Building on QP from 5 reasonable axioms

[normvcr]

Lucien Hardy's Quantum Theory From Five Reasonable Axioms has deepened my understanding of QP foundations, and motivated me to write a paper. The essence of my paper is that "connectedness" of state space (or the acting Lie group), need not be assumed, but can be deduced. Before linking to the paper, here, I want to confirm that this is OK from a forum etiquette perspective.

 

[berkeman]

Lucien Hardy's Quantum Theory From Five Reasonable Axioms has deepened my understanding of QP foundations, and motivated me to write a paper. The essence of my paper is that "connectedness" of state space (or the acting Lie group), need not be assumed, but can be deduced. Before linking to the paper, here, I want to confirm that this is OK from a forum etiquette perspective.

Per the PF rules, your paper must be published in a peer-reviewed journal before we can discuss it here. What journals are you considering submitting it to?

 

[normvcr]

The paper is a contribution, but not "cutting edge", so I am somewhat in a quandry of where to submit the paper.

 

[dextercioby]

The paper is a contribution, but not "cutting edge", so I am somewhat in a quandry of where to submit the paper.

What is the exact mathematical nature of the "state space" you consider?

 

[normvcr]

The state spaces satisfy axioms very similar to the ones that Hardy proposes e,g, level 2 state spaces sit in R^4 and level 3 state spaces sit in R^9. If you postulate connectivity of state space, you get QP, as Hardy demonstrates. If you do not postulate connectivity, it turns out the level 2 state space can consist of two parallel circles, of the same radius, on the Bloch sphere. My paper further shows that this disconnected state space does not admit a level-3 counterpart in R^9. My paper also shows that, although Lorentzian geometry needs to be considered, it does not support even a level-2 state space.

 

[Vanadium 50]

I am somewhat in a quandry of where to submit the paper.

What journals does it cite? Which ones are most likely to publish it?

 

[Swamp Thing]

I'm trying to locate a peer-reviewed version of Hardy's paper, but could only find it on arXiv so far. If it hasn't been published as a peer-reviewed work, maybe it's out of bounds for us, too.

 

[Mentz114]

@normvcr : why not send a copy to hardy@qubit.org (the domain exists )

 

[bhobba]

You can also publish it as an insights article here

I know that paper well and people have been asking me to do one for quite a while, but I never seem to get around to it.

The thing is it must not contain your own personal ideas, but rather be an analysis of the paper - you can of course have things that could be investigated, further thoughts etc at the end of the article that we can discuss here, and that is where you can mention your lie group ideas.

In doing that, and maybe helping in answering some of your 'issues' the following may help:
https://arxiv.org/abs/1402.6562

Basically all the paper does is justify in a rigorous way the following. In mathematical modelling there are generalizations of ordinary probability - they are called generalized probability models or theory. A generalized probability model makes only very simple and quite general assumptions. You have:

1. Something unspecified called states. All you are doing here is saying whatever you are modelling can be in something called a state without specifying in anyway what a state actually is. Not much of an assumption really.

2. This is the main assumption - the space of states is convex - which simply means you can apply ordinary probability theory. Specifically it says any state a can be written in the form a = ∑pi ai where ai are other possible states of the system and the pi are all positive and sum to one. Of course that sum can just contain one element so you have a = a which is trivial. If that is the only way a state can be written as such a sum then it by definition is called pure. The interpretation of the pi is as a probability ie if the system is in state a then when you do something to it, without even specifying what that something is, the pi gives the probability in will be found in state a1.

Ordinary probability theory easily fits this - in fact it's the simplest generalized probability theory. The pure states are the possible outcomes of what you are modelling and the sum ∑pi ai where the ai are other pure states ie the event space as per the Kolmogorov axioms, then pi is the probability of getting ai. In this view the pure states are usually thought of as a vector where the i'th element is the i'th event of your event space.

Now in Hardy's paper all he is noticing is in ordinary probability theory you can't continuously go from one pure state to another. But if you want to model physical systems by pure states then this is something you want to do. If a system is in pure state a at time t=0 and state b at time t = 1, then it went through some other pure state at time t=1/2. Imposing that and you are inevitably lead to QM which Hardy's paper gives the technical detail of. That's all QM is really. Formally QM is just the simplest generalized probability model where systems continuously change to other pure states.

That's why I always say formally we know very well what QM is, and why it is that way, - what it means, or even if what it means is worth pursuing ie the math is all that's required, is another matter, and we have all sorts of answers to that.

