Why algebras?

 

Eugene Wigner and Quantum Superposition

There may be other superselection rules, some of them known, some yet unknown. I do not know why so many physicists ignore this fact. As early as 1951, in Chicago, at the International Conference on Nuclear Physics and the Physics of Fundamental Particles, Eugene Wigner introduced the idea that quantum superpositions are not universally applicable.

Way to Visualize Quantum Superposition

John Earman and Superselection

John Earman, a philosopher of physics at the University of Pittsburg, in his paper “Superselection Rules for Philosophers” tells us the story of this most important issue. At the Chicago conference “some members of the audience were shocked, presumably because Wigner's proposal contradicted von Neumann's (1932) assumption”, an arbitrary mathematical assumption that has been accepted, without any questioning whatsoever, by almost all followers of the “quantum orthodoxy”.

John von Neumann, Alain Connes,  and Carlo Rovelli

John von Neumann was a genius mathematician, one of the pioneers of computer science, who also served as a consultant for the US army, and who played an essential role in the development of the American hydrogen bomb. He created the mathematical foundations of quantum theory, and played an essential role in the development of the theory of “operator algebras” – the most advanced mathematical apparatus for quantum theory even today. These operator algebras are also considered to be essential for understanding fundamental questions, such as “what is time?”; at least that is what is suggested in a number of papers by Alain Connes, Carlo Rovelli, and others. But, I am asking the question: why algebras?

Algebra and Kipling’s Six Honest Serving Men

You see, I like asking questions, simply because I am curious. I keep in my mind the poem by Rudyard Kipling that begins as follows:

I KEEP six honest serving-men

(They taught me all I knew);

Their names are What and Why and When

And How and Where and Who.

So, why do we use algebras? Usually they are called “algebras of observables”. So, I have to tell you now a little bit about these “observables” – a really confusing concept, although few physicists stop to think about this confusion. Usually they use this term “because everybody is using it” – that is the most common reason you will be given if you ask a physicist or a mathematician who is playing quantum with these algebras.

In classical physics we deal with “physical quantities”, like position, momentum, angular momentum, energy etc. They indeed do form an algebra; we can multiply two physical quantities, one by another, much like we multiply numbers. This multiplication, in classical (non-quantum) physics is commutative (like, for instance 2x3=3x2 ), and associative (like, for instance (2x3)x4=2x(3x4) ).

In quantum theory, in algebraic formulation, physical quantities are represented by certain so-called “self-adjoint” or “Hermitian" elements of the algebra. They are called “observables” for the reason that given a “state” of a quantum system, and given an “observable”, one can calculate a certain real number that (in the orthodox formulation) that should correspond to the “observed” average (or “expectation”) of the values of the physical quantity in question, when measured on an ensemble of quantum systems, all prepared in the same “quantum state”. These “observables” are then multiplied, one by another, within the algebra, but - surprise-surprise! - their product is not an observable! Standard textbooks will not give you any answer if you ask “what is the meaning of this product?”

So, we have (in the standard algebraic quantum theory) a fundamental object – the algebra, a fundamental operation – that is multiplication, but no one knows what the meaning of this operation is. And very few would even ask! Asking such questions is considered as inappropriate – unless you ask them privately, at a conference, when drinking wine or champagne in the company of an expert who will tell you sympathetically that “nobody knows.”

P.S.1.  Wigner and von Neumann were both Hungarians in origin. As well as Liszt.

Maksim Mrvica playing Franz Liszt's Hungarian Rhapsody No. 2

My wife has loved this piece since she first heard it at the age of 9. She has listened to hundreds of performances over many years and this is the one she thinks is nearly perfect. She says it is "clean and precise" and not rushed. She memorized the score many years ago and can tell instantly if there is a mistake.

P.S.2. The orientation covering from Lee, "Introduction to Smooth Manifolds" 2nd ed. - my favorite "doubled universe"

My hypothesis: the conformally compactified Minkowski space is unorientable, its double cover (as in my papers) is orientable. Probably can be seen immediately, but I need to be careful not to make a silly mistake.

P.S.3. Working on aether theory with a doubled universe I was brought to the paper "The metric aspects of a noncommutative geometry" by Lott and Connes. The References section starts like this:

It is good to know that I'm not the only one with dyslexia. And, by the way, while writing this book [CJ] we were following the Mouravieff method: each chapter starts with "exoteric" stuff (to get a general idea what the chapter is about - for a general public). Only after that follows the "esoteric" part - for advanced insiders.

P.S.4. Can time be nonorientable? It appears (see also here) that it can be:

But  I have to understand it yet.

P.S.5. Such manifolds must be non-Hausdorff, thus non-metrisable. This forgrt Engelking. Non-Hausdorff means there is at least one pair of points which cannot be separated by non-intersecting neighborhoos.

After doing some research I see that this may be a Polish speciality:

The Future of Physics - Branching Realities:

Interpreting Non-Hausdorff (Generalized) Manifolds in General Relativity
Joanna Luc & Tomasz Placek 

Philosophy of Science 87 (1):21-42 (2020)  

Figure 1: One-dimensional non-Hausdorff manifold.

Putting on my reading list.

