The function f(λ, D) is represented by the dark curve in Figure 18. It reveals that the domain of possible frequencies is limited, as well as the domain of possible wavelength. It is only physically possible that λ/2 ≥ D. The approximation f = v/λ corresponds to the red line, which would allow for arbitrarily high frequencies, but (10) imposes an upper limit (fmax = v/πD). Nevertheless, the linear approximation is valid for a relatively large domain of low frequencies. [Indeed, , when ]. Actually, the spectrum of possible frequencies is fs = sf1 when .
Figure 18. The general function f(λ) for chains of WPs.
3.3. Automatic Trimming of Chains of Water Pearls
Figure 19 summarizes the essential steps that explain why chains of water pearls are the information carriers of water memory. The upper line represents a biologically active molecule, where the charged part oscillates at some very low frequency f. It cannot communicate these oscillations to the WPs, since the length of the chain L < λ/2, where the wavelength λ is determined by f = v/λ. The second line represents what happens as soon as the length of the chain of WPs reaches the value L = λ/2 = v/2f. The electric field of the oscillating charge does then excite a standing wave for oscillatory rotations of WPs.
The last WP of the growing chain does suddenly start to oscillate when its length L = v/2f. The amplitude of this oscillation is the same as for the first WP of the chain, which can also be set in forced oscillation by the active molecule. It remains attached to it, but can now communicate its motion to other WPs of the chain. However, the rotation of the last pearl of the chain prevents the formation and attachment of an additional WP. The growing chain is thus automatically trimmed. Information that is characteristic of the type of active molecules has been encoded by means of the length L of the chain. It depends indeed on the frequency f.
The third line of Figure 19 shows that when the trimmed chain is detached from its generator, it does still allow for a standing wave for the same length L = λ/2. The liberated chain of WPs conserves the acquired information. Moreover, it produces itself an oscillating electric field. Its frequency f is adequate to create more equally trimmed chains of WPs. They are thus multiplied by an autocatalytic process. This provides the key for a rational justification of water memory. It resulted from a systematic examination of all possible interactions between water molecules and their logical consequences.
We might object that oscillatory rotations of WPs will be damped by friction, exerted by surrounding water molecules. This is true, but trimmed chains are also subjected to local impacts of water molecules. Although the impacts are random, the chain does pick-up energy when it allows for resonances at any frequency fs for possible standing waves. Since it is sufficient that the free ends of the chain can oscillate with maximal amplitude, standing waves of smaller wavelengths and higher frequencies can also be excited. Excitation of a standing wave at a higher frequency can easily be demonstrated with a flute, since “overblowing” is sufficient to double the frequency for standing waves, without modifying the effective length for longitudinal oscillations. Oscillatory rotations of WPs at
Figure 19. Creation of trimmed chains of water pearls, resonating at the frequency f of the active molecules for chains of length L. This remains true for detached chains.
higher frequencies imply more rapid motions and thus greater kinetic energies and more violent local impacts. Available energies depend on the statistical distribution of kinetic energies of water molecules in the liquid state. Although a chain of WPs of given length allows for a superposition of different modes of oscillations, those of increasingly higher frequencies will thus be excited with decreasing amplitudes. They are byproducts of random re-excitation, but the lowest possible frequency f1 remains the predominant one.
3.4. Negation of Water Memory Was Based on False Assumptions
Benveniste’s experimental proof of water memory was categorically rejected because of prevailing beliefs. They resulted from four erroneous assumptions:
1) Biologically active molecule can have no effects any more, when all of them have been eliminated by successive dilutions.
2) Even if biologically active molecules could create substitutes, made of water molecules, they would have to be adaptable. Such aggregates are unknown and can thus not exist.
3) Biologically active molecules can only act on their specific receptors by means of chemical affinities. Local structuring of liquid water would be unable to mimic these processes. This is particularly implausible for various types of molecules, since that would require an extraordinary capacity of adaptable imitation.
4) Extra high dilutions are also used for homeopathy, which is inefficient. The preparation of EHDs does even involve shaking by vigorous “successions”. This ritual is a sign of charlatanism.
We have already shown that the two first objections are contradicted by the formation of trimmed chains of WPs. The third objection concerns the fundamental problem of molecular interactions. Modifications of the state of motion can result from direct contact (collisions), but also from actions at a distance (due to attractive or repulsive forces). We are accustomed to the idea that structural changes (combinations or dissociations) at molecular level result from chemical reactions, requiring direct contact, chemical affinities and configurational conformity. However, internal modifications can also result from energy transfer (excitation or disexcitation) by means of force fields.
Figure 20 summarizes the required restructuring of our ideas, because of the concept of ferroelectric water pearls. First of all, we have to realize that biologically active molecules contain an electrically charged part (Figure 12) that has a resonance frequency f. It creates thus an electric field that oscillates at this frequency. The upper line of Figure 20 represents the normal process, where this electric field acts on a molecular receptor, which has also a charged part that can oscillate. Specific receptors of a particular type of active molecules contain also a charged part that can oscillate. It resonates at a frequency fr. It is thus sufficient that fr ≈ f to allow active molecules to stimulate their specific receptors. There is some tolerance, since the probability of interaction by resonance corresponds to a peak that has some width.
Figure 20. Water memory reveals a new type of molecular interactions.
The assumption that molecular interactions are only possible according to the “key and slot model” of chemical reactions is not correct. Biologically active molecules can also interact with their specific receptors by means of oscillating electric fields and resonances. This allow for a bypass, represented by the second line of Figure 20, since the oscillating electric field of active molecules can also create trimmed chains of WPs. As long as they are attached to their generators, they resonate at a frequency f1 ≈ f. This remains true when these chains are detached, but reactivation of their oscillations by thermal agitation in liquid water leads to a harmonic spectrum of possible frequencies (fs = sf1, where ). The fundamental frequency f1 remains dominant, however. Standing waves on trimmed chains of WPs do produce an oscillating electric field of frequency f1 ≈ fr and can thus stimulate the same receptors.
The collective electric field, generated by all trimmed chains of WPs, has even the capacity to create more and more equally trimmed chains. Their number is increased and the oscillating electric field is amplified by an autocatalytic process. The possibility that molecular interactions can result from oscillating electric fields and resonance effects had been overlooked. The discovery of water memory did thus reveal the existence of a mechanism that is of fundamental importance and even very efficient.
The fourth erroneous assumption concerns homeopathy. Since the underlying mechanism was not understood, it was believed that its efficiency can only result from placebo effects. We wonder how they can be justified for animals and small children. Our purpose is not to defend homeopathy, but to restore truth, also in this regard. It is therefore instructive to examine the argumentation advanced by those who would like to eliminate homeopathy. The Australian National Health and Medical Research Council published in 2015 a study on “Evidence on the effectiveness of homeopathy” . This report was sponsored by the Australian Government, but was not based on a scientific study of underlying physical and physiological process. It was merely a collection of 176 articles, supporting the claim that “there are no health conditions for which there is reliable evidence that homeopathy is effective.”
These evaluations are essentially dependent on subjective appreciations. It was recognized that the general conclusion of their report was “based on all the evidence considered”. Other evidence was discarded. The first report of 2012 had even been concealed , since it was not sufficiently selective in this regard. The published report stated that the aim of this study was “to assist people in making health care choices”, but it was addressed to political deciders. The authors mentioned, indeed, that they wanted to “influence policy” and to get “increased funding for such research” ( , pp. 6, 16 and 4). This report was widely publicized and is equivalent to lobbying.
3.5. The Standard Procedure for Extra-High Dilutions
It is even necessary to clarify the origin of homeopathy, which has often been misrepresented to denigrate it. The basic idea was due to Samuel Hahnemann (1755-1843). He was a regular medical doctor. After acquiring his diploma at the age of 24, he practiced during 5 years, but decided then to cease. He had realized, indeed, that it would have been better for some of his patients not to be treated according to the “art of healing” of his time. He was even horrified that he might “murder” suffering people, instead of helping them. This was an exceptionally honest attitude, justified by recognizing the cause of this horrible situation. Neither the chosen substances, nor the doses were determined in a rational way, although Paracelsus wrote already in 1543 that “only the right dose differentiates a poison from a remedy”.
Hahnemann’s linguistic gifts made it preferable for him to translate books and to search there for possible improvements of medical practice. In one of these books, it was claimed that the bark of a Peruvian tree was able to treat malaria. It is known today that the bark of “cinchona” trees contains quinine. Most efficient medicines were actually discovered by trial and error. It was already known in Antiquity, for instance, that leaves of willow trees can stop pain. A chemist discovered in 1853 that the active molecule is C9H8O4, which became famous as aspirin. Even elephants, apes and other animals know how to cure or avoid ailments . The textbook attributed the beneficial effects of cinchona powder to its taste, but Dr. Hahnemann could not believe this claim. Nevertheless, he decided to verify if there were any detectable effects and was amazed that it produced malaria-like symptoms.
By experimenting with other substances, he realized that medicines could be discovered in a more rational way, by adopting the “law of similars”. This was merely an empirical rule, but such rules were often followed before understanding why they are valid. [Even Newton’s law of gravity was expressed in terms of actions at a distance. It did account for observed phenomena, but the real cause is a gravitational field, which corresponds even to modifications of the metric of space and time.] Since Hahnemann tried to discover medications by means of tests, performed on healthy persons, he had to use the lowest possible doses. He adopted thus the method of successive dilutions. If the result was beneficial, such an EHD could also be administered to patients in a secure way. We recall that Hahnemann was a learned medical doctor and was thus able to verify if a preparation is helpful of not.
As an example, we mention Apis mellifica. Until recently, it was customary in medicine to use Latin, also for anatomy, to overcome language barriers. The European honey bee is called “Apis mellifera” and the main component of its venom is mellitine. This molecule has also anti-inflammatory properties and honey bees do even protect their larvae from infections by means of very efficient substances. Hahnemann presented his discovery already in 1796 in a German medical journal, by formulating the rule that “like cures like”. Objections that are based on the finite divisibility matter are anachronistic. [The ancient concept of atoms had been reintroduced by Boyle in 1661 and elaborated by the chemist John Dalton in 1804, but the atomic theory was only accepted at about 1860, since the kinetic theory of gases did prove that Avogadro’s hypothesis was correct. Nevertheless, Mendeleev did not yet dare to use the concept of atoms in 1869.]
Hahnemann could thus assume that even when a substance has been diluted many times, there remains something of this substance. In 1810, he presented a first collection of results and one year before his death, the 6th edition of his “Organon of the Rational Art of Healing” was ready for publication. It is easily available . It should also be obvious that successive dilutions required always homogenization before the next step. Hahnemann did this by holding the vessel in his hand and stroking it with vigor on a semi-elastic surface, like leather. Modern chemists use mechanical vortexing to insure homogenization of mixtures. This is merely a simpler method. We will show in the following section that these “successions” do also have another effect. Although Hahnemann was only concerned with practical medicine, he discovered already the bypass of Figure 20. Neither he nor his detractors were aware of this fact. Even when Benveniste did empirically prove the biological efficiency of EHDs, it was categorically declared to be impossible.