Thanks
Bill

 

[Mentz114]

Basically all the paper does is justify in a rigorous way the following. In mathematical modelling there are generalizations of ordinary probability - they are called generalized probability models or theory. A generalized probability model makes only very simple and quite general assumptions. You have:
[]
That's why I always say formally we know very well what QM is, and why it is that way, - what it means, or even if what it means is worth pursuing ie the math is all that's required, is another matter, and we have all sorts of answers to that.

Thanks
Bill

That's good summary. I just read (today) up to part 6 of Hardy's paper and enjoyed it a lot. One of thoughts was 'I wonder what BHobba thinks of it'.
Now I know.

 

[bhobba]

That's good summary. I just read (today) up to part 6 of Hardy's paper and enjoyed it a lot. One of thoughts was 'I wonder what BHobba thinks of it'.
Now I know.

Oh yes - its not really that hard.

Interesting what the OP has discovered.

Thanks
Bill

 

[bhobba]

Deleted as answered in my next post.

 

[bhobba]

I'm trying to locate a peer-reviewed version of Hardy's paper, but could only find it on arXiv so far. If it hasn't been published as a peer-reviewed work, maybe it's out of bounds for us, too.

I have just had it pointed out to me our forum rules allow that:
References that appear only on http://www.arxiv.org/ (which is not peer-reviewed) are subject to review by the Mentors. We recognize that in some fields this is the accepted means of professional communication, but in other fields we prefer to wait until formal publication elsewhere. References that appear only on viXra (http://www.vixra.org) are never allowed.

Since this paper has been cited in so many reputable sources discussing it is fine.

Now the issue is simply discussing the paper the OP has come up with. I will leave that to the mentors to decide - but if the OP, or anyone else, wants to discuss Hardy's paper its perfectly ok.

Thanks
Bill

 

[normvcr]

Thanks for all the interesting comments, and clarification of what can be discussed in this forum. My paper references, of course, Hardy's paper, another paper published in Studies in History and Philosophy of Modern Physics, and three books. I will take V50's suggestion and see if the journal is interested, and let you guys know how it turns out. In reading Hardy's paper, several questions arose. For example
1. Why is state space a closed subset of the ambient vector space? I am referring to states that actually exist in reality -- not just mathematical constructs. In terms of measurements, this can be rephrased as:
Given a sequence (Sn) of actual states, and a candidate state, S, such that for any measurement, f, f( Sn ) --> f(S)
Then S is an actual state.
2. Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S ) ( = Tr(S*T) in QP operator notation)

 

[bhobba]

1. Why is state space a closed subset of the ambient vector space? I am referring to states that actually exist in reality -- not just mathematical constructs. In terms of measurements, this can be rephrased as:

Well actually it isn't. The physically realizable states are all really finite dimensional, but sometimes of very large dimension. You can see this by thinking of an observation as being displayed on a digital readout rather that a meter etc - if it was of infinite dimension then you would need a readout of infinite length.

Hardy's paper, and many papers on the foundations of QM skirt around it by just considering the finite dimensional case.

I had given this a lot of thought many many moons ago, and couldn't really resolve our use of Hilbert spaces etc until I started investigating Rigged Hilbert Spaces. During that sojourn I came across a little book, QM for Mathematicians or something like that written in the 1950's stuck deep in the dust covered recesses of the ANU library in Canberra where I lived.

It talked about something called a Dirac Space. Its quite simple really - you take as a space all the vectors of finite length. These are the physically realizable states. Now purely for mathematical convenience you consider it dual - it in fact turns out to be any vector, finite or of infinite length and the book called it a Dirac Space - but I have never seen the terminology anywhere else. This is the archetype of Rigged Hilbert Spaces. It is often more convenient to work in the Dual than the original space - in fact there is no way to tell the difference, physically from a vector of very large dimension and of infinite dimension. So that it OK from a modelling viewpoint.

Contained in the Dirac space you have all sorts of things - a Hilbert Space, a Schwartz space - you name it, it's probably there.

But for various reasons its too big eg not everything in there can be mapped to a Schwartz space hence you can't take a Fourier Transform of it.

