_{1}

^{*}

Biologically active molecules create substitutes in liquid water by forming
*single-domain ferroelectric crystallites*. These nanoparticles are spherical and constitute growing chains. The dipoles are aligned, but can be set in oscillation at the frequency of vibration of the charged part of active molecules. They are then automatically trimmed and become information carriers. Moreover, they produce an oscillating electric field, causing autocatalytic multiplication of identical chains in the course of successive dilutions. Active molecules are thus only required to initiate this process. Normally, they excite their specific receptors by resonance, but trimmed chains have the same effect. This theory is confirmed by many measurements.

The concept of water memory is based on experimental results of measurements, published 30 years ago in the prestigious scientific journal Nature [

When they arrived at Benveniste’s laboratory, it turned out that John Maddox was accompanied by the professional magician James Randi and the debunker of scientific fraud Walter Steward. The objective was thus to detect errors or fraud. During their stay in Paris, the first experiments confirmed the published result, but not later ones. The inquisitors did immediately publish a devastating conclusion: the reported results are not reproducible and merely due to imagination [

The load of the accusation fell on Jacques Benveniste (1935-2004). He was a medical doctor, who had practiced several years before opting for research. In California, he discovered the platelet-activating factor and determined its role in immunology [

This phenomenon was totally unexpected, since successive dilutions of biologically active molecules in pure water do necessarily lead to their complete elimination. Was it really possible to create substitutes that are only constituted of water molecules? Further experimentation did prove that these structures should even be able to mimic active molecules of different types. No one knew how this might be achieved. Benveniste verified therefore if these hypothetical structures could be destroyed. It appeared that after heating extra-high dilutions (EHDs) of histamine during one hour at 70˚C, they had completely lost their biological efficiency. Exposition to ultrasound had the same effect ( [^{th} decimal dilution. Then it dropped and continued to vary in a quasi-periodic way during successive dilutions.

Since the reality of this phenomenon was tested many times for different substances, Benveniste thought that these facts had to be published, although they were unexplained. He insisted on the observed quasi-periodic variations, by providing two figures [

In their report, the group of inquisitors insisted on the variability of the peaks of activity and claimed that measurements had been treated in “disregard of statistical principles”. They declared even that “the laboratory has fostered and then cherished a delusion about the interpretation of this data”. Melinda Baldwin, lecturer on History of Science at Harvard University, identified the actual cause of this grave incident. Maddox considered that scientific journals are shaping science by controlling its quality [

Benveniste considered that his duty was only to establish the reality of these “unbelievable and fear-provoking” facts, since their meaning and the underlying mechanism could be studied later on ( [

Alain Kaufmann presented in 1994 basic facts and an analysis of the sociological context [

Francis Beauvais, a former collaborator of Benveniste, provided in 2007 much more details on events at that time [

Visceral opposition to the concept of water memory was often motivated by fear that it could justify homeopathy. The aim of the present study is merely to find out if water memory is real or not. This has to be viewed as a basic problem for condensed matter physics, since bonds between water molecules are constantly broken by thermal agitation in liquid water at the time scale of picoseconds. Martin Chaplin, specialist of properties of water molecules, proposed therefore in 2007 that water memory could result from creating statistically stable clusters of water molecules [

The structure of this article results from the itinerary that we followed. In Section 2, we examine the internal structure of water molecules and their possible interactions. This leads to the concept of “water pearls”. In Section 3, we explain why biologically active molecules can create chains of these nano-pearls and why they account for water memory. Section 4 presents more observational evidence concerning these chains. It is diverse, detailed and very remarkable, but the concept of water pearls accounts for known facts, while the alternative concept of Coherent Domains does not. In Section 5, we insist on the most important consequence of water memory: molecular interactions are not only possible by means of the “key and slot” model of chemical reactions. Intermolecular communications can also result from oscillating electric fields and resonance effects. It will appear once more that “Nature is written in Lingua Mathematica”, as Galilei stated already, but we endeavor to be understandable by non-specialists.

Martin Chaplin provides detailed information about the internal structure of water molecules [_{2}O), but the protons of both light atoms are deeply embedded in the common electron cloud of the oxygen and hydrogen atoms. Water molecules are thus practically spherical, but at close range, the protons are surrounded by a spherically symmetric excess density of electrons. They are thus equivalent to point-like charges q ≈ e/3. The core of the oxygen atom and the remaining part of the electron cloud are equivalent to a central point-like charge −2q.

The kinetic diameter (for collisions) of water molecules in the terrestrial atmosphere is 0.265 nm. H_{2}O molecules are thus smaller than O_{2}, N_{2}, CO_{2} and H_{2} [^{H}O^{H} is 104.5˚, but it is close to 106˚ in the liquid state. Vibrational and rotational spectra of water molecules disclosed that the length of OH bonds is δ ≈ 0.095 nm. Thus, δ/d ≈ 1/3. Since water molecules behave in the liquid state like hard spheres that can easily roll on one another, we adopt the model of

Because of their internal point-like charges, water molecules are tripoles, but it is customary to replace them by dipoles. They are constituted by the central charge −2q and a single charge +2q, situated in the middle between the charges +q. This dipole is represented in

The test charge +1 does then “see” the charges along parallel lines, but their distances are slightly different. They are indicated by thin red lines. Adopting also natural units (4πe^{2}/δε_{o} = 1) for electrostatic potentials, their sum is

V ( r , θ ) = − 2 q r + δ o + q r − δ + + q r + δ − = q r 2 ( 2 δ o + δ + − δ − ) + q r 3 ( − 2 δ o 2 + δ + 2 + δ − 2 ) + ⋯

