Quantum mechanical systems are not different from any other system, where dynamical degrees of freedom are coupled to a bath of fluctuations. So there doesn't exist a `` non-classical" probability theory. The only difference between a quantum system and its classical limit is the space of states and the assignment of the probability measure to each state, which involves Planck's constant.

The same axioms that define probability hold for quantum systems as for any other system. Quantum systems are, by definition, ``uncertain", they are systems in equilibrium with a particular bath of fluctuations, whose parameter is Planck's constant.

Non-traditional likelihood theories have considerably altered the representation of uncertainty in quantum mechanics, rectifying classical likelihood limitations in grasping quantum issues. Contrary to classical likelihood, dependent on Kolmogorov's axioms and assuming the events' interchangeable nature, quantum systems reveal superposition and entanglement features that break these principles (Nielsen & Chuang, 2010).

One such consequence is the acceptance of quantum likelihood, where chances are acquired from the square amplitudes of composite valued wave functions as opposed to classical frequency interpretations. This structure is delineated by Hilbert spaces and non-commutative operator algebras, making sure of a precise portrayal of measurement products and primary doubts (Holevo, 2011). Furthermore, non-traditional probability models bring in concepts like contextuality and non-locality, that cannot be illuminated by classical probability theories. Bell's theorem and the resulting studies show that quantum correlations outdo classical thresholds, requiring in-depth likelihood arguments that deal with said issues (Bell, 1964; Aspect et al., 1982). Another important facet would run with Quantum Bayesianism, which conceives of probabilities as subjective belief levels improved during quantum measurement outcomes.

This outlook reforms uncertainty as the agent's information status, changing how quantum anticipations are understood and brought into operation (Fuchs & Schack, 2013). In a nutshell, non-traditional likelihood theories deliver vital paraphernalia for depicting quantum mechanics' doubt by capturing particularly quantum features which the classical likelihood cannot articulate. These theories heighten our understanding of quantum phenomena, increase predictive accuracy, and challenge olden beliefs on probability and existence.

References

Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental test of Bell's inequalities using time-varying analyzers.

Physical Review Letters, 49(25), 1804-1807. Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics Physique Физика, 1(3), 195–200.

Fuchs, C. A., & Schack, R. (2013). Quantum-Bayesian coherence. Rev. Mod. Phys. 85, 1693 http://dx.doi.org/10.1103/RevModPhys.85.1693

Holevo, A. S. (2011). Probabilistic and Statistical Aspects of Quantum Theory: Second Edition. Edizioni della Normale.

Superposition and entanglement are not incompatible with Kolmogorov's axioms, that's, simply, wrong.

The difference between a quantum system and a classical system subject to a bath of fluctuations lies in the space of states: Take for definiteness a two-level system. The classical limit describes a space of two states, two points. To each is assigned a probability, a non-negative number between 0 and 1. The sum of the two numbers equals 1.

The corresponding quantum system, whose classical limit is the previous two-level system, has as space of states the points of the sphere of unit radius. That's the difference. The classical states are any two antipodal points, all other are non-classical. This doesn't affect how probabilities are assigned, nor Bayes' theorem.

Quantum mechanics pertains to how the probability of a state of a system is determined, it doesn't affect how probability is defined.

The major difference in the probabilistic description of quantum and classicsl systems lies in consideration of entangled particles. Classicsl particles are distinguishable, and thetefore classical logic, and classical probability theories based on it, can be applied. In quantum case entangled particles (that is, constitutive components of a quantum systems), cannot be separated and distinguished. This means that classical logic cannot be applied. Thus, all probability theories that have been developed using classical logic (that in its turn, uses FZC axiom-based theory of sets) cannot be applied.

All in all, probability theories serving quantum theory must be based on non-FZC axiom based theories of sets, varieties, etc.

To what is well clarified by Stam Nicolis, i.e. that probabilities of states/events of QM objects are quite the same probabilities as that is in “classic” mathematics let a few additional points. First of all – everything in Matter is/are completely deterministic, started from/since all QM scale objects are identical.

Why and how are identical? – that is rigorously explained in the the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- where it is proven that there exist nothing else than some informational patterns/systems of the patterns that are elements of the absolutely fundamental and absolutely infinite “Information” Set, The Set exists absolutely objectively, because of it fundamentally – logically - cannot be non-existent and so exists absolutely eternally, having no Beginning and no End;

- so all QM scale objects are identical copies of concrete “texts”, that is a banal situation in information.

