.

Professor Lu Zhixing's words were like a scalpel, slicing open this sugar-coated error.

He originally wanted to write: "Position and momentum are not two quantities."

Just as he finished writing, he felt it wasn't right again.

It wasn't that there weren't two physical quantities.

Nor was it to say that they were merely mathematical illusions.

He stared at this half-sentence for a long time, scratching the paper until it turned black, before rewriting it.

[It is not two independent distributions restricted by external forces, but a statistical expansion of the same state under different observables.]

The position representation and momentum representation are connected by a Fourier transform.

To speak more generally, the uncertainty relation comes from the inner product structure of non-commuting operators and Hilbert space.

It is not that external rules suppress two quantities from narrowing simultaneously, but that the same state itself cannot be sharp simultaneously on these two mutually non-commuting sets of observables.

This is not an external restriction imposed by the universe on particles.

This is not the particle being shackled by the outside world, but the fabric of the quantum state itself having such a texture.

Just like an audio signal, if you compress it into an extremely short pulse, its frequency components will inevitably spread very wide.

If you want its frequency to be extremely pure, it cannot possibly occupy only a single instant in time.

It's not that someone extra prescribed that it couldn't have it both ways, but that the Fourier transform itself does not allow this path.

This is not a restriction of the projector, but determined by the geometric topological properties of the sphere itself.

After a brief thought, Jiang Lin heavily added a sentence beside the third line of words: [Old images can solve problems, but later on they will bite back.]

The tone of this sentence was of a piece with the writing style in his thick Wasteland error index.

It was filled with a sense of a lesson learned.

Looking at the characters he had written, he felt not a shred of embarrassment or shame in his heart.

A long-lost sense of groundedness spread bit by bit along his spine.

Having walked alone in the Wasteland for forty years, he knew that kind of fear all too well.

What he feared most was never doing a wrong problem, nor failing to derive a formula.

What he feared most was that you got an extremely minor concept wrong, but this error could actually continue to be used.

You carried this malicious patch of error, continued to derive downward, continued to build models, and continued to use it until a very, very long time later.

Until one day, when you were about to touch the deeper and more core building of Physics, that bottom foundation suddenly collapsed silently, burying all the painstaking efforts of the first half of your life.

That was true despair.

And now, Professor Lu Zhixing, as a stranger far away on the other side of the network, had pointed out that place that was about to collapse in advance and dropped a stepping stone for him.

It was simply inestimably precious.

At the very end of the two replies, Professor Lu Zhixing did not continue at length, but abruptly asked a question.

"Where did you learn these things? If it's self-study, have you systematically studied mathematical methods of Physics and Linear Algebra?"

Jiang Lin stared at this sentence on the screen, leaned his back against the chair, pondered for a full five minutes, and only then began to deliberate on his reply.

"Thank you, Teacher Lu. The point you pointed out about the residual classical image in the uncertainty formulation has helped me a great deal. Previously, I indeed tended to understand it more as an external restriction rather than the structural relationship of the same state in different representations. Your single sentence enlightened me."

"As for mathematical methods of Physics, I did make up for them myself by following a few classic Physics textbooks and online course materials, but to be honest, I wasn't forged by the rigorous training of a classroom system. Complex variables, Fourier transforms, ordinary and partial differential equations, special functions—I've encountered all of these, and I can barely manage by following Physics textbooks for derivations and calculations. However, if we truly delve into the rigor of boundary conditions, the completeness of function spaces, or the deep correspondence between spherical harmonics and specific physical images, my foundation is still not solid enough."

He had to demonstrate that he had indeed studied them while candidly admitting his shortcomings.

Pretending to understand when he didn't would only expose his true nature even more ugly in front of an expert.

"As for Linear Algebra, the core knowledge of the lower grades is relatively solid. Finite-dimensional matrix calculations and eigenvalue problems are basically not obstacles. If I may say so without fear of your laughter, the operators and commutation relations in Quantum Mechanics actually forced me to re-understand the physical meaning of matrix composition and non-commutativity later on."

"But I would never dare claim that I've received strict professional training. What I am truly unconfident about right now is that when discussing these things within complex inner product spaces, tensor products, infinite-dimensional operators, and measurement theory, my mathematical formulation might appear insufficiently standard and could even become a laughingstock."