The first and second order approximations result from ( 1 + x ) − 1 = 1 − x + x 2 when x ≪ 1 . Since δ o + δ + = δ cos ( φ − θ ) and δ o − δ − = δ cos ( φ + θ ) , the value of 2 δ o + δ − − δ + = 2 δ cos φ cos θ , where δcosφ = a. Thus,

V ( r , θ ) ≈ p r 2 cos θ , E r ≈ 2 p r 3 cos θ , E θ ≈ − p r 3 sin θ (1)

The radial and angular components of the electric field E at the observation point result from partial derivatives of V(r, θ). At closer range, there are inevitable corrections. In liquid water, it is also necessary to account for intermediate water molecules, since they are easily reoriented by an applied electric field. The potential V(r, θ) is then reduced by the factor 1/ε_{r}, where the relative dielectric constant ε_{r} ≈ 80. When neighboring water molecules are subjected to an electric field, their effective dipoles will be aligned. These molecular chains are broken by thermal agitation when the electric field is extinguished, but they could also be stabilized by association, like sticks in a bundle.

This possibility merits attention, since it is known that electric field lines can be visualized by means of neutral particles, like grains of semolina or short plastic filaments dispersed on oil. The applied electric field does merely polarize these particles, but the induced dipoles tend then to align one another. Could similar chains be formed by means of water molecules? To answer this question, we have to examine all possible types of interactions between water molecules.

Even electrically neutral molecules attract one another in gases, because of Coulomb forces and quantum-mechanical effects. Indeed, if such a particle were subjected to an electric field, it would displace all weakly bound electrons inside this particle. This produces surface charges that create a secondary electric field inside the particle. It opposes displacements of the electrons and would restore neutrality when the applied field is switched off. For oscillating electric fields, this force leads to a resonance for the ensemble of oscillating electrons. That explains the appearance of colors and peculiar optical properties of thin granular metal films. They were said to be “anomalous” until they could be explained in terms of collective oscillations of nearly free electrons [

Nearly free electrons will be set in coupled oscillations inside neighboring particles. Their resonance frequency is then reduced, but in quantum mechanics (QM), the lowest possible energy of an oscillator is proportional to its resonance frequency. The (zero-point) energy of two neutral particles is thus reduced when they come close enough to one another. This effect can be interpreted as resulting from an attractive force. The existence of this short-range force was discovered by Van der Waals in 1873, since a dense gas does not behave like an ideal one. It corresponds to a model, where velocities are only randomized by collisions of point-like particles, but neutral particles attract already one another at some small distance. The physical origin of this force could only be understood after the development of QM.

Since Van der Waals forces are proportional to the volume of the polarizable particles, they are negligible with respect to other forces for water molecules in the liquid state. However, small metal particles that are suspended in liquid water contain nearly free electrons. They are very polarizable and big enough to attract one another by Van der Waals forces. In liquid water, small metal particles attract thus one another and constitute chains. These “necklaces of pearls” are observable by optical microscopy [

The concept of so-called “hydrogen bonds” was already introduced in 1920, since some quantum effects could be treated in a semi-classical way [_{4}, NH_{3}, OH_{2} and FH would thus result from the “tendency to complete… the octet of electrons”. This lowers the total energy, but H_{2}O molecules are special. The left part of

It was therefore proposed that the negative charge density of the free pairs “might be able to exert sufficient force” on two neighboring oxygen atoms. This would account for mutual attraction of water molecules that allows for structuring of liquid and frozen water. The right part of

“hydrogen bonds” by means of red lines, when neighboring water molecules are assumed to be situated in the same plane. QM revealed that electrons behave according to laws that apply to waves. Every oxygen atom contains two strongly bound electrons and four external electrons in (2s^{1}2p^{3}) states. Superposition of these wave functions leads to interference effects and the charge distribution of the 4 external electrons acquires then tetragonal symmetry. We might thus think that hydrogen bonds are merely due to stronger electrostatic attraction, but modern biochemistry states that “in a hydrogen bond, a hydrogen atom is shared by two other atoms” [

^{2} = 3(2h)^{2}. It follows that μ = 54.74˚.

It is important to be aware of the quantum-mechanical nature of exchange effects. They are also possible between two X and Y atoms, when an intermediate proton could belong to X or Y. Both possibilities are expressed by the notation XH… Y or X… HY. QM accounts indeed for limited knowledge. The probability distribution for possible positions of electrons is defined by means of their wave functions. Exchange effects are then due to “tunneling” through an intermediate

potential barrier. This is relevant for bonds between atoms inside molecules and in particular for H 2 + , where one electron allows for H-H^{+} or H^{+}-H. Although protons have a greater mass than electrons, they are also subjected to quantum-mechanical laws. The probability distribution for being at different places in space is then not smeared out, but reduced to needle-like (delta) functions. When a proton has two possible positions, it can be said to be delocalized, but possible exchanges are then not due to tunneling. They result from the fact that the energy of any physical system cannot be precisely determined during short time intervals Δt. There is always an irreducible uncertainty ΔE ≈ h/Δt. In semi-classical terms, the proton is able to “jump” over the intermediate potential barrier, when this happens rapidly enough.

To analyze possible effects of protons for interacting water molecules, we begin with the simplest case. _{1} that results from electrostatic interactions between two point-like particles of charge −2 and one charge +1. However, the concept of hydrogen bonds means that the intermediate proton has two possible positions. They are represented by gray dots in the upper right part of _{2}. The third configuration would be obtained if it were possible to account for hydrogen bonds in a semi-classical way, by assuming that the proton has only one well-defined position, situated in the middle. This position is represented by a black dot and the potential energy would then be U_{3}.