So what you do is cut down its size depending on your problem. For example in the Dirac space you have good functions - you can take that as your test space. Doing that its Dual is the Schwarz space so Fourier transforms are on - that in fact is the space usually used in QM. This is expressed in the so called Gelfland tripple T* ⊃ H ⊃ T where T is a test space of some sort (eg good functions), H a Hilbert space and T* the dual of the test space. This triple is the basis of Rigged Hilbert Space's - rigged not as in a game of chance - but like the rigging on a ship it provides a way to 'climb' above these issues. But you must always keep in mind all of them are just subsets of the Dirac Space that is introduced purely for modelling purposes.

You can, as Von-Neumann did in his classical treatise - Mathematical Foundations Of QM - work entirely in Hilbert Spaces, but it is neither as convenient or as beautiful as Dirac's treatment in his equally classical - Principles Of QM. Word of caution however - do not do what I did - read those books without a though understanding of modern QM as found in Ballentine. I know - it causes all sorts of issues - not the least of which is Ballentine explains, not in full rigorous detail mind you, the whole Rigged Hilbert Space vs Hilbert Space issue. You can get further detail on it all here:
https://www.amazon.com/dp/0821846302/?tag=pfamazon01-20

But again not until you are comfortable with Ballentine.

So after all that what's the answer to your question - simply mathematical convenience. The real space is the space of vectors of finite dimension which isn't complete or any of those nice mathematical functional analysis things - we just extend it to have them by means of RHS's.

Thanks
Bill

 

[bhobba]

Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S ) ( = Tr(S*T) in QP operator notation)

That my friend, is tied up with the important, but not often discussed Gleason's Theorem - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

It's the reason for Born's rule which is the underlying principle for these things.

But I get the feeling what you are asking is more than just the Born rule. - it transition probabilities in general eg spontaneous emission.

Here is the paper that expalians that:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

The calculation of the probabilities is an example of Fermi's Golden Rule:
https://en.wikipedia.org/wiki/Fermi's_golden_rule

Thanks
Bill

 

[normvcr]

Thanks for the detailed responses to my two questions. Here, I comment on the response to the one question
Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S )

That my friend, is tied up with the important, but not often discussed Gleason's Theorem - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7]
...

I read through the discussions on POVM, Gleason's Th, QFT and What is a photon, and Fermi's Golden Rule. I feel remiss in that I did not emphasise that the question was asked in the context of Hardy's paper, of deriving quantum physics from basic axioms/assumptions. For example, the concept of modelling states and measurements by operators on a Hilbert space, is a main goal of the paper. So, the formula Tr(S*T), which is clearly symmetric, can not be used to "explain" the symmetry of state transitions. I am looking for something more fundamental. For example, time reversibility does not seem to fit the bill. A special case of the question is:

If S cannot transition to T, why cannot T transition to S?​

 

[bhobba]

If S cannot transition to T, why cannot T transition to S?

Ahhh - got you now.

That I don't think anyone really knows other than the obvious - similar arguments in classical physics ie disorder is much more probable than order leads to an arrow of time. At least I don't think anyone knows - could be wrong of course..

Hardy doesn't go into it either - he simply notes what some reasonable axioms used to analyse the standard QM observational setup - preparation, transformation, measurement leads to.

Thabks
Bill

 

[normvcr]

Again, thanks for the detailed response. I appreciate the time it takes to do this. Here, I am commenting on the response to my question:

Why is state space a closed subset of the ambient vector space?​

Well actually it isn't.
...

I do not understand what is meant by, "Well actually it isn't". For example, the Bloch sphere is a closed subset of R^4. I suspect I may have, again, not been sufficiently clear in my question. The question is asked in the context of Hardy's paper, which is concerned with finite dimensional QP (although Hardy does briefly discuss infinite dimensional QP in h is paper).

 

[bhobba]

I do not understand what is meant by, "Well actually it isn't".

Sorry - I was getting my terminology confused. Yes closure - for sure - I was thinking of other things like completeness.

Closure is a trivial deduction from the principle of superposition. The space of all vectors of finite length is closed but not complete.

Thanks
Bill

 

[Fra]

Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S )
...
I am looking for something more fundamental. For example, time reversibility does not seem to fit the bill.

How about the ergodic hypothesis applied to the histories in the state of microstates? An priori transition probability between two microstates(pure states), disregarding the time would then expected to be symmetrical. But this assumption, somehow gets necessary because once decouples the hamiltonian which contains the interesting physics, from the framework. Then one is forced to make a lot of fundamentally unjustified assumptions to fill in the conceptual gaps.

