Dear Sofia,

Similar v->-v and t->-t in classical mechanics, in QM, it is expressed by “Operator”-> - "Operator”, and named as Lt- symmetry.

Dear All, in Continuum Mechanics (CM) it is well-known the global production of entropy equation. This contains a term related to the velocity of thermodynamic dissipation, referred to a volume V, where one considers the body. When this contribution is zero, then one obtains an inequality called global equation of Clausius-Duehm. This inequality encodes the second law of Thermodynamics (II-T)... but, of course it is not more an axiom but a theorem inside the (CM). Why I recalled this ? The motivation is to underline that the so-called (II-T) is a necessity in the model (CM). Well ! Now the Loschmidt's paradox arises when one aims justify (CM)-Thermodynamics with the microscopic model adopting the Boltzman point of view. Really Statistical Mechanics (SM) can justify many thinks at macroscopic level but is unable to obtain any justification ... In advance at microscopic level arise some phenomena that (SM) is inadequate to encode. In fact, at quantum level one can recognize existence of negative temperatures that are completely impossible to obtain in (CM). On the other hand, as it is well reported in a Wikipedia link, 'Johann Loschmidt's criticism was provoked by the H-theorem of Boltzmann, which was an attempt to explain using kinetic theory the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide'. This story goes back to 1876 ...

I suggest to look to some my papers reported in this Wikipedia link: http://en.wikipedia.org/wiki/Talk:Quantum_gravity,

to see what today the quantum thermodynamics is.

Thanks for your intriguing discussion.

A.P.

To all the readers who will be so kind as to consider my question, I suggest to keep in mind that there is a big difference between a quantum particle and a classical particle: the latter has simultaneously well-defined position and linear momentum, while the former, if it has a well-defined linear momentum (and in consequence, velocity), then it has no defined position, and vice-versa.

Thus, a quantum version of the Loschmidt paradox can't be a copy of the classical version.

The meaning of my talk is that the Loschmidt's paradox rises from the inadequateness of the (SM) to justify (CM). Instead the actual quantum thermodynamics can justify (CM) since this can be obtained as a classic limit. In other words there is not a quantum Loschmidt's paradox with respect to the quantum thermodynamics as I formulated in quantum super Yang-Mills PDEs.

Dear Sofia,

I agree with your opinion in message, in fact, all wave motions involve similar phenomena, “dispersion” and “distribution” are the real key words for irreversibility, statistical mechanics is the theory of such key words in a box, but, not all the physical quantities show similar dispersing behavior, entropy involves two of them, we need to distinguish these two from others. In my opinion, the Nature of the Entropy is a double distribution function.

I think, these maybe the micro sources of the second law, we need a function to describe these micro sources, which should be mutually consistent with thermodynamics.

If the micro sources of the second law can be confirmed, there will be no Loschmidt paradox.

Dear Aleksei,

First of all, thank you for giving some thought to my question. But, the picture that you describe is quite fuzzy, and I have objections. To tell you frankly, I really have doubt whether there can be a quantum version of Loschmidt's paradox.

Let's examine your idea about reversing the time. How we do that? Could it be, by reversing the linear momentum of the wave-function? If we expand the wave-function in Fourier components, we get a spectrum of values of the linear momentum. Do you agree?

And how do we reverse all these values? By making all the walls of the gas container, perfectly reflecting mirrors? Well, it won't produce the desired reversing effect. Imagine instead of the multi-particle wave-function a single-particle wave-packet, and that on its way we place a mirror. Assume that in the beginning the wave-packet is quite narrow. As it travels, it also widens because of the range of linear momenta.A t the mirror, the wave-packet would be reflected, but as it returns from the mirror it would continue to widen. This is the problem: how to stop the widening and enforce a narrowing? 

Reading the question and the above answers, I'd like to put in my two cents.

First of all, I don't think there is a paradox. Loschmidt did not present it as a paradox but as an argument against Boltzmann's derivation of the H theorem, and in particular the fact that H, according to Boltzmann a purely dynamical quantity, would always decrease in time. Loschmidt objected that Boltzmann claimed that he only used Newton's equations of motion, which are invariant for time reversal, and nevertheless was able to derive a dynamical equation for a quantity (H) that does not exhibit that property. The only thing Loschmidt pointed out is that there seems to be a contradiction in deriving a dissipative equation from a time reversible one. And he was right of course. It is now obvious that an assumption that Boltzmann made, known as the Stosszahlansatz, is equivalent to breaking the time reversal symmetry, and hence there is no paradox. The answer of Boltzmann to Loschmidt's objection, which was phrased in the form that if you reverse all velocities at a particular point in time you should go back to the initial situation (hence Umkehr), was: "go ahead, do it." (This may be a myth.) That is why it is also known as the Umkehreinwand, or reversal objection.

Quantum mechanically the situation is quite different, in that there is no Boltzmann like derivation from the  Schroedinger equation. Schroedingers equation is time reversal invariant (with a slight modification that you also need to take the complex conjugate of psi, but that does not change anything for expectation values), so if there were a derivation for dissipative terms in a dynamical equation for an expectation value derived from the Schroedinger equation, the problem would be exactly the same. But also the answer would be exactly the same: there is no paradox as long as we realize that in order to break time reversal symmetry, we need an additional assumption, which may or may not be justified.

A hell of a lot more can be (and has been) said about this and related problems. However, I don't think we should worry about Loschmidt and the problems with actually being able or not reversing the velocities.

If this does not answer your question please let us know, and we can continue the discussion.

