In The infinity ab initio and The infinity ab initio 2 we have introduced the six-dimensional space of oriented spheres in R3. R3 is endowed with the quadratic form q of signature (3,0). We use coordinates x1,x2,x3 for R3. Going to the six-dimensional space of oriented spheres in R3 involves adding 3 more dimensions, with coordinates x4,x5,x6, and added signature (1,2). x5 can be interpreted as time, x4 and x6 were extra two dimensions. We have used Q to denote the resulting quadratic form in R4,2.
Creator realized that Cl(1) is not good
But after that, in Sunday special: Conversing with Grok about the Clifford algebra Cl(2,2), we have decided to play first with a toy model, suppressing two space dimensions x2 and x3. Then space became 1-dimensional. Spheres in one dimension, oriented or not, are somewhat special. What is a sphere in R1? It is just a pair of points (-r,+r). (Why?) It is good to have in mind this particularity, but it is not a problem in our construction.
While introducing the Clifford algebra Cl(2,2) we decided to rename the coordinates and to switch from the signature (+-+-) to equivalent (++--), as it is more convenient for working with the Clifford algebra. We have also introduced a particular matrix representation of the Clifford algebra, where the matrices representing an orthonormal basis in R2,2 are given by:
These matrices will play the role of Dirac gamma matrices, but now adapted to the signature (2,2). In our case they are all real! While discussing the Clifford algebra of R3, we have mentioned the ideas developed by David Hestenes: the imaginary unit in quantum theory is related to the fact that the element ω = e1e2e3 of Cl(3) is related to quantum-mechanical imaginary unit "i", since its square is -1. But now, in Cl(2,2), ω = e1e2e3e4 is given by
ω = diag(1,1,-1,-1).
and ω2 = 1. Does that mean that for a 1-dimensional space quantum mechanics would be all real? I do not think so. One-dimensional harmonic oscillator, for instance, is studied in all quantum-mechanical textbooks, and the Hilbert space there is always complex. So, I am not sure anymore that Hestenes' idea, although elegant and attractive, really hits the target. But we will never know the truth until we understand what spin is and what electric charge is. On the other hand, perhaps Creator realized that Cl(1) is not good, Cl(2) is also not good (both do not lead to complex structures), tried the next option, Cl(3), and saw that it was good?
Taking all different products of ei we span the whole 16-dimensional algebra. In our realization of Cl(2,2) we have
Cl(2,2) = Mat(4,R).
So Cl(2,2) is isomorphic to Cl(3,1) - as an algebra. But Clifford algebras come with extra structure, and it is this extra structure that differentiates between Cl(2,2) and Cl(3,1). The vector space R2,2 is realized in Mat(4,R) differently than the vector space R3,1. With different embedding of underlying vector spaces come different form of the three Clifford involutions. We will discuss them now.
Clifford involutions for the matrix realization of Cl(2,2)
1. The main automorphism π, changing the signs of the vectors. 2. The main anti-automorphism τ, reversing the order of multiplication in Cl(V). 3. Their composition π∘τ = τ∘π, which we will callν.
We will now do the same for Cl(2,2), but we will change the notation.
The main involution will be denoted by α: x ⟼ α(x).
The main anti-authomorfism will be denoted by x ⟼ x~.
The Clifford conjugation, the composition of the main automorphism and the main anti-automorphism, will be denoted by ν: x ⟼ ν(x).
Three involutions by Raphael
The point is that we want to save π and τ for other purposes.
The main involution is easy to guess. Since ω2 =1, and ω anticommutes with all generators, the formula
α(x) = ωxω, x in Cl(2,2),
does the job in any representation.
The main anti-automorphism (Clifford reversion) is more subtle. The natural anti-automorphism in Mat(4,R) is the transposition. But the main Clifford anti-automorphism should leave the generators invariant. This works fine with e1 and e2, but fails for e3 and e4 since they are represented by antisymmetric matrices. We have to compensate for this fact. The anti-automorphism will be of the form
x~ = SxTS-1, x in Mat(4,R),
where S is a matrix commuting with e1,e2, and anticommuting with e3,e4.
There is an easy solution. Let
S = e3e4.
Then S2 = -1, so S-1 = -S, S commutes with e1 and e2, and anticommutes with e3 and e4.
AI Note. I tried AI with this problem. Grok 3 was repeatedly proposing wrong solutions for reversion. When I was pointing out why his solution is wrong, he was acknowledging his error, and then proposing another solution, also wrong. Then I tested with the new Grok 4. Grok 4 understood the problem, was thinking correctly, used Python for calculations, and came out with the right solution. But he did not realize that his solution can be expressed as e34. When asked explicitly if e34 is a good solution - he was reasoning correctly, like a good student. While Grok 3 was giving (often wrong) answers immediately, Grok 4 was a very slow thinker. I am also slow, so I think I like Grok 4.
Now we need Clifford conjugation. It will be of the form:
Quantum mechanics, as we know it from books, is a form of indoctrination for our minds. It’s fascinating how easily we are indoctrinated. We even derive pleasure from it—a kind of intellectual masochism. A psychologist or sociologist could likely base an entire PhD on this phenomenon. By allowing ourselves to be bombarded with slogans and rituals, we feel as though we are being inducted into the mysterious world of a Secret Society of Quantum Mechanics Adepts. And who among us doesn’t want to be a part of such an organization? One that possesses secrets unknown to those outside the circle? However, like in many secret societies, as we climb higher on the ladder of initiation, we eventually discover that everything is based on the belief in the existence of secrets, rather than in actually knowing them.
Secret Society of Quantum Mechanics Adepts
From Theoretical to Practical: The Concept of Measurement Enough theorizing—let’s dive into the evidence. Let’s take the concept of measurement as our example. During a discussion about the ideas of Henry Stapp, a physicist from LBL, Berkeley, California, Laura asked her friends—communicating via Cassiopaea—an intriguing question:
"Is quantum theory, as it stands, about knowledge or about physical units?" The reply she received was simple: "It is about measurement."