Using natural units for charges, distances and energies, we get

U 1 = 4 d − 2 δ − 2 d − δ = − 1.67 = U 2 and U 3 = 4 d − 2 d = − 0.69

Since U_{2} = U_{1}, a strictly classical description is sufficient, although the proton is delocalized. This would even be true (in the present case) if the uncertainty did allow for any partition q' and (1 − q') of the charge +1. The third configuration is very unstable and does not account for quantum-mechanical exchange effects. We are now ready to calculate the total potential energies V_{1} to V_{4} for different configurations of dimers, represented in _{2}O)_{2} is close to the upper left one of _{1} for the indicated configuration.

V_{2} corresponds to aligned effective dipoles. This configuration would be preferred if water molecules did only contain dipoles, but could even be privileged for effective dipoles, when the dimer is subjected to an electric field. We want thus to see if V_{2} is already close to V_{1} in the absence of an applied electric field. The tripoles should then be orthogonal to one another to minimize repulsion between protons in neighboring molecules. V_{3} is the potential energy for any pair of molecules in a nearly linear chain, where all intermediate protons are ideally situated between two oxygen atoms. The resulting zigzagging configuration is planar and in conformity with a classical representation.

An applied electric field could even allow for a perfectly linear chain, because of intramolecular exchange effects, although this was unknown. For clarity, we consider here two coplanar tripoles. One proton is always situated as close as possible to the core of the neighboring oxygen atom. The other proton has two equally probable positions, above and below the symmetry axis. This would yield the energy V_{4} for any pair of water molecules. Alternatively orthogonal tripoles would reduce repulsion between the delocalized protons. The resulting potential energy is then V_{5} < V_{4}.

To facilitate this type of calculations, we note that the total potential energies result always from adding the Coulomb potentials (V = qq'/Δ) for pairs of point-like charges q and q', separated by a distance Δ. We define thus a function S(x, y, z), where x, y and z are differences of Cartesian coordinates, respectively measured towards the right, rear and top. Thus,

V = q q ′ S ( x , y , z ) where S ( x , y , z ) = ( x 2 + y 2 + z 2 ) − 1 / 2 (2)

The values of V_{1} depends on the angle φ = 106˚ and the complementary angle ϕ = 74˚. Since δ = 1 in natural units, the distance a_{1} = cosϕ = 0.28 and b_{1} = sinϕ

= 0.96. However, a_{2} = aa_{1} and b_{2} = ab_{1}, where a = cosφ = 0.60 and b = sinφ = 0.80. Thus,

V 1 = 4 S ( d , 0 , 0 ) + 2 S ( d − 1 + a 2 , b , b 2 ) + 2 S ( d + a 1 + a 2 , b , b 1 + b 2 ) − 2 S ( d − 1 , 0 , 0 ) − 4 S ( d + a 2 , 0 , b 1 ) − 2 S ( d + a 1 , 0 , b 1 ) = − 0.116

V 2 = 4 S ( d , 0 , 0 ) + 4 S ( d , b , b ) − 4 D ( d + a , b , 0 ) − 4 D ( d − a , 0 , b ) = − 0.093

V_{1} can be slightly lowered when the repulsion between the nearest protons is reduced by a small rotation of the left molecule around its center. An angle of 1.4˚ is sufficient to reach the minimal potential energy [_{1} < V_{2}, it has been assumed that more than 2 water molecules should always be assembled according to the same rule as for the most strongly bound dimer. Water molecules could then only constitute rings or clusters of limited size, but linear polymerization is not excluded for water molecules. It would even be preferred for n_{2} water molecules, compared to clusters of n_{1} molecules, when (n_{2} − 1)V_{2} < (n_{1} − 1)V_{1}. It is thus sufficient that n_{2} > 1.3n_{1}. Of course, long chains of water molecules would be too fragile to resist thermal agitation in liquid water, but we will show (in Section 2.7) that chains of water molecules with aligned effective dipoles can be stabilized.

Moreover, V_{3} = −0.111, which is quite close to V_{1} = −0.116. Since water molecules can easily be rotated, they are aligned according to this pattern inside very narrow pores [

V 5 = 5 S ( d , 0 , 0 ) + S ( d , b 1 , b 1 ) + S ( d − 1 − a 1 , b 1 , 0 ) + S ( d + 1 + a 1 , 0 , b 1 ) − 2 S ( d − 1 , 0 , 0 ) − 2 S ( d + 1 , 0 , 0 ) − 2 S ( d − a 1 , b 1 , 0 ) − 2 S ( d + a 1 , 0 , b 1 ) = − 0.084

An applied electric field does not only align water molecules, but also polarize water molecules in such a chain. This yields stronger bonds, since all positive charges of one molecule come closer to the central negative charge of the neighboring molecule. Polarization of water molecules is possible [

When the young Theodor von Grotthuss was experimenting in1806 with a Volta pile, he discovered that pure water has a much higher electric conductivity than other liquids. The chemical structure of water molecules was not yet known. [John Dalton asserted in 1808 that it is HO, while Avogadro proposed in 1811 that it could be H_{2}O. This hypothesis was disregarded, since it resulted from the assumption that all types of particles occupy the same volume in gases, whether they are molecules or atoms. This was only accepted at about 1860, after the development of the kinetic theory of gases.] However, Von Grotthuss knew that water molecules are composed of positive and negative parts, since they can be separated by electrolysis. This had already been proven in 1800.