No, Aleksei, no,

A thought experiment should not fight a NO-GO principle. For what you say, the uncertainty principle plays the role of a NO-GO. I will explain:

 When you do Fourier decomposition to a wave-packet, you get a quantum superposition of Fourier components. If we work in 1D, e.g. the wave-packet travels along the axis z, each Fourier component covers all the z axis because it has a well-defined linear momentum. There are Fourier components with the linear momentum pointing in the direction z, and other Fourier components pointing in the direction -z. Then, where to place a mirror for reversing all the linear momenta?

Wherever you would put the mirror, you CUT the Fourier components into a part before the mirror and a part after the mirror.

With a classical particle the situation is better,in  principle, as such a particle has at each time a well-defined position. In fact, you can calculate the positions at each time, of as many particles as the computing power and memory permits you. The problem becomes technical, jus matter of technology.

Also, some general remark: you say

"Schrödinger's equation is time reversal invariant".

I apologize, but how do you reverse the time-evolution of the wave-function? See what I explained above! In physics, one has first to describe the physical experiment, and only after that resort to mathematics. Otherwise mathematics can be a false friend. For instance people speak of time-reversal operators, and all sort of mathematics, but they don't describe the experiment that implements what they propose.

Dear All, I am afraid that your discussion is enveloping on some fundamental misunderstanding. Really the Loschmidt's paradox is referred to impossibility to justify the Clasius-Duhem inequality with the Boltzman's (SM). It does not concern to the invariance of the (CM) for time-reversal ...

However, in my opinion the question 'How can we formulate Loschmidt's paradox in the quantum domain ?' should be intrepreted whether starting from the (QM) we can arrive to obtain the Clasius-Duhem inequality for (CM). Therefore to pretend to arrive to the Clasius-Duhem inequality  starting from the Schroedinger equation is simply ridiculus .... Where is the quantum thermodynamics in the Schroedinger equation ? (In this sense I agree with Sofia's remarks ...)

By the way you can look to a serious formulation of quantum thermodynamics given in my paper http://arxiv.org/abs/0906.1363. Then you can understand that this formulation recovers, at the classic limit, the Clasius-Duhem inequality, namely the so-called second laws of Thermodynamics. Let me underline 'at the classic limit'. In this sense we can answer to the Sofia's question saying that the Loschmidt's paradox does not apply to the quantum domain.

 It seems to me that the most reliable way to get closer to the answer to this question is through the pioneering work of Schrödinger A. An undulatory theory of the mechanics of atoms and molecules. Physical Review. V28. No 6. 1926. p. 1049-1070. It should be noted that the Schrodinger equation follows from the principle of least action for a material point provided that the principle of superposition of solutions have a place.

I agree with Gert Van der Zwan:

 “so if there were a derivation for dissipative terms in a dynamical equation for an expectation value derived from the Schroedinger equation, the problem would be exactly the same. But also the answer would be exactly the same: there is no paradox as long as we realize that in order to break time reversal symmetry, we need an additional assumption, which may or may not be justified.”

“there is no paradox as long as we realize that in order to break time reversal symmetry, we need an additional assumption”!

 If one made “Operator”→ - "Operator” (p→-p, or t→-t), and claim that the wave equation is reversal invariant, he had assumed that wave vector can be reversed, but this implicit assumption has never been proved, on the contrary, I found that for radiated EM field and mechanical wave, the wave vectors cannot be reversed, for a radiated EM field, the vectors  E×B and -E×B cannot coexist, the left-hand mirror system of the vectors E×B cannot be a true system in nature, and for mechanical wave, v→-v is not equivalent to t→-t for wave equation.

Sorry,  Chinese version only.

Dear Aleksei,

That Clausius-Duhem inequality seems to me an equivalent of the increasing of the entropy. I am not sure, because it's only from Agostino that I learnt of it, and I hope to get some reference from him, to learn more of that inequality.

As to what you answered to me, you say

"We don't care about practical reversing of time, it's difficult and practically impossible thing even in classical physics."

Then please DO CARE! You can't ignore a problem just because it is difficult. Next, one has to distinguish between technical difficulty - what we have in the classical case, and a theoretical NO-GO - what we have under the quantum regime. I repeat, how do you think to solve the problem in practice, assuming that you have all the needed technology, e.g. computational power, memory, suitable mirrors, etc.

As I already said already, a mirror does harm, it truncates the Fourier components.

This is why I recommend to leave aside the mathematics and think how to implement the time-reversion EXPERIMENTALLY, The technology improves all the time, but a NO-GO of the nature will NEVER be beaten.

Dear Aleksei,

If you don't describe an experiment by which you revert the time-evolution of the system, people won't know what you talk about. Just to say that the classical, or the quantum equations, are time reversible, it's not enough. Show an experiment, nobody will build an experiment in your place, it's for you to show an experiment.

In classical mechanics, I can show you such an experiment: in a system of a few particles, you can calculate the velocities and positions at any moment. So, pick an instant when the positions are so that it is easy to put mirrors which would kick back the particles with exactly opposite velocities.

In the quantum regime you CAN'T DO such an experiment, because particles don't have simultaneously well-defined positions and velocities. This is a NO-GO.

Dear Aleksey Bykov, I know the Wikipedia link that you quote ... but my criticism to your posts in this Sofia's question on a possible quantum extension of the Loschmidt's paradox, refers to the fact that we are talking about Thermodynamics.  At the macroscopic level this is encoded by (CM). The Clausius-Duhem equation for global entropy, gives as a by-product the second law of Thermodynamics (II). This has nothing to do with the time-reversal !