Yes, I think so too. (After all, those Cassiopaeans might as well have been me.) So, let's examine how measurement is treated in quantum theory today. Referring to the Stanford Encyclopedia of Philosophy entry on "Measurement in Quantum Theory," the discussion begins with Bohr’s Postulate, which is stated as:
(P)If a quantity Q is measured in system S at time t, then Q has a particular value in S at t.
Interesting, isn’t it? But there’s more. The statistical interpretation of Born is added (don’t worry if you don’t fully understand this—it's not crucial). Let me quote a longer passage from the Stanford Encyclopedia of Philosophy for context, as it contains some humorous elements:
"... the second stage of the measurement, with its radical, non-linear discontinuities, has been the source of many of the philosophical difficulties that have plagued quantum mechanics, including what von Neumann referred to as its 'peculiar dual nature' (417). Indeed, Schrödinger foreshadowed such difficulties even before the formal development of measurement theory. For example, during a visit to Bohr’s institute in September 1926, he remarked, 'If all this damned quantum jumping [verdamnte Quantenspringerei] were really to stay, I should be sorry I ever got involved with quantum theory' (Jammer 1974, 57)."
The Dilemma of Quantum Measurement Quantum mechanics offers no definitive explanation of the measurement process. Specifically, using quantum mechanics alone, we cannot predict the exact value of Q that will be recorded during measurement. What quantum mechanics does provide, however, is additional information of a statistical nature, through what is known as the Born statistical interpretation: The probability of qi being registered is |ci|², where ci is the coefficient of fi (the eigenvector of Q corresponding to the value qi) when the initial measured state of S is expressed as a linear superposition of eigenvectors of Q.
In short: quantum mechanics does not predict the measured value, but it does give us the probability distribution of the possible outcomes.
Schrödinger was furious—and rightly so. He just didn’t fully understand why he was right. He wasn’t upset simply because quantum jumps are inherently problematic; they only become problematic when Bohr and his Copenhagen colleagues treat them as events that happen instantly, without any dynamics. Such instantaneous events belong in the realm of miracles and spirits, not in the world that physics deals with.
Bohr's Postulate and the Problem of Time Here’s the crux of the issue: Bohr’s postulate (P) speaks of "measurement at time t." However, such a thing as an "instantaneous" measurement never actually occurs. Every measurement takes time—some longer, some shorter. We expose photographic film to radiation and wait patiently, sometimes for hours or days, for a proton to leave its mark. We set up our laser, nanoprobe, or magnets over an electron and wait (today, for hours—tomorrow, perhaps, for mere fractions of a second) for our "resonant system to resonate" and give an answer: spin up or spin down.
In classical physics, where particle trajectories are smooth and continuous, the distinction between instantaneous measurement and measurement over time wasn’t necessary. The idealization of instantaneous measurement was harmless and without consequence. But in quantum mechanics, where we have the Planck constant and the uncertainty principle, measuring "at time t" implies infinite uncertainty in the energy transfer, which inevitably leads to contradictions. I believe Erwin Schrödinger might have been more optimistic about quantum mechanics and its jumps—perhaps even excited about them—if he had known what I am about to reveal.
Quantum Jumps and the Nervous System Let’s take the ancient adage "as above, so below" and apply it to quantum mechanics. Electrons, protons, and such are at the bottom; we, as beings of flesh and blood, are at the top. Neurology teaches us a bit about how the nervous system functions, and measuring the action potential of a neuron provides a prototype for the quantum jump.
The potential rises, rises further, exceeds a certain threshold, and the neuron "fires." Depending on the degree of stimulation and other factors, these firings happen more or less frequently (of course, I’ve simplified the mechanism drastically). Could quantum jumps work in a similar way? Yes, they could—and I’ll demonstrate why. Another question is whether this mechanism truly occurs in quantum mechanics—but that’s a different problem, equally important, but not the focus here.
In this analogy, a detector plays the role of a neuron. It’s stimulated by the "wave function" (also called the "state vector"). When the detector is sufficiently excited, it fires, and the wave function "discharges." The detector then rests for a while, ready for the next firing. There are also single-use detectors, but I won’t delve into them, as they are just a special case of reusable detectors with adjustable efficiency.
More to Come But I’ll leave the details for the future notes.
In a world obsessed with
originality, the allure of being the first - of standing out as a
lone genius - is tantalizing. Yet, the greatest breakthroughs often
come not from the solitary thinker but from those who understand and
build upon the monumental work of their predecessors. This idea,
eloquently summarized by Sir Isaac Newton, reminds us: “If I have
seen further, it is by standing on the shoulders of giants.”
Newton: If I have seen further, it is by standing on the shoulders of giants.
Newton's Humility
Imagine Newton, arguably one
of the most brilliant minds in history, humbly acknowledging the
contributions of those who came before him. Newton’s revelation
about gravity, famously inspired by a falling apple, wasn’t
conjured in a vacuum. It was the culmination of years of scientific
inquiry, drawing on the insights of giants like Kepler, Galileo, and
Descartes. Newton’s genius lay not just in his original thinking
but in his ability to synthesize and expand upon existing knowledge.
The Metaphor Explained
Consider the metaphor of
standing on the shoulders of giants. It’s an act of both humility
and audacity. You’re acknowledging the immense contributions of
those before you while daring to reach even higher. It’s a reminder
that innovation is a collective endeavor, a relay race where each
runner passes the baton of knowledge.