As well as in the conception it is explained – why QM states are fundamentally uncertain – that is since the absolutely fundamental phenomenon/notion “Change” is logically self-inconsistent, and so on some scale [in informational system “Matter” that just is the QM scale] the states are uncertain - “illogical”.

However, Matter is completely rigorously logically organized system, which on the ultimate – Planck scale - is based on binary reversible logics [more see the Shevchenko-Tokarevsky’s Planck scale informational physical model, which is based on the conception, two main papers are https://www.researchgate.net/publication/391209088_The_Informational_Conception_and_Basic_Physics, [ in the paper secton 2.9 “Mediation of the forces in complex systems” can be passed since this is more comprehensively given in section 6.“Mediation of the fundamental forces in complex systems” in other paper below] , and

https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics]

- where the QM uncertainties are completely deterministic, and are so the same in any point in Space, at least on hundreds millions of light years from Earth, and seems in time imtervals at least in hundreds of millions of years.

The other point is to “Bayesianism”: really in every case when humas analyze something that for some reasons is random/uncertain, the analysis fundamentally is always subjective, and so is always in accordance with Bayes' theorem; if the theorem isn’t applied directly, really that means nothing else than it is really applied - at/by postulating that the prior probability is uniformly distributed.

Recent couple of SS posts in

https://www.researchgate.net/post/Wrong_calculation_of_the_possible_physical_spin_speed_of_the_Bohr_electron

- are relevant to gthis thread question

Cheers

When one speeks about logic, one must realize that there are classical and non-classical math logic theories. They constitute foundation of math. Anything else in math and its applications, and of course in natural sciences that use math, resides on math logic throry of some sort. The majority of math results (and everything in physics) is derived using classical logic (as defined in math). The major inconsistency of quantum theory is that it was developed using classical logic (and of course, FZC theory of sets) to end up with non-classical results. Probability theories used in QM is classical ones, because they were developed using classical logic. In particular, QM is a theory of linear operators in Hilbert spaces. Hilbert spaces are separable. That is, any 2 poins have non-overlaping neighbourhoods. In other words, they can be separated and thus, are distinguishable. In QM material particles are spatially non-localized. This means, that rigorously speaking, one has to introduce a spatial criterium that would help define whether or not one deals with say, one particles spread in space, or several particles. Probability theory used in QM was derived for distinguishable (spatially localized) particles. Same can be stated about Hilbert spaces of states of quantum particles - they are Hilbert ones, that is, distinguishable, separated. There is nothing further from reality for entangled states of quantum particles. One (1) cannot have those particles localized (and thus, separated), and (2) a space of states of such systems is not a Hilbert one. Thus, one cannot use classical probability theories to describe such systems.

There is a great paper by Saunders concerning non-locality of quantum particles and numerous inconsistency problems inflicted by this issue on QM, that the majority of its users do not realize.

There is no theory further from math rigor than QM.

Also, there are a power of continuum "number" of math objects that are not sets. On physics side, a theory closest to being mathematically rigorous, is classical analythical mechanics, and the least rigorous is QM. In fact, QM is an ad hoc theory that happens to describe relatively well some physics at atomic scale. However, one has to note, that it does not accurately predict even atomic electronic energy levels of a few-electron atoms (they are calculated using a number of fitting parameters gathered from experiment, such as interaction potentials, etc. that QM does not predict).

For thous of you who may be interested: please, let me know, and I will find a citetion of Saunders' paper when I return to the States in about 3 weeks.

Generaly speaking when we say about logics, there exist only one really scientific “classical” logics, while some “non-classical” ones are really defined, and remain be correct till are in accordance with, only in framework of the classical logics.

QM describes and analyze what exists and happens in Matter on QM scale, where – see the SS post above – states of objects [particles and systems of the particles] are fundamentally uncertain, however this uncertainty is completely rigorously determined, and particles are localized – that are points, though also provided “wave-particle duality”. The last is completely non-understandable in the mainstream QM, however essentially this is adequate to what really is in Matter.

This problem isn’t unique that is non-understandable in QM – really all postulates in mainstream QM [and in all other theries, though] are fundamentally scientifically non-grounded transcendent assertions, which so really often are wrong; and just this point is really the fundamental obstruction, because of which real scientific development of QM [and, again, of everything in physics else] is logically inevitably impossible.