The Lodscmidt's paradox arises just from the fact that the Boltzman's (SM) is invariant for time-reversal, hence is unable to justify (II). In other words, the time-reversal has been introduced in the Boltzman's (SM) not in (CM). On the other hand, any mathematician working with PDEs-(CM) knows very well that not all such systems are invariant for time-reversal. Well !

When one aims to extend such considerations on quantum systems, one should compare macroscopic Thermodynamics, namely (CM), with a quantum Thermodynamics. But you instead uses Schroedinger equation ! Sorry but there is a contradiction in your talk. Really the Schroedinger equation is not a quantum extension of the Boltzman's (SM)...

Let me also add that in my quantum Thermodynamics built on the quantum super Yang-Mills PDEs, (YM), you can understand that there is not contradiction at the classic limit with (II). With this respect, we can see that (YM) is invariant for time translations, hence also for time reversal, but this does not means that all nonlinear quantum propagators, encoding quantum reactions in (YM), are invariant for time reversal too ! Really we can obtain for crossed-symmetry quasi-reversed processes, but these are not simply time reversed processes

An answer can be found in Stud. Hist. Phil. Mod. Phys. 36 323-353 (2005): The Liouville von Neumann equation whcch governs quantum statistical dynamics is reversible, but the relevant entropy may increase, expressing loss of information towards inaccessible degrees of freedom over reasonable time scales

Dear All, the full history is more complex ...In fact, as  I just emphasized in some my previous talks, there is usually a confusion between the following three different steps:1) Quantum model, 2) Observed quantum model, 3) Observed quantum model at the classic limit.

Let us consider these different steps with respect to the quantum thermodynamics on the quantum super PDEs Yang-Mills PDEs, (say (YM)) as I formulated in DOI: 10.1016/j.nonrwa.2012.02.014.

1) In the quantum (YM) there is not 'time'. A nonlinear quantum propagator, V, encoding a quantum process between the Cauchy data A and B, bords A with B. From the mathematical structure, we argue also that B bords with A. But this does not mean that the process A->B is invariant for time reversal. In fact at this level the time does not exist ! 

2) The observed quantum super Yang-Mills PDEs,(say (YM)[i]), is encoded on an observed fiber bundle E[i]-> N over the proper space-time N, endowed with a time-like flow. Thus the observed (YM)[i] is encoded on N.  At this level we can talk of 'proper-time', but also of 'quantum-proper-time' on each nonlinear quantum propagator, induced by the quantum structure. It is just this quantum time that allows us to implement the Heisenberg uncertainty relation between observed quantum energy and quantum time. (Really the usual proper-time of the observer cannot work since it is a commutative function.) Thermodynamic functions can be implemented on (YM)[i]. At this level we can talk also of reversed processes. In other words whether there exists an observed nonlinear quantum propagator V bording A with B, we can also state that B bords with A. But this does not mean that we can invert the proper-time of the observer. Really this is a nonsense, since the observer's arrow-time points only into the future. The inversion of the observed quantum process can be made by means of a particular observed nonlinear quantum propagator, that I called quantum Cheshire cat propagator. This is possible just thank to the fact that (YM)[i] is invariant for time-translations. (For details it is necessary to look at the paper arXiv: 1206.4856.) Let us emphasize that at this level we can recognize also some quantum phenomena that it is impossible detect at the macroscopic level. For example negative temperatures.

3) Classic limit of (YM)[i] can be obtained, since we can recognize into these PDEs commutative (super) PDEs and commutative (super) solutions. (For details see DOI: 10.1016/j.nonrwa.2005.12.006.) Then at this level disappears the Heisenberg uncertainty relations and the thermodynamic functions are encoded on classic solutions too. Therefore, in particular, we get the usual Clausiius-Duhem inequality, hence the second laws of Thermodynamics works again.

In this sense it is also justified what Roger says in his talk.

At this point it is possible to understand that the Lodschmidt's paradox arises from a bad use of time-reversal made in the Boltzman's (SM) ... but I do not know the exact story of this misunderstanding.

            0 / 0

        ·

            2 hours ago

How can we formulate Loschmidt's paradox in the quantum domain?. Available from: https://www.researchgate.net/post/How_can_we_formulate_Loschmidts_paradox_in_the_quantum_domain [accessed May 12, 2015].

Dear Agostino,

"At this level we can talk also of reversed processes. In other words whether there exists an observed nonlinear quantum propagator V bording A with B, we can also state that B bords with A. But this does not mean that we can invert the proper-time of the observer. Really this is a nonsense, since the observer's arrow-time points only into the future."

yes in fact, the reality is not time reversible or reversed in principle,

What I can do is just to imagine to rewind the tape with some linear Transformations (LORENTZ). But it is FAKE, I just get a previous banal picture of some variables. What about the variables I didn't consider???

 it is terribly easier to treat reversible processes mathematically, it is as well easy to find in Physics irreversible ones.  The simplistic and reductionist way of the models used, though so far the only possible way, complies us to carefully focus on the limit of such models.

Spontaneous interactions like approaching opposite charges with equal mass which then undergo annihilations are not reversible processes at all. There is the ARROW OF TIME which compels the process go in a way and do not come back.

The picture according to which two fermions annhihlate and two photons come out of their global center of mass, then time reversing I get the fermions back at the expenses of the two photons, is something which is correct only in a "banal" energy-momentum balance.

I can sufficiently restrict the "field" so that the expected results come out. This can be made as a first approximation just to enter the problem, because of the intrinsic incapability to treat it in a wider perspective,  then it has to be shown that such model is like a toy of a three years old baby. There is the least action principle in the background in any case, which depends on how the space-time reacts to the mass energy at stake and this makes processes irreversible together with the maximum entropy.

