[General] De Broglie Wave

Albrecht Giese genmail at a-giese.de
Thu Feb 11 09:12:03 PST 2016


Hi Al,

your are right that we are sticking in a circle. But we can see the point.

If I look into the paper of de Broglie again (your translation), there 
is nothing of an interaction. DeB argues about the wave which 
accompanies the particle. And for a particle of a certain speed this is 
a property of the particle (in relation to some frame) but nothing about 
an interaction. Or where do you see in his text an interaction mentioned?

If we Lorentz-transform the interference pattern of an electron to the 
frame of a moving observer, there will be a change, you may call it 
distortion. But the change of the de Broglie wavelength in relation to a 
moving observer is a complete different category. I have given a 
numerical example: If an electron moves at 0.1 c and an observer moves 
as well at 0.1 c into the same direction towards the double slit, the 
Lorentz transformation of the pattern into the frame of this observer 
will have a length change of < 0.1%. But the change of the de Broglie 
wavelength is in this case from some finite lambda to /infinite/. Not 
the same, I would say.

And again a look into the use in the Schrödinger equation. The temporal 
part of this equation uses the law  E = h*frequency. That frequency is a 
property of the free moving particle. And it can be correctly 
Lorentz-transformed into any other frame. Schrödinger has then used the 
de Broglie relation lambda = h/p with the same understanding (otherwise 
his equation would be internally conflicting). So he also in this part 
describes a free moving particle. But a Lorentz-transformation will 
terribly fail in this case.

Again: Where do you see in the text of de Broglie a relation to an 
interaction?

Best, Albrecht


Am 10.02.2016 um 19:41 schrieb af.kracklauer at web.de:
> Albrecht:
> You are locked in a "do-loop."  Appropos the experiment metioned below 
> (Jönssen), you are discussing your misunderstaning not deBroglie's or 
> mine.  The deB wave that matters is not that between the 
> particle-observer or slit (crystal)-observer, but the particle-slit 
> (with registration screen).  All the observer does, no matter how fast 
> or complex his manuevers, is look at the registration to see the 
> diffreaction pattern.  What he sees, of course, will be distorted by 
> perspective, both geometric/optical and relativistic, but the rulers 
> in the frame of the slit are likewise distorted in appearance, so if 
> the observer reads the relevant displacements from comparison with, as 
> it were, the slit's rulers, the results (data) will agree with those 
> from all other observers who do the same no matter what their 
> individual motion is or was.
> Of course, the observer could, as you suggest, calculate the deB wave 
> acting between the particle and himself, but that would determine the 
> diffraction of the particle beam off the observer, not through the 
> slit!  Even deBroglie saw that.  [Actually it's the same deB wave, but 
> Lorentz x-formed to each other observer's frame.  Thus same thing, 
> looks, and acts, different.]
> Again: deB waves are NOT a characteristic of a particle, but of its 
> interaction with other objects, and for each other object there is a 
> different deB wave, because each interaction is different.
> THINK about it. best, Al
> *Gesendet:* Mittwoch, 10. Februar 2016 um 15:37 Uhr
> *Von:* "Albrecht Giese" <genmail at a-giese.de>
> *An:* af.kracklauer at web.de
> *Cc:* general at lists.natureoflightandparticles.org, "Richard Gauthier" 
> <richgauthier at gmail.com>
> *Betreff:* Re: [General] De Broglie Wave
> Hi Al,
>
> You say “DeB's formuals give results in accord with empirical 
> observations “.
>
> I am very surprised about this repeated statement. I think our past 
> discussion has shown that the concept of de Broglie is completely 
> wrong – except his statement that there exist matter waves. He has 
> postulated a wave which in fact does not exist and which does not have 
> any foundation in physics. It has a wavelength which – by his rule – 
> disappears when an observer moves at some medium speed.
>
> Electron scattering does happen, I have shown in my paper that the 
> experimental results can be quantitatively explained on the basis of 
> standard physics. Indeed very funny that also the concept of deB works 
> in a special case (but else not).
>
> Counter evidence? Assume we can perform an experiment of electron 
> scattering (e.g. the one of Jönsson in 1957) in a moving lab. And we 
> observe it from our position at rest. Then we will see that the 
> results based on the rules of deB are completely wrong. - It is of 
> course difficult to perform such experiment at high speed and at the 
> same time with high precision. But I have shown that it is a simple 
> calculation to predict this (failing) result on the basis of deB's 
> rules. Should I explain it again? (It is in my paper).