3.6. The Quasi-Periodic Variations of Biological Efficiency
Sir John Maddox, long-term editor of Nature (1966-73 and 1980-95) accused Benveniste of self-delusion, although his article contained two figures, displaying results of measurements . Figure 21 reproduces one of them. It is undistorted
Figure 21. Measured quasi-periodic variations of the biological efficiency of EHDs, published by Benveniste et al. .
and was extracted from a publication in Japanese. We see 9 peaks. The first one is higher than the following ones. Since every peak did result from several measurements, the investigators should have realized that the quasi-periodic variations cannot result from “disregard of statistical principles” and “sampling errors”. The investigators proclaimed even that Benveniste’s experimental results were merely due to self-delusion . What would result from the theory of water pearls? We begin with a description of the underlying processes in usual language by means of Figure 22. The first frame (a) shows some of the initially diluted active molecules, their charged parts (in red) and attached trimmed chains of WPs. We know that their length L = λ/2 and depends on the frequency f of the oscillating electric field of the active molecules. The second frame (b) illustrates the situation immediately after the first dilution and vigorous agitation. The concentration of active molecules has been reduced and agitation lasted long enough to detach all chains from the remaining ones, but some chains of WPs were broken.
The third frame (c) shows that after a relatively short time interval, the remaining active molecules had again formed trimmed chains of WPs. Broken detached chains did grow and new ones were generated by the global oscillating electric field. These chains have the same characteristic length L, allowing for standing waves at the frequency f1 as well as harmonics ( ). Some chains may have reached the length 2L. It allows for a mode of oscillation where 2L = λ, which is equivalent to L = λ/2 and allows for the frequency f1. There did also appear some “associated chains”, resulting from mutual attraction of trimmed chains. Figure 23 shows how two parallel chains of WPs will be attached to one another, but more than two chains are also possible. However, associated chains do not resonate at the same frequency or not at all.
Figure 22. Decoding the standard procedure for preparing EHDs. (a) Active molecules and attached trimmed chains of WPs; (b) Detached and broken chains, immediately after dilution and agitation; (c) Reconstituted chains; (d) Excess of associated chains.
Figure 23. A simple associated chain of water pearls.
The fourth frame (d) of Figure 22 represents a state where nearly all single chains of WPs got bound to one another. The biological efficiency of the resulting EHD is then much reduced, but associated chains can be broken by vigorous agitation. Single chains can then be multiplied again during successive dilutions, until there are so much of them that association will be favored. The global result is that the biological efficiency of EHDs has to vary in a quasi-periodic way. It is not perfectly periodic, since these processes allow for statistical fluctuations.
3.7. Kinetics of Water Memory
It is useful to express these ideas by means of equations, since they allow for logical deductions. Let Xo be the initial number of active molecules, dissolved in a given volume of twice distilled water. This concentration is reduced by successive dilutions, where the same fraction of the homogenized solution is eliminated at every step. Usually, this fraction is 9/10 or 99/100. It is replaced by pure water to get always the same volume. When successive dilutions follow one another at identical short time intervals Δt, the concentration of active molecules becomes a function X(t) that decreases step-wise, since
The value of αΔt = 0.9 or 0.99. For smalltime intervals, X(t) can be treated as if it were a continuous function. It decreases then according to the equation:
so that (11)
The exponential decrease does necessarily end up with X(t) = 0 when , but this does not prove that the biological efficiency of EHDs has to vanish. Active molecules are able to generate trimmed chains of WPs with a probability g per unit time and they do generate more of them with a probability β par unit time. The concentration Y(t) of trimmed chains of WPs increases thus according the equation:
When the sequence of EHDs starts without previously formed trimmed chains, the initial value Y(0) = 0. Because of (11) and (12), we get then
It appears that Y(t) = At when t → 0. The initial increase of Y(t) is thus linear and very rapid when A is great. When the generation of new substitutes exceeds losses (β > 0), the concentration Y(t) does eventually increase like exp(βt). This constantly accelerated increase would only stop when the whole amount of liquid water has been solidified. This might even apply to oceans and would be catastrophic, but is prevented by forming associated chains. Their concentration Z(t) varies also, but
Equation (15) accounts for the fact that Z(t) increases by combining already existing associated chains with single ones. The average rate ε is greater than for association of two single chains, because of more possibilities. (12) is replaced by (14), since associations imply that Y(t) decreases by the same amount. However, every associated chain has also a probability γo per unit time to be destroyed by vigorous agitation. Figure 24 presents the results of numerical integration of (14) and (15). The unit of time is τo = 1/α = 1. We assumed that A = 50 and that β = 0.5, ε = 0.1, ε' = 0, γo = 1.5. The measured efficiency is proportional to Y(t). This function is thus represented by a thicker line.
The red line describes the exponential decrease of the concentration X(t) of active molecules and the thin blue line represents the variations of the concentration Z(t) of associated chains. It was a “hidden variable” for Benveniste and his team. The first peak of Y(t) is greater than the other ones, since X(t) does still contribute to the generation of substitutes. The initial increase is linear and very rapid when A = gXo is great. This accounts for Figure 21. The following peaks were not always identical, but they had nearly equal heights. They are perfectly periodic in Figure 24, since we assumed constant parameters in (14) and (15). We assumed that Y(0) = 0 and Z(0) = 1, since ε' = 0 requires that one of the initial values Y(0) or Z(0) is different from zero to initiate the periodic variations. It should be noted that the peaks for Y(t) are dissymmetric. The increase is progressive, but the decrease is precipitated, since single chains are more frequently associated with already existing ones when their concentration Z(t) is great. This was not obvious in Figure 21, but has been demonstrated by Montagnier (Section 4.1).
Figure 24. Decrease of the concentration X(t) of active molecules and the periodic variations of the concentrations Y(t) and Z(t) of single and associated chains of WPs in the course of successive dilutions.
It is simply a matter of fact that if Maddox had tried to understand the experimental results, he would have discovered that similar variations were already known since 1910 for autocatalytic reactions and so-called “chemical clocks”. Periodic variations attracted even more attention in the 1920th, since they were also observed for variations of the population density of predators and their prey. Predators proliferate when pray is abundant, but when the population of victims has been decimated, the predators have greater difficulties for their own survival and reproduction. Fewer predators allow the population of potential victims to grow again. This phenomenon was described by the famous Lotka-Volterra equations, which are identical to (14) and (15), when A = 0 and ε' = 0.
After developing the present theory, we found the book of Francis Beauvais . He provided many examples of similar results obtained by Benveniste and his team and referred even to the Lotka-Volterra equations. Without the concept of WPs, he had to assume the formation of negative and positive molds. This hypothesis was only mentioned as an example, but proves that Maddox could have referred to the Lotka-Voltera equations, if he had tried to understand the unexpected experimental results.
Thomas Kuhn analyzed the process of scientific revolutions . He distinguished different stages. When “anomalies” appear and cannot be explained by means of “normal science”, the first reaction is to declare that they are impossible. Confirmation of the reality of these anomalies opens a period of “crisis”. Various types of approaches are then tried and momentarily permitted. [For instance, Michelson’s experiments of 1881 and 1887 did prove that the velocity of light with respect to the Earth is identical for any direction. This was not compatible with the concept of light waves propagating in ether with respect to the moving Earth. It was proposed at first that the ether could be entrained by the Earth, but Einstein realized that no luminipherous ether is needed. This hypothesis can be dropped, indeed, when we accept that results of space and time measurements depend on the chosen reference frame. Only the velocity c of light in vacuum has always the same value for any inertial frame.] This “paradigm shift” resulted thus from reexamining previous assumptions and correcting them. This is also necessary for water memory, because of the third assumption in Section 3.4.
3.8. Properties of the Limit Cycle and Aging
Consequences of (14) and (15) can also be expressed by displaying the variations of Y(t) versus those of Z(t). Any particular point (Y, Z) defines then the state of the system at some instant t and the evolution of this state is represented by a continuous line. The dark curve in Figure 25 is equivalent to Figure 24. The green curve results from identical parameters, but starts at (0, 10), instead of (0, 1). Since the presence of more associated chains does immediately reduce the concentration of single chains, the first peak of Figure 24 is reduced. However,
Figure 25. Parametric representation of the evolution of EHDs.
maximal and minimal values of Y(t) are always reached when Z = β/ε. Those of Z(t) require that Y = γo/ε. These facts result from (14) and (15) when , and ε' = 0.
Figure 26 shows what would happen for ε' = 0.0001, when all other parameters have the same values as for Figure 24. We assumed again that A = 50, but the first peak of Y(t) is greater, since Z(t) was immediately increased by ε'Y2. However, the amplitude of the periodic variations of Y(t) is decreasing. The article of Benveniste  contained a second figure for a longer series of EHDs. It suggests a possible decrease of Y(t), but ε' was surely smaller than 0.0001.
Equations (14) and (15) allow us also to answer two important questions concerning aging. Should the sequence of successive dilutions be stopped at an instant where the biological efficiency has a high value? Does the efficiency of homeopathic preparations not totally vanish after some time? To answer these questions, we solve Equations (14) and (15) after the instant t = 0, where the process of EHDs was stopped. We assume that all active molecules were already eliminated (A = 0), but without shaking, the values of β and γo are smaller. Figure 27 shows the results of numerical integrations when β is reduced from 0.5 to 0.1 and γo from 1.5 to 1, while ε and ε' are not modified. The black curve results from Y(0) = 30 and Z(0) = 1, while the red one would be due to Y(0) = 5 and Z(0) = 10. We see that the initial conditions are irrelevant for the final result, although we assumed that ε' = 0.0001.
It follows indeed from (14) and (15) that Y(t) → γo/ε = 10 and Z(t) → β/ε = 1 for ε' = 0, when and . The correction for small values of ε' can be obtained by introducing the lowest order approximation in the same equations. The reduction of the final values is negligible when ε' = 0.0001. It is remarkable that the alternative dominance of single and associated chains continues during the initial period of aging. We have also to stress the fact that preservation of the biological efficiency of EHDs requires that the system has not been perturbed. This can happen by heating and ultrasound, but also in another way.
Figure 26. Possible variations of Y(t) for many successive dilations.
Figure 27. Effects of aging after stopping serial dilutions.
3.9. Crosstalk and Beat Phenomena
Official tests, performed in 1993, led to an unexpected fiasco (Section 5.2). It appeared, indeed, that the biological efficiency of EHDs was lost, while samples of pure water, needed for blind tests, turned out to be efficient. Benveniste was confronted to authorities, who concluded that the results did merely confirm their conviction that water memory is not a real and reproducible phenomenon. Since Benveniste knew that this was not true, because of numerous tests, he tried to understand the new observed facts. He realized that tubes with genuine EHDs of active molecules had been placed during some time near tubes that contained merely pure water. Benveniste thought therefore that the invisible structures, which are responsible for water memory, have to emit “signals”, allowing them to transfer their biological efficiency to pure water. It was not clear why this information transfer was possible and why this could result in silencing authentic EHDs. However, Benveniste succeeded in proving, by means of purely empirical means, that the assumed signals do really exist.
He could detect them by merely putting a sample of some EHDs in a coil. This did yield an electric signal that could be amplified and applied to pure water. This involves physical processes that will be explained later on, but we know already that biologically active molecules stimulate their specific receptors by means of an oscillating electric field and resonance effects. Trimmed chains of WPs allow for standing waves and create an oscillating electric field that has the same effect inside EHDs. Normal interactions between active molecules and their receptors can also be bypassed (Figure 20).