Dear Stefano, my compliment for well understood that the observed arrow-time points into future ... But let us underline that there are some other observed nonlinear quantum propagators, that nothing have to do with the observed time reversal and that are quantum processes mathematically foreseen and also experimentally verified. In this category, for example, there are ones obtained for crossing symmetry. The photo-production of massive particles from energetic photons is a sector of advanced experimental research. I am instead surprised of your superficiality that pretends to claim sentences on what the world should be according to ' your 'Philosophy' that instead is not supported by any serious mathematical model ... The history of Physics and Mathematics should teach to people like you more modesty ad a more serious approach in discussing when one considers new serious mathematical models in Physics but also in Mathematics ... Please think on the concept of antimatter and a lot of particles like neutrinos, W, Z, Higgs, ...  all foreseen only at the mathematical level .... Mathematics is the only serious tool that we have to understand our world ... I am not like Godel that arrived to state that Mathematics is the unique reality ... ( ... of course this is another fundamentalism ...) but luckily we have had Galileo's legacy ...

.Dear Agostino,

should be according to ' your 'Philosophy' that instead is not supported by any serious mathematical model .

I don't know which 'Philosophy'. And I was not criticizing you answer, I was just underlying the fact that

1) the models so far adopted are mostly time-reversible in Quantum Physics

2) it is not so easy to model reality and there is something which does not work.

I didn't criticize you models either, because I don't know them.

"Please think on the concept of antimatter and a lot of particles like neutrinos, W, Z, Higgs, ...  all foreseen only at the mathematical level ....

1) Yes for sure, it is very powerful and has to be used, because it is also a very accurate Language and  the experiments test models which were created with math. In such case there is no alternative.

Mathematics is the only serious tool that we have to understand our world "

I don't agree sorry, it is not the only serious tool, it is a serious tool we have to use to make suitable predictions guided by intuition.

Math is not Physics, Physics is a grasp on something which has to be expressed in some Language to be accurate and shared, and Math is good at that, if this is philosophy, what to say, I'm doing philososphy...

First understand how things work then write down math, my professor "Fernando Ferroni" president of INFN used to say.

It happened sometimes in QM : first write down math then check if the predictions are experimentally true.  And in GRT too, now we are aware that there are serious problems in GRT.

We are not in the realm of ideas of PLATO, this is reality

" teach to people like you  more modesty ad a more serious approach in discussing when one considers "

about "modesty" I would argue the opposite again sorry. I didn't formulate laws myself or given math models. I Just expect that, what so far written reports "modestly" a grasp on the limits of the models written. Because every model will be overcome by other models .The ability of the Physicists has to be expecially measured to predict the range of validity of their models.  This "modesty" to state such limits on their predictions state their real value.

"The photo-production of massive particles from energetic photons is a sector of advanced experimental research"

It is a branch of research but what are the results?

Do you really think that same photons meeting from different directions with approximately the same  energy-momentum with which they exit from annihilation but opposite direction, can make up a fermion??

Maybe with far stronger interfering beams it can be a bit reasonable that it may happen.

The photo-production I just checked is a tecnique that is not relevant to photon-photon interference to build up matter, but to photon-fermions to build up different elements.

The problem you are referring as result from this question, apparently has still to be experimented..

https://www.researchgate.net/post/Do_you_think_that_a_two-laser-beams-in-vacuum_collision_will_produce_a_positron_electron_pair

Dear Stefano,

unfortunately you continue to claim sentences that are neither related to what I said.

Photoproduction of matter is not a process obtained by reversing the time ! Really it can be obtained by crossing symmetry.

Before to spend words ... I suggest you to be more informed on the subject that you are discussing. Crossing symmetry is well experimentally proved in particles reactions, and I also proved it in general in my formulation of quantum (super)gravity, founded on a strict mathematics of quantum super PDEs geometry. Such processes was first conjectured also by very good mathematical physicists, e.g., Breit and Wheeler ...

I skip on the other your sentences ...

Dear Sofia, what you call the reversed of (1) is really realized by crossing symmetry, that is not a process where we are reversed the time ! ...  even if someone could believe  so !  But you know to reverse time is a nonsense. Instead crossing symmetry works well whether in my formulation of quantum supergravity, as well in the phenomenology of particle physics ...

My best regards

Agostino

Dear Agostino,

"Photoproduction of matter is not a process obtained by reversing the time ! ."

This has nothing to do with what I said. It seems you really didn't get the gist of what I was saying. I usually don't attribute so obvious and stupid things to  people I'm taiking to, otherwise what is the point to continue talking if you think that people you adress to are so naive?

And I tell you again, Photoproduction so far has not been obtained from boson interferences.

Dear Sofia, about the usual definition of crossing symmetry in Particle Physics, you can look to the following Wikipedia link:

http://en.wikipedia.org/wiki/Crossing_(physics)

To understand more, related to the quantum geometry of quantum super PDEs, you can look to my paper: arXiv: 1205.2894.

Then you can also understand that what you believe about decays:

'I have experience with the α decay of radioactive nuclides, which I studied in detail, and I know that it is irreversible.'

is not completely true. In fact you can consider the well known beta neutron decay

                                                  n  ->  p + \bar\nu_e + e^-

that for crossing symmetry gives the electron capture of proton

                                                    p + e^-  -> n + \nu_e.

These are  well known phenomena in particle physics.

Dear Stefano, you claim:

'Photoproduction so far has not been obtained from boson interferences.'