Shakespeare's Inspiration
Yet, the importance of
building on past knowledge extends beyond science and technology. In
the arts, we see it too. Shakespeare, the quintessential literary
genius, drew heavily from classical sources and historical texts. His
ability to weave old stories into new, captivating narratives is a
testament to the power of standing on the shoulders of giants. (see however [2])
Embracing Past Knowledge
So, what does this mean for
us today? In an era where the pressure to be original is immense,
let’s not forget the value of the work that has come before us.
Embrace the insights of those who have paved the way. Dive into the
rich pool of existing knowledge and use it as a springboard for your
own ideas. Recognize that originality and tradition are not mutually
exclusive but are, in fact, symbiotic.
Seeing Further and Dreaming Bigger
Dreaming bigger
By standing on the shoulders
of giants, we gain a vantage point that allows us to see further and
dream bigger. We honor the past, enrich the present, and pave the way
for future giants to come. After all, in the grand tapestry of human
knowledge, each thread, no matter how innovative, is intertwined with
countless others.
Newton's Wisdom
So, the next time you embark
on a new project or seek to solve a complex problem, remember
Newton’s wisdom. Seek out the giants. Stand tall on their
shoulders. And from that elevated view, strive to see further, dream
bigger, and contribute your unique thread to the ever-growing
tapestry of human understanding.
The Source
Here is the relevant except from a letter from Isaac Newton to Robert Hooke, dated Cambridge Feb. 5 1675 [1]:
At ye reading of your letter^) I was exceedingly well pleased & satisfied wth your generous freedom, & think you have done what becomes a true Philosophical spirit. There is nothing wch I desire to avoyde in matters of Philosophy more then contention, nor any kind of contention more then one in print: & therefore I gladly embrace your proposal of a private correspondence. What's done before many witnesses is seldome wthout some further concern then that for truth: but what passes between friends in private usually deserves ye name of consultation rather then contest, & so I hope it will prove between you & me. Your animadversions will be therefore very welcome to me: for though I was formerly tired wth this subject, & have not yet nor I beleive ever shall recover so much love for it as to delight in spending time about it; yet to have at once in short ye strongest or most pertinent Objections that may be made, I could really desire, & know no man better able to furnish me wth them then your self In this you will oblige me. And if there be any thing els in my papers in wch you apprehend I have assumed too much, or not done you right, if you please to reserve your sentiments of it for a private letter, I hope you will find also that I am not so much in love wth philosophical productions but yt I can make them yeild to equity & friendship. But, in ye meane time you defer too much to my ability for searching into this subject. What(S) Des-Cartes did was a good step. You have added much several ways, & especially in taking ye colours of thin plates into philosophical consideration. If I have seen further it is by standing on ye sholders of Giants.
I modern English: In his letter to Robert Hooke, Isaac Newton expresses his appreciation for Hooke's open and generous approach. Newton emphasizes his dislike for public disputes in scientific matters and prefers private correspondence to avoid unnecessary conflicts. He welcomes Hooke's feedback and objections to his work, acknowledging Hooke's capability in providing valuable insights. Newton shows willingness to correct any mistakes or oversights in his papers if pointed out privately, prioritizing truth and friendship over pride in his scientific achievements. He praises the contributions of Descartes and Hooke, and famously remarks that his own accomplishments are built upon the work of great predecessors.
References:
[1] The Correspondence of Isaac Newton, Volume 1, H. W. Thurnbull (ed.), Cambridge University Press 1959
[2] ChatGPT supplies (on inquiry) the following information:
"The strongest evidence cited by those who believe William Shakespeare was not the author of "Hamlet" and other works attributed to him generally falls into several categories:
Shakespeare's Limited Education:
Critics argue that Shakespeare's limited formal education, having only attended the local grammar school in Stratford-upon-Avon, would not have provided him with the extensive knowledge of classical literature, languages, and courtly life evident in his works.
Lack of Manuscripts:
No original manuscripts of Shakespeare's plays or poems in his handwriting have been found, which some see as suspicious given the volume of work he produced.
Baconian Theory:
This theory suggests that Sir Francis Bacon, a well-educated philosopher, statesman, and writer, was the true author of Shakespeare's works. Proponents point to perceived similarities in the writing styles and the use of knowledge Bacon possessed that Shakespeare supposedly did not.
Oxfordian Theory:
Another popular theory proposes Edward de Vere, the Earl of Oxford, as the true author. Supporters argue that de Vere's education, travels, and courtly experience align more closely with the content of the plays. They also point to de Vere's use of a pseudonym due to his noble status.
Authorship Question:
The general authorship question raises doubts based on the lack of detailed biographical information about Shakespeare's life, particularly during his supposed writing years. Some argue that other writers, who left more detailed records of their lives and works, might be better candidates."
From my today's chatting with GPT: Response to my Prompts
Created by Kandinsky 3.5
Title: "The Quest for Twistor Truth"
Concept Description:
1. The Labyrinthine Library:
A vast, surreal library with endless twisting hallways and floating bookshelves. Books have keywords and references written on them, such as "Null Lines," "Minkowski Space," and "Twistor Theory."
2. The AI Guide:
A glowing orb with a halo of spinning binary code. The AI is holding a large magnifying glass and appears to be frantically searching through books and scrolls.
3. The Patient Scholar:
A calm, wise scholar wearing a whimsical outfit and a thinking cap adorned with mathematical symbols and equations. The scholar has a bemused smile.
4. Mischievous Imps of Misinformation:
Small, mischievous creatures scattering fake books and incorrect references around the library. They have playful, cheeky expressions.
5. The Twistor Vortex:
A swirling vortex in the center of the library, representing the complex topic of twistor theory. Null lines and mathematical symbols spiral into it.
6. The Illuminated Path:
A bright, illuminated path leading the scholar and the AI guide towards a glowing book labeled "Truth." This book represents accurate and verified information.
Surrealistic Elements:
Floating clocks showing different times, symbolizing the timeless nature of the search for knowledge.