Von Grotthuss thought therefore that water molecules are held together in liquid water by mutual attraction of positive and negative parts. An electric field should align them. The high electric conductivity of liquid water could then be explained, if water molecules did contain tiny charge carriers that move more easily inside these chains than outside them [

The right part of _{3}O^{+} ion became a normal H_{2}O molecule, while the neighboring H_{2}O molecule was converted into an H_{3}O^{+} ion. The proton did advance without being deviated by collisions. [Ohm’s law is still valid for relatively small electric fields, as for ionic conduction in solid state physics]. We could equally well consider H_{2}O molecules with one delocalized proton on the left side and an adjacent HO^{−} ion. It contains only one proton, attracted toward the center of the next molecule. The electric field E would then cause a jump of the intermediate proton towards one of the two empty places. This would be equivalent to opposite motion of a proton hole. The essential result is that intramolecular exchange effects are realistic, since they explain the high electric conductivity of liquid water in a more detailed way.

The story of this discovery is similar to that of water memory. It began with an unexpected observation, made by the Russian chemist Nikolai Fedyakin. He condensed water in thin capillary quartz tubes and found that its physical properties are different from those of ordinary water. This phenomenon was totally unexpected, but arose at first much attention and scientific curiosity. Lippincot and Stromberg combined, for instance, routine infra-red spectroscopy with an improved method for producing this type of water. They confirmed that it has peculiar properties and proposed an explanation [

2D polymerization of water molecules would thus yield a “new state” of liquid water, since it is partially crystallized. However, other persons declared that this is impossible, since they preferred to stick to customary ideas. The media propagated the slogan of “bad science”, which had great impact. It led even to total prohibition of research on “polywater”. Academic careers would have been broken for anyone who might dare to be involved in such “pathological” science. Even Stromberg, interviewed some 40 years later, accepted that researchers might be misled by unconscious bias [

Actually, it is well-known in material science that crystallization can be influenced by the substrate, because of local attractions. Rostrum Ray proposed that this phenomenon of epitaxy might explain water memory, because of the “extreme structural flexibility” of water molecules [

2D polymerization of water molecules was rediscovered by the bioengineer Gerald Pollack. He wondered why particles that imitate red blood cells can easily move through narrow capillaries. Trying to understand this fact, he realized that some materials create an “exclusion zone” near their surface [

Pollack’s discovery and empirical investigations were outstanding achievements, but

In interstellar space, there are ions that attract water molecules and align their effective dipoles, since the configuration V_{2} of

Although positive and negative ions are strongly bound to one another in ionic crystals, they are easily dissolved in liquid water, since the small water molecules are more attracted. They penetrate inside ionic crystals and dissolve them. In liquid water, ions will thus usually be isolated and surrounded by a “hydration sphere”, where the effective dipoles of water molecules are oriented towards the central charge. This polarization decreases further away, because of thermal agitation. However, molecular chains could also be formed and rapidly stabilized by attracting one another.

According to the dipole approximation (1), the electric field is −p/r^{3}, when θ = 90˚. Real dipoles would thus be antiparallel in lateral positions, but we have to consider chains of tripoles. Two parallel molecular chains will thus be shifted by the distance a with respect to one another. This allows for parallel or antiparallel effective dipoles. We expect that parallel ones are preferred, since the two protons of the lower molecule are then closer to the negative center of the upper right molecule. However, it is useful to verify if this leads to significant differences for the resulting potential energies V_{a} and V_{b}. Since the potential energy of the upper pair is V_{2} = −0.093 in natural units, we get

V a = V 2 + 4 S ( a , 0 , d a ) + 2 S ( a , b , d a ± b ) + 4 S ( D − a , 0 , d a ) + 2 S ( d + a , 0 , d a ) + 2 S ( d + a , 2 b , d a ) − 2 S ( 0 , 0 , d a ± b ) − 4 S ( 2 a , b , d a ) − 4 S ( d , b , d a ) − 4 S ( d − 2 a , b , d a ) = − 0.28

V b = V 2 + 4 S ( a , 0 , d a ) + 2 S ( a , b , d a ± b ) + 4 S ( d − a , 0 , d a ) + 2 S ( d + a , 2 b , d a ) + 2 S ( d + a , 0 , d a ) − 2 S ( 0 , 0 , d a ± b ) − 4 S ( 2 a , b , d a ) − 8 S ( d , b , d a ) = − 0.09

The ± signs mean here that we have to sum two different terms. It appears that V_{a} is 3 times lower than V_{b}. Agglomerations of parallel chains of water molecules are thus preferred. Moreover, the effective dipoles are already oriented in nearly the same way by the electric field of the ion. Molecular chains with parallel effective dipoles get spontaneously assembled and stabilized.

It is very important to realize that biologically active molecules contain electrically charged parts. They explain why these molecules are easily dissolved in liquid water and require at least contact with saliva.

they can be set in oscillation by thermal agitation of water molecules in the surrounding liquid. They resonate at a particular frequency, which is characteristic of the active molecule. Its value is much lower than for vibrations of strongly bound charged particles inside molecules of any type. For active molecules, the resonance frequency is determined by the effective mass of its charged part and a weak restoring force. It results from deformations of the soft cocoon of polarized water molecules. These ideas are essential to unravel the puzzle of water memory.