With this respect it is necessary to distinguish ... In fact photoproduction of matter is a very hold quantum process .. (photoproduction of electrons, photoproduction of pions, ...) There is also the so-called Breit-Wheeler process 

                                         \gamma + \gamma  ->  e^+ +  e^-

that is under experimental investigation. To be informed about please visit the following Wikipedia link:

                http://en.wikipedia.org/wiki/Breit%E2%80%93Wheeler_process

However I am sure that these experiments will have a full success since photoproduction of matter is well justified by my quantum super gravity theory. In fact it is related with the geometric structural understanding of how matter is created or destroyed. The well known Higgs-mechanism is only a part of a more complex and general mechanism related to the geometric structure of the quantum super Yang-Mills PDEs. 

What's the problem, dear colleagues,

in quantum mechanics any measurement (observation) is irreversible.

Dear Agostino,

"There is also the so-called Breit-Wheeler process                                          \gamma + \gamma  ->  e^+ +  e^-"

exactly, that somebody made such Hypothesis I already knew as I told you.

Andrei Seryi at the University of Oxford said :"Theoretically, however, it would be great if we are able to create particles from only light."

 Light for sure creates mass, can create also new matter or destroy it (photofission) with the help of other fermions. That interfering radiation can create fermions is quite difficult to accept ...what are the boundary conditions???

"justified by my quantum super gravity theory"

You mean  gravitation has to be accounted  too when there is annihilation or not??

Producing of matter from only light is forbidden by energy and momentum conservation laws. Third body (except two photons) is necessary to produce electron-positron pair.

Dear Eugene, please consider that quantum conservation laws applied to nonlinear quantum propagators, encoding quantum reactions, hence their global effects, can change the global evaluations of the initial and final quantum contents. This is related to the contribution of the boundary of such nonlinear quantum propagators. In order to well understand what I mean please look to these my papers:   arXiv: 1205.2894 and arXiv: 1206.4856 

Thanks for your contribution.

For what overcomplicate, Agostino? Simple kinematics is complicated enough.

Dear Eugene, kinematics is not enough when one considers extended objects in the category of quantum manifolds and solutions of quantum super Yang-Mills PDEs ...

Dear Sofia,

The opposite process does not satisfy the "maximum entropy principle". Expect to perform the matter creation  just by interference of equal beams to the ones resulting  from the annihilation, would make the  "fermion" a very simple particle. The fermion is a singularity with infinites in it, for which a renormalisation was performed.

It is born out of entities totally unknown which resonate with the Higgs Field according to the theory of the Drac spinors. The mass-charge we experience is just the result of a resonance giving birth to the SPIN h/2  the ZPE. That it is possible to produce two half integer spin particles out of two bosons, I'm sorry no math in the world can convince me, just experiments could.

It is like saying that I can extract energy from vacuum, might be, but serious sets of experiments have to show that incontrovertibly,  math can just suggest it, affirming that it is not impossible.

Dear Sofia,

first of all in linear Maxwell's theory photons don't interect at all. In quantum electrodynamics lowest order process is photon-photon scattering, i.e. virtual production of two electron-positron pairs and their annihilation.

You must keep in mind, that production and annihilation of virtual particles in diagrams is not the same as in reality (experiments). In the last case conservation laws must be fulfilled.

What is electron-positron annihilation in reality? To annihilate they must be close to each other for some time, i.e. form positronium. Positronium may have total angular moment (spin) ground state 1 or 0. There can not be state of two photons with total angular moment J=1 (L.D. Landau, 1948), therefore two photon annihilation of positronium state 1 is forbidden (must be as minimum three photons). Yes, singlet positronium state can annihilate in two photons, but to produce electron-positron pair from two photons you must create their J=0 state with equal and opposite impulses. How can you achieve this in practice? If you achieve, you will receive positronium in singlet state. And from where will you get energy to desintegrate it?

Regards,

Eugene.

P.S. Who has seen electron-positron pair production by two photons?

Dear Stefano, you continue to claim things that nothing have to do with scientific facts just well known ...

Now you claim:

'That it is possible to produce two half integer spin particles out of two bosons, I'm sorry no math in the world can convince me, just experiments could.'

But bound states of quantum fermionic particles can be bosonic particles (molecules) and by using collisions of such particles one can produce fermionic ones ...

Again one can obtain fermionic particles by photoproduction from bosonic bound states ...

All these facts are experimentally well known other than theoretically described just in the classic quantum mechanics from long time ! ...

I remember another your ridiculous statement that at low temperatures Heisenberg uncertainty relations do not works more ... And Bose-Einstein condensates how it is possible to justify without quantum mechanics ? ...

I do not think that ResearchGate is a good place where send such stupidities  ...

Dear Agostino,

"I remember another your ridiculous statement that at low temperatures Heisenberg uncertainty relations do not works more "

As usual this tells that you don't understand what I mean. I said that "only" the Heisember uncertainty principle works at absolute zero against the argument of Gurjati who came out with  heat fluctuations to justify QM.

Please read carefully and your are kindly invited not address offensively to people.

The attitude you have is to address to people superficially as if they are almost idiots, I've already told you  in the other case of the "TIME ARROW" it already revealed in your response a very superficial way of treating other people's thought, jumping to conclusions without reading or asking clarifications.

Personal Attacks furthermore reveal  a very poorly minded person and irrespectful. When people do not share you thoughts, you attack personally, this is not an ARENA

"Again one can obtain fermionic particles by photoproduction from bosonic bound states All these facts are experimentally well known other than theoretically described just in the classic quantum mechanics from long time ! ......"

