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The physics and chemistry of high-dilutions

The physics and chemistry of high-dilutions

Marc Henry, Association Natur’Eau Quant, Strasbourg e-mail :
Abstract The original suggestion of Samuel Hahnemann of the existence of an immaterial dynamical force for explaining homeopathy is analyzed within the frame of quantum theory. Based on quantum field theory and the concept of 2D coherence domains, a plausible model of action of homeopathic remedies is proposed allowing to discuss the concept of water memory and information transfer through electromagnetic signals within a scientific rigorous quantitative frame. This quantum-mechanical viewpoint points to the crucial role played by lipidic membranes, a universal component of any living system. Attempts to assimilate homeopathy to a mere placebo effect and trials to block research funding on homeopathic remedies should be viewed as a conservative attitude stemming from a sticking to a materialistic philosophy based upon classical physics laws. Time is now ripe for physicians and biologists to realize that quantum physics applies at all scales, giving to homeopathy its credentials as a rational medical approach for healing people.
Keywords Life, homeopathy, water, Samuel Hahnemann, high-dilutions, Quantum field theory, phase coherence, information, electromagnetism.
Figure 1: Homeopathy was originally designed as a rational medicine based solely on observable symptoms and not of various theories pretending that it was possible to penetrate in the body through speculative lines of thought in order to discover the causes and the essence of various illnesses.

