Dear Stefano,

Thanks for an interesting question. Since I am a bit careful when it comes to spaces with indefinite metrics, I have looked at the group theory formulation from a simple linear algebra point of view.

If you look at the appendix of the paper enclosed, you will see that the transformation where time "reverses" gives the formulas as you write in the introduction of your paper "The Lorentz Transformation and the Doppler loop of light". This corresponds to the transformation T in my appendix. 

Including a simple metric matrix, see appendix, I can "divide out" the metric in the secular equation obtaining the complex symmetric matrix L. As you can see in this formulation L=L-1 and hence L2=I and the paradox should be resolved.

Best

erkki

Dear Erkki,

you are Always welcome,

my formulas of the LT begins simply with the + instead of the minus, just a matter of defininig the positive direction of the x and so v.

I could have put the - in the forward trasformation and the + in the reverse nothing would have changed basically.

The thing is that time does not reverses in the Doppler loop, how can it? The Doppler loop is a sequence of transformations so there no way time reverses.

You can reverse all the doppler loop like in a movie just with L-2 but that would not mean much..only bring the system to its initial conditions at the emission.

If I get the identity I, the Doppler loop does not have anymore that form,

that ratio of the frequency f1/fo=(1+beta)/(1-beta) is the one predicted by the Doppler Radar, moving mirrors, Gravity Probe A experiment and so on. So with the identity it would not fit the experimental evidence anymore..

Maybe  my paper was obscure then...

The complex transformation Matrix is interesting anyway..

Best

Stefano

Dear Stefano,

Your LT formulas are simply T and T-1 in my notation. Since the LT makes a coordinate transformation between an emitter RFA, we will for simplicity call this the system S1, moving with velocity v in the observers system S.

Now for the Doppler loop (the beam goes back) one has that the system S moves in the system S1 with velocity –v. To understand the results of the RDE loop it is convenient to introduce two systems S’ and S’1 with the x-axis reversed (x=-x, x’1=- x’1). Since now S’ moves in S’1 with the velocity v then one can see that the RDE loop corresponds to T2 and not TT-1=I.

It is unfortunate that I used the notation L for T 𝚫-1 , where the metric is indefinite. Nevertheless it is interesting that in the latter case L=L-1. Coincidently reversing time (and v) gives a “similar result” even it of course is not meaningful in the Doppler case.

Dear Erkki,

"Your LT formulas are simply T and T-1 in my notation"

"Now for the Doppler loop (the beam goes back) one has that the system S moves in the system S1 with velocity –v. To understand the results of the RDE loop it is convenient to introduce two systems S’ and S’1 with the x-axis reversed (x=-x, x’1=- x’1). Since now S’ moves in S’1 with the velocity v then one can see that the RDE loop corresponds to T2 and not TT-1=I."

Yes.

 The ways to understand it are also:

a) the Doppler formula comes from (1+beta*cos(theta)) where theta=180 in the reverse path, cos(theta)= -1 , it is a rotation of the vector, and  since as you said I have to consider -v, then the + sign is maintained (minus times minus) 

b) you have to consider that for reciprocity the beam starting from the system on the left and arriving at the sistem on the right, will see when it is sent back the same configuration as before. Since in the RDE the only thing that counts is that two systems approach or depart the two paths have to be identical in the approaching or departing situation (which are alternative of course).

Einstein solved the mirror problem, which is a double application of the RDE which brings to the same relation between frequencies

f1/fo=(1+beta)/(1-beta) which is the Doppler RADAR frequency ratio also.

Einstein went like this in his paper on electrodynamics of 1905

started with LT --> RDE --> DOUBLE RDE PATH (MIRROR) -> Radiation pressure on a perfect mirror.

Unfortunately he didn't go deeper and realize that the Mirror corresponds to L2 . But this is  obvious because the time in such problem does not reverse. It is not a coordinate transformation, where I can go back and forth in space and time like in a simulation.

In the attached paper (which was published then in EJP) directly derived from Einstein the frequency ratio with the unitary cosine, brings to f1/fo=(1+beta)/(1-beta), considering that beta for me (as you noticed too) is the normalized speed with versus opposite to the positive x axis.