I do not like that he hamiltonian is left out. I think in a proper inference the state spaces and the space of laws, should be unified and defined "in principle" operationally from the point of view of an inside observer. And axiomatisation for THAT, but in the same spirit of hardys papes is what i would like to see. This should have emergent evolving state space and emergent evolving law. The decoupling of the flow of time already makes me think axiom 1 is not acceptable.

I like the ambition of these things, but i think we need to incorporate more of the real conceptual issues.

/Fredrik

 

[bhobba]

I do not like that he hamiltonian is left out. I think in a proper inference the state spaces and the space of laws, should be unified and defined "in principle" operationally from the point of view of an inside observer. And axiomatisation for THAT, but in the same spirit of hardys papes is what i would like to see. This should have emergent evolving state space and emergent evolving law. The decoupling of the flow of time already makes me think axiom 1 is not acceptable.

The Hamiltonian comes from symmetry considerations - specifically the rather obvious idea probabilities of observational outcomes is frame independent. Strictly speaking though you are invoking the POR. See chapter 3 Ballentine.

It was Wigner who first realized the central importance of symmetry in QM - basically QM is the fundamentals of that paper by Hardy (or the two axioms found in Ballentine) and symmetry.

It is hinted at in a 'correct' treatment of classical mechanics such as found in Landau - Mechanics which I think should be compulsory reading for anyone before seriously studying QM. And Physics From Symmetry wouldn't hurt either:
http://physicsfromsymmetry.com/

Thanks
Bill

 

[Fra]

The Hamiltonian comes from symmetry considerations - specifically the rather obvious idea probabilities of observational outcomes is frame independent. Strictly speaking though you are invoking the POR. See chapter 3 Ballentine.

It was Wigner who first realized the central importance of symmetry in QM - basically QM is the fundamentals of that paper by Hardy (or the two axioms found in Ballentine) and symmetry.

It is hinted at in a 'correct' treatment of classical mechanics such as found in Landau - Mechanics which I think should be compulsory reading for anyone before seriously studying QM. And Physics From Symmetry wouldn't hurt either:
http://physicsfromsymmetry.com/

Thanks
Bill

Yes, we discussed this before, and I do not expect that we agree here. I of course know about the symmetry principles. They are powerful constraints, but - limited as well. And that is even WHY they are powerful.

Once we "know" the symmetry, we can use it. But what is the empirical justification of the symmetry? Which physical process "informs" the observer about this symmetry? this is what i am worrying about. To think of symmetries as constraints written in the sky are to me too metaphysical, I want an empirical inferential justification. All we need is to axiomatise the inference system, not the symmetries. The symmetries should follow from interaction with the environment only.

/Fredrik

 

[bhobba]

But what is the empirical justification of the symmetry?

Are you asking how the POR is tested? Well for one thing classical mechanics obeys it. How do you test F=MA - it's a definition. There are things in science that can not be tested directly - its just a fact - you obviously think that's a worry - no actual physicist (modern ones please and scientists - not philosophers) I am aware of does.

Thanks
Bill

 

[Fra]

Are you asking how the POR is tested? Well for one thing classical mechanics obeys it. How do you test F=MA - it's a definition. There are things in science that can not be tested directly - its just a fact - you obviously think that's a worry - no actual physicist (modern ones please and scientists - not philosophers) I am aware of does.

Thanks
Bill

No, I was asking something deeper in the context of trying to axiomatize physics.

If physicists are expected to have fatih in this axiom system - enough to use it at its full power by constraining how a theory of nature must look like, then it is essential that we at least try to approximately attach the key abstractions in theory to something in reality.

Obviously you do not prove an axiom, you just need to prove consistency with the prior axioms. But this is not what i mean either.

The ideal case is that a clever theory should build on axioms that are close to easy to accept, this is what hardy also tries. And maybe most people accept the axioms, but i think that there is still too much baggage. Axiom1 is nontrivial if you really consider the physical limis of memory capacity and computating, from history or actual ensembles, to arrive at an expectation. This is indeed at the root of probability theory itself, not just physics. But probability theory as in mathematics, is one thing, but when you start to apply it to physics, the plausability requirements of axioms become stronger IMO.

If you think this is not worth analysing, then let's note that probability is somehow quite central to physics, classical and QM. I think its been taken too lightly. And let's distinguish between mathematics and physics.

/Fredrik

 

[bhobba]

If physicists are expected to have fatih in this axiom system - enough to use it at its full power by constraining how a theory of nature must look like, then it is essential that we at least try to approximately attach the key abstractions in theory to something in reality.

Ok - let's try different approach then.