>
> Or alternatively we have to give up the Symmetry of Space - believed 
> unrestrictedly since Newton. Give it up just to save de Broglie? For 
> no other use?
>
> E&M waves on the other hand are fully consistent with the standard 
> rules for waves. No E&M wave will disappear just because there is an 
> observer moving at some medium speed.
>
> Ciao, Albrecht
>
> Am 09.02.2016 um 20:46 schrieb af.kracklauer at web.de:
>
>     Hi Albrect:
>     DeB's formuals give results in accord with empirical
>     observations---your claim notwithstanding. (BTW, what are you
>     refering to as counter evidence?) Thus, they are useful and in
>     this sense correct.  The story he told himself and used to derive
>     his formulas is, actually, immaterial insofar as he got a useful
>     conception and useful formulas.  Stories are a dime-a-dozen,  you
>     have some that many consider as off-track as you appear to
>     consider DeB's.  That matters only as "philosphy" but not as
>     techinical physics.  Anyway, I suspect that your deep
>     antiaffection for this "wrong" deB wave is grounded on the notion
>     that this wave is a characteristic of the particle instead of its
>     interaction with the rest of the universe as described by the SED
>     background (AKA: the 1/h h-bar x omega of the quantized free E&M
>     wave).
>     The stories told by conventional physicists to motivate QM are of
>     course just so much blather.  Mostly also inconsistent too---a
>     capital crime for those bragging about their rational thinking!
>      And, obviously, that is the push behind my efforts leading to #7
>     on www.nonloco-physics.0catch.com!
>     In any case, your fixation with a fictitious wave should be
>     extended to all E&M waves.  None of them exist as they are
>     described---there is no media.  Here DeB is much less the offender
>     than Bohr, Bell, Heisenberg, Von Neumann, and whole flock of 2nd
>     generation QM enthusiasts.  Still, QM works.  To me that means
>     there is a coherent story to tell for the math, we just have to
>     find it.
>     ciao, Al
>     *Gesendet:* Dienstag, 09. Februar 2016 um 19:18 Uhr
>     *Von:* "Albrecht Giese" <genmail at a-giese.de>
>     *An:* af.kracklauer at web.de
>     *Cc:* general at lists.natureoflightandparticles.org, "Richard
>     Gauthier" <richgauthier at gmail.com>
>     *Betreff:* Re: [General] De Broglie Wave
>     Hi Al,
>
>     the choice of de Broglie is not suboptimal, it is clearly wrong.
>     Badly wrong. The wave he has introduced does not exist, and if it
>     would exist its behaviour would cause a physical behaviour which
>     is in conflict with measurements (if those are comprehensively done).
>
>     I agree with you that the main object now is to move forward. But
>     we will not move successfully forward if we carry millstones with
>     us. De Broglie's wave is a millstone. I just had a look into a new
>     textbook about QM, which was highly recommended by our university.
>     It makes full use of de Broglie's relation between momentum and
>     wavelength, so this is unfortunately not just history.
>
>     But looking into the history: Bohr, Sommerfeld and others have
>     used the result of de Broglie to explain quantum numbers.
>     Particularly the quantisation of the angular momentum on atomic
>     shells is explained by "standing waves" where the wavelength is
>     the one defined by dB. This obviously hides the true reason of
>     this quantisation, but as anyone believes that the Ansatz using de
>     Broglie is right, nobody is looking for the correct cause. - This
>     is one of the reasons for our sticking physics.
>
>     Tschüss back
>     Albrecht
>
>     Am 09.02.2016 um 14:57 schrieb af.kracklauer at web.de:
>
>         Hi Albrecht:
>         As you fully know, the very same idea can be expressed in
>         various languages.  This is true of physics also. The very
>         same structure can be attached to variuos words and images.  I
>         do not defend deBroglie's choice of words and images. I too
>         find his choice suboptimal and somewhat contrdictory.  So
>         what?  He was playing his hand at that time with the hand he
>         was delt at that time.  Since then, other ideas have been
>         found in the deck, as it were.  I find that, without changing
>         any of his math, one can tell a story that is vastly less
>         etherial and mysterious and, depending on the reader's depth
>         of analysis, less self-contradictory.  I think my story is the
>         one DeBrogle would have told if he had been inspired by some
>         facits of SED.  And, some people have a greater affinty and
>         interest in abstract structures, in particular when their
>         mathematical redintion seems to work, that for the stories
>         told for their explication.  This is particularly true of all
>         things QM.