So the loop of radiation cannot belong to a transformation Group.

No scandal about it so far..but there is more...

Dear Stefano,

It is a good observation that you have made! Will you publish it – where?

Thanks also for the reference – I have also seen similar formulas given by my teacher Per-Olov Löwdin, one of the founders of quantum chemistry but originally a theoretical physicist.

Another point: when talking about group theory and LT there is quite a complex hiearchy of groups, from the restricted LG, the proper orthochronous LG the full inhomogeneous LG, the Poincaré group etc.– a point that your country man Santilli elevates to a massive critique of contemporay physics as a conspiratory activity manipulated by the establishment. Sorry for this "political statement".

Anyway, my point regarding time reversal (works in LT) but not in Doppler, unless one translate (no pun intended) this into exchange of emitter and receiver.

Dear Erkki,

I don't think it is enough for a publication, if you help me we can make one

adding more aspects, I would be glad.

Better not to talk about Santilli otherwise we risk to be sued....:-)

I think too that Groups have something too Mathematical in them, too limited.

They are models where physics restricted itself instead of integrating

and opening I think.

The LT are coordinate transformations and as such

they have to be consdered, they cannot in themselves restrain physics

but be able to describe and help in the description according to the motion

of objects.

In principle according to my humble opinion, it is a sort of an error to derive the RDE from the LT . Einstein in his paper of 1905 when he found the results of the radiation pressure missed the chance to notice it. He derived so many results that maybe it has been difficult to for him to take time to rest and think about the important implications of what he was finding.

These points emerge:

1) The IRF is an inertial reference frame, and as such cannot receive from without any momentum, it is a totally isolated system and people like you prepared in thermodyn. understand it very well. Something lost in deep space thermically isolated. (NO INTERACTION)

2) The Em radiation is the only effective communication system we know to transfer infos (maybe the smoke is more effective)  but as we know very well,  radiation disturbs systems... this is  especially found in Quantum Mechanics . Radiation has a potential momentum, exerts a pressure when it is absorbed, momentum only when it is absorbed. In other cases we cannot know, but in the instant in which it exerts momentum is not radiation anymore, it is transforming itself in some additional mass at the absorber. (RADIATION PRESSURE)

3) There is no way to make 1 and 2 agree rather than using approximations.

Physically it is reasonable to find a RF which is an IRF to a good approximation. If systems Exchange radiation won't ever be inertial in the proper sense, but only to a certain extent...and this is obvious. But if they don't exchange radiation nothing can be communicated, or the radiant energy cannot be communicated.

Then if I build a theory based on IRFs it will contain in itself the PITY of being approximate and you don't know to what extent and if you really want its parts to be inertial, these won't be able to Exchange any radiation, They become a sort of MONADS..

So you  insert in the theory an approximation (Inertiality, LOCAL LORENTZ invariance)  which  you cannot control in any way. Nothing in principle is inertial, only things that don't Exchange anything are.

Do you agree???

Dear Stefano,

You are right - the Doppler set up is an idealization as we often do in physics and particularly in connection with deductive reasoning. Nevertheless one should not forget the weakness and approximate character of any model.

I think your question is related to the problem of quantum jumps. As you know Schrödinger was very unhappy about the jump scenario that quantum mechanics assumed without any deeper thought. In a recent paper, see enclosed, I discussed this problem from the point of view of an extended open system variant of quantum mechanics (yet mathematical rigorous). The particular case of a Rubidium atom emitting an 8 m long photon during 26ns is discussed in detail.

Maybe you can see some analogies between this and your initial question. In fact after emitting a wave train we discuss the first part of the "Doppler" in a system where the emitter is moving with velocity v after the proper momentum correction has been taken into consideration. The second part should be discussed under the proviso that the system causing the reflection is so heavy (clouds of water) that no measurable momentum change arises.

Will this be on OK solution?

Very interesting.. I will have a look..

I think I have already uploaded this paper of Fermi about Electrodynamics. Fermi demonstrated that in the doppler effect and Compton effect the phases of the Em waves are not fixed as required by the Lorentz Transformations but are variables..

Dear Erkki,

it seems that for Shrodinger the quantum jumps were undigested...

the Doppler Effect is of paramount importance for the macro and micro world and I think in many cases it has been highly underestimated.