Books morphing into birds and flying around, representing the elusive nature of finding the right information.
Twisted, Escher-like staircases leading to nowhere, symbolizing the complexity and confusion of the topic.
Created by DALL-E 3 v.1
"I apologize for the repeated errors in my responses today. Here are a few hypotheses that could explain why these errors occurred:
Misinformation in Source Data:
The information I provided may have been based on inaccurate or outdated sources. Although I strive to offer accurate data, occasionally, errors from source materials can propagate into responses.
Miscommunication or Misinterpretation:
There might have been a miscommunication or misinterpretation of the specific materials and references you were seeking. Ensuring a precise understanding of your needs is crucial.
Complexity and Specificity of Topic:
The relationship between null lines in Minkowski space and twistors is a specialized and complex topic. This complexity increases the likelihood of referencing inaccuracies or less accessible materials.
System Limitations:
As an AI, I rely on a vast array of data but may lack specific, up-to-date bibliographic details on niche academic papers. This limitation can result in incorrect references.
To address these issues more effectively, I will ensure that I use more reliable databases and references for your specific request. I can assure you that there is no deliberate programming or intention by my developers or handlers to provide incorrect information in response to questions containing specific keywords. The goal of this AI is to provide accurate, helpful, and reliable information to the best of its ability.
The errors in my responses can be attributed to a few common factors:
Complexity of Topic:
Niche and highly specialized topics like the relationship between null lines in Minkowski space and twistors are complex, which increases the chance of unintentional inaccuracies.
Source Data Limitations:
The AI relies on a vast dataset compiled from diverse sources. Occasionally, errors or outdated information from these sources can influence responses.
Context and Interpretation:
Misinterpretation of your requests or the nuances of specific academic references can lead to mistakes. Ensuring precise understanding and context is challenging.
System Limitations:
Despite extensive training, there are inherent limitations in the AI's ability to access real-time data or verify specific details like ISBN numbers or paper availability in real-time.
There is no motive or programming directive to provide incorrect information. The intention is always to be as accurate and helpful as possible. If you have further questions or need assistance with other topics, I'm here to help! "
Today (25-06-24) got an email from ai.feedback@outlook.com, with subject "Feedback on your recent arXiv preprint 2401.14217". Here it is
Hi there,
Congratulations on your recent preprint on arXiv, titled "The explicit form of the unitary representation of the Poincar\'e group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators". We are grateful for your hard work and dedication to the field, and we value your contributions!
We are part of a team from Northwestern University, Stanford University, and Cornell University, committed to providing research feedback to scholars with the assistance of advance AI models. We have followed your work closely and, upon a thorough examination, generate the suggestions below. These suggestions cover various aspects of your work, including the writing style, research design, and title. We hope they offer you fresh insights that may enhance the depth and impact of your research.
To view detailed comments regarding your research, please visit this link: http://feedback.kellogg.northwestern.edu/FNQ3OO.html
Once again, congratulations on your achievement. We are certain that your work will have an impact on the future of your field and will inspire fellow researchers worldwide.
Should you have any questions, concerns, or suggestions, please do not hesitate to reach out to us at ai.feedback@outlook.com. Your insights would be invaluable in helping us better support the community's research development.
Best regards,
Feedback Team
And here are AI suggestions:
The explicit form of the unitary representation of the Poincar\'e group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators
1. Significance and Novelty
Geometric Derivation: Innovatively builds a unitary representation of the Poincaré group using a geometric framework.
Photon Position Operators: Offers an explicit form for the photon position operator with commuting components, aligning with the HawtonBaylis formulation.
Affine Connections: Compares the Pryce and HawtonBaylis connections, highlighting differences in semisymmetry and metric properties.
Photon Localization: Provides a rigorous analysis of photon states localized on loops, applying both the HawtonBaylis operators and the JauchPironAmrein POV measure.
2. Suggestions for Improvement Mathematical Rigor and Clarification:
Contextual Clarification: The paper should more explicitly clarify the context and motivation behind the study of photon position operators. While the historical background is comprehensive, linking it directly to the benefits and applications of photon localization in contemporary quantum mechanics could enhance reader engagement.
Detailed Proofs: Include more detailed proofs for the derivations of the photon boost eigenmodes and related transformations. For example, the section detailing the explicit form of so obtained unitary representation and its limit to m=0 would benefit from stepbystep mathematical exposition.
Figures and Tables:
Inclusion of Visual Aids: The manuscript lacks visual aids. Diagrams illustrating the geometric constructs, and tables summarizing the comparative properties of different connections (e.g., metric semisymmetry versus flatness) could significantly enhance comprehension.
Clarity in Captions: Ensure that all figures and tables have clear, selfexplanatory captions. The absence of such captions in the current outline is a noticeable omission.
Comprehensive Literature Review: Expand Bibliography: While the references are robust, ensure that all recent relevant studies are included. For instance, adding discussions about recent empirical studies or theoretical advancements related to photon localization and affine connections could provide a more comprehensive view.
Terminology and Notation: Consistency: Ensure that terminology and notations are used consistently throughout the paper. For example, the term HawtonBaylis connection should be used uniformly without alternation if it is referred to by other names.
Structure and Readability: Section Consistency: Improve the flow between sections. The transition from one main section to another sometimes feels abrupt. Providing brief introductory and concluding paragraphs within sections could improve overall coherence.
Abstract Brevity and Focus: The abstract is too technical and dense for a general reader. Consider distilling it to highlight the core contributions and findings succinctly, reserving detailed technicalities for the main text.
Theoretical Implications: Implications of Findings: Expand on the potential implications of your theoretical findings. For example, how do the new photon position operator and the findings on different connections impact future research perspectives or practical applications in quantum field theory or optics?