>         Anyway, the main object now (2016) is to move forward, not
>         critique historical personalitites.  So, I'm trying to
>         contribute to this discussion by adding what I know now, and
>         what I have found to be useful.  We are "doing" physics, not
>         history.  Let's make new errors, not just grind away on the
>         old ones!
>         BTW, to my info, both Dirac and Schrödinger would agree that
>         deBroglie proposed some not too cogent arguments regarding the
>         nature of QM-wave functions. Still, the best there at that
>         time. All the same, they too went to their graves without
>         having found a satisfactory interpretation.  SED throws some
>         new ingredients into the mix.
>         Tschuss, Al
>         *Gesendet:* Dienstag, 09. Februar 2016 um 13:41 Uhr
>         *Von:* "Albrecht Giese" <genmail at a-giese.de>
>         *An:* af.kracklauer at web.de
>         *Cc:* general at lists.natureoflightandparticles.org, "Richard
>         Gauthier" <richgauthier at gmail.com>
>         *Betreff:* Re: [General] De Broglie Wave
>         Hi Al,
>
>         I have the impression that you have a solution for particle
>         scattering which is in some way related to the idea of de
>         Broglie. (I also have of course a solution). But was this the
>         goal of our discussion and of my original contribution? It was
>         not! My objection was de Broglie's original idea as stated in
>         his thesis and as taken over by Schrödinger and Dirac.
>
>         You have a lot of elements in your argumentation which I do
>         not find in the thesis of de Broglie. (There is e.g. nothing
>         at dB about SED ore background.)
>
>         The essential point of our discussion is the meaning of his
>         wave - and his wavelength. I think it is very obvious from his
>         thesis (which you clearly know) that his "fictitious wave"
>         accompanies a particle like the electron/all of the time/.
>         There is no interaction mentioned except that there is an
>         observer at rest who measures the frequency of the particle.
>         But without influencing the particle.
>
>         Now it is normal knowledge that a frequency and as well a
>         wavelength appears changed for an observer who is in motion.
>         This is caused by the Doppler effect. But the Doppler effect
>         will never cause that a finite wavelength changes to Infinite
>         if an observer moves at some speed unequal to c. But just that
>         happens to the wave invented by de Broglie. It follows the
>         equation
>
>         lambda = h/(m*v)    where v is the speed difference between
>         the particle and the observer (to say it this time this way).
>         And this is in conflict to any physics we know.
>
>         Best, Albrecht
>
>         Am 08.02.2016 um 17:20 schrieb af.kracklauer at web.de:
>
>             Hi Albrecht:
>             Your challenge is easy!  In fact my last responce covered
>             it.   The RELEVANT velocity is the relative velocity
>             between the particle and the slit; not that between the
>             observer-particle or observer-slit.   An observer will see
>             all kinds of distortions of the events, starting with
>             simple persepctive due to being at some distance from the
>             slit and its registration screen.  In additon this
>             observer will see those deB waves affecting the particle
>             (NOT from the particle, nor from the slit, but from the
>             universal background there before either the particle or
>             slit came into being)  as perspectively-relativistically
>             distorted (twin-clock type distortion).  BUT, the observer
>             will still see the same over-all background because the
>             totality of background signals (not just those to which
>             this particle is tuned), i.e., its spectral energy
>             density, is itself Lorentz invariant.  That is, the
>             observer's  motion does not  enable it to empirically
>             distinguish between the background in the various frames,
>             nor does the background engender friction forces.
>             You have got to get your head around the idea that deB
>             waves are independant of particles whatever their frame.
>             Schrördinger did toy with some aspects that deBroglie
>             used, but never did succeed in rationalizing his eq. in
>             those or any other terms.  For him, when died, wave
>             functions were ontologically completely mysterious.  From
>             SED proponents, I'm told, my thoughts in #7 on
>             www.nonloco-physics.0catch.com, are unique in formulating
>             S's eq. in terms of deB concepts.  Try it, maybe you'll
>             like it.
