Yesterday I decided that using html codes for writing Clifford algebra math makes little sense. Math should be written using TeX, using all its advantages. It is also easier for my Readers to save pdf as a file, and peruse it during sleepless nights at ease.
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| Peruse it during sleepless nights at ease |
So, here is the new note. You can download it from here.
22-07-25 7:48 Playing with AI. At the end of the document I have added the Summary and the solution to Exercise 2 as it was found and nicely written for me in .tex by DeppSeek AI. The Summary also written by AI.
22-10-17 Renamed the summation index "i" to "mu". It is better to use "i" for 1,2 and "mu" for 1,2,3,4. Probably it would be less confusing to denote vectors by x, and matrices representing these vectors by x^\hat. Perhaps I will do it in the next post.
23-07-25 8:30 Added Exercise 5: Find the errors in AI's "reasoning".
Using Mathematica notebook code you provided;
ReplyDeleteEx.1
x = x1 e1 + x2 e2 + x3 e3 + x4 e4 = {{x1 - x3, x2 + x4}, {x2 - x4, -x1 - x3}};
=> 0.5*Tr[x] = -x3 = 0.5*Tr[t[x]].
(Transpose of a 2×2 matrix with real entries does not change the value of the trace of that matrix)
y = y1 e1 + y2 e2 + y3 e3 + y4 e4 = {{y1 - y3, y2 + y4}, {y2 - y4, -y1 - y3}};
x*y = {{(x1 - x3) (y1 - y3), (x2 + x4) (y2 + y4)}, {(x2 - x4) (y2 -
y4), (-x1 - x3) (-y1 - y3)}} = y*x;
=> 0.5*Tr[x*y] = x1*y1 + x3*y3 = 0.5*Tr[y*x].
In general,
Tr[a*b] = a11*b11 + a12*b21 + a21*b12 + a22*b22;
Tr[b*a] = b11*a11 + b12*a21 + b21*a12 + b22*a22;
=> Tr[a*b] = Tr[b*a] for a_ij and b_ij real.
Ex. 3
B(x,y) = 0.5*Tr[x*\[Nu][y]] = -x1*y1 + x3*y3 = 0.5*Tr[y*\[Nu][x]] = B(y,x).
Ex. 4
Simplify[\[CapitalDelta][x]] = {{-x1^2 - x2^2 + x3^2 + x4^2, 0}, {0, -x1^2 - x2^2 + x3^2 + x4^2}} = Det[x] * Id;
thus the group G is the set of all 2×2 matrices x in our representation of the Cl(2) with their determinant equal to 1, i.e. -x1^2 - x2^2 + x3^2 + x4^2 = 1.
Thanks. Will check later. Right now I solving my own :exercises", and they do not want to come out as expected! And I was hoping that my expectations were good.
DeleteWell, something similar is happening in my case with Mathematica calculations of simple squares of our e_mu; doing it by hand gives e1^2=e2^2=e3^2=Id and e4^2=-Id as expected, while Mathematica stubbornly gives e2^2=e4^2={{0, 1}, {1, 0}}. Go figure...
DeleteAsk AI to help. Often simply formulating the question helps in spotting the reason for our problems.
DeleteNo need to bother AI with that, found the issue in my code - was using "*" for matrix multiplication, instead ".", and everywhere where "*" was used got the wrong results, including the solutions to exercises in the first comment.
DeleteSo, the correct solution to Ex. 1:
In general,
Tr[a.b] = a11*b11 + a12*b21 + a21*b12 + a22*b22;
Tr[b.a] = b11*a11 + b12*a21 + b21*a12 + b22*a22;
=> Tr[a.b] = Tr[b.a] for a_ij and b_ij real.
Tr[a] = Tr[a^T], because transpose of the matrix does not change values on the matrix main diagonal and thus does not change the value of its trace.
Ex. 3
TrueQ[Simplify[0.5*Tr[x.\[Nu][y]]] == Simplify[0.5*Tr[y.\[Nu][x]]]]
True
0.5*Tr[x.\[Nu][y]] // Simplify
-1. x1 y1 - 1. x2 y2 + 1. x3 y3 + 1. x4 y4
Ex. 4
What's written above in the previous comment is OK, with a small addition that 0.5*Tr[\[CapitalDelta][x]] = Det[x].