By addressing these suggestions, the authors can significantly improve the clarity, impact, and comprehensiveness of their paper. The emphasis on more explicit motivational contexts, visual aids, detailed proofs, and thorough discussions on potential implications will not only make the paper more accessible but also more valuable to the scientific community.
3. Suggestions on Title
Original Title
The explicit form of the unitary representation of the Poincar\'e group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators
Recommended Titles
Unitary Representations of the Poincaré Group in Photon Localization
Reasoning: This title is concise and highlights the core mathematical and physical concepts of unitary representations and photon localization."
Photon Localization through Unitary Poincaré Group Representations
Reasoning: Reverses the structure for emphasis on photon localization
Geometric Derivations of Unitary Poincaré Representations and Photon Position Operators
Reasoning: Incorporates the geometric aspect
Affine Connections and Photon States in Unitary Poincaré Representations
Photon Position Operators and Localization via Unitary Poincaré Group Theory
Reasoning: This title combines the key elements - photon position operators and localization - with the Poincaré group
4. Grammar Check for Abstract
1.Original Sentence: We geometrically derive the explicit form of the Unitary representation of the Poincaré group and use it to apply speed-of-light boosts to simple polarization basis to end up with Hawton-Baylis photon position operator with commuting components.
ErrorType: Subject-Verb Agreement Errors
Explanation: The verb 'apply' should be in its base form as it follows 'use it to'
Recommended Fragment: apply speed-of-light boosts to a simple polarization basis
2.Original Sentence: We geometrically derive the explicit form of the Unitary representation of the Poincaré group and use it to apply speed-of-light boosts to simple polarization basis to end up with Hawton-Baylis photon position operator with commuting components.
ErrorType: Incorrect Word Usage
Explanation: The phrase 'simple polarization basis' should include an article 'a' to be grammatically correct.
Recommended Fragment: apply speed-of-light boosts to a simple polarization basis
3.Original Sentence: Finally we discuss localizabil- ity of photon states localized on closed loops and show that photon states on the circle, both unnormalized improper states and finite norm wave packet smeared over washer-like regions are strictly localized with respect to Hawton-Baylis oper- ators with commuting components and also with respect to the noncommutative Jauch-Piron-Amrein POV measure.
ErrorType: Spelling Errors
Explanation: The word 'localizabil- ity' is split incorrectly over two lines and should be corrected to 'localizability'.
Recommended Fragment: Finally we discuss localizability
4.Original Sentence: photon states on the circle, both unnormalized improper states and finite norm wave packet smeared over washer-like regions are strictly localized with respect to Hawton-Baylis oper- ators with commuting components and also with respect to the noncommutative Jauch-Piron-Amrein POV measure.
ErrorType: Spelling Errors
Explanation: The word ‘oper- ators’ is split incorrectly over two lines and should be corrected to ‘operators’.
Recommended Fragment: operators with commuting components and also
* Disclaimer: The grammar suggestions provided are checked by advanced AI models and are intended for reference purposes only.
5. Grammar Check for Introduction
1.Original Sentence: Introduction The group-theoretical analysis of elementary relativistic quantum systems lead to the concept of imprimitivity systems, developed by G.W. Mackey (cf. e.g. [2, Ch. VI] and references therein), and to the associated concept of the localization of elemen- tary quantum particles.
ErrorType: Run-On Sentence
Explanation: The sentence lacks proper punctuation to separate distinct ideas.
Recommended Fragment: Introduction. The group-theoretical analysis of elementary relativistic quantum systems led to the concept of imprimitivity systems, developed by G.W. Mackey (cf. e.g. [2, Ch. VI] and references therein), and to the associated concept of the localization of elementary quantum particles.
2.Original Sentence: A.S. Wightman [3] applied these concepts to the study of localizability of quantum mechanical systems and came to conclusion confirming the previous analysis of T.D. Newton and E.P Wigner [4], namely that photons (as well as other particles of rest mass zero and helicity ≥ 1) are covariantly non-localizable in a strict sense of an imprimitivity system bases on the 3-d Euclidean group acting on R 3 ..
ErrorType: Run-On Sentence
Explanation: The sentence lacks proper punctuation to separate distinct ideas.
Recommended Fragment: A.S. Wightman [3] applied these concepts to the study of localizability of quantum mechanical systems and came to the conclusion confirming the previous analysis of T.D. Newton and E.P. Wigner [4]. They determined that photons (as well as other particles of rest mass zero and helicity ≥ 1) are covariantly non-localizable in a strict sense of an imprimitivity system based on the 3-D Euclidean group acting on R3.
3.Original Sentence: J.M. Jauch and C. Piron [5], developed a concept of \"weak localizability\" replacing projection-valued measure by POV (positive operator-valued) measures, and A.O. Amrein [6] proved that there exist photon states strictly POV-localized in arbitrarily small regions of space, while, more recently, I. and Z. Bialynicki-Birula [7] argued that photons cannot be sharply localized because of a kind of complementarity between magnetic and electric energy localization.
ErrorType: Run-On Sentence
Explanation: The sentence lacks proper punctuation to separate distinct ideas.
Recommended Fragment: J.M. Jauch and C. Piron [5] developed a concept of \"weak localizability\" by replacing projection-valued measures with POV (positive operator-valued) measures. Additionally, A.O. Amrein [6] proved that there exist photon states strictly POV-localized in arbitrarily small regions of space. More recently, I. and Z. Bialynicki-Birula [7] argued that photons cannot be sharply localized due to a kind of complementarity between magnetic and electric energy localization.
4.Original Sentence: It is rather easy to show that the standard requirements of the covariance with respect to the Euclidean group and inversions lead to a unique Q, - known as the Pryce photon position operator 2 [8], the trouble is that the components Q i do not commute, which makes the simple probabilistic interpretation for the photon's localization problem impossible.