>             There are other SED-type stories too, but as they are
>             based on diffusion (parabolic, not hyperbolic) precesses,
>             I find them self contradictory.
>             ciao, Al
>             *Gesendet:* Montag, 08. Februar 2016 um 141 Uhr
>             *Von:* "Albrecht Giese" <genmail at a-giese.de>
>             *An:* af.kracklauer at web.de
>             *Cc:* general at lists.natureoflightandparticles.org,
>             "Richard Gauthier" <richgauthier at gmail.com>
>             *Betreff:* Re: [General] De Broglie Wave
>             Hi Al,
>
>             if you follow de Broglie, you should have an explanation
>             for the following experiment (here again):
>
>             Electrons move at 0.1 c towards the double slit. Behind
>             the double slit there is an interference pattern
>             generated, which in the frame of the slit follows the rule
>             of de Broglie. But now there is an observer also moving at
>             0.1 c parallel to the beam of electrons. In his frame the
>             electrons have momentum=0 and so wavelength=infinite. That
>             means: No interference pattern. But there is in fact a
>             pattern which does not disappear just because there is
>             another observer. And the moving observer will see the
>             pattern. - This is a falsification of de Broglie's rule.
>             What else?
>
>             The understanding that the de Broglie wave is a property
>             of the particle (even though depending on their speed, but
>             not on an interaction) was not my idea but the one of
>             Schrödinger and Dirac and many others. Also by de Broglie
>             himself.
>
>             Ciao Albrecht
>
>             Am 08.02.2016 um 03:30 schrieb af.kracklauer at web.de:
>
>                 Hi Albrecht:
>                 BUT, the laws of Physics for "being" in a frame are
>                 not the laws for interacting between frames!  The deB.
>                 wave is not a feature of a particle in its own frame,
>                 but a feature of the interaction of such a particle
>                 with at least one other particle in another frame.
>                  When the two frames are moving with respect to each
>                 other, then the features of the interaction cannot be
>                 Lorentz invariants.  When one particle is interacting
>                 with another particle (or ensemble---slit say) the
>                 relevant physics is determined by the deB wave in that
>                 sitation, whatever it looks like to an observer in a
>                 third frame with yet different relative velocities.
>                  It is a perspective effect: a tree is the same
>                 ontological size in fact no matter how small it
>                 appears to distant observers.  Observed diminished
>                 size(s) cannot be "invriant."  Appearances =/= ,,so
>                 sein''.
>                 You have gotten your head stuck on the idea that deB.
>                 waves are characteristics intrinsic to particles in an
>                 of themselves.  Recalibrate!  DeB waves are
>                 charactteristics of the mutual interaction of particles.
>                 Best, Al
>                 *Gesendet:* Sonntag, 07. Februar 2016 um 22:10 Uhr
>                 *Von:* "Albrecht Giese" <genmail at a-giese.de>
>                 *An:* af.kracklauer at web.de
>                 *Cc:* general at lists.natureoflightandparticles.org,
>                 "Richard Gauthier" <richgauthier at gmail.com>
>                 *Betreff:* Re: [General] De Broglie Wave
>                 Hi Al,
>
>                 at one of your points I really disagree. The physical
>                 laws have to be fulfilled in every frame. That means
>                 that all physical processes have to obey the same laws
>                 in all frames. So also the process at the double slit.
>                 But the rule given by de Broglie looks correct in only
>                 one frame, that is the frame where the double slit is
>                 at rest. For an observer in motion the diffraction
>                 pattern looks very similar as for the observer at
>                 rest, but for the observer in motion the results
>                 according to de Broglie are completely different,
>                 because the momentum of the particle is different in a
>                 wide range in the frame of a moving observer and so is
>                 the wavelength assigned to the particle.
>
>                 The specific case: At electron scattering, the
>                 observer co-moving with the electron will see a
>                 similar pattern as the observer at rest, but de
>                 Broglie says that for this observer there does not
>                 exist any pattern. That is strongly incorrect.
>
>                 The Schrödinger equation and also the Dirac function
>                 should have correct results in different frames, at
>                 least at non-relativistic speeds. This requirement is
>                 clearly violated through their use of de Broglie's rule.
>
>                 Grüße
>                 Albrecht
>
>                 PS: Your article refers to "Stochastic
>                 Electrodynamics". That is in my knowledge not standard
>                 physics and so a new assumption.