Look up Euclid's axioms. Then look up Hilbert's axioms.

Do you see a difference? If so describe it - if their is no difference then explain why - that would mean, for example, explaining how something with length but no width can exist.

One is the kind of axioms physicists use - the other mathematicians. The difference lies at the foundation of the two disciplines.

Thanks
Bill

 

[Fra]

Ok - let's try different approach then.

Look up Euclid's axioms. Then look up Hilbert's axioms.

Do you see a difference? If so describe it - if their is no difference then explain why - that would mean, for example, explaining how something with length but no width can exist.

One is the kind of axioms physicists use - the other mathematicians. The difference lies at the foundation of the two disciplines.

Thanks
Bill

I am not sure I understand your presumed argument from this comparasion and to the foundations of physics?

The difference between Euklides and Hilbert is around 2000 years, so the late 19th century critique against Euklides non-stringent meothds, such that arguments proofs where based on drawing geometrical figures - while valid critque today - seems a bit unfair to be honest and Euklides isn't here to defend himself ;)

Hilberts ideas holds a higher deductive level. But what can we expect even from a bright mind that lived over 2000 years ago? We might instead wonder what a person like Euklides might have done if he was born in the 20th century.

Also the critique of Euklides was not based on physics, its was more on the logical system level. I also see neither of Euklides nor Hilbert as physicists (or natural philosphers). I am not critiquing the foundations of probability theory - as a mathematical theory.

In physics OTOH, the probloem is different, so its hard to compare. The kind of postulates that are the assumed connection between model and experiment are fuzzy. We can never analyze that purely in terms of deductive logic. I think of physics as a tool to predict and control our environment in order to survive. Here development of matehmatics indeed goes hand in hand with physics. Ideally thinking as a physicists, it see the development mathematical theories here because they are of utility to us. This does not justify that we loose contact with nature, and mistake mathematical possibility for physical possibility.

But i think you mean that there are different levels of stringency in the logical system used. Then I agree.
And in physics there is a different level of "analysis" for how deep into the mud we need to attach our "postulates"?

Maybe we just disagree on how deep into the "physical mud" we need to have our foundation?

/Fredrik

 

[bhobba]

I am not sure I understand your presumed argument from this comparasion and to the foundations of physics?

It's based on the level of abstraction used.

Euclid's axioms speak of things that don't really exist as if they actually do. Yet nobody, even 12 year old's its taught to, has any problems drawing diagrams and making deductions from it. Hilbert is entirely abstract - no attempt at all in made to make it an actual model of something.

That's the difference between axioms used in physics and those in math. Those in physics are written in a form that are directly applicable and assume a bit of 'common sense' on the part of the student, those in pure math are simply deductions from assumed axioms.

Probability is different again - both pure and applied mathematicians have the same axiom's - the Kolmogorov axioms. Applied mathematicians have zero problems at all applying it because as part of a probability course they gradually build up the idea, via example, what an event etc is, just like in Euclidean geometry in the form presented by Euclid the teacher guides you to that understanding. Philosophers argue what it means, but in practice its of zero concern.

Maybe we just disagree on how deep into the "physical mud" we need to have our foundation?

Or maybe there is a lack of understanding of the difference between pure and applied math. In applied math you have a model that is mapped in some way to intuitive ideas we have of things like events, points, line, observations etc. It is part of applying it developing that intuition. Most of the time its so obvious nobody worries about it. When you studied Euclidean geometry did you say to your teacher - hey you said points have position and no size - these things you draw are not like that - how can that be? I think the piece of chalk that would be chucked your way would require considerable reflex to dodge, as the rest of the the class gave you a strange look. Of course actual points have size - but its obviously irrelevant to proving theorems etc. You are concerned about things that in practice are of no concern. Now in philosophy they likely worry about this sort of stuff - they worry about all sorts of strange things there - but in physics we use a bit of common-sense just like the teacher expected the class to use and without doubt you did without complaint when you were taught it.

If such things interest you - that's fine - but its not science - its philosophy and not what we worry about here.

I think most physicists are not concerned about such things, I certainly am not, and hold views similar to Weinberg:
http://emilkirkegaard.dk/en/wp-content/uploads/Steven-Weinberg-“Against-Philosophy”.pdf

You may say - that what's wrong with modern physics or similar things. I have been posting on science forums for years and have heard it all.

I can't prove it of course - but history shows the kind of things that worry you basically lead nowhere.