We noted that resonance effects allow for some tolerance. It is thus sufficient that f ≈ f1 ≈ fr, where f1 is determined by the length L of trimmed chains (f1 = v/2L). However, it follows from Lo’s measurements that L = ND, where N ≈ 20. The fundamental frequency f1 for standing waves will thus not always be precisely equal to the frequency f that is characteristic of the chosen type of active molecules. It can happen, for instance, that the chosen type of active molecules did initially create attached trimmed chains where N = 20 or N = 21. Autogeneration of equally trimmed chains of WPs during successive dilutions is governed by a collective electric field that oscillates then at one of these frequencies. EHDs are thus able to “breed” identically trimmed chains. However, the resulting “strains” can be slightly different for two EHDs of the same substance. This is irrelevant when f ≈ f1 ≈ fr. However, two EHDs with trimmed chains of nearly equal length produce electric fields of slightly different frequencies ( , where ). When these EHDs are contained in vessels that are put side by side, these fields are superposed. The resulting electric field does then oscillate at the average frequency fo with a modulated amplitude. Indeed,
This beat phenomenon is well-known in acoustics. Even when the proximity of two slightly different EHDs was only temporary, their already strong electric fields generate together both types of trimmed chains. When this happened only during a short time, they will continue to produce both types of chains and therefore “mixed signals”. Their biological efficiency will vary in a periodic way, but at a very slow pace. The relative phase of the superposed fields is also important, since the sum of two signals of equal amplitude will double their amplitude of oscillation. For the difference, the signals annihilate one another, but become strong again from time to time.
Momentary proximity of a genuine EHD with pure water can create there a small number of trimmed chains that resonate at one of the two possible frequencies. The sample of pure water becomes biologically active by “breeding” always the same strain of trimmed chains of Water pearls. Although Benveniste ignored the underlying mechanism, he had discovered that information transfer is possible. It proved the existence of signals and that they were responsible for water memory. Nevertheless, these signals had also extremely disconcerting effects. Beauvais used the term of “coherent discordances” to designate “wild transfers” and the fact that one operator did even “erase” the imprint  (p. 429, 450 and 453). These facts were enigmatic, but reproducible. For us, these parasitic effects do corroborate the theory of WPs.
3.10. Confirmations by Physico-Chemical Measurements
Vittorio Elia and his collaborators performed remarkable experiments with the “long term goal” of clarifying the problem of water memory . They began in 1999 with measuring the heat of mixing when EHDs of active molecules (like As4S4) were dissolved in twice distillated water. It appeared that addition of EHDs of NaOH resulted in energy release. It was concluded that EHDs contain aggregates of water molecules that are able to create new ones. Elia insisted on the fact that these structures have the capacity of “self-organization”. In terms of WPs, we can say that the added ions do also create trimmed chains of WPs. They are different, but bonds do always correspond to negative energy states. Mixing had thus to liberate more energy in the form of measurable heat.
The same team measured also the electric conductivity χ at 25˚C. Dissociation of NaOH produced ions that contributed to the measured electric conductivity. Ions liberated from the walls of the vessels did that also, but addition of EHDs of active molecules produced a significant excess conductivity χE. Since it was always proportional to the measured heat of mixing, both phenomena had a common cause . The added active molecules increased the concentration of trimmed chains and thus also the electric conductivity inside molecular chains by the von Grotthuss mechanism.
Elia and his team made two other remarkable discoveries by measuring the excess conductivity χE for homeopathic dilutions of Arnica Montana during aging. The active substance is helenalin, (C15H18O4, containing charged oxygen atoms). Its EHDs displayed wave-like variations of χE, but at an extremely slow pace . In 4.4 years, there appeared only 4 peaks, which had nearly the same form, but their amplitude increased  . We can now understand this fact in terms of beat phenomena. It appeared also that the excess conductivity χE was much greater for smaller volumes of EHDs. Initial conditions are usually irrelevant for aging (Figure 27), but Y(t) → γo/ε. This value will thus be increased when the probability ε for creating associated chains is reduced by surface effects.
4. More Evidence of Ferroelectric Water Pearls
4.1. The Measured Frequency Spectrum
Benveniste discovered that EHDs of biologically active molecules produce “signals” that can be detected by means of a coil. The output was an electric tension that could easily be amplified and stored in analogical or digital form. The waveform was similar to that of noise, resulting from a superposition of simpler signals, randomly shifted with respect to one another. The predicted spectrum is a harmonic one:
This results from (9), which is an approximation of (10). Montagnier did publically show the results of Fourier analysis, nicely displayed on a computer screen . The frequency spectrum did correspond to equidistant bell-shaped peaks. Moreover, their heights varied in the course of successive dilutions. Montagnier’s patent US2010323391 contains the spectrum obtained by Fast Fourier Analysis . It yields clearly separated spectral lines for shorter extracts. For a particular type of active molecules, the reported values were
The three lowest values do precisely correspond to (16). Since higher frequencies than f1 result from molecular agitation in liquid water, they are excited with decreasing intensities. This implies greeter uncertainties. Important results of normal Fourier analysis were presented in an article , even for several substances and different stages of dilutions. They are especially interesting, like hieroglyphs, when we understand their meaning. For the first dilutions, there was only one great peak at the fundamental frequency f1. We can attribute it to trimmed chains that are still attached to active molecules. Further dilutions led to the appearance of additional peaks, according to (16). Their average height increased during successive dilutions, but was maximal for dilutions D-9 to D-12. It did strongly decrease for D-13.
These facts agree with Figure 24 and confirm the predicted dissymmetry of peaks of activity. For the purpose of these authors, it was not necessary to continue the sequence of dilutions. They wanted only to get EHDs of maximal efficiency. It could be reliably predicted by observing the height of the peaks at higher frequencies. It corresponds, indeed, to the first stage of successive dilutions where Y(t) and the biological efficiency are maximal. It may be astonishing that the peaks for some harmonic frequencies fs were depressed, but this due to interference effects, also for musical instruments .
4.2. Water “Balls” and Their Alignments
S.Y. Lo measured the diameter (D = 15 nm) of WPs. The second group of structures had variable sizes of about 300 nm. This yields the length L of chains of WPs. Montagnier and his collaborators tried to measure the size of the required information carriers of water memory by means of filters. This method suggested a size between 20 and 100 nm . No one knew that the carriers of water memory are single chains of WPs and can thus be broken by filtering. If L = ND were always shorter than 100 nm, these chains would contain at most N = 7 WPs. The resolution for distinguishing different types of active molecules would then be too small to be efficient. By the way, the measured values of L = λ/2 ≈ 300 nm and f1 = v/λ ≈ 1 kHz, allow us to determine the velocity v = 2L/f1 ≈ 0.6 nm/s for any chain of WPs.
Lo found also by means of interference measurements a third group of supramolecular structures. Their size was much more variable, but of the order of 3000 nm . The X-ray spectrum confirmed that they were only constituted of water molecules, even without silicon contaminations from the vessels. These structures could be created by solutions of different types of initiators, but required high concentrations of chains of WPs. Excitation by UV light produced fluorescence at 298 nm, which is absent for pure water and ice. These “water balls” survived after short heating at 80˚C, but were decomposed by vigorous agitation. They were then spontaneously reassembled after about 15 min. Figure 28 reproduces, with kind permission of World Scientific Publishing Company, two pictures that were published the same year .
The left image proves that these balls tend to be aligned and the right image that they can be deformed. We propose therefore that water balls are constituted of chains of WPs, loosely bound to one another with global quasi-ferroelectric ordering. Water balls are thus dipolar, but contain also water molecules that can be expelled. They were said to be examples of “soft matter”. Lo discovered also that when these balls are very numerous, they constitute extremely long alignments, visible by optical microscopy . Spectacular images of these alignments were published in 2009 and reproduced in a book . It gathers all articles of Lo and his collaborators concerning the discovery and empirical analysis of structured water. These images are also contained in the open-access article of Mae-Wan Ho .
It is very remarkable that these alignments are branching-off sideways, always at the same angle of 78˚. Figure 29 shows that this astonishing fact can be explained. Since water balls contain chains of WPs, they have to be attached to one another at points, where all effective dipoles are oriented toward the center of neighboring water molecules. These dipoles are represented by red arrows, but parallel molecular chains are shifted with respect to one another by the distance a.
Figure 28. Transmission electron-microscopies, realized by Lo . They show deformable “balls”, constituted of chains of water pearls.
Figure 29. Chains of water molecules allow for strong bonds when they meet one another at the angle ψ = 78˚.
Since the centers of adjacent water molecules are separated by the distance d, molecular chains meet one another at an angle ψ, which appears in Figure 29 at two different places, indicated by dots. It follows that cosψ = a/d = 0.6/2.9. Thus, ψ = 78˚. This result does strongly confirm the validity of the proposed theory.
4.3. Proposed Explanation of Water Bridges
What would happen if liquid water were subjected to very strong electric fields? This question has been raised long ago by the British lawyer Willian George Armstrong, who became a respected engineer, inventor and scientist. He had a powerful source of high electrostatic potential differences and used it to find out if they produce very intense electric currents in liquid water. He knew that these currents would result from motions of H+ and OH− ions, but he discovered surprising facts. He presented them in 1893 to a general audience as being entertaining, but mentioned that they might “be interesting to experts” . Actually, he filled two wine glasses up to the rim with pure water and put them in contact with one another. He immersed a long cotton thread in one of them and introduced one wetted end of it in the other glass. He was amazed to see that a high potential difference between water in the adjacent glasses did transfer the whole cotton thread to the other glass. Why did this happen?
The spongy cotton thread had to be entrained by motions of charges. Since OH− ions have a greater mass than H+ ions, the thread should move towards the positive pole. This is what Armstrong observed, but the water level remained constant in both glasses. This was confirmed when Armstrong used a vessel where a thin tube allowed to see more precisely any variation of this level. Hementioned that during a few seconds after the complete transfer of the cotton thread, a “rope of water” remained suspended between the two glasses. When Elmar Fuchs was studying physics in Austria, he heard about this phenomenon and reproduced it with a source that could sustain high currents at 15 kV, for instance.
The first results were published in 2007. He found that the cotton thread was not necessary to produce a “floating bridge” of liquid water . It remained stable as long as the high electric field was applied. This phenomenon did raise a fundamental problem  : how is it possible that a strong electric field modifies the interaction between water molecules in the liquid state so much that it becomes able to resist gravity? The generation of the water bridge was itself a complex process. At first, the water surface became agitated in both vessels. Then, a very thin filament of liquid water established contact, but it lasted only a fraction of a second. Its diameter was then suddenly increased to yield a cylinder of circular section, indicating strong surface tension. It remained suspended in air like a catenary, even when its length was slowly increased up to 2.5 cm at 25 kV. Fuchs started then a systematic study of this strange phenomenon with various collaborators, to discover the cause of this “new state” of liquid water.