When the electric field of a biologically active molecule has started to assemble water molecules, it becomes a germ of ongoing crystallization. More and more water molecules are attracted and align their effective dipoles. This yields closely packed molecular chains, like that of

To discard unnecessary objections, we mention that the solid state physicist Kittel realized already in 1946 that molecules of magnetite (Fe_{3}O_{4}) create single-domain ferromagnetic crystallites [_{3}O_{4} or Fe_{3}S_{4} molecules. They are then spontaneously assembled and constitute single-domain ferromagnetic crystallites [

To determine the radius R of these ferroelectric crystallites is a tricky problem. It has been tackled for the most common ferroelectric material (BaTiO_{3}) by means of the theory of phase transitions [

We use polar coordinates (r, θ) with axial symmetry. On the average, the WP is electrically neutral, because of the closeness of the charges ±2q inside all water molecules. The internal surface charge density results from the charges ±2q at the extremities of every molecular chain. It occupies the surface d^{2} in planes that are perpendicular to these chains, but the surface of the sphere is inclined. The intersected surface is thus increased and the internal surface charge is σ_{i}(θ) = (2q/d^{2})cosθ at the positive side. The external surface charge density σ_{e}(θ) is lower, but proportional to the internal one. The total surface charge density is thus σ(θ) = σ_{o}cosθ. Positive and negative surface charges on opposite sides of a WP create a homogeneous electric field E_{i} inside this sphere, as if it were composed of many very thin condensers. The electrostatic potential inside the WP is therefore ϕ_{i}(r, θ) = E_{i}rcosθ.

_{o}) that would be produced outside the sphere by the total charge Q = N2q of all effective positive poles of water molecules, if this charge were situated at the center of the sphere. The usual Coulomb potential (Q/r) is modified by screening effects, resulting from positive and negative charges in the polarized region of the liquid water. The radial decrease is characterized by the Debye length λ_{o}. _{e}(r, θ) in the external medium. It is due to the charges ±Q of all positive poles and negative poles, separated by the distance a. The external potential depends then on Δr = (a/2)cosθ, since

ϕ e ( r , θ ) = ϕ ( r − Δ r ) − ϕ ( r + Δ r ) for r ≥ R

ϕ_{i}(r, θ) and ϕ_{e}(r, θ) are subjected to boundary conditions, which determine the values of R and E_{i}. This more technical problem is solved in the short appendix, but all required physical concepts have been explained here and the result is that R ≈ 10λ_{o}. The value of the Debye length λ_{o} depends on the concentration of ions in the surrounding water. It is also proportional to the square root of the absolute temperature T, but this factor is nearly constant between 20 and 30˚C. The value of λ_{o} has been measured at 25˚C for water with different concentrations of dissolved NaCl [_{o} ≈ 1 nm at 0.1 M (mol/liter), but increases for lower and higher concentrations. Actually, λ_{o} ≈ 3 nm at 0.01 M and 5 M, which is only slightly higher than for Dead Sea water. For pure water, λ_{o} would depend on the concentration of H^{+} and OH^{−} ions. The radius R of WPs is then somewhat smaller than 10 nm.

When the particle physicist Shui-Yin Lo was visiting professor at the famous California Institute of Technology in 1996, he adopted a research project concerning properties of liquid water. He was surprised to discover that EHDs of HCl, NaOH or HNO_{3} molecules in very pure water led to the formation of “novel stable structures”. Lo thought that they result from crystallization of hydration spheres [

When S. Y. Lo determined the sizes of various types of structures by means self-interference of scattered laser light, he found 3 distinct groups. The smallest particles had a diameter D ≈ 15 nm with very low dispersion. We interpret this result as meaning that D is the diameter of WPs in pure water. Thus, R ≈ 7.5 nm and the Debye length λ_{o} ≈ 0.75 nm. Since the volume occupied by every water molecule is d^{3}, where d = 0.275 nm, WPs contain N ≈ 85,000 molecules. This huge number justifies the assumption that they are spherical. Nevertheless, WPs are nanoparticles, since water molecules are very small.

The second group of structures, discovered by S.Y. Lo, had a size of about 300 nm. We consider that this group corresponds to the length L = ND of chains of WPs, containing N ≈ 20 water pearls. We will explain (in Section 3.3), why their length L has to be limited. Its value depends on the mutual attraction between positive and negative surface charges on adjacent hemispheres. It is thus useful to replace the distributed surface charges of WPs by point-like poles. They are situated inside the sphere, like those of magnetized steel balls, but we can be more explicit. The total charge Q_{o} on the surface of the positive hemisphere, is the integral of 2πrσ(θ)rdθ, where r = Rsin(θ) and σ(θ) = σ_{o}cosθ, while the angle θ varies from −π/2 to + π/2. This yields Q_{o} = (4π/3)R^{2}σ_{o}. We can also calculate the electrostatic potential V(x) for a test charge +1 that is situated on the symmetry axis at the distance x from the center O of a WP. ^{2}dθ divided by the distance R'. This allows us to define the effective charge Q(x) of the positive pole, if it were situated at the distance R/2 from the center O. This pole is represented by a black dot and

V ( x ) = 3 Q o 2 ∫ − π / 2 + π / 2 sin 2 θ cos θ d θ [ R 2 + x 2 − 2 R x cos ( θ ) ] 0.5 = Q ( x ) x − ( R / 2 )

The result of numerical integration is shown in _{o} when x > 3R, but when the test charge is close to the surface of the WP, it does mainly interact with the nearest surface charges. This reduces the value of Q(x). On the surface, Q(R) = 0.68Q_{o}. We neglected all screening effects, but it is only important that neighboring poles are separated by the same distance R and carry charges ±Q.

Because of the rapidly decreasing Coulomb forces, it is sufficient to consider the mutual attraction of neighboring positive and negative poles. At rest, they are aligned and their poles are separated everywhere by the same distance R, but small oscillatory rotations of WPs around their center will lead to transverse displacements of the poles. They are represented in ^{th} water pearl by the variable u_{n}(t).