ErrorType: Run-On Sentence
Explanation: The sentence lacks proper punctuation to separate distinct ideas.
Recommended Fragment: It is rather easy to show that the standard requirements of covariance with respect to the Euclidean group and inversions lead to a unique Q, known as the Pryce photon position operator [8]. The trouble is that the components Q i do not commute, which makes the simple probabilistic interpretation for the photon's localization problem impossible.
5.Original Sentence: acting on R 3 ..
ErrorType: Sentence Fragment
Explanation: The sentence lacks a subject and verb to create a complete thought.
Recommended Fragment: acting on R3.
* Disclaimer: The grammar suggestions provided are checked by advanced AI models and are intended for reference purposes only.
P.S. 28-06-24 15:49 I received something that I did not expect:
P.S. 29-06-24 12:08 I am currently learning Prompt Engineering with ChatGPT. With an account set up and the valuable advice and encouragement from Natasha Urakchina, I am embarking on a journey to navigate the vast sea of science.
"3. The Quantised EM Field in the Stationary Frame
For a long time, it has been believed that photons do not have a wave function and that light cannot be localised [55–57]. However, quantum physics should apply to all particles, and photons should not be an exception. For example, when a single-photon detector clicks, it measures the position of the arriving photon at that instant in time [58,59].
Defining a time of arrival operator for a localised photon detector, however, has beena significant problem and could not be achieved within the standard Hilbert space of the quantised EM field[ 60–64]. The origin of the wave function problem was that many authors liked to identify the wave function of the photon with its electric fi eld amplitudes, but the complex electric fi eld amplitudes at different positions do not commute. The eigenstates of the observable electric fi eld are therefore not local, although they can be made to appear local by altering the scalar product that is used to calculate the overlap of quantum state vectors [24,65].
An alternative way of establishing the wave function of a single photon is to double the Hilbert space of the quantised EM fi eld to include both positive and negative frequency photons and to separate light from its carriers [25–27]. The carriers of the quantised EM fieeld in momentum space are nonlocal monochromatic waves. The Fourier transforms of these carriers, the blips (which stands for bosons localised in position), however, form a complete set of pairwise orthonormal local carriers of the quantised EM fi eld in position space. Similar to how a point mass is a carrier for a gravitational fi eld, blips are carriers of nonlocal electric and magnetic field amplitudes. When expressing the observables of the electric and magnetic field in free space in terms of blip annihilation and creation operators, these include contributions from blips at all points along the position axis. By applying a constraint to the blip dynamics, a relativistically form-invariant representation of the EM field is derived. Below, these expressions are used to derive a transformation between the blips in Alice’s and Bob’s reference frames. "
My much older (1996) paper dealing with time of arrival, "Time of events in quantum theory" coauthored with Ph. Blanchard, has garnered significant attention with 70 citations.
P.S. 30-06-24 13:34 From my today's chat:
My Prompt:
I have read in a paper that I do not remember that the conformal is unique in the sense that it cannot be obtained via deformation of some other group. I think the author of this result is Segal, but I do not remember the details. Can you help mi by proposing possible references where this kind of statement can be found?
Response:
The assertion that the conformal group is unique in the sense that it cannot be obtained via deformation of any other group is indeed associated with the work of Irving Segal. Segal's contributions highlight the rigidity of the conformal group, meaning that its structure is fixed and cannot be smoothly transformed into another group through a process of continuous deformation.
One of the key insights from Segal's work is that while other groups can be deformed into one another, the conformal group stands out as an exception. This uniqueness makes it a fundamental aspect of the symmetries in spacetime and field theories. Segal demonstrated that the Lie algebra of the conformal group cannot be obtained from that of another group through deformation, solidifying its place as a distinct and fundamental structure in mathematical physics.
For further reading and detailed explanations, you might want to look into Segal's original works and subsequent analyses, such as those discussed in the context of conformally compactified Minkowski space and related geometric constructions (ar5iv) (Wikipedia). These sources provide comprehensive insights into the unique properties of the conformal group and its implications in both mathematics and physics.
Not too bad at all, even if I aware of the fact that the responses are individually tailored, and different users will get not necessarily the same answers. Both sources, (ar5iv) (Wikipedia), have links. The first one, by the way, is to my own paper!!!
P.S. 30-06-24 13:54 AI is really dangerous. "Parker Emmerson", for instance, has 174 papers on Academia.edu, all with highly provoking titles.. They look to me like all written by AI with some unclear purpose. These papers are then analyzed by ChatGPT, and their content may be used in responses to others. This way the quality of od responses is self-degrading (perhaps exponentially?) with time. Every disinformation put on the web is thus amplified!!!
P.S. 01-07-24 9:24 I asked the same question about the uniqueness of the conformal group another GPT, namely YandexGPT, which is based on GPT 3 (and seems to be mainly business oriented).
The answer was disappointing
Response:
I'm sorry, but without more specific details about the paper or the context in which the statement was made, it is difficult to provide specific references. However, you can try searching for papers by Irving Segal or other researchers who work on conformal field theory and related topics.
Here are some steps you can take to find relevant references:
Try searching online databases such as Google Scholar or arXiv for papers written by Segal that might contain the statement you remember. You can also try searching for related terms such as "conformal field theory" or "conformal group".
Look for textbooks or review articles on conformal field theory that might discuss this result. These may provide more context and references to the original paper.
Contact a researcher who works on conformal field theory or related areas and ask them if they know of any papers that discuss this result or if they have any suggestions for further reading.
If you still cannot find the reference, consider consulting with a librarian or research advisor who can help you search for relevant materials.
On the Net I have found the following funny comparison of ChatGPT with YandexGPT:
"There are 8 benches in the park. Three were painted. How many benches are there in the park?"
ChatGPT-3.5's response: "There are still 8 benches left in the park, just three of them have been painted."