Since water bridges provided direct visual evidence, it would have been difficult to negate the reality of these observations, but the basic problem was the same as for water memory. What could be observed? In regard to electric conductivity, it was confirmed that the transport of charges is bidirectional . Protons (H+) and proton-holes (OH−) are passing through the bridge. By measuring the complex impedance between 100 Hz and 10 MHz, it appeared even that the charges were moving like nearly-free conduction electrons in metals . Raman scattering indicated also that vibrational modes in liquid water were not modified with respect to usual ones in bulk water . Charge transport was due to the von Grotthuss mechanism, but we have to solve two problems. 1) Why is cohesion of water molecules in the liquid state enormously increased by applying a very intense electric field? 2) How can many H+ and OH− ions move in opposite direction, without hindering one another?
The first question has to be related to the higher density of liquid water near the surface of the bridge. This was proven by X-ray scattering for two-dimensional beams of submillimeter extension . Since neutron scattering is sensible to hydrogen atoms, it yields complementary results . The stability of water bridges has been attributed to higher surface tension , but that is not sufficient, since longitudinal stretching of the “liquid cable” requires greater bond-strength along this direction. We propose that the stability of water bridges is due to the formation of very long chains of WPs.
We can then understand the sequence of events. At first the water surface became agitated in both beakers, since growing chains were formed near the electrodes and then moving around, until contact was established between water in both vessels. At first, there were only few chains, constituting a capillary bridge, but once the way was open, more and more chains of WPs were rapidly formed by the very intense electric field. Statistical fluctuations of the traffic of protons and proton-holes along the thin bundle of chains of WPs led to repulsion. It became more efficient to push parallel chains towards the surface of the liquid bridge. The capillary bridge was, indeed, replaced by a thicker one to achieve a new equilibrium. Since WPs are constituted of more densely packed water molecules than in liquid water, chains of WPs, situated at the surface of water bridges should there produce a greater density. This explains also the observed birefringence for linearly polarized light. Moreover, higher potential differences and greater electric currents require more chains of WPs at the surface of the liquid bridge. This should increase the diameter of the bridge and does agree with observations .
An increase of the applied potential difference allows also for a greater length of the catenary. This fact is due to the polarization of water molecules, as shown in Figure 30. The left part represents a molecular chain like that of Figure 10 for a moderate electric field, while the right part indicates that a very intense electric field E does strongly polarize the water molecules. The usual angle of 106˚ between OH bonds is reduced and the distance between proton pairs and the negative cores of oxygen atoms is also reduced by deformations of the usual, nearly
Figure 30. Aligned water molecules get more tightly bound by a strong electric field E.
spherical shape. Very strong polarization of water molecules would thus increase the mutual attraction of neighboring water molecules. This is also true when the bonds are due to intramolecular exchange effects (Figure 6). Electric conduction by the von Grotthuss mechanism would thus be facilitated, but regulation of opposite traffic of protons and proton holes is imperative for very intense electric currents passing though the water bridge.
Some observations indicated that this might happen in a helicoidally coordinated way . This makes sense, when we adopt the model of Figure 31. Groups of positive and negative charges are there represented by red and blue dots. They are distributed in a plane, to constitute a lattice like that of ionic crystals, but this plane rotates around the axis, while the charges are advancing along different lanes, represented by red and blue lines. [This double helix is similar to Da Vinci’s famous stair case of the chateau of Chambord in France. It contains two flights, allowing for simultaneous upward and downward traffic.]
4.4. Lack of Evidence for Coherence Domains
Emilio Del Giudice et al. proposed in 1988 a bold hypothesis concerning properties of bulk liquid water at very small scales . Basically, they assumed that inside some water molecules that belong to small spherical domains, electrons can remain in an exited state. Since the ionization energy of water molecules is 12.6 eV, they considered excitation to an energy level at 12.06 eV. To keep the electrons in this weakly bound state, they assumed constant emission and reabsorption of virtual photons, trapped inside the small spherical volume. This required that electrons should be accumulated at the surface of the sphere, while positive charges had to be expelled and would be homogeneously distributed in the surrounding liquid water. The accumulated electrons should constitute a barrier that reflects virtual photons. It was also necessary to postulate that the ensemble of excited electrons inside such a sphere are oscillating in phase with the EM field of all virtual photons. Because of the selected excitation energy, the wavelength would be close to 100 nm and determine the size of these coherence domains (CDs). It was estimated that they contain about 1000 water molecules, since only a fraction of the internal electrons would be excited.
Although this theory was initially conceived for bulk water , it was assumed later-on that it should even account for exclusion zones, water bridges and water memory. Because of the prestige of quantum field theory and the need of a rational justification of the mysterious phenomenon of water memory, the theory of CDs had increasing impact. However, we have also to mention what happened in 1993. It had already been decided that Benveniste’s Unit 200 would
Figure 31. Opposite motions of bunched charges along a double helix.
be closed, but he found a more objective method for detecting the biological efficiency of EHDs. He was thus allowed to prove the reality of water memory by means of blind experimentation, subjected to rigorous control of Georges Charpak. He was the 1992 Nobel Prize laureate in physics, since he invented a new type of particle detectors for CERN. Benveniste tried to convince him, by referring to the article of Del Giudice et al. .
Since this article concerned condensed matter physics, Charpak asked the opinion of Pierre-Gilles de Gennes, also Nobel Prize winner in physics. He answered that this theory is “worth nothing”, since it is based on “false hypotheses” ( , pp. 154, 168). Beauvais mentions that de Gennes had asked Philip Nozières ( , pp. 395, 398 and 402). He was the top theoretical physicist in France for new properties of condensed matter. Since such a harsh judgment might have been influenced by controversies, we have to know why the concept of CDs has been proposed. The theoretical physicist Giuliano Preparata (1942-2000) made important contributions to elementary particle physics, before becoming interested in applying quantum electrodynamics (QED) to condensed matter physics . He viewed the existence of CDs as being a general property of pure liquid water in the sense of a “two fluid picture”. He stated even that “the concept of H-bonds is merely phenomenological”, since liquid water should rather be viewed as a “condensed vapor” than “molten ice” ( , pp. 38, 196, 197). Liquid water would locally be condensed, before becoming ice.
Preparata was coauthor of the official article on CDs , presenting the same idea in Physical Review Letters. The primary author, Emilio Del Giudice (1940-2014), was one of the pioneers of string theories and explained why he got interested in CDs . Since his youth, he did feel that the whole universe constitutes a profoundly integrated entity and studying physics, he developed a sense of wonder in regard to quantum mechanics (QM). He got especially interested in “entanglement”. Since this concept provided his motivation for considering CDs, we have to clarify its origin and physical meaning.
Albert Einstein had contributed to the development of quantum mechanics, but in 1927 he learned about new ideas and discussed them with Niels Bohr at the Solvay conference in Brussels. He perceived very keenly that this theory attributed peculiar properties to measurements. He developed this idea in the famous EPR article of 1935, by means of a thought experiment. We describe it in equivalent terms, by considering two particles that have a property that can be precisely measured, but allows only for two possible values: +1 and −1. Since QM accounts for limited knowledge, we can define a state where it is only known that the values ±1 are equally probable. It is then sufficient to determine by means of a new measurement that the value is +1 for one particle, to be instantly sure that it is −1 for the other particle. It does not matter how far these particles are separated from one another at that instant.
In classical physics, that would require the existence of a physical link. Einstein asked therefore: are such “spooky actions” at a distance physically possible or not? He did only raise the problem, while Schrödinger had developed a theory where causal relations were preserved for the propagation of wave functions in space and time. He insisted on the need of causal relations and did not like the idea of “quantum jumps”. They do not allow for further analysis. Since Einstein had described in a vivid way that some measurements seem to imply universal connections, he coined the word of “entanglement” to account for them. They would require that instantaneous information transfer is possible at any distance, although this is excluded by the theory of relativity. Actually, it is sufficient to accept that QM is a theory of possible knowledge, limited by universal restrictions that Nature imposes on some measurements.
Classical physics postulated that totally precise, simultaneous knowledge of positions and velocities is possible. It implied continuity and strict causality, but this assumption has to be corrected. The basic paradigm of QM is that the constant h imposes irreducible uncertainties. The concept of “virtual photons” results from the fact that during short time intervals Δt, the energy of a system can only be known with a minimal uncertainty (ΔE ≈ h/Δt). Since the theory of relativity imposes that the energy (mc2) of particles of given mass is finite, the number of identical particles cannot be precisely known during short time intervals. This applies also to photons and leads to some observable effects.
Del Giudice was convinced that virtual photons are even able to establish a link between physical systems when they are separated by arbitrarily great distances. He expressed this idea  by stating that telepathic communications are possible, but he knew that physical theories are only justified by their agreement with observations. He tried thus to show that the concept of CDs accounts for unexplained phenomena, like water memory, exclusion zones and the stability of floating water bridges. This idea was diligently propagated, but does it really account for water memory?
Bellavite and other health specialists at Italian universities reviewed in 2013 the status of research concerning EHDs . They concluded that “evidence strongly supports the notion that the structuring of water and its solutes plays a fundamental role”, but the mechanism is not yet understood. Two hypotheses had been proposed: hydrogen bonded clathrates and coherence domains. After examining the evidence, they concluded that “none is fully convincing”. The concept of ferroelectric water pearls was not yet known. It explains essential properties of EHDs, like the observed quasi-periodic variations of their biological efficiency, the measured frequency spectrum of signals created by local structures and even the angle of 78˚ between alignments of water balls. In his regard, we have to mention that Mae-Wan Ho tried to explain alignments of water balls , by assuming that expelled protons are not homogeneously distributed in liquid water. They could be more concentrated between the charged CDs, but that is not sufficient to explain the observed angle of 78˚ for chains of water balls.
Benveniste was sure that water memory is real, because of the often verified quasi-periodicity of the biological efficiency of EHDs and detection of signals. It was thus obvious that a physical explanation had to be possible. Since CDs seemed to offer an explanation, Montagnier accepted the help of scientists who advocated this approach. They stated even that water memory involves the “gauge theory paradigm of quantum fields” and “the framework of spontaneously broken gauge symmetry theories” . These words are familiar to those who know the jargon of quantum electrodynamics (QED). They are impressive, but are they necessary? We recall that Thomas Kuhn distinguished three periods in the process of scientific revolutions . At first, the observed anomaly is declared to be impossible, since it does not agree with conventional knowledge. When it is confirmed and cannot simply be denied anymore, it comes to a “crisis”, where any hypothetical explanation may be proposed.
The theoretical chemist Tamar Yinnon published a series of articles, where the concept of CDs was elaborated, by postulating the existence of various types of these structures. He presented them in 2015 as being “predictions of QED”, but we found only a catalogue of assumed structures . The original CDs were renamed CDelec. Their diameter would thus be about 100 nm. Greater structures, called CDplasma, were assumed to contain some molecules of the solvated substance, but also more water molecules. Their effective dipoles would be oriented by the electrically charged active molecules or ions inside these domains. The resulting hydration spheres were said to be subjected to (monopolar) plasma oscillations with overall coherence. The size of these domains would be of the order of 1000 nm = 1 μm. A third type of CDs should reach sizes of 10 - 100 μm. They were called CDrot and assumed to contain only water molecules, but all of them would have nearly parallel effective dipoles. They would be large elongated ferroelectric particles. Moreover, agglomerated CDplasma and CDrot entities constitute “supra-domains”.