P.S.6. Continuing my search. Another find (with a Polish trace):

New Foundations for Branching Space-Times

N. Belnap, T. Müller & T. Placek 

Studia Logica volume 109, pages239–284 (2021)

Abstract

The theory of branching space-times, put forward by Belnap (Synthese 92, 1992), considers indeterminism as local in space and time. In the axiomatic foundations of that theory, so-called choice points mark the points at which the (local) possible future can turn out in different ways. Working under the assumption of choice points is suitable for many applications, but has an unwelcome topological consequence that makes it difficult to employ branching space-times to represent a range of possible physical space-times. Therefore it is interesting to develop a branching space-times theory without choice points. This is what we set out to do in this paper, providing new foundations for branching space-times in terms of choice sets rather than choice points. After motivating and developing the resulting theory in formal detail, we show that it is possible to translate structures of one style into structures of the other style and vice versa. This result shows that the underlying idea of indeterminism as the branching of spatio-temporal histories is robust with respect to different implementations, making a choice between them a matter of expediency rather than of principle.

and this:

Defining a Relativity-Proof Notion of the Present via Spatio-temporal Indeterminism

Thomas Müller 

Foundations of Physics volume 50, pages644–664 (2020)

Abstract

In this paper we describe a novel approach to defining an ontologically fundamental notion of co-presentness that does not go against the tenets of relativity theory. We survey the possible reactions to the problem of the present in relativity theory, introducing a terminological distinction between a static role of the present, which is served by the relation of simultaneity, and a dynamic role of the present, with the corresponding relation of co-presentness. We argue that both of these relations need to be equivalence relations, but they need not coincide. Simultaneity, the sharing of a temporal coordinate, need not have fundamental ontological import, so that a relativizing strategy with respect to simultaneity seems promising. The notion of co-presentness, on the other hand, does have ontological import, and can therefore not be relativized to an observer or to an arbitrarily chosen frame. We argue that a formal representation of indeterminism can provide the structure needed to anchor the relation of co-presentness, and that this addition is in fact congenial to the notion of dynamic time as requiring real (indeterministic) change. The resulting picture is one of an extended dynamic present, implying a formal distinction between static (coordinate) simultaneity and dynamic co-presentness. After working out the basics of our approach in the simpler framework of branching time, we provide our full analysis in the framework of branching space-times, which allows for a formal definition of modal correlations. The spatial extension of the dynamic present can reach as far as the modal correlations do. In the limit, the dynamic present could extend across a maximal space-like hypersurface.

P.S.7. Posts about Science, so far

1.  Talking about Science: 1 Boys and Frogs 
2. Talking about Science: 2 Poincaré and The Search for Truth
3. Talking about Science: 3 Tony Smith and “arXiv.org” 
4. Carlos Castro Perelman and the tide 
5. Bertrand Russell and Independence in Science 
6. Questions About Science: Is Science rational?
7. The Taboo of Subjectivity 
8. Can Science be just?
9. Einstein and Klein, Plagiarism 
10. Religion and Science – cruel Gods 
11. Bertrand Russell and “A Way of Feeling" 
12. Plato and The Value of Myths and Parables 
13. Cronus and Uranus 
14. Defining "Science" 
15. Wrong use of Science 
16. Curiosity, intellectual freedom and Science 
17. The Curiosity of Alfred Russel Wallace 
18. The Encyclopedia Universalis Twists the Truth 
19. Clifford’s Solution
20. Language Barriers Make Knowledge Barriers 
21. Forbidden Science 
22. You Shall Know Them by Their Fruits 
23. No True Science Allowed! A Priori Assumptions Prevail 
24. William Crookes and the Paranormal: True Science 
25. Ray Hyman and Modern Apathy: To Explain Away and Dismiss
26. Dangerous to be Curious? Quantum Future - Gossip and Censorship 
27. A Brush With the Dark Side of Science 
28. Brian D. Josephson on Censorship in Science 
29. Silence is the greatest persecution
30. The case of Grigori Perelman and When bad men combine, the good must associate
31. Criticism is easy
32. Trinh Xuan Thuan - "The Quantum and the Lotus" - Part I
33. Trinh Xuan Thuan - "The Quantum and the Lotus" - Part II
34. Trinh Xuan Thuan - "The Quantum and the Lotus" - Part III
35. Real Scientists Do Speculate!
36. Quantum Magic - Incoherent Decoherence
37. Psychological interlude - Authority in Science
38. Thinking With a Forked Brain
39. Schrödinger’s Cat
40. Why algebras?

P.S.8. About EPR and Bell's theorem:

Kowalski, Placek
Outcomes in Branching Space-time and GHZ-Bell Theorems

Brit. J. Phil  Sci.  50  (1999),  349-375 

1  Introduction 

2  The algebra  of outcomes  in branching  space-time 

3  Physical  interpretation 

4  Common causes  in branching  space-time 

5  Is there a common cause  in the GHZ-Bell  setup with three particles? 

6  Is there a common cause  in the GHZ-Bell  setup with four  particles? 

7  Is there a common cause for  the EPR (perfect) correlations? 

8  Is there a common cause for the GHZ-Bell  setup with three particles  if directions  are fixed? 

9  Conclusions 

P.S.9. From David Gould, "Non-metrisable manifolds", Springer 2014, p. 153

Non-Hausdorff manifolds also appear as possible models of space-time in ‘many-worlds’ interpretations of quantum mechanics, relating to time travel and as reduced twistor spaces

in relativity  theory (see, for example, [5], [11, pp. 594–595], [12, pp. 249–255] and [14]).

P.S.10.🙋

"We have ALIEN craft in our possession" Govt. UFO whistleblower admits BOMBSHELL | Redacted News