Q^{2}/(u^{2} + R^{2}). The transverse component F is reduced by the factor u/R. It follows that when u ≪ R , the restoring force is

F ( u ) = − K u where K = Q 2 R 3 (5)

This force is proportional to the relative displacement u, as for any elastic system. When M is the effective inertial mass of poles, the equation of motion for the n^{th} water pearl is

M u ¨ n = K ( u n + 1 − u n ) − K ( u n − u n − 1 ) (6)

Every dot stands for a time derivative. Since this equation is identical for all WPs, it describes the behavior of the whole chain. It can be simplified when the displacements u_{n} are smoothly varying along the chain, which is equivalent to saying that the diameter D of WPs is small compared to the distance where the relative displacements u_{n} are notably varying. We can then replace u_{n}(t) by u(x, t), where the coordinate x is treated as if it were a continuous variable. Actually, u n ± 1 = u n ± D u ′ + D 2 u ″ / 2 , where u ′ and u ″ designate first and second order partial derivatives with respect to x. Equation (6) is then reduced to

u ¨ = v 2 u ″ where v 2 = D 2 K / M (7)

This is the usual wave equation for vibrating strings. An infinite chain would allow for u ( x , t ) = u ( x ± v t ) . This corresponds to a function of any shape, moving at the velocity v towards the right or the left. Possible attenuations of oscillatory rotations have been neglected in (6) and (7), but will be discussed later on.

For a chain of finite length L, we have to know the boundary conditions at x = 0 and x = L. When both ends are free, the first and last pearls are not subjected to any force. Thus, u ′ ( x , t ) = 0 for x = 0 and x = L. In other words, u(x, t) has to reach maximal values at both extremities. This allows for a particular solution of well-defined frequency f and well-defined wavelength λ:

u ( x , t ) = A cos ( k x ) sin ( ω t ) where ω = 2 π f and k = 2 π / λ (8)

Since u ′ ( x , t ) is proportional to sin(kx), the boundary condition u ′ ( 0 , t ) = 0 is satisfied, but u ′ ( L , t ) = 0 requires that kL = sπ, where s = 1 , 2 , 3 , ⋯ . It follows that L = sλ/2 and because of (7), that the spectrum of possible frequencies is defined by

f = v λ = s v 2 L = f s where s = 1 , 2 , 3 , ⋯ (9)

The only possible frequencies are thus integer multiples of the fundamental frequency f_{1} = v/2L. For sound waves, any pair of such frequencies would produce an impression of harmony. The spectrum f_{s} = sf_{1} is therefore said to be a “harmonic” one. These properties are well-known in physics, but everyone should see why a chain of WPs with free ends does only allow for standing waves. This means that for any particular solution (8), all WPs are set in oscillatory rotations at the same frequency f, but the amplitude of these oscillations varies along the chain.

However, the approximation (7) is of limited validity, since it requires that D ≪ L . The measurements of Lo imply that chains of WPs contain a relatively small number of WPs (N ≈ 20). To see how far the approximation (9) is realistic, we have to solve the general equation (6). This is easy when we use complex notations, since standing waves are then defined by

u n ( t ) = A e i ( k x n − ω t ) where ω 2 = K M ( e i k D + e − i k D − 2 )

Thus,

ω ( k ) = v sin ( k D / 2 ) D / 2 and f ( λ , D ) = v π D sin ( π D / λ ) (10)

The function f(λ, D) is represented by the dark curve in _{max} = v/πD). Nevertheless, the linear approximation is valid for a relatively large domain of low frequencies. [Indeed, sin ( x ) = x − x 3 / 6 ≈ x , when x ≪ 2.5 ]. Actually, the spectrum of possible frequencies is f_{s} = sf_{1} when s ≪ N .

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

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 f_{s} 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

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 f_{1} remains the predominant one.

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.

_{r}. It is thus sufficient that f_{r} ≈ 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.

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 _{1} ≈ 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 (f_{s} = sf_{1}, where s = 1 , 2 , 3 , ⋯ ). The fundamental frequency f_{1} remains dominant, however. Standing waves on trimmed chains of WPs do produce an oscillating electric field of frequency f_{1} ≈ f_{r} 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” [

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 [

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 C_{9}H_{8}O_{4}, which became famous as aspirin. Even elephants, apes and other animals know how to cure or avoid ailments [

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 6^{th} edition of his “Organon of the Rational Art of Healing” was ready for publication. It is easily available [

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 [

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 [

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 f_{1} as well as harmonics ( 2 f 1 , 3 f 1 , ⋯ ). 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 f_{1}. There did also appear some “associated chains”, resulting from mutual attraction of trimmed chains.

The fourth frame (d) of

It is useful to express these ideas by means of equations, since they allow for logical deductions. Let X_{o} 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

X ( t + Δ t ) = X ( t ) − α Δ t X (t)

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:

X ˙ = − α X so that X ( t ) = X o exp ( − α t ) (11)

The exponential decrease does necessarily end up with X(t) = 0 when t ≫ 1 / α , 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:

Y ˙ = g X + β Y (12)

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

Y ( t ) = A α + β [ e β t − e − α t ] where A = g X o (13)

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

Y ˙ = A exp ( − α t ) + β Y − ε Z Y − ε ′ Y 2 (14)

Z ˙ = ε Z Y + ε ′ Y 2 − γ o Z (15)

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 γ_{o} per unit time to be destroyed by vigorous agitation. _{o} = 1/α = 1. We assumed that A = 50 and that β = 0.5, ε = 0.1, ε' = 0, γ_{o} = 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 = gX_{o} is great. This accounts for

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 1920^{th}, 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 [

Thomas Kuhn analyzed the process of scientific revolutions [

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

maximal and minimal values of Y(t) are always reached when Z = β/ε. Those of Z(t) require that Y = γ_{o}/ε. These facts result from (14) and (15) when Y ˙ = 0 , Z ˙ = 0 and ε' = 0.