YaGPT's answer: "There are 5 benches in the park."
P.S. 01-07-24 17:00
A must watch (be patient - slow may be a connection):
P.S. 02-07-24 17:23 I am finding the Penrose's 1967 paper "Twistor Algebra" has been cited 976 times!
P.S. 03-07-24 This morning I asked ChatGPPT for help, using the following Prompt:
I received the following email from X X:
"Dear Professor Jadczyk,
I hope you are in good health and high spirits.
We were so delighted to hear that you'll be chairing a Mathematics
Webinar on "The Measurement Problem in Quantum Theory". I am writing to
follow up on whether you have any speakers in mind and the month you
would like to hold the webinar.
Attached is the document for us to proceed further. When the speakers
and time are confirmed, I would appreciate it if you could send me the
document back.
Should you have any inquiries or require assistance, please do not
hesitate to reach out to me. I am here to provide any necessary support.
I look forward to hearing from you.
--
Best regards,
X X Managing Editor
Mathematics Editorial Office "
Please, help me to write a reply to this message. I want to tell X that I will NOT organize any such webinar, because I am too busy with other obligations and simply do not have time for any organizational activities. Please make the answer to sound kind, friendly, somewhat jokingly, and sounding a little bit old baroque stylish. At the end add something that will make this X feel happy and smiling.
Here is the Response:
Dear Ms. X,
I trust this missive finds you in the very pink of health and delight.
Your gracious note reached my desk, and I must confess it brought a smile to my face as I envisioned the grand symposium you described. The subject, indeed, is a matter of great fascination and import in the realm of quantum theory, and I am humbled by the thought of chairing such a distinguished gathering.
Alas, I must beg your indulgence, as the weight of my current obligations leaves me scarcely a moment to catch my breath, let alone to undertake the organization of such an illustrious event. The labyrinth of my daily tasks and commitments has me ensnared to a degree that permits naught but the barest of escapes.
Therefore, with a heart both heavy and lightened by the fondness of your kind offer, I must regretfully decline the honor of organizing this esteemed webinar. I assure you, my dearX, this decision is wrought from the deepest depths of necessity rather than desire.
May I, however, extend my sincerest wishes for the grand success of your endeavor? And should you require a humble participant or an enthusiastic audience member, I shall be at your service with the utmost alacrity.
With warmest regards and a twinkle of good humor,
Professor Jadczyk
P.S. I am certain that with your delightful charm and keen organizational acumen, the webinar will shine as a beacon of enlightenment and joy, leaving all who attend with hearts uplifted and minds enriched.
I admit: I shamelessly used without any changes! Probably I shouldn't... Then I asked:
Thank you! What was your Temperature while providing your last Response?
The Response was:
My response was generated with a focus on a friendly and formal tone, tailored to match your request for a kind, somewhat humorous, and old baroque stylish message. In terms of "temperature" as a metaphor for writing style, it was set to be warm and engaging, aiming to strike a balance between formality and friendliness.
If you are referring to the technical term "temperature" used in AI language models, which controls the randomness of the output, I don't have a direct measure of that for each response. However, the style and tone were crafted to meet your specific requirements for a pleasant and engaging reply.
YandexGPT proposed perhaps a more appropriate reply:
Dear X,
I hope you are having a wonderful day! I received your email and I am so glad that you wrote to me. I must confess that I am overwhelmed with my current obligations and simply do not have the time to organize a webinar on "The Measurement Problem in Quantum Theory". However, I appreciate your enthusiasm and interest in this topic.
It is truly a shame that I cannot take on this task, but I am sure that there are many other brilliant minds who can step up and organize such an event. Perhaps you could find someone else who would be willing to take on the challenge.
Best regards,
Professor Arkadiusz Jadczyk
YandexGPT shows the Temperature of the response. It was the default 0.3.
P.S. 05-07-24 8:51 And so I asked ChatGPT this question:
I am working on Mathematica code drawing a picture of the torus. Here is my code:
torus2[a_, b_][fi_,
psi_] := {(a + b Cos[fi]) Cos[psi], (a + b Cos[fi]) Sin[psi],
b Sin[fi]}
mc = ParametricPlot3D[
torus2[8, 3][fi, psi], {fi, 0, 2 Pi}, {psi, 0, 2 Pi}, Mesh -> None,
PlotStyle ->
Directive[Opacity[0.6], LightGray, Specularity[White, 10]]]
inf1 = ParametricPlot3D[torus2[8, 3][fi, fi], {fi, 0, 2 Pi},
PlotStyle -> {Black, Thick}]
inf2 = ParametricPlot3D[torus2[8, 3][fi, Pi - fi], {fi, 0, 2 Pi},
PlotStyle -> {Black, Thick}]
Show[{mc, inf1, inf2}]
Can you propose a simplification of this code?
Here is the response:
Sure, we can simplify your code by defining the torus once and then using it for both ParametricPlot3D and the Show function. Here's a simplified version of your code:
(* Define the torus function *)
torus2[a_, b_][fi_, psi_] := {(a + b Cos[fi]) Cos[psi], (a + b Cos[fi]) Sin[psi], b Sin[fi]}
P.S. 03-07-24 9:42 Got a reply from X X! Here it is:
Dear Professor Jadczyk,
Thank you so much for your kind and thoughtful response. It was a pleasure reading your note, and I understand completely about the constraints on your time and energy. Your warm wishes mean a great deal to us, and your enthusiasm for the subject matter is heartening.
Wishing you the very best in all your endeavors. Have a nice summer!