>
>                 Am 07.02.2016 um 19:03 schrieb af.kracklauer at web.de:
>
>                     Hi Albrecht:
>                     In my view the story in my paper has no new
>                     assunptions, rather new words for old assumptions.
>                      As I, along with most others, see it, there is no
>                     conflict with experiment, but a less than fully
>                     transparent explantion for experimental
>                     observations (particle beam diffrction) otherwise
>                     unexplained.  At the time of writing, and nowadays
>                     too (although I'd to think that my paper
>                     rationalizes DeB's story) it was the most widely
>                     accepted story for this phenomna.
>                     The only entities that logically need to be
>                     Lorentz invariant are the particle.  I the deB
>                     wave is not a 'Bestandteil' of the particle, but
>                     of its relations with its envionment, then
>                     invariance is not defined nor useful.
>                     M.f.G.  Al
>                     *Gesendet:* Sonntag, 07. Februar 2016 um 14:39 Uhr
>                     *Von:* "Albrecht Giese" <genmail at a-giese.de>
>                     *An:* af.kracklauer at web.de
>                     *Cc:* general at lists.natureoflightandparticles.org,
>                     "Richard Gauthier" <richgauthier at gmail.com>
>                     *Betreff:* Re: [General] De Broglie Wave
>                     Hi Al,
>
>                     thank you for your reference. Your paper has a lot
>                     of intelligent thoughts but also a lot of
>                     additional assumptions. With reference to the de
>                     Broglie wave, I think, is the situation much
>                     simpler on the level of conservative knowledge. De
>                     Broglie has misunderstood relativity (particularly
>                     dilation) and so seen a conflict which does in
>                     fact not exist. He has solved the conflict by
>                     inventing an additional "fictitious" wave which
>                     has no other foundation in physics, and also his
>                     "theorem of harmonic phases" which as well is an
>                     invention without need. And his result is in
>                     conflict with the experiment if we ask for Lorentz
>                     invariance or even for Galilean invariance. - If
>                     we follow the basic idea of de Broglie by,
>                     however, avoiding his logical error about
>                     relativity, we come easily to a description of
>                     matter waves without logical conflicts. This does
>                     not need new philosophy or other effort at this level.
>
>                     Best, Albrecht
>
>                     Am 06.02.2016 um 03:15 schrieb af.kracklauer at web.de:
>
>                         Hi Albrecht:
>                         DeBroglie's verbage is indeed quite rococo!
>                          Nonetheless, his machinations, although
>                         verbalized, in the true tradtion of quantum
>                         mechanics, mysteriously, can be reinterpreted
>                         (i.e., alternate verbage found without
>                         changing any of the math) so as to tell a
>                         fully, if (somewhat) hetrodoxical, story.  See
>                         #11 on www.Nonloco-Physics.0catch.com.
>                         cc:  Waves are never a characteristic of a
>                         single, point-like entity, but colletive
>                         motion of a medium.  IF they exist at all.  My
>                         view is that E&M waves are a fiction wrought
>                         by Fourier analysis.  The only real physical
>                         part is an "interaction", which mnight as well
>                         be thought of an absract string between
>                         charges.  Also, neutrons have electric
>                         multipole moments; i.e., they are totally
>                         neutral but not charge-free.
>                         Best,  Al
>                         *Gesendet:* Freitag, 05. Februar 2016 um 21:43 Uhr
>                         *Von:* "Albrecht Giese" <genmail at a-giese.de>
>                         *An:* af.kracklauer at web.de,
>                         general at lists.natureoflightandparticles.org
>                         *Cc:* "Richard Gauthier" <richgauthier at gmail.com>
>                         *Betreff:* Re: [General] De Broglie Wave
>                         Hi Al,
>
>                         true, in the frame of the particle the dB
>                         wavelength is infinite. Because in its own
>                         frame the momentum of the particle is 0. The
>                         particle oscillates with the frequency of the
>                         particle's Zitterbewegung (which background
>                         fields do you have in mind? De Brogie does not
>                         mention them). This oscillation is in no
>                         contradiction with this wavelength as the
>                         phase speed is also infinite. For the
>                         imagination, the latter means that all points
>                         of that wave oscillate with the same phase at
>                         any point.
>
>                         Which background waves do you have in mind?
>                         What is the CNONOICAL momentum? And what about
>                         E&M interactions? De Broglie has not related
>                         his wave to a specific field. An E&M field
>                         would anyway have no effect in the case of
>                         neutron scattering for which the same de
>                         Broglie formalism is used. And into which
>                         frame do you see the wave Lorentz-transformed?
>
>                         So, an electron in his frame has an infinite
>                         wavelength and in his frame has the double
>                         slit moving towards the particle. How can an
>                         interference at the slits occur? No
>                         interference can happen under these
>                         conditions. But, as I have explained in the
>                         paper, the normal wave which accompanies the
>                         electron by normal rules (i.e. phase speed =
>                         c) will have an interference with its own
>                         reflection, which has then a wavelength which
>                         fits to the expectation of de Broglie. But
>                         that is a very local event (in a range of
>                         approx. 10^-12 m for the electron) and it is
>                         not at all a property of the electron as de
>                         Broglie has thought.
>
>                         To say it again: The de Broglie wavelength
>                         cannot be a steady property of the particle.
>                         But Schrödinger and Dirac have incorporated it
>                         into their QM equations with this understanding.
>
>                         If I should have misunderstood you, please
>                         show the mathematical calculations which you mean.
>
>                         Ciao, Albrecht
>
>                         Am 05.02.2016 um 19:20 schrieb
>                         af.kracklauer at web.de:
>
>                             Hi: Albrecht:
>                             Your arguments don't resonate with me.
>                              The deB' wave length is infinite in the
>                             particles frame: it is the standing wave
>                             formed by the inpinging background waves
>                             having a freq. = the particle's
>                             Zitterbewegung.  If these TWO waves are
>                             each Lorentz x-formed to another frame and
>                             added there, they exhibit exactly the DeB'
>                             modulation wavelength proportional to the
>                             particle's momentum.  The only mysterious
>                             feature then is that the proportionality
>                             is to the CNONICAL momentum, i.e.,
>                             including the vector potential of whatever
>                             exterior E&M interactions are in-coming.
>                              Nevertheless, everything works our
>                             without contradiction.  A particle
>                             oscillates in place at its Zitter freq.
>                             while the Zitter signals are modulated by
>                             the DeB' wavelength as they move through
>                             slits, say.
>                             ciao,  L
>                             *Gesendet:* Freitag, 05. Februar 2016 um
>                             12:28 Uhr
>                             *Von:* "Albrecht Giese" <genmail at a-giese.de>
>                             *An:* "Richard Gauthier"
>                             <richgauthier at gmail.com>,
>                             general at lists.natureoflightandparticles.org
>                             *Betreff:* Re: [General] De Broglie Wave
>                             Hi Richard and Al, hi All,
>
>                             recently we had a discussion here about
>                             two topics:
>
>                             1. The functionality of the de Broglie
>                             wave, particularly its wavelength
>                             if seen from a different inertial system.
>                             Such cases lead to illogical
>                             situations.
>                             2. The problem of the apparent asymmetry
>                             at relativistic dilation.
>
>                             I have investigated these cases and found
>                             that they are in some way
>                             connected. Relativistic dilation is not as
>                             simple as it is normally
>                             taken. It looks asymmetric if it is
>                             incorrectly treated. An asymmetry
>                             would falsify Special Relativity. But it
>                             is in fact symmetrical if
>                             properly handled and understood.
>
>                             It is funny that both problems are
>                             connected to each other through the
>                             fact that de Broglie himself has
>                             misinterpreted dilation. From this
>                             incorrect understanding he did not find
>                             another way out than to invent
>                             his "theorem of phase harmony"; with all
>                             logical conflicts resulting
>                             from this approach.
>
>                             If relativity is properly understood, the
>                             problem seen by de Broglie
>                             does not exist. Equations regarding matter
>                             waves can be derived which
>                             work properly, i.e. conform to the
>                             experiments but avoid the logical
>                             conflicts.
>
>                             As announced, I have composed a paper
>                             about this. It can be found at:
>
>                             https://www.academia.edu/21564534/The_Conflict_with_the_De_Broglie_Wavelength
>                             .
>
>                             I thank Richard Gauthier for the
>                             discussion which we had about this
>                             topic. It caused me to investigate the
>                             problem and to find a solution.
>
>                             Albrecht
>
>
>
>
>
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