What really logically inevitably follows from that in mainstream philosophy and sciences , including physics, all fundamental phebomena/notions, first of all in this case “Matter” , “Consciousness”, “Space”, “Time”, “Energy”, “Information” [and so everything in Matter, i.e. “particles”, “fundamental Nature forces” – and so “fields”, etc.] are fundamentally completely transcendent/uncertain/irrational, and so in every case when the mainstream addresses to any really fundamental problem, the result completely inevitably is transcendent/mystic something;

- when in any theory main fundamental problem is “for what reason, why and how the theory’s postulates are as they are?”

The phenomena/notions above can be, and are, rigorously scientifically defined only in the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- and more concretely in Matter case, i.e. what are “particles”, etc., in the Shevchenko-Tokarevsky’s Planck scale informational physical model, refs to two main papers see SS post above.

Including in the model it is rigorously shown that Matter is based on binary reversible logics, and so everything in Matter, is/are specific disturbances in the Matter’s ultimate base - (at least) [4+4+1]4D dense lattice of primary elementary logical structures – binary reversible fundamental logical elements [FLE], which have (at least) [4+4] independent degreases of freedom at changing state [“FLE binary flip”] ;

- so the lattice is placed in the corresponding Matter’s fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at l east) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), where ct is the time dimension, the 8 others are space dimensions; FLE “size” and “FLE binary flip time” are Planck length, lP, and Planck time, tP.

I.e. really description and analysis of what is in Matter can be made by using a 8-dimensional Boolean algebra version; i.e. completely in framework of classical logics and probability theories, and basing on/developing the Planck scale model above.

The post is long now, so

Cheers

Sergey Shevchenko:

1. you are completely wrong regarfing non-classical logics. They have different systems of axioms and predicate structure, and are developed without any reference to the classical one, apart from comparison with it.

2. The math structure of QM is a theory of linear operators in a Hilbert space. Obviously, this is overly simplified description of nature, considering that even classical mechanics is a non-linear theory. There are many nonlinear quantum properties of natural systems that are not described by QM, such as, once sgain, entangled states of particles, particle non-locality and potentials of their interactions. QM does not provide any systematic approach to derivation of any realistic potential of interactions. They come into QM from classical theory and are considered classically in QM. If one goes further to QED, one immediately realizes that the QED theory does not offer any logical results, unless one disregards all divergencies and keeps only convergent diagrams after renormalization (that itself has little to do with axiomatic structure of any math theory used), and even those are chosen intuitively.

Generally, all theoretical physics relies on applying math methods (most of them simplistic) to desribe complex natural systems that cannot be described by such primitive methods (because natural systems are intrinsically non-linear), unless ad hoc and experiment-based fitting "parameters" are introduced and many "axiomatic" assumptions are used. In particular, who ever proved that QM is the only theory to describe physics at atomic scale? Who ever proved that all quantum observables must be described by Hermitian operators, and not by, say, superoperators that may have Hermitian components? Who ever proved Noether throrems (conservation laws) in QM without implicitly using classical d'Alembert's PRINCIPLE - that is, a mathematically unproven assumption that comes from physics intuition and has nothing to do with axiomatics of math theories used in physics, QM including.

QM is a physics theory, which automatically means it includes many assumptions and parameters that have nothing to do with math rigor.

That's why physics and related sciences are called natural sciences, and math is mathematics, the only real science, that is, it does not play guessing games. And on the top of it, no one has ever proved rigorously that math can adequately describe natural systems. We expect it so because we do not have any better way, but that does not make any physics theory mathematically rigorous, because components of physics systems are far from being "points in space", or being always separable, etc., or otherwise equated to the corresponding constituents of math theories and axioms.

We do not even know what is time at atomic scale, and whether it is quantized or not! We are using classical differential calculus that includes pointwise limiting procedures as applied to the description of dynamics of presumably quantized systems in QM! Who ever proved that quantum space is not quantized itself? Is there anything further from being even logical in the classical logic sence, let alone rigorous!

So let continue [see two SS posts on page 1], firstly a few notes to last Liudmila Pozhar post on page 1: Liudmila Pozhar:

- you correctly write about a number of real mainstream QM/QFTs evidently strange points,

- though say, that

“…. QM does not provide any systematic approach to derivation of any realistic potential of interactions. They come into QM from classical theory and are considered classically in QM.…….”