Thanks
Bill

 

[Fra]

You are concerned about things that in practice are of no concern.
...
If such things interest you - that's fine - but its not science - its philosophy and not what we worry about here.

As I see it, these things are of no major concern for just maintaining status quo ocurrent established scientific knowledge.

But I think they are of deepest concernt for making progressions in the foundations of physics.

I think most physicists are not concerned about such things, I certainly am not, and hold views similar to Weinberg:
http://emilkirkegaard.dk/en/wp-content/uploads/Steven-Weinberg-“Against-Philosophy”.pdf
...
I can't prove it of course - but history shows the kind of things that worry you basically lead nowhere.

Unfortunately I suspect a causal relation between these two statements of yours, which i also mentioned in the other thread ;-)

You may say - that what's wrong with modern physics or similar things. I have been posting on science forums for years and have heard it all.

Yes, I extracted this point of you of yours from the other thread, and i certainly respect your perspective. Indeed I do realize that my perspective is in minority.

However, no researcher should be seriously discouraged by the failure of others. The optimistic attitude is obviously that all the other failed because they didnt do it the right way. Statistically most who think like that fail, but if none think like that progress will be stalled. What you label philosophy is IMO essential for progress of foundational physics.

But I am as sure that i will not convince you to you change your understand no more that you will change mine :)

/Fredrik

 

[bhobba]

But I am as sure that i will not convince you to you change your understand no easier that you will change mine :)

Lets leave it at that then.

If the words of Weinberg are not enough, and the utter failures he mentions of similar approaches to what you want to do, are not enough, then I think that's where we have to leave it.

Thanks
Bill

 

[normvcr]

Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S )

How about the ergodic hypothesis applied to the histories in the state of microstates? ...

This got me to thinking. There is some principal that a system should be able to evolve to maximum entropy. Perhaps, this can form the core of an argument that the transition probabilities are symmetric. This might also not pan out at all ...

 

[Fra]

This got me to thinking. There is some principal that a system should be able to evolve to maximum entropy. Perhaps, this can form the core of an argument that the transition probabilities are symmetric. This might also not pan out at all ...

[bayes theorem] P(T|S) = P(S|T) * P(T)/P(S)
and
[ergodic hypothesis] P(S) = P(T)
=> P(T|S) = P(S|T)

or were you seeking something more subtle?, this applies only to pure microstates though.

/Fredrik

 

[normvcr]

[bayes theorem] P(T|S) = P(S|T) * P(T)/P(S)
and
[ergodic hypothesis] P(S) = P(T)
=> P(T|S) = P(S|T)
... this applies only to pure microstates though.
/Fredrik

Clever argument. The fly in the ointment is that since P(T|S)=P(S|T), then we always will have P(T)=P(S), even when not at equilibrium. The analysis needs to take place in a somewhat more complex context: P(T|S) manifests itself when a measurement is done for T, in initial state S, and vice versa for P(S|T).
It is sufficient to work with microstates (pure states). Symmetry for mixed states would then follow from linearity of the transition probabilities.

 

[Fra]

Clever argument. The fly in the ointment is that since P(T|S)=P(S|T), then we always will have P(T)=P(S), even when not at equilibrium.

Yes if we are introducing another state (some macrostate) which "equiliirium" refers to, then one needs to make that explicit in the formulas, ie we are then not just talking about an uncertaint microstate, but also an uncertain macrostate. Its also along these paths, and when associating the macrostate to an observer as an inferential agent in its envirnoment, and ponder about its "expectations on evolution", then you can connect arrow of time to the observer depedent directions
that you start to get into the objections i had before.But i will not ramble too much about that, becauase is am quote sure most arent going to see my point anyway and this is the wrong place to explain my whole idea anyway. But i think there is a lot of interesting stuff in these inferential structures! So good luck with your paper, without know what stance you take!

/Fredrik

 

[normvcr]

I will ... see if the journal is interested, and let you guys know how it turns out.

The paper was turned down by the journal, for reasons that I accept -- "this is an interesting mathematics article, but does not contain sufficient philosophical/conceptual insights to be publishable in ...". This is quite reasonable, given the nature of the journal. I raised two such insights in this thread

1. Symmetry of probability of state transitions.
2. The physical states (states that exist in reality) are topologically closed.

However, I did not make a point of underlining these, and other issues in the paper, as I am not trained in physics, and feel it would be presumptive of me to espouse physics to physicists. So, I am in something of a quandary ...