Re Note on p.2 in the file;
ReplyDelete"In the Clifford algebra of space, Cl_(3,0)(R), the pseudo-scalar was of square −1. It also commuted with all elements in the algebra. Thus, we had a natural complex structure.
[...]
Now we are dealing with the Clifford algebra of the 2D Euclidean plane. The pseudo-scalar is also of square −1, but it anti-commutes with the generators. Therefore all must stay real."
How do we know that omega^2=-1 in case of commutativity leads to complex numbers or structure and in case of anti-commutativity to real numbers?
In the case of omega^2=-1 and commutativity we INTERPRETED omega as I, and such an interpretation was allowed because numbers, real, or complex, should commute with everything. In the case of anticommutativity such an interpretation would be incompatible with numbers' properties. So, we have to stay with what we have, and we have algebra over reals.
DeleteThank you!
DeleteSurprisingly, just two days ago I asked similar question to Vadim, and here is his answer: "Algebra cl(3,0), whose division ring is complex and volume element belongs to the center with square - 1, is isomorphic to the complex algebra C(2) (the algebra of biquaternions). Algebra cl(0,3), whose ring is quaternionic and volume element squares to +1, is isomorphism to the double quaternion algebra, etc.
DeleteBy the way, there are some strange things in the Proof of Ex. 2 provided by AI on p.5 in the note file; do not know enough to say right away they are wrong or not though.
ReplyDeleteFor example, Line 3 of the Proof says B(a,b) = tau(nu(a) nu(b)) which is not what Line 2 says and neither corresponds to definition of B(a,b) in Line 1, while Line 7 says tau(b nu(nu(a))) = B(b,a), but from the definition of the bilinear form B we have B(b,a) = tau(b nu(a)).
FWIW.
Similarly in the Alternative Proof at the end of p.5 of the note where Line 4 says tau(b^T nu(a)) = B(b,a), while by definition we have B(b,a) = tau(b nu(a)), which implies that b^T=b.
DeleteFWIW, simply using the results from Ex. 1, the proof for Ex. 2 seems rather straightforward and much cleaner than what's provided by AI in both cases on p.5 of the note file, in the sense that no strange things and claims are needed to prove B(a,b)=B(b,a).
DeleteDef. B(a,b) = tau(a nu(b));
From Ex.1, tau(a b)=tau(b a) and tau(a)=tau(a~)=tau(a^T),
=> tau(a nu(b)) = tau((nu(b))^T a^T);
Using nu(b)=-omega b^T omega, omega^T=-omega and (b^T)^T=b,
=> tau(a nu(b)) = tau(omega b (-omega) a^T);
And using again tau(a b)=tau(b a),
=> B(a,b) = tau(a nu(b)) = tau(b (-omega) a^T omega) = tau(b nu(a)) = B(b,a).
QED.
"By the way, there are some strange things in the Proof of Ex. 2 provided by AI on p.5 in the note file"
DeleteYes, agreed, in going from point 2 to point 3 AI lost one "nu". It's strange that it still got the correct result.
When you check Line or point 7, it's evident that AI didn't get the proof done or correct result, but just said it did. The same happened in Alternative proof also in last line or point of the proof.
DeleteIn other words, this time AI didn't make things up or halucinate, but just plainly and simply lied, saying that something is what it is not.
DeleteThanks for your careful checking it all. I was evidently over-enthusiastic and naively assuming superiority of AI. So much for DeepSeek! Grok 4 seems to be better, but also not reliable. But we are at the very beginning of AI. I like to compare AI of today to first cars that were running on steam engine, and required a person walking in front of the car and warning people to step away.
Delete"For example, Line 3 of the Proof says B(a,b) = tau(nu(a) nu(b)) "
ReplyDeleteSaša. what I see is
B(a,b) = tau(nu(nu(a)) )nu(b))
and it is correct.