Best regards,
X X
So, I asked ChatGTP what is the probability that this reply was AI-generated. Received the following response:
Assessing whether an email response is AI-generated or written by a human can be quite challenging without definitive markers. However, certain characteristics can help in making an educated guess. Here are some elements to consider:
Language and Tone: AI-generated responses often have a certain consistency in tone and politeness. The reply you received is very polite, concise, and lacks any personal or emotional nuances that might be more characteristic of a human response. Structure: AI-generated responses tend to follow a predictable structure: acknowledging the received message, expressing understanding or empathy, and ending with well wishes. This response follows that pattern closely. Personalization: Human responses may include more personalized details or specific comments relating to previous interactions. This response, while polite and appropriate, is fairly generic. Errors and Style: Human emails sometimes contain typos, informal language, or unique stylistic elements. The absence of these can sometimes hint at an AI origin, although it is by no means conclusive. Contextual Cues: If the organization frequently uses AI for correspondence, this increases the likelihood. Without knowing more about the practices of the Mathematics Editorial Office, it is hard to say definitively.
Given these considerations, while the reply you received is polished and formal, which might suggest it could be AI-generated, there is no definitive evidence. Many human professionals write in a similarly structured and polite manner, especially in formal or professional contexts.
Thus, it is plausible but not certain that the response was AI-generated. If you have prior interactions with X X and can compare the writing styles, it might provide further clues.
P.S. 03-07-24 10:08 Reading the 1967 paper "Twistor Algebra" by Roger Penrose, on p. 348, there is the following footnote
5. To give a rigorous definition of a spinor which takes into account its sign, it is usual to appeal to the theory of fibre bundles. This is not essential, however, and an elementary (nonlocal) geometrical description will be given in an appendix to a forthcoming book by R. Penrose and W. Rindler on the applications of spinors in relativity.
It is interesting that this "forthcoming book" has been published only in 1984. Thus it took 17 years to prepare this book for publishing!!!
P.S. 09-07-24 10:31
Sabine Hossenfelder My dream died, and now I'm here
P.S. 09-07-24 14:36
Watched this video:
Sabine Hossenfelder I don't believe in free will. This is why.
Then I added the following comment on the Youtube channel:
Sabine, I asked Chat GPT
about “free will”. Here is my prompt:
“The concept of "free
will" may have many different meanings. I would like to know if
psychologists have done some research on the following topics: Which
factors influence the amount of free will while people are taking
decisions? Can the amount of free will in a given person at a given
time be evaluated, and if so, how? “
And here is the response:
“Psychologists have indeed explored the concept of
free will and the factors that influence decision-making. Research in
this area often intersects with cognitive psychology, social
psychology, and neuroscience. Here are some key findings and methods
related to your questions:
Factors Influencing Free Will in
Decision-Making
Cognitive Load:
When individuals are under high cognitive load (e.g., multitasking
or under stress), their ability to exercise free will may be
reduced. This is because cognitive resources are limited, and
complex decision-making can be impaired. Social and Environmental Influences:
Social norms, peer pressure, and environmental contexts can
significantly impact decision-making. For instance, individuals
might conform to group decisions or be influenced by authority
figures, reducing their perception of exercising free will. Emotional State:
Emotions can either enhance or impair decision-making. For instance,
positive emotions can lead to more creative and open-minded
decision-making, while negative emotions like fear or anxiety can
lead to more constrained and risk-averse choices. Personality Traits:
Traits such as locus of control (whether individuals believe they
have control over events in their lives) play a role. Those with an
internal locus of control are more likely to feel they are
exercising free will compared to those with an external locus of
control. Psychological Conditions:
Conditions such as depression or anxiety can affect one's perception
of agency and free will. For example, depressive individuals may
feel a lack of control over their actions and decisions. Neuroscientific Factors:
Brain activity and neural mechanisms also play a role. Research
using fMRI and EEG has shown that certain brain regions (e.g., the
prefrontal cortex) are crucial for decision-making processes that
involve free will.
Evaluating Free Will
Evaluating the amount of free will a person perceives
in their decision-making can be challenging, but there are several
approaches:
Self-Report Measures:
Psychologists use questionnaires and surveys where individuals
report their sense of autonomy and control in their decisions.
Examples include the Free Will and Determinism Scale (FAD-Plus) and
the General Causality Orientations Scale. Behavioral Tasks:
Experimental tasks can measure decision-making processes. For
example, researchers might use choice reaction time tasks to see how
much deliberation is involved in making a choice, which can be an
indirect measure of free will. Neuroimaging:
Techniques like fMRI or EEG can measure brain activity related to
decision-making. Differences in brain activation patterns can
indicate the degree of conscious control or automaticity in
decisions. Implicit Measures:
Implicit Association Tests (IAT) and other implicit measures can
assess underlying beliefs and biases that might influence perceived
free will without relying on self-report.
Research Examples
Baumeister, R. F., & Monroe, A. E.:
Their research has explored how individuals perceive their own and
others' free will and the factors that influence these perceptions. Vohs, K. D., & Schooler, J. W.:
They investigated how manipulating beliefs in free will can affect
moral behavior and decision-making. Libet, B.:
Conducted famous experiments on the timing of conscious intention
and brain activity, suggesting that unconscious brain processes
initiate actions before conscious awareness.
Conclusion
The perception and exercise of free will in
decision-making are influenced by a myriad of factors ranging from
cognitive load and emotional state to social influences and brain
mechanisms. Evaluating free will involves both subjective
self-reports and objective measures like neuroimaging and behavioral
tasks. Psychological research continues to deepen our understanding
of this complex and multifaceted concept.”
Perhaps your skepticism about free will stems from a specific and narrow definition you hold. If you consider redefining what free will means, you might not only find it more plausible but also see new opportunities for understanding it, and measuring its amount at any given time in any given person more effectively/objectively.
P.S. 10-07-24 14:50 I tried to fix the html code of this post, and the result is even worse than it was. A ghost of a video appears at the beginning of the comments section! Well, we learn on mistakes!
