The photon wave function

Buzz about a quantum gravity theory that sends space and time back to their Newtonian roots I read that autumn begins this year September 23. I am forced to make a break in my post until at least this day. Here is the reason:

I am interested in understanding the phenomenon of light propagation. I am not completely sure what light is, but it is somehow related to electromagnetism, quantum theory  and the concept of a photon. At least so they say. So I read what is known about the subject. Actually I am studying the monograph "Geometry of Quantum Theory" by V.S. Varadarajan, 2nd edition Springer 2007. 

This is not exactly "Geometry of Quantum Theory".

But there are connections - see below.

The last chapter there:

Here is content of this chapter:

CHAPTER IX 

Relativisitic Free Particles  322

1.  Relativistic Invarince  322

2.  The Lorentz Group  330

3.  The Representations of the Inhomogeneous Lorentz Group  343

4.  Clifford Algebras  348

5.  Representations in Vector Bundles and Wave Equations 356

6.  Invariance Under the Inversions  372

7.  Localization  377

8.  Galilean Relativity  391

Notes on Chapter IX  399

Photons are discussed on p. 371. The last paragraph of this chapter, p. 371,  reads as follows

I think Varadarajan has found that he is not yet ready to finish this chapter the same way as he did it for other particles. He did not go into these idease ideas neither here nor anywhere else that I know about. But I have to go into these ideas. It's my duty. There is one more mention of the photon in this book:

"The photon is  not localizable". This problem also need to be dealt with. So it becomes my priority. I am having problems with connecting the loose ends of the photon's saga. Therefore I am forced to take a break from posting on this blog - until the loose ends connect. Until I make them connect in my mind.

The beginning of the preface to the second edition  (and the end of the preface to the first edition) of "Geometry of Quantum Theory" has  this quotation:

The footnote explains that the quotation is taken from 

* Bhagavadgita,  2 :47a. 

So I check. And here is the translation taken from the book edition displayed at the top of this post:

Your right is to work only and never to the fruit thereof.  And this becomes my motto too. I hope to be able to come back with Pythagorean stuff at the beginning of this autumn.

P.S.1 22-09-23 Here is the expanded reply to a comment by Bjab, September 20, 2023

at 9:50AM

 

P.S.2. 25-09-23 Worth to know and to think about. Tucker Carlson interview for Die Weltwwoche: 

“They’re all afraid”

view and listen also here.

P.S.3. 26-09-23 And so we have autumn now and still didn't solve the problem that Varadarajan left unfinished. I feel like there is a dark force acting and preventing me from finding the solution. I give myself time till next week to resolve it. After that I do what I Ching (I have consulted yesterday)tells me to do: "To bring oneself to put aside pride and to follow good man ...". Good men are in this case the experts. John Baez, Peter Woit, Jan Derezinski come to my mind. Each of them wrote a book on quantum field theory and dealt with the subjects related to my problem. Perhaps someone already did it and they know. But first I have to do my homework diligently.

P.S.4. 26-09-23 The preprint by G.M. Koczan, "Physical unambiguity of the definition of the photon position operator and its special eigenstates" mentioned in  my comment three days ago, is now available as pdf on arxiv (it was not available before because of some bug in the processing). The Author informed me kindly yesterday that soon there will be a new improved version of this paper. This time my active help in working on  this paper is going to be acknowledged (or so I am told). Nevertheless we still disagree on a number of important issues.

P.S.5. 26-09-23 17:47 The dark force that I have mentioned in P.S.3 must have got scared by the fact that I was contemplating asking "the experts". Since then everything started working and, miraculously, all lose ends come now together. Everything works as I have predicted/expected. Even better. Test after test - they all end with a success. It is a very good day today! There is a little technical question that I would like to know the answer to, but that concern only "longitudinal photons", not of my concern right now.

" For all of us seeking to find the true nature of photons, this invites us to look ever deeper into the mysteries of light. In 1917, Albert Einstein said, “For the rest of my life, I will reflect on what light is.” Likewise, light continues to motivate us to take on new challenges."

Mr. Takhashi - The mysterious behavior of individual photons

P.S.6 27-09-23 09:11 Prompted by the discussion with Bjab in the comment section below, this morning I ordered the book

The cover tells us:

"This book shines bright light into the dim recesses of quantum theory, where the mysteries of entanglement, nonlocality, and wave collapse have motivated some to conjure up multiple universes, and others to adopt a "shut up and calculate" mentality. After an extensive and accessible introduction to quantum mechanics and its history, the author turns attention to his transactional model. Using a quantum handshake between normal and time-reversed waves, this model provides a clear visual picture explaining the baffling experimental results that flow daily from the quantum physics laboratories of the world. To demonstrate its powerful simplicity, the transactional model is applied to a collection of counter-intuitive experiments and conceptual problems."