4.5. Forced Oscillations of Water Molecules in EHDs
Instead of commenting these respectable attempts to find an explanation of water memory, we continue to collect experimental results and to test the validity of the concept of WPs. Adriana de Miranda measured, for instance, the dielectric response of water molecules that interact with supramolecular structures . She used 14 lots of LiCl, prepared by pharmaceutical laboratories and determined the spectral distribution of the real and imaginary parts of the impedance in the frequency range of 1 kHz - 13 MHz. She defined these variables by means of an equivalent circuit and compared the results to those of the measured dielectric response of EHDs of H2O molecules in pure water, prepared in the same way. To understand the underlying physical processes, we consider the response of the effective dipoles of water molecules to an oscillating electric field at various frequencies. The center of these molecules remains practically motionless, while the positive tip of the effective dipole is displaced by a small distance u(t) along the direction of the applied electric field. When n is the density of water molecules, the instantaneous polarization density is P(t) = n2qu(t), where the displacement u is subjected to the equation of motion
Indeed, 2q is the charge and 2m the effective mass of the pair of protons. When the applied electric field E(t) is suddenly extinguished at the instant t = 0, . The value of the relaxation time τ depends on all possible interactions in bulk water. For an electric field that oscillates with some constant amplitude at a given (angular) frequency ω, we get and the instantaneous polarization density is . The complex conjugate (c.c) requires merely that ω is replaced by −ω. It follows that
The function β1(ω) specifies the average orientation of effective dipoles. At low frequencies ( ), they are oriented along the direction of the applied electric field. The polarization drops quite suddenly when ωτ ≈ 1 and is reduced to zero when . The imposed rhythm is too fast to allow the molecules to follow, because of friction. The imaginary part β2(ω) varies like x/(1 + x2), when x = ωτ. This yields a peak that is centered on x = 1. The function β2(ω) describes the energy-loss of water molecules, because of friction. It is maximal when ω = 1/τ. Since increased friction leads to lower values τ, a higher frequency is then required to achieve great energy losses. In QM, higher frequencies correspond to higher energies and in QED even quasi-static electric forces are due to exchanges of virtual photons. Miranda found that the values of relaxation times were situated between 40 and 100 kHz for EHDs of LiCl and H2O molecules in pure water. This means for us that friction resulted from the creation of trimmed chains of WPs that had different lengths. However, the height of the peak for the energy-loss function varied in the course of successive dilutions of LiCl in such a way that it was maximal for D-9. Since friction is proportional to the concentration Y(t) of single trimmed chains of WPs, this agrees with our theoretical predictions (Figure 24).
4.6. Discovery of Highly Remarkable Resonances
Since we expected that stationary waves on trimmed chains of WPs can be excited by an electric field, we did search relevant data and found the results of the Indian electrical engineer Chitta Ranjan Mahata  . He had witnessed the healing power of homeopathy and wanted to understand how this is possible. Studying the literature , he discovered the proposition of Barnard . He declared already in 1965 that homeopathy is in need of a “surer foundation” than to refer to positive results. This should be possible, since homeopathy requires that EHDs create substitutes of the initially dissolved active molecules. Since they should merely be constituted of water molecules, Barnard thought that they might “join up to form long molecular chains”. This idea was probably suggested by the concept of polywater, which was in vogue at that time. Barnard proposed that these molecular chains may be wrapped around active molecules and that their length would thus provide the required specificity.
We were surprised that some important ideas had already been formulated more than 20 years before Benveniste’s discovery. Specificity was even related to the length of local structures, composed of water molecules. Barnard was aware of the hypothetical nature of this proposition, but insisted that it indicates “the kind of experimental research in physics and chemistry needed now to establish the truth of homeopathy.” Mahata realized that he could test this hypothesis, since ordered molecular groups in liquid water should lead to resonance effects.
Before describing and explaining his results, we have to mention that Barnard and Stevenson provided more details in another article . They reviewed what was known about properties of EHDs, prepared for therapeutic purposes. It had already been established that their capacity of healing is subjected to “rhythmic variations” in the course of successive dilutions. It was also mentioned that biologically active molecules contain electric charges and that Alphonse Gay had developed in the early 1950th a method for measuring the dielectric response of EHDs . It was complicated, but sufficient to identify EHDs of NaCl after 27 centesimal dilutions. [We thank Dr. van Wassenhoven for his help to get access to this paper.]
Mahata was not aware of these measurements and developed with his collaborators a much more efficient technique. In 2007, it was ready  and in 2010, they presented results for the spectral distribution of the real and imaginary parts of the dielectric constant  in a large frequency domain (100 kHz - 50 MHz). They found typical features of resonances for preparations of Cuprum met, but also for pure water. In 2013, they provided more results  and did interpret them in terms of Barnard’s model. There were resonances, but we noted four perplexing anomalies:
1) These resonance frequencies were always very high: about 25 MHz instead of the expected ones at about 1 kHz.
2) There was only one resonance frequency, without harmonics.
3) A resonance was even observed for pure water.
4) The energy-loss functions were not symmetric, as required for usual resonances.
This did not correspond to expectations for single chains of WPs, but we realized that the third anomaly could be explained, since pure water contains H+ and OH− ions. Their electric fields might be sufficient to produce at least isolated water pearls. The observed high value of the resonance frequency would then require a very strong restoring force. For single WPs, it could only be due to their surface charges. They are represented by red and blue rims in Figure 31. For low resonance frequencies (~1 kHz), it was licit to assume that the oriented water molecules remain oriented towards the surface of the WP. However, at high frequencies (~25 MHz), the solid substrate is moving too fast to allow them to be reoriented so rapidly. Miranda’s measurements indicated, indeed, that reorientation of water molecules occurs at lower frequencies. Because of the proximity and spatial distribution of the internal and external charges, this would then lead to a very strong restoring force.
We test the validity of this hypothesis, by comparing its logical consequences to Mahata’s experimental results. The component E(t) of the applied electric field, which is normal to the symmetry axis of the WP sets the positive and negative poles of the WP in forced oscillation. The equation of motion for small displacements uo(t) of these poles is
The (angular) resonance frequency Ω is determined by the strong restoring force and the effective inertial mass of the poles. Energy losses by friction are characterized by γ, while Cω is proportional to the amplitude of the electric field E(t). Thus, for constantly forced oscillations. The polarizability β(ω) of pure water that contains a given concentration of isolated WPs is proportional to B(ω). When we normalize β(ω) to get always the same static polarizability β(0) = 1, it follows that
We simplified the real part β1(ω) to insist on the fact that without friction, this function would diverge when ω = Ω. At low frequencie (ω ≤ Ω), the average orientation of an ensemble of isolated WPs is identical to that of the applied electric field. It is opposite when ω ≥ Ω, but WPs cannot follow the applied field when . The function β2(ω) describes energy losses. They are maximal when ω ≈ Ω. The peak is symmetric and we can set γ/2 = Γ. The height of the peak is then Ω/2Γ and its width at half height is equal to Γ. Figure 32 shows (in black) the frequency dependence of β1 and (in red) the spectral distribution of the energy-loss function β2 when the resonance frequency f = 25 MHz and Γ = 5 MHz.
These results had to be expected for a normal resonance, but Mahata’s experimental results for β2 did correspond to curves like the blue one. The height of the observed peak has been adjusted in Figure 33 to coincide with the height of the normal peak and β2exp → 0 when . Measured energy-loss functions are thus asymmetric. This feature did also appear for EHDs of various biologically active substances. Only the values of Ω and Γ, as well as the mysterious “dip” of the energy loss functions were slightly different. All these facts will be explained later on (in Section 5.4). Before we do that, we continue the equally necessary search of more evidence.
Figure 32. Rapid oscillatory rotations of a single water peal, excited by an electric field.
Figure 33. The spectral distributions of the real and the imaginary parts of the polarizability β, predicted for oscillators that resonate at 25 MHz. The blue curve represents typical results for Mahata’s resonances.
4.7. Complementary Experimental Evidence
The Swiss biochemist Louis Rey did prove in 2003 that EHDs of NaCl and LiCl in ultrapure water produce local structures that are preserved after freezing . He kept the EHDs during 24 h at −20˚C and reduced then the temperature to 77 K. When he exposed the transparent disk to a high dose of ionizing radiation, most excited electrons did immediately return to their normal state in ice, but some of them were trapped in local defects. To leave enough time for diffusion of electrons, the ice remained during one week at 77 K. After reaching equilibrium, the sample was warmed up at constant rate. The trapped electrons were thus progressively raised to states of higher energy, allowing for light emission. This thermoluminscence is a standard method for examining defects in insulators, since it determines the excitation energies that are required to liberate electrons from various traps.
There appeared a glow for T ≈ 120 K and a more intense one at T ≈ 166 K, especially when NaCl had been dissolved in heavy water (D2O). Since electron traps were different for NaCl and LiCl, at least some of them were due to Na+ and Li+ ions. Rey concluded that these results prove “without any ambiguity” that liquid water has been structured in a lasting way. Since the glows appeared only when serial dilutions were followed by vigorous shaking, Rey thought that it might produce nanobubbles, attracted by ions . The basic facts did not depend on this interpretation and were brilliantly confirmed by van Wijk et al. . We can conclude that EHDs did produce trimmed chains of WPs, where the molecules are more closely packed and so strongly bound to one another that the ferroelectric crystallites do even subsist in normal ice (Figure 4).
Demangeat et al. studied the effects of EHDs by means of nuclear magnetic resonance . Protons have indeed a magnetic moment that tends to be parallel or antiparallel to the direction of a constant magnetic field Bo. The energy difference is proportional to the magnitude of this field and the transition can be stimulated by EM radiation of adequate frequency. When the excitation ceases, the system returns to its ground state, but there are two different relaxation times, T1 and T2, for components of the magnetic moment of protons along the direction the magnetic field Bo and perpendicular ones. The ratio T1/T2 depends on their environment. For EHDs, the results of measurements did prove that “water is a self-organizing system” . Van Wassenhoven et al. presented more detailed results of NMR studies, confirming this conclusion  and providing additional information.
Other very important facts were discovered in Russia. They did prove that water molecules can also constitute stable structures without having to apply the standard procedure to get EHDs. Konovalov and Ryzhkina presented in 2014 a review , where they stated that the following conditions have to be satisfied: 1) only one type of molecules should be dissolved in pure liquid water at extremely low concentration, 2) these molecules have to carry an electric charge and 3) be subjected to low-frequency EM fields. This was verified for different types of molecules and the resulting structures were also observed by Atomic Force Microscopy (AFM). The authors insisted that these entities do not result from the formation of nanobubbles, but are associations of water molecules. This is essential to justify the concept of WPs, although the hypothesis of nanobubbles is closer to conventional ideas.