It should represent the key interaction between matter and radiation

and it should be treated like a quantum effect the Quantum Doppler Effect as Others proposed in the past.

The Raman effect , according to which the absorption frequency of atoms can be changed according to their speed respect to the source, is of Paramount importance in many disciplines including chemistry.

I think Bohr in such case was right...since the quanta are emitted and absorbed undivided and as such there have to be sort of discontinuities  "jumps"...

Dear Stefano,

Thanks for sending the excellent Fermi paper. Yes the phase is invariant under the LT. Of course treating the quantum case you need to distinguish mathematically waves with group velocity (i) smaller than c, (ii) equal to c and (iii) larger than c (tachyons) with em waves (ii) leading to the relativistic Doppler.

Dear Erkki,

 in the case of the scattering of radiation in principle even with ..one photon... the change in frequency during the scattering occurs. Part of the energy of the moving surface is provided to the asborbed Photon, raising its energy in the case of approaching configuration. For this reason its actual frequency will raise, not only the one perceived by the observer but the same energy/frequency as if it was perceived by the emitter...

The Maxwell equations describing correctly  the generation of waves through unsteady fields, account basically for the characteristics of the sources. The structure of the wave is a consequence of the fact that the photons were generated by the same source. The wave talks about the source it was generated from, and the wave by itself  is a collective phenomenon correlated by the source and to the source. The photon is something which does not necessarily depend on a EM wave but can travel alone and is detected undivided.

In your interesting paper on the quantum jumps, the instantaneous absorption of quanta versus the finite time which takes for an atom to reset its orbitals during the energy shift , both contain some truth.

For sure it is a mistery, but intuitively for me the only way to make become massive something which is distributed in a finite volume of space-time but does not have mass, travelling at the maximum possible speed,  is to make it collapse it in a Planck time.

So maybe in the shortest possible time the quantum is absorbed by the atom which makes it collapse in tied energy, but there are cascading processes to be performed and the reconfiguration takes place in a finite time as a consequence of the absorption already occured.

Dear Stefano,

In the extended quantum theory any spectral line has a (natural) width. This imparts that states are resonances with a given life time. A transition takes time and the energy uncertainty is given by the Heisenberg uncertainty relations.

Hence for instance a spontaneous excitation of a photon by an atom takes time with the energy uncertainty, associated with borrowing a virtual photon (to be returned at the end of the process), being commensurate with the whole excitation phenomenon. There is no collapse of the wavefunction in this picture of an atomic transition.

The good thing with this picture moreover is that all the physical quantities mentioned above are both measurable and calculable and no ad hoc assumptions are necessary.

Dear Erkki,

"In the extended quantum theory any spectral line has a (natural) width. This imparts that states are resonances with a given life time."

Yes I agree if you talk about eigenstates of systems which have to adapt to an external perturbation and take a finite "transitory" time to do it.

But are we sure that these aren't just a cascade phenomenon of the already occured "condensation" of the radiation?

"Hence for instance a spontaneous excitation of a photon by an atom takes time with the energy uncertainty, associated with borrowing a virtual photon (to be returned at the end of the process), being commensurate with the whole excitation phenomenon."

Yes the atomic phenomena interact with the quantum vacuum (virtual particles),

but what puzzles me is that the photon already had become mass..or not??

There should be a instant in which such thing occurs...a Planck time...

"There is no collapse of the wavefunction in this picture of an atomic transition."

I referred to a more real problem eventually correlated with the wave function,  the creation of the mass from radiation. The collapse I intended is the transformation of something  undefined (not tied) in something more definite as having a measurable mass and be localized (tied) in a constant volume for a suitable time.

Anyway we began with the Lorentz Transformations... and I think we went far from them because real problems are actually far from them...

the wonderland depicted with LT is useful to know but fortunately we have got much more...

Dear Stefano,

Just some short comments on what you said.

First I think "borrowing a virtual photon" is the weakest perturbation of the atom by an em field you can think of. The fact that this imparts a resonance with a complex energy (however small) was already investigated by the matematician Titchmarsh in the 50:ies, see also paper and references therein, enclosed.