Typing up to this point, Jiang Lin sighed and added the final heartfelt sentence.

"What I worry about the most right now is actually not failing to calculate a result, but whether the concepts have been placed in the correct positions. I'm not afraid of calculating wrongly; I'm only afraid of thinking crookedly."

After writing it, he carefully read it through from beginning to end before clicking send.

Thinking that early in the morning it was impossible to get an immediate reply, he put down his phone and went to wash up.

The moment he unscrewed the stainless steel faucet, a slight whistling sound came from the tap water pipe, and immediately afterward, a clear, cool stream of water gushed out noisily, impacting the white ceramic washbasin and splashing fine droplets.

Jiang Lin looked at that ceaselessly flowing clear water, his gaze dazed for a split second.

Subconsciously, he reached out to turn down the valve of the faucet a bit, making the water flow into a thin trickle.

After eating the breakfast his mother had cooked early on, he went downstairs for a stroll, and only after driving away some of the oppressive feeling brought by the Wasteland did he sit back down in front of the desk.

The phone still had no movement.

He wasn't in a hurry either, planning to first dissect Professor Lu Zhixing's reply and correct his old understanding item by item.

The heading of the first page read: Bell's Experiment.

The heading of the second page read: Interpretation Boundary.

The heading of the third page read: Uncertainty and Preparation.

At the bottom of each page, he used vertical lines to divide it into two, drawing two columns.

The title of the left column was: [My Original Understanding]

The title of the right column was: [Professor Lu Zhixing's Correction]

Then he began to recall his past derivation process, writing those rough understandings smelling of the Wasteland on the left, and then extracting Professor Lu Zhixing's precise modern Physics language to write on the right.

This kind of left-and-right sparring comparison could let the brain most intuitively feel the pleasure of cognitive elevation.

Just as he was writing the second page, trying to describe in mathematical language how the wave function branches under the many-worlds interpretation maintained orthogonality, the mobile phone screen on the side suddenly lit up.

Professor Lu Zhixing had replied.

With a heart of gratitude bordering on pilgrimage, Jiang Lin immediately put down his pen and grabbed his phone.

"Since your finite-dimensional Linear Algebra layer has no problem, that shouldn't be the main obstacle. Later, if you continue studying downward and encounter angular momentum, spin, two-particle systems, and quantum entanglement, you must pay attention to switching the Linear Algebra model in your brain from mere matrix arithmetic to the language of linear operators on complex inner product spaces."

"Hermitian operators, unitary transformations, spectral decomposition, commutation relations, tensor products—these five words, these few things, you must repeatedly compact them in your brain until they become subconscious intuition."

"Especially the tensor product; many self-learners die right here. It is not about crudely piecing two matrices together into a big matrix, but combining the state spaces of two systems into a new state space. The reason entangled states are unavoidable is precisely because some states in the composite system cannot be decomposed into simple products of two subsystem states. As long as you haven't truly thoroughly understood this point, looking at Bell's inequalities, spin coupling, and two-particle systems later on will feel like walking in a fog."

"To put it bluntly, the fundamental reason why many places in Quantum Mechanics make people feel so mystical is simply that Linear Algebra has changed into a suit of Physics clothes. In finite-dimensional models, you can first crudely view operators as matrices, state vectors as column vectors, and time evolution as unitary transformations. This image is incomplete, but it is enough to help you pass the first door. When you walk into infinite-dimensional Hilbert space, continuous spectra, and unbounded operators, you can then dismantle this set of scaffolding piece by piece. But if you haven't seen through this process of changing clothes, you will be confused by those strange Physics terms and think it's metaphysics."

Jiang Lin looked at the last few sentences on the screen; his originally tense facial muscles relaxed, the corners of his mouth couldn't help but curve upward, and he let out a dry, amused chuckle.

Linear Algebra is changing clothes.

This metaphor was so down-to-earth.

Of course it wasn't rigorous, but for a self-learner stuck right in a cognitive bottleneck, this sentence was simply a searchlight in the dark night.

It was so interesting that it hit the nail on the head.

His fingers flew as he quickly replied: "Thank you, Teacher Lu, I have been instructed. I'll go over these five concepts again."

After sending it, he wisely did not continue asking further questions.