^{2}. However, the amplitude of the periodic variations of Y(t) is decreasing. The article of Benveniste [

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 γ_{o} are smaller. _{o} 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) → γ_{o}/ε = 10 and Z(t) → β/ε = 1 for ε' = 0, when Y ˙ = 0 and Z ˙ = 0 . 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.

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 (

We noted that resonance effects allow for some tolerance. It is thus sufficient that f ≈ f_{1} ≈ f_{r}, where f_{1} is determined by the length L of trimmed chains (f_{1} = v/2L). However, it follows from Lo’s measurements that L = ND, where N ≈ 20. The fundamental frequency f_{1} 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 ≈ f_{1} ≈ f_{r}. However, two EHDs with trimmed chains of nearly equal length produce electric fields of slightly different frequencies ( f ± = f o ± Δ f , where Δ f ≪ f o ). 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 f_{o} with a modulated amplitude. Indeed,

cos ( f o − Δ f ) t + cos ( f o + Δ f ) t = 2 cos ( f o t ) cos ( Δ f ) t

cos ( f o − Δ f ) t − cos ( f o + Δ f ) t = 2 sin ( f o t ) sin ( Δ f ) t

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 [

Vittorio Elia and his collaborators performed remarkable experiments with the “long term goal” of clarifying the problem of water memory [_{4}S_{4}) 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 χ^{E}. Since it was always proportional to the measured heat of mixing, both phenomena had a common cause [

Elia and his team made two other remarkable discoveries by measuring the excess conductivity χ^{E} for homeopathic dilutions of Arnica Montana during aging. The active substance is helenalin, (C_{15}H_{18}O_{4}, containing charged oxygen atoms). Its EHDs displayed wave-like variations of χ^{E}, but at an extremely slow pace [^{E} was much greater for smaller volumes of EHDs. Initial conditions are usually irrelevant for aging (_{o}/ε. This value will thus be increased when the probability ε for creating associated chains is reduced by surface effects.

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:

f s = s f 1 where s = 1 , 2 , 3 , ⋯ ≪ N (16)

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 [

f s = 1000 , 2000 , 3000 , 4100 , 5100 and 5500 Hz

The three lowest values do precisely correspond to (16). Since higher frequencies than f_{1} 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 [_{1}. 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 _{s} were depressed, but this due to interference effects, also for musical instruments [

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 [_{1} = v/λ ≈ 1 kHz, allow us to determine the velocity v = 2L/f_{1} ≈ 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 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 [

It is very remarkable that these alignments are branching-off sideways, always at the same angle of 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

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” [

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 [

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 [^{+}) 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 [^{+} 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 [

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

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 (

Some observations indicated that this might happen in a helicoidally coordinated way [

Emilio Del Giudice et al. proposed in 1988 a bold hypothesis concerning properties of bulk liquid water at very small scales [

Although this theory was initially conceived for bulk water [

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” ( [

Preparata was coauthor of the official article on CDs [

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 (mc^{2}) 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 [

Bellavite and other health specialists at Italian universities reviewed in 2013 the status of research concerning EHDs [

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” [

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 [_{elec}. Their diameter would thus be about 100 nm. Greater structures, called CD_{plasma}, 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 CD_{rot} 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 CD_{plasma} and CD_{rot} entities constitute “supra-domains”.

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 [_{2}O 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

u ˙ = − u / τ + ( q / m ) E (t)

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, u ( t ) = u ( 0 ) exp ( − t / τ ) . 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 E ( t ) = 2 E ω cos ( ω t ) = E ω exp ( − i ω t ) + c . c and the instantaneous polarization density is P ( t ) = β ( ω ) E ω exp ( − i ω t ) + c . c . The complex conjugate (c.c) requires merely that ω is replaced by −ω. It follows that

β ( ω ) = n ( 2 q ) 2 / 2 m γ − i ω = β ( 0 ) 1 − i ω τ = β ( 0 ) ( 1 + i ω τ ) 1 + ( ω τ ) 2 = β 1 ( ω ) + i β 2 (ω)

The function β_{1}(ω) specifies the average orientation of effective dipoles. At low frequencies ( ω τ ≪ 1 ), 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 ω τ ≫ 1 . The imposed rhythm is too fast to allow the molecules to follow, because of friction. The imaginary part β_{2}(ω) varies like x/(1 + x^{2}), when x = ωτ. This yields a peak that is centered on x = 1. The function β_{2}(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 H_{2}O 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 (

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 [

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 [^{th} a method for measuring the dielectric response of EHDs [

Mahata was not aware of these measurements and developed with his collaborators a much more efficient technique. In 2007, it was ready [

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

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 u_{o}(t) of these poles is

u ¨ o + Ω 2 u o + γ u ˙ o = C ω exp ( − i ω t ) (17)

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 C_{ω} is proportional to the amplitude of the electric field E(t). Thus, u o ( t ) = B ( ω ) E ω exp ( − i ω t ) 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

β ( ω ) = Ω 2 Ω 2 − ω 2 − i ω γ ≈ Ω 2 Ω 2 − ω 2 + i Ω Γ / 2 Ω 2 − ω 2 − Γ 2 (18)

We simplified the real part β_{1}(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 ω ≫ Ω . The function β_{2}(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. _{1} and (in red) the spectral distribution of the energy-loss function β_{2} 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 β_{2} did correspond to curves like the blue one. The height of the observed peak has been adjusted in _{2exp} → 0 when ω ≫ Ω . 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.