 Prof. Eugene F Kislyakov  who answered the questions said that it is impossible and experiments were not performed and he is expert in QM, so what else do you want?

Please provide reference to experiments then

"I do not think that ResearchGate is a good place where send such stupidities  ..."

I agree...

and PS. I wasn't even addressing to you but talking with Sofia, so this reveals an even more  irrespectful behaviour.

Dear colleagues!

Quantum electrodynamics has no logically correct foundation, therefore new ideas are welcomed. Be carefull to each other!

Regards,

Eugene.

Dear Stefano,

I did not claim that you are stupid !

I referred to some your statements that are evidently unfounded ...

Instead I see that you spend some more words against my person ! However I consider you too young to be seriously considered ...

______________________________________________________________

Dear Eugene,

about the Breit-Wheeler process, I proved that it is possible to realize it. My proof is not occasionally or built to this purpose, namely it is placed in a general new theory of quantum supergravity. Furthermore, this last is placed in a general algebraic topologic theory of quantum super PDEs.

In conclusion I have given a robust proof that Breit-Wheeler process works well.

Moreover the fact that there are experimental laboratories in the world that work for obtain such a process, should suggest to any serious scientist to assume a more careful approach on this subject ...

In fact I see that you have adopted this line.

My best regards

Agostino

But they have not seen it till now, Agostino. We may create many different theories, but You know very well about the problems of  "seeing"...

This is not a problem !... Let us only remark that also the celebrated Standard Model in Particle Physics, formulated in 1970, has seen its Higgs particle only after 45 years ... The importance of a theory is not only the problems that is able to solve, but rather whether it contains new insights ...

Dear Agostino,

"Instead I see that you spend some more words against my person ! "

are you are referring to this?

"Personal Attacks furthermore reveal  a very poorly minded person and irrespectful."

So please stop saying that  things I say are "ridiculous" unless you don't get  the proper clarifications, because for the second time, as I told you before, you misinterpreted what I said.

Dear Stefano, I am glad reading that I misunderstood your statements, ... by the way you have not yet contradicted the following:

'That it is possible to produce two half integer spin particles out of two bosons, I'm sorry no math in the world can convince me, just experiments could.'

Sorry, but I cannot define this your claim different from a stupidity ! I hope to be clear enough !

Dear Eugene, I do not understand to what you refer ! Could be more clear please ?

Agostino,

the famous hero of russian tails is Иван дурак (Ivan fool). He is the most clever of all the other actors of tails.

For example, when bosons are photons we have problems.

Eugene, you continue to be cryptic ... you talk of Ivan fool ... you talk about problems with photons ... could be more explicit ?  Otherwise I cannot continue this duscussion ...

Dear Stefano, unfortunately you go on with producing posts, in any direction, without some scientific sense, completely empty, and in advance you pretend to consider yourself as an arbiter that can judge what can be made and what instead must be forbidden. In the following I report a recent your ridiculous sentence:

'PS. I wasn't even addressing to you but talking with Sofia, so this reveals an even more  irrespectful behaviour'

(sic) ... at the same time you pretend also to know what Eugene thinks about my post ...

Sorry, but I have not so time to lose with you !

Dear users,

The question here is about Loschmidt's paradox in the quantum regime. Unfortunately, more than half of the pages with comments address the topic of the annihilation, which is alien to the question here.

Therefore, I opened a question about annihilation, see the site

https://www.researchgate.net/post/Is_the_particle-antiparticle_annihilation_a_reversible_process#5558eb1f60614beb478b456c ,

and I would be very grateful if people would agree to move all the comments on annihilation to that site. I apologize, but I would be glad that at the present site be posted only comments that address Loschmidt's paradox.

With a lot of thanks in advance, and apologies for disturbing,

Sofia

Dear Eugene,

What you say is general things, everybody knows them. But, specifically to the Loschmidt paradox, there is a concept of time, see later in this comment.

A first question is whether at low temperature the gaseous state, on which Loschmidt made his judgements, still holds. It is questionable whether the quantum regime holds at high temperatures, it seems to me that even at room temperature. (Well, Agostino seems to have another opinion, the truth is that until this moment I didn't have time to read his works.)

Now, if the gaseous state holds in the quantum regime, then there exists a time in this problem. Assume together with Loschmidt that we have a container with two partitions, one filled with gas, A, and one empty, B, and suddenly we remove the wall between them. The gas will first of all expand into B with a group velocity. But, until the thermodynamic equilibrium is reached, there will take some time, so I think.

A 3rd issue: how we define entropy in the quantum regime? If we define it as  proportional to  -n ln n,  where n is the number of states, in a closed container in which the gas is completely isolated from the environment, the number of quantum states doesn't change, s.t. with the  -n ln n  definition the entropy doesn't increase and doesn't decrease.

Kind regards,

Sofia

Dear Sofia,

the ideology of quantum mechanics is: If nobody observes, then nothing happens.

How to extract thermodynamics from this?

And observer must be classical...

Dear Eugene,

How to extract thermodynamics from this?

Schrodinger equation corresponds to the first law of thermodynamics, i.e., the law of conservation of energy.

The distributions of some physical quantities correspond to the second law of thermodynamics.

Dear Tang,

in classical and quantum mechanics they have different senses and can not be extracted from each other. Classical and quantum mechanics have different sets of axioms.

Regards,

Eugene.

Dear Eugene,

Schrodinger equation is reversible, but this reversible equation can only be verified by irreversible measurements; and by definition, Schrodinger equation can hardly describe these irreversible measurements. This is a trouble involved irreversibility.