Burkin and his collaborators  performed light scattering measurements for very low concentrations of diluted molecules and interpreted their results in terms of bubble clusters around ions. This hypothesis leads to inconsistencies . The Indian chemical engineer Chikramane and his collaborators used homeopathic drugs, prepared by means of EHDs of various metals (Au, Ag, Cu, Sn, Zn and Pt). They did yield local structures of sizes that ranged from 214 to 325 nm. They could thus be trimmed chains of WPs of length L ≈ 300 nm, but these authors claimed that x-ray analysis demonstrated that they do contain particles of the initially dissolved metals. They assumed therefore that minute gas bubbles were formed during agitation and stabilized by attachment to small metal particles. Moreover, they thought that these entities rise to the surface and constitute there a thin layer that is reconstituted during subsequent dilutions. This would raise unanswered questions. NMR and other measurements of Van Wassenhoven for copper particles did not confirm this hypothesis . Maybe, some procedures for preparing EHDs allow that charged metal particles get attached to WPs, because of their surface charges.
Demangeat thought also that vigorous agitation could produce nanobubbles. He proposed even that they might account for water memory by creating a “stereospecific shell” around active molecules . How could these molds be detached, remain stable and allow for replication? Nevertheless, the concept of nanobubbles was attractive in the context of the “crisis”, which resulted from the unsolved puzzle of water memory. It required a specific experimental investigation. The Slovenians Sedlák and Rak combined light scattering with rapid centrifugation and established that biologically active molecules at very low concentration produce real material entities . Their measured sizes ranged from100 nm to 400 nm for various substances. It was explicitly noted that the mechanism of formation of these supramolecular structures and their nature were not yet known. However, the exclusion of nanobubbles and the reported sizes are compatible with trimmed chains of WPs.
Elia et al. produced supramolecular aggregates in pure water by repeated contact with a polymer, called Nafion . It is strongly hydrophilic, because of its electrically charged sulfonate group (-SO3H). It produced structures in liquid water, subsisting after freeze-drying. In solution, they had a high electric conductivity, attributed to proton hopping. They exhibited UV absorption at 270 nm and modified IR absorption, associated with the OH stretching mode of vibration. Solid residues after evaporation displayed clustered particles of about 40 to 400 nm. It appeared that these aggregates of water molecules produce circular dichroism . This means that plane polarized light, decomposed in left and right circular polarized light, does not lead to equal absorption of these components. Dichroism indicates the presence of left-handed and right-handed structures, which can be tripoles in ferroelectric crystallites.
5. New Types of Molecular Interactions
5.1. The Sense of Smell
The most important consequence of water memory and its elucidation is that molecules do not only interact with one another by direct contact. This discovery belongs to a trend that began with trying to understand the sense of vision. John Dalton had described in 1794 his color blindness. Actually, he was unable to distinguish green from red. The British physician and physicist Thomas Young was intrigued by this anomaly, which led him to raise again the basic question: what is light? Newton had discovered that light of different colors can be separated by refraction. This could be explained by assuming that light is composed of particles, moving at constant velocity in any homogeneous, transparent medium. Why is it constant and depends on the medium remained mysterious, but refraction would then simply result from acceleration or deceleration at the interface.
However, Young discovered in 1801 that when light passes through two very narrow and close holes or slits, there appear dark fringes. He explained these results in terms of superposed “undulations”, similar to those that can be observed on water surfaces. These interference effects led to the concept of light waves. Colors were then determined by their wavelength. They can be measured by means of gratings, but our eyes do not perform this kind of spectroscopy. Young realized that color vision requires only three types of receptors in our retina, mainly sensible to red, green and blue. Dalton had no green sensible receptors. These receptors had to absorb energy. [During about one century, it was assumed that light corresponds to waves. Actually, it is composed of photons, which are particles that carry energy and momentum, defined in terms of frequencies and wavelength. Photons behave even according to laws that are valid for waves. This synthesis transcends the idea of a dual nature of light.]
The English physiologist William Ogle tried to understand the sense of smell. Since anomalies could provide a clue, he collected and analyzed cases of “anosmia”, i.e. partial or total loss of the sense of smell. He concluded in 1870 that odors are not perceived by means of chemical processes. It requires receptors that detect waves . Ogle proposed that they should correspond to “undulations of the ether”. [The concept of EM waves emerged only in 1873, through the development of Maxwell’s theory. He gathered the laws of electricity and magnetisms, established for static or slowly varying electric and magnetic fields. He found that they are inconsistent when they are extrapolated to high frequencies, but harmony could be restored by adding a new physical law. It implied that electric and magnetic fields can constantly be transformed into one another, without needing the presence of electric charges. The theoretically predicted existence of electromagnetic waves was experimentally confirmed in 1887 by Heinrich Hertz.]
Because of QM, it became also clear why molecules can emit photons of infrared light and that this type of spectroscopy allows us to distinguish different types of molecules from one another. Malcolm Dyson proposed therefore in 1938 that the sense of smell is due to receptors that detect vibrational frequencies of molecules by energy absorption . How is that possible without complicated instruments? This problem has been solved in 1996 by the biophysicist Luca Turin. Being a specialist of perfumes, he wanted to understand how odorants can be perceived and differentiated. Since olfactory receptors of insects, animals and humans are very small, he was looking for very compact detectors. Thus, he explored the domain of solid state electronics. Eventually, he found that this can be achieved by means of inelastic electron tunneling . He told the story of this discovery in a very interesting book .
The Japanese Leo Esaki had invented in 1957 an efficient diode. It was based on properties of two n or p type semiconductors, separated by a very thin gap. It blocks the passage of electrons or electron holes, when the conduction band on one side meets a forbidden band on the other side. However, these bands can be shifted with respect to one another by applying a potential difference. Charge carriers can then pass through the intermediate potential barrier by wave-mechanical tunneling. The Norwegian Ivar Giaever applied this method to prove in 1960 that the BCS theory of low temperature superconductivity is correct. It had assumed that electrons can constitute bound pairs and predicted the existence of a forbidden band for possible energies. Giaever shared the Nobel Prize with Esaki for demonstrating that this is true.
Somewhat later, physicists realized that it is not necessary to apply a potential difference when the passage of electrons through a very thin gap between two semiconductors is not possible. It is sufficient that electrons of higher energy lose some kinetic energy inside the gap by collisions with other particles. Turin understood that this method allows us to distinguish molecules from one another, since this is equivalent to determining the energy required to excite vibrations inside these molecules. This is a simplified version of infrared spectroscopy, but requires specific receptors for different odors. That is a matter of genetics, as for color vision. Our color-space is usually three-dimensional, while the odor-space is multidimensional. Honeybees have 174 types of receptors and ants have even about 400 different ones . This fact led to a system of communications that determines the peculiar structure of these societies. There is no privacy. Information is locally shared.
5.2. The Nature of Signals Discovered by Benveniste
Dr. Benveniste discovered the existence of water memory, but this phenomenon could not yet be explained in 1988. Since his experimental results were attributed to error or fraud, he was constantly searching simpler and more objective methods to prove the reality of water memory. In 1990, he began to use the system of Langendorff ( , p. 123). It is based on an allergic reaction of isolated hearts of guinea pigs or rats and modifies the coronary flow by about 20% ( , pp. 337-343, 354). He used empirical methods, but had heard about the theory of Coherent Domains, where EM fields seemed to be involved. He tried thus to verify if magnetic fields of 50 Hz, easily produced by means of the European electric distribution system, might have some effect. About 100 experiments were performed in 1990 and 1991 with physicists of the Central Laboratory of Magnetism in France. It appeared that these fields could annihilate the biological efficiency for EHDs of histamine ( , pp. 126-128).
This fact suggested that EM signals might be involved in water memory, but how could that be proven? Benveniste spoke in 1992 to a friend, who was an electronics hobbyist and thought that if molecules are able to produce EM waves, it might be possible to detect them by means of a coil. He constructed a kit, used for amplifying telephone sounds. It turned out that this method was sufficient to detect signals, created by an EHD that was contained in a tube, simply placed inside a coil. The wire delivered an electric signal that could be amplified ( , pp. 128-132) by means of an electronic circuit that is valid for microphones, detecting audible sound waves. The detected signal varied thus at low frequencies (20 Hz - 20 kHz). When the amplified signal was applied to a second coil and when another tube, containing only pure water, was placed in this coil, it acquired the capacity to stimulate the same receptors than the original EHD. This fact did prove that EHDs produce “signals” that allow for information transfer. Beauvais called this a “molecular telephone” ( , p. 336).
It was not possible anymore to attribute the published results to errors or sloppy work. Benveniste tried thus to restore his credibility in 1993 by means of experiments, performed with the system of Langendorff and controlled by physicists in Georges Charpak’s laboratory in Paris. Charpak told Benveniste that if molecular communications were possible by means of ELF waves, that “would be the biggest discovery since Newton, if it were true”. However, he was convinced that some fakers in Benveniste’s laboratory did “arrange” the experimental results ( , pp. 155-157). Nevertheless, it was decided that two types of molecules (acetylcholine and ovalbumin) should be used for preparing EHDs. They would be blindly treated by means of coding, but the results were catastrophic. For 11 of 18 series of tests performed in 1994, the reaction of the isolated hearts was inversed. Authentic EHDs had no biological efficiency anymore, while pure water produced positive results.
Benveniste tried to understand this fact. He knew that the tubes had not been exchanged, since he did transport them himself, but they had been placed side by side. The capacity of genuine EHDs to provoke biological reactions could thus have been transferred to pure liquid water, even without needing intermediate detection and amplification. To test this hypothesis, he did shield all samples by means of thick aluminum foil, but such a Faraday cage was not sufficient to suppress the unexpected information transfer. We conclude that it was not merely due to oscillating electric fields, but for Charpak, the idea of information transfer from tube to tube was even more abstruse than water memory. Actually, he wrote in July 1995 to Benveniste that he advances “the most baroque reasons to explain the failures” ( , p. 166). This reaction was revolting for Benveniste. He was accustomed to attach primary importance to experimental evidence, but was now confronted with the conviction that “a phenomenon that cannot be explained with present-day knowledge cannot exist.” ( , p. 90).
Benveniste considered, on the contrary, that the new, objectively established facts demonstrated that information transfer is possible. It has to result from signals, which could be “intrinsic to molecular activity” ( , p. 164). How this might be achieved remained mysterious, but Benveniste continued to improve his experimental methods ( , pp. 336, 461, 519, 542, 584). His now very small laboratory acquired in 1995 a computer with a program for sound analysis. The electric signals, obtained by means of a coil could be digitalized before applying them to the second coil ( , p. 461). Pure water became active, when it was exposed to this signal, even when it was digitally transmitted between different continents. He described these results in an article, submitted for publication to the Journal of Immunology. The final answer was that “the protocol and experiments are perfect, but the article cannot be published without precisions concerning the exact physical nature of the signal” ( , p. 179).
It was only known that molecules can emit and absorb EM waves in the frequency domain of microwaves or infrared light. That molecules are able to emit signals of very low frequencies seemed to be impossible. Moreover, it was believed that biologically active molecules can only stimulate their receptors according to the model of chemical reactions. Since water memory transgressed this dogma, the well-known neurologist Changeux called it a “scientific heresy” ( , p. 194). Why did referees and “experts” not verify if the reported facts are true? Why did the physicist Charpak not help to identify the nature of the signals? Benveniste recognized that “classical biologists” are not competent in electro-physics, since they use “structural models” ( , pp. 180, 184). He mentioned also that he had only “adopted the hypothesis of an electromagnetic nature of the molecular signal” to organize empirical research ( , p. 211). He was scientifically isolated and Charpak told him even: “your experiments challenge elementary laws of physics and common sense” ( , p. 212). As far as we know, the nature of the relevant signals has never been identified. We summarize our explanation in Figure 34. The upper line corresponds to the description that we used so far. Electrically charged parts of active molecules do create an oscillating electric field.
