Two space and two time dimensions - simplified

 Whenever we start working on some project, at first everything we do, seems to be, and often is, very complicated. It is only with time, and with experience, we learn how to simplify our task, sometimes in an extraordinary way. We find shortcuts, we invent tricks, we learn from other experienced people whom we meet due to our unceasing efforts.

I asked AI to provide an example illustrating my preamble above, and here is the result:

"Historical Example: Richard Feynman and the Manhattan Project

Feynman simplified one group’s approach

When Richard Feynman joined the Manhattan Project at Los Alamos during World War II, he was of the brightest minds of the century. At first, the work was incredibly complicated: the project involved calculations of neutron diffusion, nuclear reactions, and bomb design—concepts that were at the cutting edge of physics, and far from straightforward.

Feynman found himself and others bogged down with repetitive, error-prone hand calculations. Every problem seemed daunting. But over time, through relentless experimentation and observation, Feynman began finding clever shortcuts.

One breakthrough came when he noticed that some of the mathematical tables used for calculations were riddled with mistakes. Instead of doing everything manually, he started looking for patterns in the errors and designed faster ways to check and simplify the calculations. He even taught himself how to use mechanical computing devices better than the engineers assigned to operate them, often streamlining the whole process.

In a particularly telling anecdote, Feynman simplified one group’s approach to neutron diffusion by realizing they were unnecessarily solving a full partial differential equation when an approximation and symmetry argument would give an answer almost instantly. This insight not only saved time but brought clarity to the underlying physics.

What had started out as complex and overwhelming became manageable—even elegant—through experience, effort, and creative shortcuts."

In a series of previous posts we were discussing the space R^2,2, two dimensional space and two-dimensional time, to study the conformal compactification of R^1,1, a toy spacetime with only one space dimension. We played with the Clifford algebra Cl(2,2), and we have a somewhat strange doubling. Matrices representing R22 vectors were block-off diagonal, matrices representing Spin(2,2) were block diagonal. They contained a lot of zeros! Why do we need all these zeros? Can't we get the desired result without all these zeros? The Eureka came onto me only yesterday. I checked if my discovery has no errors, and it seems that all works as desired. At the same time we are getting a new insight into the internal machinery of the whole structure. Which makes me happy. Doing all this I recalled one of the songs that I like. It has these words.

And happiness is close, happiness is far.
It is difficult and easy to find

It sounds much better in the original Russian (Роксана Бабаян):

А счастье близко, счастье далеко.
Его найти трудно и легко.

You can find the song online.

There is a movie "Maestro". We have another annoying doubling there: "American composer Leonard Bernstein (Bradley Cooper) lives a double life." The same with our Cl(2,2). We do need all this suspicious doubling. We do not need all these unnecessary zeros. So let us simplify everything from scratch. What we need is R^2,2 and Spin(2,2) isomorphic to SL(2,R) x SL(2,R).

The solution.

R^2,2 is already a Clifford algebra! Namely it is the Clifford algebra Cl(2,0) aka Cl(2). Excellent notes  "Clifford algebra, geometric algebra, and applications" by Douglas Lundholm and Lars Svensson provide the hint in Exercise 2.5, p. 13:

Exercise 2.5. Find an R-algebra isomorphism
G (R²)) → R^{2 × 2} = { 2 × 2 matrices with entries in R } .
Hint: Find (preferably very simple) 2 × 2 matrices γ¹, γ² which anti-commute and satisfy γ₁² = γ₂² = 1_{2 × 2} .

Note. Sec. 2.2 of this paper describes "Combinatorial Clifford algebra" - the concept I was not aware of before.

We will do it in details in the next post.

Quantum Sins and the Art of Being Confused: A Journey Through the Minds of Giants

 

Introduction: Quantum Mechanics – A Beautiful Mess

In my last post, Quantum Sins: Why I’m Not Sold on the Uncertainty of It All, I addressed the “sinful” nature of quantum theory—its deeply probabilistic essence, as lamented in the book The Emerging Quantum: The Physics Behind Quantum Mechanics by Luis de la Peña, Ana María Cetto, and Andrea Valdés Hernández. One could argue that quantum theory has more than its fair share of sins, but the authors have generously narrowed it down to six. Honestly, even that feels like an understatement.

Before diving into these so-called sins, the book curates a delightful assortment of quotes from the scientific elite, those who, while creating the theory itself, seem to share a not-so-secret discomfort with its foundational quirks. So, let's summon the titans of physics and hear their thoughts on this cosmic enigma we call quantum mechanics.

Feynman’s Fog of Quantum Confusion

Let’s kick things off with Richard Feynman, the physicist equivalent of a rock star. He famously said:

“I think I can safely say that nobody understands quantum mechanics.”

