The last post, "Flow of time and flow of space",
ended with "In the next post we take a look at what kind of life can
reside on the two infinity circles?". That was my plan. Trying to
bring it to life I started calculating and researching. One thing was
leading unavoidably to another, while I was intensively using AI (mainly
Perplexity) for digging into "what has been already published on the
subject?" My search was mostly unsuccessful. Perplexity could not help
itself but to start hallucinating. It was referring me to sources,
papers, textbooks, online resources, that did not contain the stuff it
was claiming to have. So I was changing my prompts until finally I was
led to a paper that made me very very happy. It was like discovering a
wonderful meadow with lots of daisies and grazing unicorns. The title of
the paper is: "A primer on the (2 + 1) Einstein universe".
A primer on the (2 + 1) Einstein universe
It was exactly what I needed. The paper is some fifty pages long, and
I will be studying it slowly while at the same time describing what I
have already learned here. After that we will return to the point where
we have left, but with a much broader perspective. The paper is written
by five mathematicians, T. Barbot, V. Charette, T. Drumm, M. Goldmann,
K. Melnick, while they were visiting the Schrodinger Institute in
Vienna. You can download it from arxiv here.
It was published in print in"Recent Developments in Pseudo-Riemannian
Geometry", (ESI Lectures in Mathematics and Physics), Dmitri V.
Alekseevsky and Helga Baum eds., EMS - Publishing House, Zürich 2008,
pp. 179-229.
Keywords: Minkowski space, spacetime, Lorentzian manifold, conformal structure, Lie algebra, symplectic vector space
"(...) The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure..."
I will start with the math in the next post. This post is just an
announcement. But I also want to tell you that my view on Artificial
Intelligence has evolved. The AI is not "intelligent" at all. It is a
useful companion, it has access to huge databases, but occasionally it
is simply dumb! It can be also dangerous when you try to rely on it.
Just yesterday I asked Perplexity to sketch a proof of a certain result
quoted in the paper I am advertising, but without providing a proof.
Perplexity happily "proved" the statement and suggested a reference to
look at. I looked and there was nothing even roughly similar there. Then
I thought for a while, and decided to ask if another statement,
contradicting the first one is true. The answer was: yes, the second
statement is true, which means that the first one, the "proven" one, is
false. Sure, both statements look "similar", but we are dealing with
mathematics. The statements x=y and x≠y look similar! But, while using
two-valued logic, only one of them can be true at a time.
Here is another example. I asked Perplexity to analyze the picture below and describe what kind of optical illusion it provides:
Here is the reply:
The attached picture shows a classic impossible object illusion,
specifically related to the genre of the Reutersvärd Triangle or Penrose
Triangle.
In the image, you see a group of cubes arranged in a triangular
configuration on an isometric grid. At first glance, each cube's shape
appears normal, but as you mentally connect the arrangement as a 3D
object, you realize the structure can't exist in real 3D space: the
cubes fit together in a way where the geometry of connections between
them is self-contradictory.
This is an example of an impossible figure illusion: the brain tries to
interpret the arrangement as a three-dimensional object, but the spatial
relationships between the cubes force contradictory perspectives. The
drawing leverages local plausibility with global impossibility—meaning
each small part looks realistic, but the whole configuration violates
Euclidean geometry when viewed as a solid object.
Such illusions are used to illustrate how perception can be tricked by
carefully designed two-dimensional images that depict impossible
three-dimensional shapes—one of the pioneering styles introduced by
Oscar Reutersvärd and later made famous by Penrose and Escher.
Well, I do not see anything "impossible" in the picture!
So I asked the same question Yandex AI, Alice. The answer was much much more satisfactory:
The optical illusion in the attached picture is an example of a 3D
isometric illusion (also known as a multistable perception illusion or
illusion of depth).
Explanation:
The image shows a group of cubes drawn on a grid using isometric
perspective — a technique where parallel lines remain parallel, creating
a 3D effect on a 2D surface.
The shading and positioning of the cubes trick the brain into perceiving depth and volume, even though the drawing is flat.