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 [

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 (D_{2}O). 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 [

Demangeat et al. studied the effects of EHDs by means of nuclear magnetic resonance [_{o}. 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, T_{1} and T_{2}, for components of the magnetic moment of protons along the direction the magnetic field B_{o} and perpendicular ones. The ratio T_{1}/T_{2} depends on their environment. For EHDs, the results of measurements did prove that “water is a self-organizing system” [

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 [

Burkin and his collaborators [

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 [

Elia et al. produced supramolecular aggregates in pure water by repeated contact with a polymer, called Nafion [_{3}H). 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 [

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 [

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 [

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 [

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 ( [

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 ( [

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 ( [

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” ( [

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” ( [

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” ( [

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

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

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

The second line of ^{8} m/s.

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 ( [

Montagnier and his collaborators [

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 × 10^{14} 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 (^{−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 ( [

We come now back to the important and still unsolved problem of the asymmetry of the energy-loss functions for Mahata’s resonances (

To acquire more physical insight, we consider two identical pendulums (

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 _{+} > 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.

_{+}. 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 β_{2}(ω) displayed always at the side of lower frequencies (

u ¨ 1 + Ω 2 u 1 + ω o 2 ( u 1 − u 2 ) + γ 1 u ˙ 1 = C ω exp ( − i ω t ) (19)

u ¨ 2 + Ω 2 u 2 − ω o 2 ( u 1 − u 2 ) + γ 2 u ˙ 2 = − C ω exp ( − i ω t ) (20)

They apply to the left part of _{o} ≈ 1 kHz accounts for interactions. They correspond to opposite forces. For the right part of _{2}. 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 u_{1} > u_{2} implies that γ_{1} < γ_{2}. When we set γ_{1} = γ − η and γ_{2} = γ + η, the sum and the difference of (19) and (20) yield two equations for u_{±} = u_{1} ± u_{2}:

u ¨ + + Ω 2 u + + γ u ˙ + − η u ˙ − = 0 (21)

u ¨ − + Ω + 2 u − + γ u ˙ − − η u ˙ + = 2 C ω exp ( − i ω t ) (22)

The pair of WPs behaves thus as if there were two oscillators that resonate at the frequencies Ω or Ω_{+}, where Ω + 2 = Ω 2 + 2 ω o 2 . However, only the mode u − is directly excited by the oscillating electric field of angular frequency ω. The mode u + is then excited by entrainment, but forced oscillations require that

u ± = A ± 2 C ω exp ( − i ω t )

Because of (21) and (22), the amplitudes A ± are determined by the equations:

a + A + + i ω η A − = 0 and a − A − + i ω η A + = 1

where a + = Ω 2 − ω 2 − i ω γ and a − = Ω + 2 − ω 2 − i ω γ . Thus,

A + = − i ω η a + A − and [ a − + ( ω η ) 2 a + ] A − = 1

Forced oscillatory rotations of pairs of WPs involve both modes of oscillation. The polarizability is thus

β ( ω ) = Ω + 2 ( A − + A + )

It is normalized to get β(0) = 1. When Ω and ω_{o} 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 β_{1}(ω) and β_{2}(ω) are shown in

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.

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

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 [

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.

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

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 [

Since oscillating electric charges do also produce oscillating magnetic fields (

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 [

The basic claim of Montagnier’s patent US2010323391 [

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 [

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 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 [

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.

The author declares no conflicts of interest.

Meessen, A. (2018) Water Memory Due to Chains of Nano-Pearls. Journal of Modern Physics, 9, 2657-2724. https://doi.org/10.4236/jmp.2018.914165

The basic ideas were presented in Section 2.8. Since ϕ ( r ± Δ r ) = ϕ ( r ) ± Δ r ϕ ′ ( r ) , where the prime indicates derivation with respect to r, the potential in the external medium (r ≥ R) is

ϕ e ( r , θ ) = − P ( 1 r 2 + 1 r λ o ) cos θ e − r / λ o while ϕ i ( r , θ ) = E i r cos θ

is the potential in the internal medium (r ≤ R). P is the total dipole moment of the WP and E_{i} the electric field inside this nanoparticle. In the external medium it is

E e ( r , θ ) = − ϕ ′ e ( r , θ ) = P ( 2 r 3 + 2 r 2 λ o + 1 r λ o 2 ) cos θ e − r / λ o

The radius R is then determined by the boundary conditions:

ε e E e ( R , θ ) + ε i E i ( R , θ ) = σ ( θ ) and ϕ e ( R , θ ) = ϕ i ( R , θ ) + w (θ)

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) = σ_{o}cosθ. The second relation accounts for the dipole density w(θ). It results from the inner and external surface charge densities σ_{i}cosθ and −σ_{e}cosθ. They are separated by a distance Δ R ≪ R , but σ_{e} = ϰσ_{i} and σ_{o} = σ_{i} − σ_{e}, while w(θ) = σ_{e}ΔRcosθ is negligible. Moreover, ε_{i} = 0. The boundary conditions yield then the following relations:

E i = − P ( 1 R 3 + 1 R 2 λ o ) e − R / λ o and ( 2 R 3 + 2 R 2 λ o + 1 R λ o 2 ) e − R / λ o = σ o ε e P

We know that P = Nq2a, where N = (4πR^{3}/3)/d^{3} and σ_{i} = 2q/d^{2}. Thus, σ_{o}/ε_{e}P = (1 − ϰ)(3d/4πε_{e}aR^{3}) and

( 2 + 2 η + η 2 ) e − η = 0.015 ( 1 − ϰ ) where η = R / λ o

We ignore the value of ϰ = σ_{e}/σ_{i}, 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 ≈ 10λ_{o}.