Best Regards,

Tang

Of course, Tang. To my mind there is no Loschmidt's paradox in QM because of irreversibilty of measurements. Contrary, quantum idealogy may be one of the possible resolutions of this paradox.

Regards,

Eugene.

P.S. This concerns also the arrow of time. Other question, how this ideology is justified itself.

I disagree with Eugene. 

Assume that we have a container with two chambers A and B separated by a wall, and the chamber A filled with a gas in thermodynamic (T.D.) equilibrium, while in the chamber B is void. Assume that the container walls and also the separating wall, as perfectly reflecting mirrors for the gas.

If we suddenly remove the separating wall, the gas from the chamber A rushes to fill also the chamber B. This is a transient state, during which we don't have T.D. equilibrium in the container, because the gas doesn't have infinite velocity and doesn't reach the end of the chamber B instantaneously. Even after the gas hits the end of the chamber B, still a time elapses until all thought the container the T.D. equilibrium is installed.

Now, the irreversibility of the measurements is not a problem here. If the concept of T.D. equilibrium exists in the quantum regime, (and it probably exists since the concept of temperature exists under this regime), then this equilibrium exists before any measurement. 

The measurement can only confirm or disconfirm the T.D. equilibrium state, and for a measurement, one gas to have very many containers identically prepared and that underwent the same process. 

No, no, Sofia,

there is no any thermodynamics in QM and mearsurment (observer) must be classical. Thermodynamics appears only on the level of measurments. There are no information, entropy and so on without measurments.

Regards,

Eugene.

P.S. Also, thermodynamics is only about quasi-equilibrium states.

Dear Sofia,

this last Eugene's remark is correct ! Thermodynamics of a quantum system refers to the observed quantum system. In fact its entropy is related to the spectrum of the observed quantum Hamiltonian evaluated at the considered solution of the observed quantum super Yang-Mills PDEs.

By the way I confirm that at the classic limit this quantum thermodynamics justifies the Clausius-Duhem inequality ... Of course to become convinced it is necessary to look to my work.

With respect to the Eugene's PS. I must add that instead quantum thermodynamics does not necessarily encode solutions in equilibrium with heat bath.

My best regards,

Agostino

Dear Eugene and Agostino,

I read what you said, and I have some remarks. The results of a measurement are encrypted in the wave-function. To learn the state of a quantum system we don't do a single measurement, we do many. In this way we get all the possible results, with their probabilities. I explain myseld more clearly (I hope) below.

Although I am no specialist in thermodynamics, let me though speculate that for the simple experiment described in my previous answer we can know the state of the multi-particle system at the time t0 , before removing the separating wall W. Now, for a container with perfectly reflecting walls, the boundary conditions of the Schrodinger equation are simple. So, I again speculate that given the initial wave-function, we can find the wave-function at any time after removing the wall W (Agostino, you are surely more competent than me in this.)

But, going on with my optimistic assumptions that we can calculate the wave-function at any time, then we can also expand it in a superposition of linear momentum eigenfunctions of all the particles. Thus, we will obtain the probability of each configuration of the linear momenta of all the particles. At a single measurement, i.e. opening at a given time ONE SINGLE container, we get one single configuration from the many possible. But opening all the identically prepared containers (which must be many), we obtain the distribution of the configurations.

Thus, we can find out how many different microscopic states there are, and what is the probability of each one. In this context let's remind that the particles are identical, i.e. two microscopic states differing by interchanging two particles, may differ only by the leading sign, s.t. we don't consider them different. Thus, we can calculate the entropy.

Secondly, we can check whether the system is at equilibrium at that time, and fits a certain value of the temperature.

Thus, I return to my beginning words, what we are bound to observe at a measurement is encapsulated in the wave-function, and a measurement at a given time t after removing the wall W, must consist in very many trials (on very many containers identically prepared).

Well, I would appreciate to know from you if this time I was clearer, and I am sending you my kind regards,

Sofia

(P.S. Agostino, I can't deal with a heat bath at this stage, let's first elucidate the things at the simplest level.)

"We can know the state of the multi-particle system at the time t0"

Are You sure that this state exists, Sofia?

For example, the state of free moving particle is wave packet, but there can not be stable wave packets in QM. They are spreading.

Dear Sofia, I understood your speculative design, but there is a fundamental problem in this virtual architecture. I mean the Schroedinger equation. It appears a too naive mathematical tool to encode a complex system, like a quantum gas of all equal particles. Furthermore, since your ideal experiment design wants to suddenly boundary conditions change, such a Schroedinger equation should be submitted to a singular boundary value problem.

You know, even if the Schroedinger equation is considered a master equation in QM, it is really a classical PDE. Therefore a suddenly change of its boundary conditions should produce a shock wave. In other words this ideal experiment is not so smooth as it should appear to you. Furthermore, even if one can mathematically dominate such a problem for the Schroedinger PDE, at the end in my opinion its result is too far to be realistic. In fact such a transient process at the quantum level cannot exclude reactions, hence production of energy inside the gas.

Let also add that the classic assumption in QM to see a quantum system encoded by a probabilistic wave it corresponds just to the statistical interpretation encoding a quantum process as a statistical sample of experiments. In this sense one can answer in the affirmative to your question.

However, my geometric formulation of quantum-gravity, has proved that just this classic point of view can be nowadays beyond. Then, to remain in your ideal experiment, one can ask wether, for example, an integral sphere of the observed quantum super Yang-Mills PDes, having some thermodynamic properties, can evolve to another sphere having some other thermodynamic properties ... and if this evolution can be obtained with a smooth observed nonlinear quantum propagator, or with a singular (weak) one. This ideal experiment is more directly related to the quantum world of any-other one founded on the Schroedinger PDE.