The illusion works by:
Using consistent angles (typically 30° or 45° from horizontal) to represent the three dimensions (length, width, height).
Applying shading to simulate light and shadow, which enhances the 3D effect.
Arranging the cubes in a way that mimics spatial relationships (some appear in front, others behind).
Key characteristics:
Ambiguity: The viewer’s brain struggles to reconcile the 2D lines with the 3D interpretation.
Forced perspective: The grid lines help maintain the illusion by suggesting a consistent spatial grid.
Depth perception: The shading and cube arrangement create a convincing sense of depth.
Similar illusions: This type of illusion is related to:
Necker cube: A classic example of a reversible 3D cube drawing.
Impossible objects: Like the Penrose triangle, which uses similar
isometric techniques but creates geometrically impossible shapes.
Conclusion: This is a classic example of how 2D drawings can deceive the
visual system into perceiving three-dimensional space, relying on
perspective, shading, and spatial arrangement.
In the early 1960s, Penrose was deeply preoccupied with a fundamental
problem in physics: how to describe the geometry of spacetime in a way
that naturally incorporated quantum mechanics and the behavior of light.
He was frustrated with the standard mathematical tools and felt there
must be a more profound, elemental description of reality.
The pivotal moment came in 1963. Penrose was a visiting professor at the
University of Texas at Austin. He was not in his office, but was
driving with a colleague (some accounts say it was the physicist Ivor
Robinson) outside of the city.
As he was gazing out the car window at the flat, featureless Texas landscape, his mind began to wander.
The long, straight highway and the vast, open horizon triggered a
shift in his perspective. He started thinking about the paths of light
rays—how they could be seen as fundamental, and how points in spacetime
might be a derived concept from the way these light rays intersect.
The key insight was this: Instead of thinking of space as the primary
concept and light rays moving through it, what if he reversed the roles?
What if the light rays (the "null lines" or paths of photons) were the
primary objects, and a "point" in spacetime was defined as the set of
all light rays passing through it?
This was the genesis of twistor theory. In that moment, he realized he
needed a new mathematical space—what he would later call twistor
space—where each point represents a light ray in our physical spacetime.
The geometry of our universe could then be encoded in the complex
geometry of this twistor space.
The Humorous Aftermath
The anecdote often includes a charmingly human detail. The flash of
inspiration was so intense and all-consuming that Penrose, excitedly
trying to explain his radical new idea to his colleague, began
scribbling equations and diagrams on the car's dashboard.
One can imagine the driver's mixed feelings about having their car used as a blackboard for groundbreaking theoretical physics!
We continue from An SO(2,2) Iterated Function System Part 2. The two-dimensional real vector space R2 of Weyl spinors for Spin(2,2) carries an SL(2,R)-invariant
bilinear form ε. In order to distinguish spinors from vectors, from now
on, we will use Greek letters ψ etc. to denote the elements of R2 endowed with this form. Thus
so that the invariant bilinear form ε can be written as
ε(φ,ψ) = φbar ψ. (3)
Exercise 1. Verify that φbar φ = 0 for all φ.
Null Vectors from Weyl spinors
Now we we are ready redefine the construction of determinant zero matrices from spinors. We define now
X(φ,ψ) = φ ψbar = φ ψTω. (4)
Now, for (S,T) in SL(2,R)⨉SL(2,R), we have
X(Sφ,Tψ) = S X(φ,ψ) T-1, (5)
the standard transformation law of vectors x represented by 2⨉2 matrices x^.
Exercise 2. Verify Eq. (5).
Note.Our "spinors" are spinors of the group SO(2,2), which is the conformal group of R1,1.
Therefore we should, in fact, call them "twistors". They are of course
our toy baby twistors. The "adult" twistors of Roger Penrose are
spinors of SO(4,2).
Spinors from null vectors.
In the construction below we will first take a purely pragmatic approach, without discussing its geometrical meaning. So, let
A={{a,b}, (6)
{c,d}}
be any nonzero matrix of determinant zero. We will show that A is necessarily of the form
A = φ ψbar (7)
for some φ, ψ.