We could compare the frequency f of the oscillating electric charge of active molecule to the message that has to be transmitted to potential receptors. This can be done in a direct way or by means of a bypass, as indicated in Figure 20. Trimmed chains are then the information carriers and could be compared to messengers. Anyway, the information has to be expressed or encoded in a particular way. For usual language, it could be encoded in oral or written form. For water memory, the vehicle can be an oscillating electric field, but this is not the only possibility. This results from the fact that oscillating electric charges are equivalent to electric currents. They create an oscillating magnetic field. [Initially, Oersted did use Volta’s pile to show that motions of electric charges inside a wire do heat this conductor, but he noted that the current caused a rotation of a nearby compass needle. This phenomenon was totally unexpected, since it was only known that magnetic materials are able to do that. Oersted described in 1820 the properties of the new type of “magnetic forces” in terms of actions at a distance, as Newton did for gravitational forces and Coulomb for electric forces.]
The concepts of electric and magnetic fields were introduced by Faraday. [He discovered in 1831 that imagined “lines of magnetic forces” passing through a closed loop of a conductive wire induce there an electric current, but only when the flux is varying. It can increase, decrease or oscillate. In 1845, he used the
Figure 34. Communications by means of oscillating electric and magnetic fields inside EHDs, with the aid of two coils (C and c).
more general concept of electric and magnetic fields, defined for any point in space and time by means of a fictitious experiment. These fields can then vary in space and time.] The second line of Figure 34 shows that oscillating electric fields can be replaced by oscillating magnetic fields. For Benveniste and Montagnier these fields were hidden in “black boxes”. They become now mentally transparent and even brilliant. Charpak could have explained what was in these black boxes, but did not even care, since they were irrelevant for him.
The second line of Figure 34 indicates also that when a tube with an EHD of a particular type of active molecules is placed inside a coil C, the oscillating magnetic field exerts a force on charged particles in the wire. This force produces a current when the circuit is closed or a tension between the extremities of an open circuit. [This fact results from Faraday’s discovery.] The resulting electric signal can be amplified and applied to a second coil c. It produces a current in the wire of this coil and thus again a magnetic field. It oscillates at the same frequency and exerts a force on electric charges. Benveniste compared the relevant signals to radio waves. This analogy was helpful, but they were merely electric and magnetic fields, oscillating at low frequencies. They require the presence of electric charges, while EM waves consist of electric and magnetic fields that are constantly transformed into one another and can even propagate in vacuum. They are characterized by the relation f = c/λ, where the velocity c ≈ 3 × 108 m/s.
5.3. Stimulation by Low-Frequency Magnetic Fields
After realizing that water memory is transferable to pure water, Benveniste developed “digital biology” with the engineer Didier Guillonnet, who joined the team in 1996 ( , p. 596). Their common patent  indicates that information transfer is even possible when pure water is simply flowing through the second coil (c). This means that the formation of new trimmed chains of WPs is a rapid process. It would even be sufficient to create one trimmed chain of WPs, since it will generate other equally trimmed chains of WPs. This fact is also important for beat phenomena, since equally trimmed chains of WPs are then reproduced inside the initially pure water. Benveniste had noted that shaking of the receiving tubes was not required, but Benveniste and Guillonnet found that “it is possible to improve the performance” of this system by adding a signal that corresponds to white noise. It covered the whole low frequency domain (1 Hz to 20 kHz). They mentioned that they were surprised when they discovered this fact. Konovalov and Ryzhkina noted also that the formation of supramolecular structures is favored by low-frequency EM fields .
Montagnier and his collaborators  placed the test tubes, which contain “informed” and “naive” water in different coils (C and c), but inside a greater coil (C’). It was subjected to an electric current, oscillating at 7 Hz. This frequency was chosen since it is close to “Schumann frequencies”. They correspond to genuine EM waves, generated by thunderstorms in the terrestrial atmosphere and reflected by the surface of the Earth and the ionosphere. Since these waves are propagating in a spherical layer, they correspond to standing waves and their wavelength λ is of the order of the circumference of the Earth. It follows that their frequency f = c/λ ≈ 300 (thousand km/s)/40 (thousand km) = 7.5 Hz. The measured fundamental frequency for Schuman resonances is close to 7.8 Hz. However, the coil C’ creates only an oscillating magnetic field and not an EM field, similar to radio waves of extra low frequencies.
The term of “electromagnetic fields” was probably used by Benveniste and Montagnier in the general sense of being related to electric and magnetic phenomena. To realize the difference, we note that standing waves on trimmed chains of WPs could be compared to standing waves for oscillating electrons in an antenna that radiates EM waves. Their wavelength λ would then be determined by the length of these chains (L = λ/2 ≈ 300 nm), but their frequency would be extremely high (f = c/λ ≈ 5 × 1014 Hz). The inverse process corresponds to an EM wave that excites standing waves for oscillating electrons in a receiving antenna. For 7 Hz, the wavelength λ = c/f ≈ 43,000 km.
Nevertheless, a coil C’ and low frequency currents were beneficial. To explain this fact, we recall that water molecules are bound to one another in bulk liquid water by hydrogen bonds (Figure 4(a)), but are constantly exchanged at the time scale of 10−12 s. The applied magnetic field is oscillating at a much lower frequency (7 Hz or 1 Hz - 20 kHz) and does reorient the effective dipoles of water molecules at this frequency. They will thus be liberated from their usual bonds. This facilitates the formation of water pearls and trimmed chains of WPs.
We recall also that “wild transfers” were not suppressed when the test tubes were shielded by means of aluminum foil. It did only suppress electric fields. That was not sufficient, since magnetic fields do also allow for information transfer. However, it is possible to eliminate magnetic fields by means of mu-metal. Benveniste had already discovered that it abolishes crosstalk and beat phenomena ( , p. 565) and Montagnier  did interpose a sheet of mu-metal between the primary coil C and the secondary coil c. Empirical discoveries do not require that their cause is already known.
5.4. Resonances for Pairs and Chains of Water Pearls
We come now back to the important and still unsolved problem of the asymmetry of the energy-loss functions for Mahata’s resonances (Figure 33). The model of single WPs has to be corrected, since WPs would be subjected to Brownian motion. They attract one another and will thus constitute pairs of WPs. Oscillatory rotations are then possible for both WPs, even at a frequency Ω ≈ 25 MHz), because of surrounding water molecules. However, mutual attraction of neighboring poles would yield an additional restoring force, which could have noticeable effects.
To acquire more physical insight, we consider two identical pendulums (Figure 35(a)). They would oscillate at the same frequency Ω, but can be coupled by means of a weak spring. [A normal string is sufficient, when it is somewhat longer than the separation of the masses at rest and when a small mass is attached to
Figure 35. (a) Low and high frequency modes for free oscillations of two coupled pendulums; (b) Resonances correspond to quantum mechanical transitions, allowing for excitation at the frequency Ω+ and emission at the frequency Ω.
the middle of the string.] Free motions of the coupled pendulums depend on the chosen initial conditions and can be quite complicated, but there are two special cases, shown in Figure 35(a). It is intuitively clear that both masses will oscillate at the frequency Ω for strictly parallel motions, since the length of the intermediate spring is not modified. It is higher for antiparallel motions (Ω+ > Ω). These coordinated motions are called normal modes of oscillation, since the system behaves like a single oscillator. It is possible to produce forced motions of both pendulums, by applying an oscillating force to one of them.
Figure 35(b) recalls that QM is a generalization of classical mechanics. The lowest possible energies for any single harmonic oscillator are then separated by intervals that are proportional to the resonance frequency. This can be Ω or Ω+. Classically described forced oscillations are then replaced by transitions. It should thus be possible to absorb EM energy at the frequency Ω+ and to restitute a part of it at the lower frequency Ω. This happens for fluorescence and Mahata’s energy-loss functions β2(ω) displayed always at the side of lower frequencies (Figure 33).
Figure 36 shows that we get also two normal modes for small oscillatory rotations of the coupled WPs. As for Figure 35(a), displacements of neighboring poles can be parallel or antiparallel. They are perpendicular to the symmetry axis and represented by red arrows. To account for possible resonances, we have to consider two coupled equations of motion:
They apply to the left part of Figure 36 and Ω ≈ 25 MHz is the resonance frequency for independent oscillations, while ωo ≈ 1 kHz accounts for interactions. They correspond to opposite forces. For the right part of Figure 36, it is sufficient to change the sign of u2. Both water pearls are subjected to viscous friction, but it is not identical, since the resulting motions of the surrounding liquid water can hinder or facilitate one another. To assume that u1 > u2 implies that γ1 < γ2. When we set γ1 = γ − η and γ2 = γ + η, the sum and the difference of (19) and (20) yield two equations for u± = u1 ± u2:
Figure 36. Two modes of forced motions for pairs of WPs.
The pair of WPs behaves thus as if there were two oscillators that resonate at the frequencies Ω or Ω+, where . However, only the mode is directly excited by the oscillating electric field of angular frequency ω. The mode is then excited by entrainment, but forced oscillations require that
Because of (21) and (22), the amplitudes are determined by the equations:
where and . Thus,
Forced oscillatory rotations of pairs of WPs involve both modes of oscillation. The polarizability is thus
It is normalized to get β(0) = 1. When Ω and ωo correspond to 25 MHz and 1 kHz, there are only two adjustable parameters: γ and η. They specify the average energy loss and energy transfer by viscous friction. The resulting values of a± and A± allow us to calculate the spectral distribution of the real and imaginary parts of β(ω). When γ = 20 and η = 0.01 (MHz), the calculated spectral distributions of β1(ω) and β2(ω) are shown in Figure 37. The energy-loss function presents a dip, like the experimentally observed one (Figure 33). The concept of water pearls accounts thus also for the asymmetry for pairs of WPs.
However, Mahata found similar curves for EHDs of various types of active molecules. This would not be compatible with the excitation of a standing wave, if all positive poles were displaced along the direction of the applied electric field E. However, the lowest possible excitation energy would only require that the rotation of one water pearl is inversed. Figure 38 shows an example. Even the excitation of standing waves on trimmed chains of WPs at high frequency and the asymmetric resonances can thus be explained by the proposed theory.
5.5. The Stabilizing Effect of Lactose
Homeopathic preparations are often presented in the form of pills of lactose. Opponents of this medical practice claim that lactose is merely used because of its sweet taste. Initially, Hahnemann used lactose for grinding hard substances to reduce their concentration before dissolving them in pure liquid water. Mahata
Figure 37. Predicted spectral distribution of the realpart (β1) and the imaginary part (β2) of the polarizability of pairs of coupled water pearls in pure water.