- really isn’t some QM principal fault. Matter is fundamentally rigorously and simply organized informational system, where utmost fundamental and universal laws/links/constants act essentially identically on both, i.e. macro and QM, scales, including this is true for fundamental Nature forces fields potentials; so, say, Schrödinger equations with “classical” E-potential well adequately describe light atoms; Dirac’s 1928 equation with “classical” E-4-potential and E-momentum, resulted, including, in prediction of positron, etc.

That is another thing that attempts to describe QM scale objects /events /processes more correctly – classical and QM scales nonetheless are of course different, QFTs, firstly QED, were developed, which – as you quite correctly write – are based on evidently transcendent postulates, first of all that the Forces mediators are “virtual”, particles are “excitations” of some transcendent “fields”, so in QFTs evidently transcendent “renormalization” technique is principally applied, etc.

All this really essentially is legitimate in mainstream physics in accordance with the principle that mathematics is so mighty, that if something exists in mathematics, then this also exists in Matter, and with the corresponding tenet “shut up and calculate”,

- but if someone to remedy this situation will use in physics some “non-classical logics”, etc., that will result only in quite more transcendence. Matter, again – see SS page 1 posts - is based/organized on a simple binary reversible logics, and description and analysis of what exists and happens in Matter should be made by using quite classical math, including 8D Boolean algebra.

Just so application in physics of [classical] math is completely correct; this

“…And on the top of it, no one has ever proved rigorously that math can adequately describe natural systems. ….”

- is principally incorrect, the proof see above.

Though yeah, math is principally first of all only a physical tool, but in some cases it indeed “dictates” some physics; first of all that happens in the angular momentum case – AM as cross-product mathematically doesn’t exist in 4D space as vector, while, as that is rigorously shown in the SS&VT Planck scale informational physical model [more see SS posts on page 1], Matter’s spacetime has utmost fundamental and universal “kinematical” [5]4D metrics (cτ,X,Y,Z,ct), where ct is the time dimension, cτ,X,Y,Z are 4 space dimensions;

- while in the model it is rigorously shown that everything in Matter, since are only disturbances in the FLE lattice, constamtly and always propagate/move in the 4D space above only with the 4D velocities that have absolute values c=lP/tP. Since in mainstream physics the Minkowski space has metrics (ict,X,Y,Z), i.e. in QM/QFTs the really space cτ-dimension is postulated as the time dimension, just this [one of a lot of others] is the reason why, say, really QM is adequate to reality relating to antiparticles only provided really evidently transcendent assumption that somewhere some evidently transcendent “sea of negative energies” exist.

Etc…, but the post is long. More see the SS posts on page 1 and links in the posts

Cheers

Sergey Shevchenko:

I humbly remind you that you are not the Creator of the Universe. How do you know that the matter is "rigorously organized"? What this term really means for you? There is no notion of any rigor in organization of matter. There are natural laws we learn from ecperience, and only from ecperience. Indeed, those natural laws seem to organized matter in some order in our tiny part of the Universe we can observe. We have no idea of any properties of matter in those parts of the Universe that are located millions light years from us. Moreover, any phys theory existing at present uses potentials of intetactions (including gravitation one) obtained from experiment. Do

We have no idea what those potentials are into other parts of the Universe distant from ours. All cosmology is simply a result of extrapolation of our knowledge applicable to our tiny portion of cosmos applied to the entire Universe, which is logically wrong.

Matter does not know meaning of the term "rigor", and does not know math. The term "rigorous" means "logical, mathematically correct, without any assumltions, apart from axioms of the theory or method considered".

I am sorry, but I have no time or interest to continue this discussion. I suggest you refer to math definitions and dictionaries to make sure you use math terms and words as they are defined, rather than put into standard terms your own meaning they do not really have.

Good luck!

Liudmila Pozhar & Sergey Shevchenko I am appealing to both of you in researchGate to bury their hatchets. We are all Academicians in this forum. We have young researchers in this forum that are counting on both of you to learn and grow in their research we must respect each other because we are all top researchers in the world today, so we must act with decorum to understand we are all trying to live a legacy for future generations! Most of us here are retired or close to retirement. We are here to support young researchers to gain insights into better methods of researching and to ignite their curiosity. Most of us have long ago gotten our Ph.Ds. and many of us have produced many Ph.D. Candidates we must not stoop low.