Should come the first week of October. Will be waiting impatiently. 

P.S.7. 9:40 Which reminded me of the books by Eric Berne on Transactional Analysis that I have avidly studied in my youth. Dealing with longitudinal photons now.

P.S.8. 11:50

""I'm a time traveler," said George. "I came from the future, or perhaps I should say one possible future."

John G. Cramer "Einstein's Bridge",

P.S.9 12:13 Have calculated the longitudinal photon boost cocycle. It came out unbelievingly  simple! It is just the redshift ratio E'/E where E' is the energy of the photon emitted by the moving source. The cocycle property translates simply into

(E''/E')(E'/E)=E''/E. Can't be anything simpler!

P.S.10 09:14 Space and time (together with momentum and energy) need to be split apart!

• Jantzen, "Spacetime splitting techniques and gravitoelectromagnetism in general relativity"
• Horawa, "Topological Quantum Gravity of the Ricci Flow"
• Hoprava, "Lifshitz Gravity for Lifshitz Holography"
• Scientific American 2009: "Splitting Time from Space—New Quantum Theory Topples Einstein's Spacetime - Buzz about a quantum gravity theory that sends space and time back to their Newtonian roots" (except: change Hoava into Horava)
• Horava, "The Geometry of Time in Topological Quantum Gravity of the Ricci Flow"
• Horava, "Surprises with Nonrelativistic Naturalness"
• Blas, "Models of non-relativistic quantum gravity: the good, the bad and the healthy"
• Jacobson, "Extended Horava gravity and Einstein-aether theory"
• Landsman - "The 3+1 split of space-time", Chapter 8 from: Landsman - Foundations of General Relativity_ From Einstein to Black Holes-Radboud University Press (2021). in particular: 8.11 Epilogue: The problem of time. P. 217:
  
  "This Pre-Socratic opposition between “becoming” and “being”, or “change” and “existence”, continued with Aristotle. This had disastrous consequences for mathematical physics. In his Metaphysics, Aristotle organized knowledge into something like a 2×2 matrix, where the axes are “changing/permanent” and “dependent/independent”(that is, of man). He put physics in the change & independent entry, whereas mathematics was supposed to be permanent & independent (the latter against Plato). See e.g. Gaukroger (2020). This classification held back the interaction between physics and mathematics for 2000 years, until initially Kepler and Galilei and subsequently Huygens and especially Newton recombined them and thus provided the basis for modern science.
  
  These implication were all proposed by McTaggart (1908, 1927). See also Dainton (2010). The only implication that really counts for our technical discussion is “time ⇒ B-series”, or rather its contrapositive “no B-series ⇒ no time”, but the chain in (*) is convenient in order to frame the overall problem of time. The first implication goes back at least to Aristotle (Physics, Book IV, chapter 11), see Shoemaker (1969) for a nice philosophical analysis. It would be denied by Newton (Rynasiewicz, 2014), but GR can deny it, too, as it admits static solutions (see §8.4).
  The point, however, is that according to the arguments reviewed and critiqued below GR admits no flow of time whether or not time requires change. Similarly, the second implication needs to be argued for, as McTaggart does at some length, but his target is the A-series, whose alleged incoherence allows him to disprove the existence of time.
  Instead, the argument in our main text concerns the B-series. It is remarkable that of the two great twentieth-century philosophical treatises about existence and time, both of which are hard-core specimens of “armchair” philosophy based on pure speculation, McTaggart (1921, 1927) has been very influential on discussions that are informed by modern science, whereas Heidegger (1927) has, rightly, been completely sidelined in the philosophy of science.
  
  This is the version of the problem addressed in Callender (2017), whose opening sentences deserve to be quoted: ‘Time is a big invisible thing that will kill you. For that reason alone, one might be curious about what it is.’"

Also relevant for lose ends: 

• Landsman: "Massless particles, electromagnetism, and Rieffel induction"
• Buchholz, "The universal C*-algebra of the electromagnetic field"

P.S.11. 11:07 -  8 HOUR Sleep Is The WORST

00:04:10,199 --> 00:04:14,878

disoriented I mean remember in sleep

00:04:12,299 --> 00:04:17,160

space and time are are totally uncoupled

P.S.12 13:59 Hitzer, "Special relativistic Fourier transformation and convolutions"

P.S.13 29-09-23 8:01 Next on my reading list (after Cramer's "Einstein's Bridge" and "Twistor"):

I like this lady! She is so much better than I am, in all respects. Errata to her "Analysis, manifolds and physics Part 1", that is at the end of Part 2, takes the whole 11 pages! I am not alone making so many mistakes! I like it!