Second, the virtual photon as well as the excited photon do not have any restmass. The process has a well-defined lifetime and width. I do not see any need for a Planck time here. This is the playground of the cosmologists – no derailing intended.

Thirdly, the "creation" of mass, being one of the hard problems of high energy physics and presently stated to have been solved by the Higgs, is not a problem in this excitation process. However, if you want to extend the discussion to include pair production and annihilation it comes out as a consequence of the formulation I portrayed in the "background paper".

Dear Erkki,

I say these things from my ignorance of QM because I never sufficiently focus on it.

Thanks for the referece I will see what I can understand from it..

"Second, the virtual photon as well as the excited photon do not have any restmass."

I agree, but the final result is that there is some mass-energy (tied) in the atom..

 the energy was travelling at the speed of light and was captured, confined.... and this "seizure" has to be performed in a terribly short time due to the speed of the perturbation...

"Thirdly, the "creation" of mass, being one of the hard problems of high energy physics and presently stated to have been solved by the Higgs, is not a problem in this excitation process."

I find it difficult to divide the things The excitation includes the mass increase of the atom due to energy..

But I say I'm ignorant...so I might rise umproper doubts...

Going back the RDE, you said in 1952 Shrodinger was still puzzled about quantum jumps... well..

Einstein in a letter to von Laue in 1952, described  what he called a  second type of radiation pressure made of   “indivisible point-like localized quanta of energy which are reflected undivided”.

Dear Stefano,

Usually the non-relativistic Schrödinger equation is enough to formulate the atomic structure for reasonably heavy atoms. However, reaching metals like silver and gold relativistic corrections become essential. My colleague Pekka Pyykkö has demonstarted that the essential difference between silver and gold (as far as their surface appears to the admirer) is a relativistic effect!

Nevertheless the general formulation in the "jump paper" is Lorentz invariant. Energy and momentum before and after the emission process are conserved, however during the emission the atom is not in isolation – it is an open dissipative system interacting with the environment as viewed by an observer in a system where the atom after emission is at rest.

Dear Erkki,

"Nevertheless the general formulation in the "jump paper" is Lorentz invariant."

Basically Lorentz invariant means isolated I agree...

"Energy and momentum before and after the emission process are conserved, however during the emission the atom is not in isolation – it is an open dissipative system interacting with the environment as viewed by an observer in a system where the atom after emission is at rest."

You mean that in the GAP [emission-before, emission-after] it can be seen not in isolation...but it is in communication with the quantum vacuum for the Exchange of these virtual particles...but for the Heisenberg uncertainty prinicple the Exchange with the vacuum is reversible (BORROW something--- Give that back) the way the conservations hold??

Dear Stefano,

Good remarks. Still I would say that LT, concerning SR goes beyond isolated systems. The Ukranian Academician Alexander S. Davydov said in the preface to the second edition of his book Quantum Mechanics "we show the inapplicability of the concept of an essentially relativistic motion of a single particle"! A deeper reason follows from the consequences of the indefinite metric of the Minkowski space, se also my "background paper".

Regarding the exchange with the vacuum. I utilized the principle of micro-reversibility in the "jump paper" to infer absorption from emission, but this assumption, often utilized, does not of course hold in general, since strictly speaking this principle will not work in GR.

Dear Erkki,

thanks...

"Still I would say that LT, concerning SR goes beyond isolated systems. "

SR goes certainly in the realm of non inertiality, or rather interactions.....but it does it within the Newton's II law which is transformed with the relativistic momentum.... but the LT are not necessary and should  not be applicable in such case..

Dear Erkki,

SR respect LT but relativistic dynamics is not built on LT.

In an accelerator the momentum of the accelerated body is transformed according to relativistic dynamics in the RF of the laboratory. In such case the momentum is (classical momentum) * gamma. Where the classical momentum is due to the speed  of the object registered at rest in the Labframe.

In any RF moving at constant speed respect to the LABframe the accelerated body would be seen as transformed according to the LT. But I cannot apply LT between the rest frame and the accelerating  RF.

Dear Stefano,

In practice accelerator physicists have a good control of the relativistic corrections of a circulating beam, e.g. the synchrotron etc. When the beam is released for the scattering experiment one usually employs the center of mass of the system, which means that one has to find the relation between the centre of mass- and the lab frame which is related to LT.