And just like that, Feynman perfectly encapsulates the mystique of quantum mechanics. Imagine building a house but never quite figuring out how the plumbing works. Sure, the water flows, but ask how, and you’re met with shrugs. That’s quantum mechanics for you: the water flows, but no one can tell you how the pipes connect.

Referring to matter diffraction, Feynman added:

"A phenomenon which is impossible, absolutely impossible, to explain in any classical way... It contains the only mystery."

Yes, you heard that right. This isn’t just a complicated puzzle; it’s the puzzle. And the best part? No one has a clue about the machinery behind the magic. Quantum mechanics, ladies and gentlemen—where the rabbit hole is both endless and inexplicable.

Gell-Mann’s Grim Acceptance

Next up is Murray Gell-Mann, who tosses his hat into the “we don’t get it, but it works” ring. He describes quantum mechanics as:

“... that mysterious, confusing discipline, which none of us really understands but which we know how to use.”

It's like using your smartphone without ever peeking at the user manual. You don't know why it works, but it does, and that’s good enough. Gell-Mann even went so far as to call quantum mechanics a “framework,” rather than a theory. It’s not a complete explanation, but more of a container, like a philosophical Tupperware. You can stuff your theories into it, but good luck explaining how the lid stays on.

Dyson: Embrace the Mystery, Just Do the Math

Freeman Dyson, ever the pragmatist, had this to say:

“If you want to understand quantum mechanics, just do the math.”

In other words, don’t waste your time trying to interpret what’s going on. Get out your calculator and power through it. According to Dyson, all the poetic language we spin around quantum theory is just that—fluff. The math is where the magic happens, and everything else is window dressing.

Dyson’s philosophy is akin to saying, “If you want to enjoy a good meal, don’t ask what’s in the sausage. Just eat it.” No need to complicate things with big questions. Just trust the process and let the equations do the heavy lifting.

Bell: Weekday Pragmatism, Weekend Dreams

John Bell, known for his groundbreaking work on quantum theory, also found himself straddling the line between practicality and idealism. During lectures, he famously said that he spent his weekdays using the “FAPP” theory (For All Practical Purposes), but on weekends, he returned to his principles and searched for something better.

Bell’s approach suggests that quantum mechanics works fine for the day-to-day grind, but when you get a chance to sit back and ponder life (say, over a Sunday coffee), you can’t help but wonder: Is this really all there is? It’s like living in a city you know well, but every weekend you yearn for the mountains.

Quantum Mechanics: Good, But Dangerous?

Now, what’s my personal take on this? It’s complicated, to say the least. There’s an old saying, “Don’t fix what ain’t broken,” paired with another, “Perfect is the enemy of good.” Quantum mechanics is, without a doubt, good. But is it perfect? Far from it. Is it broken? Well, that depends on how philosophical you’re feeling and which day of the week it is.

For most physicists, quantum mechanics is annoyingly good—like an irritatingly effective app that does the job without letting you peek behind the code. But here’s the rub: its success stunts our ability to push the boundaries of our understanding. It’s so successful that it feels like a roadblock rather than a stepping stone.

In that sense, quantum mechanics isn’t just imperfect; it’s dangerous. It’s the flashy magic trick that distracts us from what’s really going on behind the curtain. And unless we figure out how to peek behind that curtain, our understanding of the universe will remain frustratingly incomplete.

Conclusion: The Quantum Dilemma

At the end of the day, quantum mechanics works—and it works really well. But if you're looking for clarity, don’t hold your breath. As our physicist heroes have lamented, it's a framework, not an explanation. It's a tool we wield with precision, but one we don't truly comprehend.

We can keep plugging away, content with the fact that it works. Or, like Bell on the weekends, we can keep searching for something better. One thing's for sure: the quantum puzzle isn’t going anywhere anytime soon. So buckle up, do the math, and enjoy the ride—however bizarre it may be.

References

[1]Popper, K.:The Logic of Scientific Discovery. Basic Books, New York (1959)

[2] Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. III. Addison-Wesley, Reading, Mass (1965)

[3] Gell-Mann, M.: Questions for the future. Series Wolfson College lectures, 1980. Oxford University Press, Oxford (1981). Also in the collection The Nature of Matter, Wolfson College Lectures 1980. J. H. Mulvey, ed. (Clarendon Press, Oxford, 1981)

[4] Dyson, F.J.: Innovation in Physics. Sci. Am. 199(9), 74 (1958). Quoted in Landé 1965, p. 148, and requoted in Selleri, Quantum Paradoxes, p. 2

[5]Dyson, F.J.: Interview with Onnesha Roychoudhuri, Sep 29, 2007 (in Atoms & Eden)

[6]Gisin, N.: Sundays in a Quantum Engineers’s Life, in Bertlmann and Zeilinger (2002)