P.S.14 8:08 My lose ends came nicely together. Krishna must have helped me. Still one little cloud, but that is probably my Mathematica code, not the idea itself. Have to rewrite the code from scratch. Will fix it today. Then I have to write it all down before I forget the details of the construction that still resides in my mind only. After that will try to rewrite it using Clifford algebra. Projecting on transversal photons should be much easier using gamma matrices. Photons are gamma matrices algebra generators! BTW Mathematica is also featured in "Einstein's Bridge".

P.S.15 19:18 The nasty glitch persists. I am close but no cigar yet.

P.S.16 8:38 Glitch resolved! Krishna helped. Got the ball this morning! Longitudinal photons under control. Now I need to understand what I have done!

And have to write it all down in a nice, clear way. Then move to connections and "structured light".

P.S.17 9:11 

"What did this enzyme do? She considered this, and the answer became startlingly clear. In normal cells each strand of DNA had a special noncoding segment on each end, like the plastic tips of a shoestring. She could see that each time a cell divided this special end segment became shorter. And finally, when the segment length went to zero, the cell could not divide again, the natural cell renewal processes stopped, and the body began to age. The new enzyme systematically restored the end segments to human DNA. It did not require cells to divide, but it allowed them to when the body's repair mechanisms made the request. "Wow!' said Alice aloud. "It's the Fountain of Youth!"

John G. Cramer, "Einstein's Bridge"

P.S.18 15:45

""You mean that we can go back to a time when Alice is still alive?" George asked. He looked a bit wild-eyed, Roger thought.

"Yes," Iris replied, "of course."

"Wait a moment," said Roger. "Are you talking about moving back to some alternate-branch Everett-Wheeler universe?"

Iris laughed. "Since we established contact with your culture, many Individuals of our world, particularly our science meta-historian specialists, have derived great amusement from your  quantum mythology, that area which you call the 'interpretation of quantum mechanics.' They were particularly amused by your Copenhagen interpretation, with its state vectors that are altered by the thoughts of intelligent observers, and by your Everett- Wheeler interpretation, with its splitting and resplitting into multiple universes. In this regard, your culture is unique among  those that we have encountered. No other has provided such a remarkable demonstration of fertile creative desperation in seeking to understand physical behavior at the quantum level. We find these myths of yours quaint and charming."

"In other words, 'wrong'?" asked Roger.

Iris looked troubled. "No more wrong, say, than your Greek  or Norse myths. Your excursions of scientific fantasy are an interesting manifestation of your culture, but they are not an accurate portrayal of the behavior of the universe. Human observers, for example, are not demigods with the ability to collapse a wave function with an act of measurement or of insight. It is better that they are not, believe me." "

John G. Cramer, "Einstein's Bridge"

P.S.19. 12:06 Studying now in depth two old papers by A. Staruszkiewicz:

1) On Affine Properties of the Light Cone and Their Application in the Quantum Electrodynamics,

Acta Physica Polonica B 4, pages 57-63 (1973)

2)

Pythagorean triples and 12 rules for life - from things to actions

 I do like Jordan Peterson. I his "12 rules for life" you will find many goodies. Let me quote just one:

"Scientific truths were made explicit a mere five hundred years ago, with the work of Francis Bacon, René Descartes and Isaac Newton. In whatever manner our forebears viewed the world prior to that, it was not through a scientific lens (any more than they could view the moon and the stars through the glass lenses of the equally recent telescope). Because we are so scientific now—and so determinedly materialistic—it is very difficult for us even to understand that other ways of seeing can and do exist. But those who existed during the distant time in which the foundational epics of our culture emerged 

were much more concerned with the actions that dictated survival 

(and with interpreting the world in a manner commensurate with that goal) than with anything approximating what we now understand as objective truth."

I would like to stress the word "action". Actions, processes, connections are more fundamental than "things" subjected to actions. Of course at the "beginning" where there were only actions, they were feeling lonely, so the need was generated for "things" to act upon. So "matter" was created out of the dense vortices of actions.

 That is my own view of Genesis.

But now I have started with "things", even though in the Platonic world of numbers. These "things" under our scrutiny are Pythagorean triples.  

We need to compensate for the lack of actions so far. How these triples were "created"? In what kind of a process? By what kind of actions? We will deal with it in the coming posts. For now we need some vocabulary.