However, the scattering experiment requires quantum mechanics to handle everything from elastic to all kinds of inelastic processes. An excellent book that explains the various complexities of the area without becoming forbiddingly thick and detailed is "Lectures in Scattering Theory" by Ukrainian Academician A.G. Sitenko, Translated by P.J. Shepherd, Pergamon (1971).

Dear Erkki,

"When the beam is released for the scattering experiment one usually employs the center of mass of the system, which means that one has to find the relation between the centre of mass- and the lab frame which is related to LT."

  It is like using the galielan transformations when setting a fictitous force as the centrifugal one...because the coordinate system is accelerating...

"However, the scattering experiment requires quantum mechanics to handle everything from elastic to all kinds of inelastic process"

No doubts

"Lectures in Scattering Theory" by Ukrainian Academician A.G. Sitenko, Translated by P.J. Shepherd, Pergamon (1971)."

Thanks I will have a look.

Dear Erkki,

I studied a lot the RDE and LT by now. The problem of the closed path of radiation is just a dobule application of the RDE or a double application of the Lorentz transformations. There is nothing particularly strange in that application, it would be invertible by applying L-2.

The RDE is just what occurs when I assume that the radiation Exchange occurs between inertial reference frames. That the IRF is an ideal (never occuring) condition is true, it can be a very good approximation for a Physical system though.  In the case where the radiation recoil is large, then the system lose its inertiality at least during the absorption or emission of radiation.

It is the problem of the infinite massive wall, such system absorbs the momentum and reflects the ball. If the scattering is inelastic, the momentum is transmitted to the wall but its effects cannot be detected. If the wall is not so massive the effect can be detected in a variation the status of the wall. This basically alters the inertiality of the object like in the case of the radiation.

Guiseppe,

I have not down voted your posting! I understand your irritation.

I might understand someone down voting me after a long and fruitless debate, where nobody is listening to the arguments of the other. Automatic down voting without arguments is counterproductive and threatens scientific communication and in effect scientific development as a whole.

Dear Erkki J. Brändas ,

after a lot of time spent to study Physics, I go back to this apparently trivial topic...

Although the Doppler Radar formula f/f0 = (1+beta+)/(1-beta)

and the Doppler formulas can be derived through LT, they have a main issue:

they are non-reachable limits of a realistic scenario of mirrors and light: mirrors are objects with mass and light is endowed with recoil.

Those formulas are just good approximations of real phenomena in particular conditions.

https://arxiv.org/pdf/1912.11726.pdf

the general formula for the mirror is f/f0 = (1+beta)/(1-beta+2hf0/E0)

where E0=mc2 , with m the mass of the mirror

On the contrary the formulas derived from LT had the following assumptions:

a) light recoilles or a mirror of infinite mass

and

b) variation of frequency of light does not involve variation in its energy content in an inertial frame.

WHY?

Given two frames approaching from opposite directions with relative speed v, equipped with ideal mirrors, the light emitted, bounces back and forth.

Due to the Doppler effect, the frequency raises after a bounce (experimentally verified in Doppler RADAR to very good approximation) .

If the frames are inertial, v is unchanged, while the energy detected increases due to frequency increase, no other change occurs.

That literally creates “energy” as measured from each inertial frame, a “perpetuum mobile” of the first kind which invalidates at least the hypothesis of constant speed. That is to say that inertial motion never occurs in presence of light transmission, it is only a good approximation in most cases.

The root of the issue stays on the assumption of invariant speed (inertiality): systems should have an infinite mass to keep invariant their speed in presence of external forces (Leon Brillouin) , hence, in such cases, the use of inertial frames to find exact solutions of physical problems, is singled out.

A realistic scenario involves relations between non-inertial frames, with radiation recoil of light (experimental evidence) and finite masses: to allow to decrease their relative speed. That allows the transfer of the kinetic energy, from the mirrors, finite mass objects, into the increasing energy of radiation (experimental evidence) avoiding a perpetuum mobile of the first kind. The only way to provide a sound physical framework for the Doppler effect (Leon Brillouin, Fermi, Schrodinger).