Since it is at non-zero matrix, at least one of its elements must be
non-zero. Suppose it is the first row, first column, element A11 = a. Let us define φ' to be the column vector equal to the first column of A:
a
φ' = ( ), (8)
c
and let ψ'bar be the first row
ψ'bar = (a,b). (9)
Construct A' =φ' ψ'bar . This the matrix
A' ={{aa,ab},{ca,cb}}. (10
The zero determinant condition implies ad = bc. We can thus substitute cb in A' by ad. Then A' becomes
A' = {{aa,ab},{ca,ad}} = a {{a,b},{c,d}} = a A. (11)
Therefore setting
1
φ = ( ), (12)
c/a
ψ'bar = (a,b), (13)
solves our problem.
Exercise 2. Can a similar reasoning be used assuming, for instance, that it is b ≠ 0, instead of a ≠ 0 as above?
Exercise 3. Show that the decomposition (7) is
essentially unique, that is that if φ', ψ' is another solution of (7),
the there is a constant λ≠0, such that
φ' = λφ, ψ'= (1/λ)ψ. (14)
Hint: the exercise may need the concepts of a kernel and a range of a matrix considered as a linear operator. Using AI for help is allowed, and even encouraged.
In the next post we will solve the same problem but replacing real numbers with integers. Since division is not allowed within integers, we have have to use a more sophisticated approach in that case.
There’s no shortage of theories trying to explain the mysteries of quantum mechanics, and consciousness often gets dragged into the mix. Some even claim it has a special role in state vector reduction—those quirky “quantum jumps.” Roger Penrose is one such advocate, but he’s not alone. When you ask these consciousness enthusiasts if physics behaves differently on the far side of the Moon (where no one’s watching) or if atoms over there just stop jumping, the answers get...let’s say, "evasive."
Who's Consciousness Is It Anyway?
Now, I'm not here to knock anyone’s ideas. Everyone is entitled to their opinions. But, honestly, this view feels a bit like trying to peer through a foggy window. I mean, whose consciousness are we talking about here? When John Wheeler, another big name in the "consciousness club," was pushed on this point, he eventually said it was “consciousness in general.” Sounds poetic, right? The problem is, no one knows what “consciousness in general” actually means.
And let’s be real—science knows more about quarks and gluons than it does about human consciousness. So, no, I don’t think consciousness is irrelevant, but I do think it’s being overhyped in quantum mechanics.
Is the Universe Watching Itself?
Measurements and observations in the universe happen, whether we’re looking or not. The cosmos doesn’t wait for us to wake up in the morning to start its business. Atoms are constantly observing other atoms, cats are keeping tabs on mice, and things happen regardless of our awareness.
This brings me to my simple, no-nonsense perspective: consciousness doesn’t create reality, it just interprets it. Consciousness, in my view, is there to give meaning to what we observe. The more complex the consciousness, the deeper and richer the meaning.
The meaning of it all?
The Meaning of Consciousness
Here’s the kicker: the higher your level of consciousness, the more meaning you can extract from an experience. A cat sees the world differently than bacteria, and we humans have the potential to see meaning on levels a cat can’t even dream of. This ability to find meaning ties into free will—we see meaning in something, and it allows us to make conscious decisions, changing ourselves and reality. It’s like designing your own narrative rather than just playing out some mechanical script.
So, no, I don’t believe in consciousness collapsing wave functions. But that doesn’t mean it’s useless—it’s just not responsible for playing referee in quantum mechanics.
Gravity vs. Consciousness: The Final Showdown
Now, if consciousness isn’t out there reducing wave packets, then what is? Gravity, maybe? Some theories suggest gravity might be responsible for collapsing quantum states. But here’s the twist—what if wave function collapse actually explains gravity? It’s one of those chicken-and-egg scenarios, except the chicken is quantum mechanics and the egg is the entire fabric of spacetime.
I don’t claim to know the definitive answer to what reduces wave packets, but I do know how to describe the process. And that’s what matters. A simple, quantitative description that satisfies both quantum theory and the observations of a humble peasant (equipped with the finest metaphorical pitchfork, of course).
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!