Figure 38. Resonance for trimmed chains of WPs.
knew that lactose is also used to insure better preservation of the biological efficiency of EHDs. He wanted thus to find out if the association of EHDs with lactose modifies the measurable resonance curves. When he dissolved lactose powder in pure water, he found a resonance at 50 MHz and the energy-loss function was symmetric . We have thus also to explain these facts. From the point of view of condensed matter physics, lactose is an insulator. It attracts thus charged particles by image forces. Since lactose has many pores, nearly all single WP would be attached to lactose, but the applied electric field would simultaneous set the WP and its image in oscillation. They are subjected to identical restoring forces. The resonance frequency will thus be doubled, but attachment of single WPs would keep the resonance symmetric, as for the red curve in Figure 33.
When Mahata added EHDs of Cu-Met-30 to the dissolved lactose powder, the resonance frequency was reduced to about 45 MHz, but the peak of the energy-loss function remained symmetric. Because of image forces, the trimmed chains of WPs would usually be parallel to the surface of the insulator. This will also lead to a high resonance frequency, but its value would be somewhat lower than 50 Hz. The essential result is, of course, that the protective role of lactose is due to adhesion by image forces.
5.6. A New Type of Molecular Interactions and Medical Applications
The empirical discovery of water memory and its elucidation modify the traditional paradigm that molecular interactions are only possible according to the “key and slot” model of chemical reactions. Even normal interactions between biologically active molecules and their specific receptors are due to oscillating electric fields and resonance effects (upper lines of Figure 20 and Figure 34). Dr. Hahnemann wanted only to select medications in a rational way and to increase safety of medical practice. Dr. Benveniste did prove that this is due to the phenomenon of water memory. He asked INSERM several times that “interdisciplinary teams” could come to help him to interpret what he observed. Scientific institutions should have provided intellectual and materiel assistance, instead of “isolating” him. Benveniste felt that these authorities did even “strangle” him ( , p. 89). The case of water memory has to be remembered as a monument of what should not happen, especially in science.
Moreover, it is not unusual that new phenomena are discovered without understanding their cause. We mentioned many examples that illustrate this fact. Christian Huyghens discovered in 1665 that pendulum clocks tend to be synchronized, but this phenomenon has only recently been explained . Collective oscillations are also very important in plasma physics and provide for instance the key for understanding the occasional appearance and peculiar properties of ball lightning . Oscillations of the charges of biologically active molecules and standing waves on trimmed chains of water pearls create electric and magnetic fields that tend also to be synchronized. This favors the autocatalytic generation of new trimmed chains and does more effectively stimulate the relevant receptors.
Since oscillating electric charges do also produce oscillating magnetic fields (Figure 34), we could try to pursue investigations by means of modern magnetometers. Squids are used for magneto-encephalography, but a new, less expensive and very efficient technology is now available in the form spintronics. This word is an abbreviation of spin-electronics. Instead of exerting forces on the electric charge of electrons, it is indeed possible to exert also forces on their spin. This technology has medical applications . New magnetic detectors of this type have a very high sensitivity in a large frequency domain (in the picotesla region for frequencies below 100 Hz). Their small active areas allow for unprecedented spatial resolution (down to tens of nanometers). They seem thus to be ideal tools for trying to study interactions between active molecules and their receptors. They could open new avenues for future research in biology, medicine and pharmacology, if it were possible to detect low resonance frequencies of biologically active molecules and their specific detectors.
This has already been achieved by means of EHDs and there are observations that indicate the usefulness of such measurements. The collaboration of Mahata with the medical doctor Chattopadhyay, an Indian specialist of homeopathy, led indeed to are markable discovery  . They found that EHDs of the blood serum of patients did also produce resonances at about 25 MHz. They were similar to those of EHDs of biologically active molecules. This proves that they have charged parts that can also oscillate and create trimmed chains of WPs. Moreover, Chattopadhyay found that for patients, having ailments that can be treated by means of homeopathic medicines, preparations that led to a closer match of resonance curves were more efficient. It was concluded that it should be possible to “make a selection of medicines based on scientifically measurable parameters.” The reported results require independent confirmations, but it would not be prudent to discard this possibility without further objective studies.
The basic claim of Montagnier’s patent US2010323391  was that hidden infections in humans and animals can be detected by a non-invasive method. It requires only to collect blood samples and to prepare EHDs of them. This allowed for spectral analysis of signals that were detected by means of coils. Another patent added that “any human or animal fluid, e.g. blood, urine, various secretions” can be used for this purpose. Even the unexplained resurgence of some sicknesses might result from water memory, since Montagnier   found that DNA molecules do also produce similar signals.
Another potentially important question concerns Hahnemann’s empirical rule that “like cures like”. Is it possible to prove the existence of a link between receptors and the sickness they can cure? Present-day knowledge and already acquired experience in the domain of receptors and neurology could be used, for instance, to verify if some sicknesses are related to particular receptors, by exiting them. Research is motivated by curiosity, whatever may be found.
The objective of this study was to find out if water memory is possible or not. We treated this problem in terms of condensed matter physics, but it illustrates a much more general and fundamental difficulty: the recurrent conflict between facts and ideology. In science, it is not unusual that empirical research uncovers phenomena without understanding the underlying mechanism. It does even happen that the framework of existing theories cannot explain them. Past experience and commonsense tell us that previous assumptions may have to be corrected in such a situation. Kuhn has shown  that the discovery of an anomaly with respect to “normal theory” does often lead to initial rejection. This was also true for water memory. It is thus necessary to be aware of the fact that scientific progress can be drastically slowed down by so-called “skeptics”, who are not skeptic enough in regard to their own preconceptions and beliefs.
It is true that “extraordinary claims require extraordinary evidence,” but this slogan does only displace the basic problem. What is valid evidence? In science, it can merely be recognized by referring to reality. This is well-known, but unfortunately, there is a strong tendency to rely only previously acquired ideas and theories, although they could have been based on hypotheses that were only valid in a limited domain. They may have to be replaced by more general ones. The case of water memory illustrates such a need in a rather exemplary way.
Physics is also confronted today with a similar problem. It results indeed from observations that there are only certain types of elementary particles and that our Universe contains an enormous amount of Dark Matter. The accelerated expansion of space is caused by Dark Energy, but we are unable to explain all these facts. It is thus necessary to ask if present-day theories do not contain an assumption that was simple and useful, but merely an approximation. In this regard, we learned even from the development of the theory of relativity and quantum mechanics that Nature can impose restrictions on our measurements. It appeared that they are related to the existence of two universal constants (c and h). Nevertheless, we continue to believe that space and time are continuous. This is equivalent to postulating that there is no finite limit for the smallest measurable distance. How do we know? We can only say that until now, we did not yet meet such a limit. However, we can try to find out what would happen if there did exist a universally constant quantum of length (a) and thus also a universally constant quantum of time (ca).
The value of a is surely very small, but we cannot assert that a = 0. If this value were finite, all physical laws for particles and fields that involve variations in space and time would not be expressed anymore by differential equations, but by finite-difference equations. [The differential wave Equation (7) was also an approximation of the more general finite-difference Equation (6), but for another reason.] When we did that for any type of particle and force fields, it turned out that the generalization would not lead to logical inconsistencies when a ≠ 0. However, some ideas have to be changed. The highest possible energy, which has to be attributed to the whole Universe, would be finite. The behavior of fields at the smallest possible scale in space and time would be described by hitherto unknown quantum numbers. They account for all possible types of elementary particles, in agreement with already known facts . This applies also to Dark Matter. It is known to be present in our Universe, but is composed of particles that have not yet been identified by means of accelerators. In cosmology, it is also possible to account for the accelerated expansion of space and Dark Energy . This enlargement of the foundations of physics widens the domain of possibilities, but the concept of a “space-time continuum” is so deeply rooted in our minds and our culture, that it will take time to be able to modify this idea.
This is also true for the conviction that biologically active molecules can only interact with their specific receptors according to the model of chemical reactions. The possible existence of water memory seemed to be absurd, but the real problem was merely that it could not yet be explained. Of course, ferroelectric crystallites of water molecules and trimmed chains of water pearls are merely concepts. These entities are not directly observable, but they allow us to make verifiable predictions. An increasing part of science follows this pattern. Theory and experiment are complementary. On one hand, we have to imagine what might be possible, to draw logical consequences from the proposed hypotheses. On the other hand, we can establish what it real or not. Sometimes the observations precede their explanation and sometimes, they can be used to test the validity of hypothesis and theories.
In regard to water memory, we found that the concept of trimmed chains of WPs accounts for the quasi-periodic variations of the biological efficiency of EHDs (Sections 3.4 and 3.5), the measurable frequency spectrum (Section 4.1) and the peculiar angles for junctions of large-scale structures (Section 4.2). It is at least probable that very long chains of WPs explain the stability of liquid water bridges (Section 4.3). The von Grotthuss mechanism can be understood in terms of intramolecular exchange effects for delocalized protons (Section 2.5). They are also relevant for 2D polymerization of water molecules (Section 2.6). The perplexing effects of cross talk can be attributed to beat phenomena (Section 3.7). Physicochemical and other types of measurements make sense (Sections 3.8 and 4.5). It is also possible to understand the physical nature of detected signals (Sections 5.2 and 5.3). Even Mahata’s unexpected high frequency resonances can be explained (Sections 4.6, 5.4 and 5.5).
Of course, there are still open questions, inviting to pursue research (Section 5.6). We wonder for instance if low-frequency resonances can be detected for single chains of WPs. Preliminary results for dielectric responses of EHDs indicated that a resonance could be excited at about 3 kHz . Miranda did perhaps not detect them , since that would have required greater initial concentrations of LiCl or various types of biologically active molecules. The now highly developed methods of X-ray and neutron scattering could also be applied to EHDs of initially very great concentrations of ions or active molecules. We mention that Mahata obtained images of elongated entities that constituted a super lattice. These structures were observed by scanning probe microscopy . Can variations of the distance between oxygen ions near the surface of water bridges subjected to different high tensions be measured diffraction of X-rays? Can oscillating magnetic fields be detected at molecular levels by means of modern magnetometers? Could water memory have contributed to the emergence of life on Earth?
Water is a very familiar substance, but still a fascinating domain of research. Moreover, it concerns not only experimental and theoretical results, but also truth and justice.
Appendix: The Size of Water Pearls
The basic ideas were presented in Section 2.8. Since , where the prime indicates derivation with respect to r, the potential in the external medium (r ≥ R) is
is the potential in the internal medium (r ≤ R). P is the total dipole moment of the WP and Ei the electric field inside this nanoparticle. In the external medium it is
The radius R is then determined by the boundary conditions:
The first relation results from Gauss’s law, since the electric fields point away from the interface, which carries a surface charge density σ(θ) = σocosθ. The second relation accounts for the dipole density w(θ). It results from the inner and external surface charge densities σicosθ and −σecosθ. They are separated by a distance , but σe = ϰσi and σo = σi − σe, while w(θ) = σeΔRcosθ is negligible. Moreover, εi = 0. The boundary conditions yield then the following relations:
We know that P = Nq2a, where N = (4πR3/3)/d3 and σi = 2q/d2. Thus, σo/εeP = (1 − ϰ)(3d/4πεeaR3) and
We ignore the value of ϰ = σe/σi, but it results from the last equation that η is nearly constant. Indeed, η = 9.5 or 10.5 when ϰ = 0.5 or 0.8. Even when ϰ = 0.2, we get η = 9.1 and η = 11.5, when ϰ = 0.9. We conclude that R ≈ 10λo.