Where did you get B(a,b) = tau(nu(a) nu(b)) from?
That's Line 2 you're looking at; next Line, Line 3 is where AI substitutes nu(a) and nu(b) to write B(a,b)=tau(nu(a) nu(b)) that I was talking about.
DeleteAnd Line 7, where AI claims that tau(b a)=B(b,a), which is evidently not what definition of B in Line 1 says.
DeleteYou are right! And this a good exercise: "Find the errors in the AI's reasoning below.". Thanks!
DeleteSo, it seems DeepSeek is smarter than Grok; it still brings you hen when asked for a duck, but disguises it to appear like a duck if not carefully inspected. And allegedly does it in just a percentage of the effort Grok and others would use. Nice! ;)
DeleteYou are right. Perhaps Chinese products are naturally better than any other? Cost less, look nice, and give you more satisfaction!
DeleteIn that respect, an idea crossed my mind that might be handy for history and chronology entusiasts.
DeleteIt was mentioned some time ago that tracking re-appearances of some more or less regular well-known comet with longish period like Haley's could be used for chronology purposes. Chinese historical records allegedly go back cca 3000y, and they (Chinese) are known to had been keenly observing the skies too.
Maybe using Chinese product like DeepSeek to comb through Chinese records for occurences of Haley's comet in their chronological order could provide like a baseline, onto which known events and figures in the West that were mentioned in the Chinese records could be placed by comparison.
For example, Gengis Khan was very likely mentioned in the Chinese records and by associating him with specific appearance of Haley's comet also in these records, the AD year period known in the West for him could be kinda crosschecked with this sort of astronomical cometary clock.
Same could be done for example for Alexander the Great if Chinese ancient historians took notice of his incursions into Central Asia and (North) Indian subcontinent.
If there were some misteriously added years into Western chronology as some "conspiracy" theorists posit, then in case of Alexander the Great, Chinese records would have him mentioned cca 25 Haley's comet appearances ago, and not cca 30 as official chronology claims, or something on that ballpark.
Do you think this Chinese cometary chronological 'baseline' and comparison to Western chronology could be done by an AI like DeepSeek?
Perhaps you should try, and let me know what you find, what DeepSeek knows about Chinese history and philosophy. I doubt that it knows a lot, because it is mainly for a commercial success....
DeleteYeah, probably I've been a bit naive to think that DeepSeek would know anything more about Chinese history and chronology, that other AIs wouldn't know or been able to dig out, just because it's a Chinese product.
DeleteHaven't used any AI so far, maybe this would be a nice way to start and check it's capabilities. It's only that I'm not so keen in running in front of it and warning everybody that an AI is about to roll in after me. Seems that the energy and time invested in that ordeal might be better spend in doing the research on my own.
Regarding the last part of my previous comment;
DeleteI think that using AI by student is not only not beneficial, but even detrimental, one of the reasons being that AI usage by-passes the natural learning process in the sense of disruption of transforming the information obtained into real knowledge. For teachers, AI could be a tool to improve their delivery style and ease the transfer of knowledge when met with genuine student's interest and efforts invested.
At the end of the day, it's like what the text next to your profile avatar says; Knowledge protects, Ignorance endangers; with a small addition from the same source; False knowledge is more dangerous than no knowledge.
And it is clearly evident by now that AI is no stranger when it comes to spreading false knowledge. The teacher in principle would be in position to notice when something like that happens, if he's attentive enough to what AI is spewing out, but the student being a student and only just learning about the subject would naturally be in much more vulnerable position and basically at the mercy of AI's whimps. In that sense, he might be better without an AI than with it.
I think AI can also help improve the delivery style of one's own ideas even well off the beaten path ideas. You have to train the AI on your ideas but that seems surprisingly easy to do. It can sort of be a helpful echo chamber where if you are wrong, you are equally as wrong with it or without it. If it for example goes with you when you say 4-vector but not when you say 4-form, it can help you to be less sloppy.
DeleteWith exercises I am done more or less, but I cannot convince myself that delta(a^-1)=1/delta(a)
ReplyDelete