Pythagorean triples can be primitive or not. The triple of positive integers  (a,b,c) is primitive if the pair (a,b) is "coprime" or "relatively prime" if a and b have no common (natural number divisor different from 1. So (3,4,5) is a primitive triple, while (6,8,10) is not.

One easily proves that if (a,b,c) is a primitive triple, then one of (a,b) is odd and the other one is even. Like it is with (3,4). On the net, or, more seriously, in

"Elementary Theory of Numbers" by W. Sierpinski, North Holland 1988, 

we can find the following theorem (known apparently already to Euclides)

And then the book goes with some kind of an action:

"In order to list systematically all the primitive solutions of equation (1) we take values 2, 3, 4, ... for the number m successively and then for each of them we take those numbers n which are relatively prime to m, less than m and being even whenever m is odd. Here is the table of the first twenty primitive solutions listed according to the above-mentioned rule:"

So, there is some action, but is kind of dull, not "real". No drama. And what is life worth without a drama? Not exciting. 

We will meet action and drama in the coming posts. 

P.S.1. 11-09-23 A new paper appeared in  Siberian Advances in Mathematics

Published: 01 September 2023 Here is the first page:

and here is the last

Very soon we will be plotting Pythagorean quadruples on the Dupin cyclide of the conformal infinity - as in my papers cited by Berestovskii and as in my book on quantum fractals. Probably already Euclid knew it all, but then it was all forgotten!

P.S.2. 11-09-23

On the same wave the song and the voice that I like very-very much

Ирина АРХИПОВА (меццо-сопрано) и Ансамбль скрипачей БТ под управлением Ю. Реентовича. Ф. Лист "Канцона". (Из телефильма "Незабываемые голоса")

Matrix=womb

 Wikipedia has an article "Pythagorean triple". 

It contains a lot of useful information, but what draws my particular attention is this sentence in the section "Spinors and the modular group":

The group Γ(2) is the free group whose generators are the matrices

Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices U and L.

We will call these matrices U₂ and L₂. I really like the matrix U₂. Here is a piece from page 96 of my book Quantum Fractals.

The matrix U2 actively participated in creation of the cover image of my book

It deserves our special attention. And yes, it has to do with spinors, and it has to do with light. And, of course, with Pythagorean triples and quadruples, and even quintuples. There is a whole saga that needs to be told. But first things first. Step by step.

We will be dealing with matrices. While it is a plausible hypothesis that God created the integers (see my previous post "The magic of numbers 3,4,5", it is not clear who invented matrices  with their noncommutative multiplication rule. Till now nothing is known about matrices being featured in the Bible, Talmud or Babylonian tablets. The online paper "When was Matrix Multiplication invented?" informs us that:

1850 Sylvester first use of term "matrix" (matrice=pregnant animal in old french or matrix=womb in latin as it generates determinants)

1858 Cayley matrix algebra [7] but still in 3 dimensions [14]

1888 Giuseppe Peano (1858-1932) axioms of abstract vector space [12]

The Matrix is a womb. Billy Carson

I will assume that the Reader is acquainted with matrix multiplication. We will be dealing here first with 2x2 matrices: two rows and two columns. Then with 3x3 matrices, then with 4x4 matrices, then ... the future will tell. Matrices will have real or complex entries, but we will will be particularly interested in matrices with integer entries. The matrix U₂ above  has integer entries 0,1,2. Our cat Pikabu knows this matrix pretty well. 

Neo's Best Moments| The Matrix Trilogy | Max

There is a particular matrix with integer entries, we call it E. Wikipedia calls it the identity matrix and denotes it by the capital letter I:

The identity matrix is often denoted by ${\displaystyle {�}_{�}}$, or simply by ${\displaystyle �}$ if the size is immaterial or can be trivially determined by the context.^[1]

If A and B are square (the same number rows and columns) their matrix product (in the order written) is denoted AB. In general AB is not the same as BA. For an arbitrary matrix A and the identity matrix we have though AE=EA=A. If A,B are two matrices and AB=E, one can prove that then also BA=E. The matrix B with this property is unique and called the inverse A.  Quoting Wikipedia again:

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that

where I_n denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.^[1] If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A⁻¹. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

 Here Wikipedia is using bold letters to denote matrices. We will use normal letters. We will also need the concept of the determinant and trace. Exploiting the handy  Wikipedia again:

The determinant of a 2 × 2 matrix is

The trace Tr(A) of the matrix A is simply the sum of all its diagonal elements. In the acse of the 2x2 matrix above

Tr(A ) = a+d. 

There is a beautiful little paper by Roger C. Alperin "The Modular Tree of Pythagoras", p.807,  we notice the following interesting sentence