Joseph O Akomodi: I believe there were no hatches or disrespect. It has been a research discussion. There was no personal issues there at, until you brought your moral standing into it. However, I believe that any one MUST understand math and phys therminology before entering any such discussion. Otherwise ResearchGate platform may deviate from its research orientation. Old or young a scientist is, she/he must understand definitions of basic terms he/she is talking about. Otherwise a research discussion becomes a meaningless chat. I refuse to participate in meaningless chats, regardless whether you like it or not. Such chats are wasting time of other people and lead to misunderstanding of really deep research problems.

Also, I suggest we do not enforce our opinion about how to teach young scientists on each other. Young scientists, in my opinion, should not be screened from real research discussions. They are not babies, they are scientists. Let's respect their intelligence.

Also, I would appreciate any one entering a discussion follow its subject and not bring into it his/her political and/or moral issues. Thank you for understanding.

Liudmila Pozhar I am only pleading to both of you if the hostility discussion you and Sergey is having is normal to you guys that is fine with me, I do not see it as healthy discussion. I sincerely appreciate your contribution to the ResearchGate. My believe is sharing our knowledge is a healthy thing to do, and we can agree and disagree that is what a healthy discussion is all about. I sometimes, learn from the students I teach. I am not condemning your knowledge and contributions to ResearchGate. Thank you again!

Jodeph Ozigis Akomodi: please, do not bring to a research discussion irrelevant non-research problems created by your imagination. Thank you.

[Joseph Ozigis Akomodi, my posts contain only really scientific clarifications of mostly fundamental points that are fundamentally transcendent in the mainstream philosophy and sciences, including physics; and never contain any hatchets. Besides discussion of hatchets as a rule has no relation to science, and really often causes really, in many cases deliberate, spamming].

A few notes else to the QM probability/ uncertainty problem, here some continuation of the last passage in last SS post:

- since in QM – and in all mainstream physics – Matter’s spacetime is really strange Minkowski 4D space with metrics (ict,X,Y,Z) where time coordinate is [mathematically] imaginary one [as a rule, in this case completely equally time coordinate can be real, while XYZ coordinates imaginary],

- in mainstream physics antiparticles really scientifically rationally cannot exist. So, besides the transcendent Dirac’s sea of negative energies, in QED the “Feynman–Stueckelberg QM interpretation” that antiparticles move “back in time” is postulated, what is again quite transcendent, “back in time” motion is fundamentally absolutely impossible.

But the interpretation really works correctly in QED, what happens since – as that is rigorously shown in the SS&VT Planck scale informational physical model [links to couple of main papers see in SS post on page1], particles are specific disturbances in the FLE lattice – copies of cyclic algorithms that run on FLE “hardware”, at that given type antiparticles algorithms run in inverse command order,

- so antiparticles really move partially in negative cτ-axis direction, which, again, in the mainstream is postulated as the time dimension.

Again - disturbances in the FLE lattice appear if a FLE is impacted by some 4D momentum, P=mc, [“bold” means 4D vector], which, in contrast to scalar “energy”, has 4D direction in the space, so for antiparticles existence there is no any necessity in seas of negative energy and motion “back in time”. So, say, if Dirac instead of Hamiltonian operator would use the non-existent now in the mainstream 4D momentum operator, by Pythagoras theorem P2=p02+p2, p0 is the momentum’s cτ-component, p0=m0c , p is XYZ component,

- √ of which so differs from mainstream Hamiltonian in evidently inessential the c-constant, there would not be necessity to invent some sea.

Note also that the above is true not only in this case - in whole physics the Hamiltonoan really is the 4D momentum absoulute value that is mostly unnecessarily multiplied by c.

Including while in the mainstream evident QM observable “time”, despite that is evident dimension, so rather strangely isn’t – in contrast to XYZ – operator, and so hasn’t corresponding conjugate momentum operator ∂/∂t, while that evidently strangely is energy operator in the QM.

Let hope that further in the thread the hatchets and something like, topics will not be discussed.

Cheers

Please be aware, Joseph in discussions has apparently used AI and faked citations to support his bias.

For example

Google Scholar found none of your cites and searches specific articles failed Search of the Disability and Society journal volume 36 issue 4 table of contents at pages 567-582 failed to find the article you cited from this journal - Miller 2021.

https://www.tandfonline.com/toc/cdso20/36/4?nav=tocList

AND

Reading of the Psychology in the Schools journal volume 58 issue 7 table of contents at pages 1234-1247 failed to find the article you cited from this journal - Johnson et al. 2021.

https://onlinelibrary.wiley.com/toc/15206807/2021/58/7