Dear Eugene,

I am not sure whether I understood you. Why shouldn't exist a wave-function for an isolated system? Well, it may be a mixture of wave-functions, but an isolated system is supposed to have a Hamiltonian Ĥ, and the Hamiltonian to have solutions. Even if there is a gravitational field, which is known as conservative, still a Hamiltonian should exist.

The fact that the wave-packet spreads is well-understood. The evolution of a quantum state which is the solution of a Hamiltonian is given by  

(1) ψt = exp(iĤt/ħ) ψ0, 

where ψ0 is the solution at some time that we take as t=0.  For instance, if Ĥ is the free particle Hamiltonian, i.e. no field of forces, one can expand the wave-function as

(2) ∫F(p1,...pN) Πj exp[i(pjxj - pj2t/2mj)/ħ] Πj dpj ,

where F(p1,...pN) is the Fourier transform of ψ0. The expansion (2) encapsulates the spreading that you say.

Unfortunately, in Loschmidt's problem the Hamiltonian is not of free particles, because a) there are collisions, b) there is a gravitational field. Agostino is right, the problem is not simply tractable mathematically, but in principle a Hamiltonian should exist, the difficulties are of technical order - computing power.

Best regards 

In late fiftieth L.D. Landau had said, that Hamiltonian is dead. There are a lot of problems with QM, concerning continuos spectrum.

Agostino, and not only he, is trying to solve (or at least formulate) these problems.

The dificulties are of principle (fundamental) order. Computing power is not the solution.

Regards,

Eugene.

Sofia,

"Unfortunately, in Loschmidt's problem the Hamiltonian is not of free particles, because a) there are collisions, b) there is a gravitational field. "

Isn't it possible to imagine such problem in free space where gravitation is negligible, in order at least to eliminate one complication?

__________________________________________________________-

"However, as the container is completely isolating, the gas is a completely isolated system and it evolves unitarily. The number of states remains the same all the time. So how can the entropy increase?"

In this case it cannot increase if the container is perfectly reflecting, such that also no radiation can come out... either we have to suppose that no radiation is produced by the gas, (which is not easy in a realisitic way because scattering always produces transitories which produces free energy).

Of course, in this case gravitation is not a problem, but collisions are.

They destroy coherent quantum picture.

You mean destroys entangled states?

Dear Stefano,

As far as I saw in Loschmidt'd text, he needed the gravitation not for the paradox, but for arguing with Maxwell on whether in a gravitational field, the temperature depends on height. Maxwell held that there is no dependence - so I infer from Loschmidt's article, while Loschmidt held that there is dependence.

Let me explain: for questioning whether a system of particles may return during its free evolution to the initial state, Loschmidt considered a set of identical, spherical balls, in a container. He showed in the article a picture with a vertical section through the container, and its form is rectangular. The initial state of the balls is that all the balls except one are at rest on the bottom of the container, and the remaining ball, that Loschmidt named m1, is held (in some way) near the ceiling of the container. At a time t0 the ball m1 is released to fall. It hits eccentrically another ball which begins to fly, and after being reflected by the container wall hits in turn another ball, and so on, until all the balls are in movement. Well, due to the gravitational field, the fly of the balls is ballistic, but in order to put all the balls in movement, one could have simple thrown the ball m1 onto another ball so as the hit be eccentric.

I saw a commentary on Loschmidt's paradox, (http://www.loschmidt.cz/  see the chapter Biography) which said that Loschmidt was displeased with the prediction of the heat death of the universe that results from the 2nd law of the thermodynamics, and for this reason, among others, he began to search for possible mistakes in Boltzmann's works. I didn't understand from his article why considering the gravity could beat the 2nd law. As of the heat death, he said:

"The famous problem of undoing what was done has after all no solution indeed, s.t. retaining a plain formulation consisting in a simple statement, reverse suddenly the momentary velocities of all the atoms of the universe."

It's quite a sybilic sentence but in the literature it is interpreted that Loschmidt considered that since the laws of the Newtonian mechanics are time-reversible, then if the evolution of a set of bodies from a state A to the state B is possible, then the reverse evolution is also possible, from B to A, by reversing all the velocities.

All these old paradoxis are oversimplified by modern interpretators, Sofia, and don't have original meaning in modern interpretation.

Concerning gravity, it actually beats 2nd law globally in cosmology. See Landau's "Statistical Physics" 5-th volume of course.

Regards,

Eugene.

I agree with Eugene F Kislyakov.

But Sofias question pointed to some fundamental problems.

In reality is no gas (or particle-collection) with such attributes you have mentioned. No particle is absolutely spherically symmetrical. Particles are on the one hand entangled otherwise independent, depended on the size of the container .... Gravity is also a quite significant, etc.

What predictions can you make about the reality with such idealised models and their mathematical equations? If one or more parameters are imprecise, the projection differ more and more from the facts, finally all predictions are wrong.

At the present time in physics are to many theories with uncertain assumptions. A few examples: quarks, gluons - unproven; dark matter - unproven, gravity - whats the cause? These theories are the basis of our world view!?

I can add to your list, Hans, that we can be sure only about inner structure (degrees of freedom) of molecules and atoms, that they are quantum. We can not be sure, that all the gas is quantum, not speaking about thermostat and so on.

Regards,

Eugene.

P.S. At very low temperature, when quantum effects are possible, any gas is liquid.

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity