[General] De Broglie Wave

Albrecht Giese genmail at a-giese.de
Wed Feb 10 09:47:42 PST 2016


Hello Richard,

good question!

My model works originally for leptons and for quarks. The photon is 
somewhat different visible through the different spin. So, I am not sure 
that the photon can be described by two sub-particles, maybe the 
description needs four sub-particles. But this means only a limited 
correction factor.

For the determination of the mass, the model needs the size of the 
particle. For the electron I have determined the size from the magnetic 
moment. In case of the photon the wavelength can be taken as a measure 
for the size. If the wavelength is inserted into the equation for mass, 
then the dynamical mass (equivalently the energy) is the result. And 
that fits well.

I have attached two pages of my power-point presentation in San Diego. 
There you can see the calculation. The calculation is done there the 
other way around. I start with the energy, convert it to the mass and 
show that the resulting size of the photon corresponds to its 
wavelength.  - In case of any questions, please ask.

Albrecht


Am 10.02.2016 um 17:41 schrieb Richard Gauthier:
> Hello Albrecht,
>      If you have solved the problem of inertia, then the inertial mass 
> of a photon should also be explainable by your two-particle electron 
> model. Is it?
>          Richard
>
>> On Feb 10, 2016, at 5:13 AM, Albrecht Giese <genmail at a-giese.de 
>> <mailto:genmail at a-giese.de>> wrote:
>>
>> Hi Wolf,
>>
>> why do you think that I am frustrated? Why should I? Since I found 17 
>> years ago the mechanism of inertia, which functions so straight and 
>> logical with precise results, I am continuously happy. And the 
>> appreciation by interested physicists is great. Since 14 years my 
>> site about mass in internationally #1 in the internet. Only sometimes 
>> the mass site of Nobel Prize winner Frank Wilzcek is one step higher. 
>> But that is good companionship.
>>
>> True that it is a problem with Main Stream. They do not object but 
>> just do not care. They love the Higgs model even though it is proven 
>> not to work. - It just need patience. I still have it.
>>
>> Yes, quantum numbers work fine, but they are physically little or not 
>> founded. It is similar to the known Pauli Principle. That also works, 
>> but nobody knows why. And the bad thing is that nobody from Main 
>> Stream concerned about this non-understanding. That is the biggest 
>> weakness in today's physics in my view.
>>
>> Albrecht
>>
>>
>> Am 09.02.2016 um 20:35 schrieb Wolfgang Baer:
>>> I can feel your frustration, Albrecht,
>>> The oldies are probably all wrong, but it's important to remember 
>>> that right or wrong they give us the platform from which to see farther.
>>> "standing on the shoulders of others", and right or wrong they give 
>>> us something tangible to argue about
>>> and what quantum numbers have done for us to organize chemistry is 
>>> amazing.
>>>
>>> wolf
>>>
>>> Dr. Wolfgang Baer
>>> Research Director
>>> Nascent Systems Inc.
>>> tel/fax 831-659-3120/0432
>>> E-mailwolf at NascentInc.com
>>> On 2/9/2016 10:18 AM, Albrecht Giese wrote:
>>>> 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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