Introduction Since its introduction in 1810 by Samuel Hahnemann [1], homeopathy has always been the subject of deep controversies (figure 1). The main reason seems to lie in the postulate that the lower the dose of a given remedy, the higher its therapeutic action on the body. Accordingly, such a principle is in direct contradiction with the postulates of materialism that claims that any thing is made of material particles and that any effect should be directly proportional to the cause that produced this effect. A direct consequence of such a materialistic viewpoint for medicine is the contraria contrarii curantur principle, the credo of any allopathic approach for curing illness. Basically, faced to a given pathology, allopathic remedies tries to annihilate the primitive cause by using weighable doses of drugs. In deep contrast, the homeopathic credo is based on the similia similibus curantur principle, where the sole indication for a remedy is to select the one that shares the totality of symptoms with the illness. As one use a substance that provokes the same symptoms of sickness, it is mandatory to use it under a highly diluted form. The strange thing is that the more the dilution, the higher seems to be the therapeutic activity, provided that the remedy has been correctly selected from the similarity principle. In particular, dilutions far beyond Avogadro’s limit are routinely used in homeopathy, calling for a scientific explanation that cannot be based on weighable amounts of the therapeutic agent, as in allopathic medicine. This lack of scientific explanation on how the homeopathic remedy works is the main reason for the considerable skepticism met in the allopathic medical community against homeopathic practices. In order to explain the real effects of homeopathic remedies, Hahnemann was obliged to invoke the existence of an immaterial dynamic force whose strength increases as physico-chemical forces decrease due to the dilution. Its great idea was to discard material properties in order to keep only therapeutic properties, exalting the dynamic virtue of homeopathic remedies through trituration and succussion. The basic aim of homeopathy is thus to remove all the symptoms of sickness by provoking a healing response in the body itself. Consequently, it is not the drug that heals, but the organism itself, which – thanks to «pathogenic» information received by the medicine – can find the road to recovery. In modern language, a homeopathic medicine allows the body to evolve from a dynamic attractor representative of an illness state to another dynamic attractor more in line with healthiness. It is the whole body, through its complex metabolism involving modulation of gene expression that reacts to a pathogenic information responsible for the sickness state coming both from the outside and the inside of the body. As both pathogenic kinds of information are in conflict for the same channels of expression leading to the same symptoms, one should win while the other is obliged to regress. Owing to the high dilution, the pathogenic information coming from the outside is usually more active than the pathogenic information rooted inside the body. Consequently, provided that the dilution factor is correctly chosen, the body continues to express the same symptoms, being now tuned to the homeopathic remedy coming from the outside. Once this tuning is perfect, the old pathogenic information rooted in the body becomes erased. The only thing to do is now to stop the homeopathic treatment, leading automatically to the full simultaneous disappearance of all the symptoms. As an illness state is usually defined by a full set of symptoms, removal of all these symptoms means that the body is definitively healed of the corresponding sickness. The closer the similarity of symptoms between the
homeopathic remedy and the sickness, the better the tuning and the faster the recovery of healthiness state. With this mechanism of action in mind, it should become obvious that the worse way of healing should be to give a non-diluted substance that have no symptoms in common with the sickness. In that case, instead of competing for the same channels of expression by the body, both pathogenic information have their own channels, adding an outside stress to the inside one already present. This overload of the body may well temporarily remove one or two old symptoms at the cost of a full set of new symptoms that may be more invalidating than the previous ones. Moreover, as soon as the remedy is discarded, the old symptoms usually reappear, but now on a more degraded ground, meaning a more difficult recovery of healthiness. In fact, a striking feature of the homeopathic approach is that it is perfectly in line with a quantum way of thinking. Accordingly, let’s recall that at the beginning of the twentieth century, physicists were confronted with the troublesome behavior of the electron revolving around the atomic nucleus. Polarized by the classical way of thinking in terms of well-defined trajectories, they thoughts were based on the concept of electronic orbits around the nucleus in full analogy with the concept of planetary orbits around the sun. But a puzzling feature was that electronic transitions between different orbits was responsible for emission of electromagnetic waves by the atoms, whereas electronic movement around the same orbit gives absolutely no light emission. This was in full contradiction with Maxwell’s electromagnetic theory that was stating that electromagnetic waves should be emitted as soon as an electrical charge is moving whether on the same orbit or not. In order to solve the puzzle, a young German physicist named Werner Heisenberg propose a revolutionary way of thinking by stating that electronic orbits have no physical reality. Sticking to a strict positivist philosophic position he argues that an atom should not be characterized by its material content (electronic orbits around nuclei) but by the only thing that was really observable: a full set of immaterial electromagnetic frequencies. Based on this brilliant idea, he was the first to develop a coherent quantum mechanical frame for explaining atomic properties based on non-commuting transcendental matrices. The key point was that one has to consider the full matrix and not the individual elements for describing the electronic movement, which was unobservable by principle. Basically, how do we know that atoms contain electrons? Just because we are able to dissect them, a process called ionization. But, as soon as we have a unionized neutral atom, we have no means at our disposal to know that there are electrons inside it. The only thing that we can observe is the set of electromagnetic frequencies that the atom is able to emit upon excitation. We infer the existence of electrons inside an atom because we can observe them when they are outside. This is just a logical inference definitively not based on an observation. Now, it should be clear that this idea of defining a reality by an indivisible immaterial reality was invoked by Hahnemann one century before Heisenberg. Before Hahnemann, there were several medical schools pretending to know how the human body was working. Such knowledge was a consequence of the dissection of corpses and was definitively not based upon the observation of living bodies. Hahnemann was the first to stress the fact that it was impossible by principle to really know how a living body was working, as one century later, Heisenberg was the first to stress that it was impossible to observe an atomic orbit [2]. The only thing that we can learn from a living body is a full set of symptoms defining either a state of illness or of healthiness. An isolated symptom has absolutely no meaning by itself, the biological reality lying in a more or less deep entanglement of symptoms. If you know the full set of symptoms, you
know the sickness. Similarly, if you know the full set of electromagnetic frequencies, you know the atom. Knowing just a frequency, does not allow you to identify the atom and similarly, knowing just a symptom does not mean that you know the disease. It is this entanglement in the living body that prevents its analysis as a mechanical machine with well-defined interacting parts. For instance, if you move your leg, can we be sure that only the leg is moving and not the whole body? In fact, the only way to be sure to move only the leg is to cut it away from the body. This inability to cut something in two separated parts is a mere consequence of parts being entangled together. This is clearly a quantum property that applies to atoms as well as living bodies. Entanglement is thus at the very root of the homeopathic doctrine and this was clearly recognized more than two centuries ago by Hahnemann in its “Organon”, a term meaning an instrument, a practical method and not a system of thoughts. Again, this purely epistemological position of Hahnemann concerning treatment of diseases fits perfectly well with the Copenhagen interpretation of quantum mechanics. This is the reason why any attempt to understand homeopathy and high-dilutions effects without a quantum mechanical background is doomed to fail. If classical mechanics is the natural language for allopathic supporters, quantum mechanics is the natural language for homeopaths. This is one of the main reasons of the big clash between the two medical approaches. Unfortunately for homeopathy, quantum mechanics principles are by essence non deterministic, by opposition to the clear deterministic nature of classical physics principles. Consequently, it is easy to put forward detailed mechanistic explanations for the action of a drug-based allopathic treatment, whereas action of homeopathy is based on a non-predictable reaction of a living body involving an immaterial dynamic force. Consequently, the very first step allowing understanding how homeopathy may work is to become acquainted with quantum mechanics principles. This logically stems from the fact that quantum mechanics uses non-observable complex numbers mixing a length with a phase angle, and not observable real numbers that have only a length. Consequently, one finds at the very root of any quantum world, material things that are observable and immaterial waves that are not observable, but nevertheless carry all the information necessary to predict what material things will do or not. Similarly, in homeopathy, we have material living bodies with their observable symptoms, and immaterial information transfers that even if they are not directly observables, govern nevertheless the state of illness or healthiness of the living entity.
Quantum mechanics for biologists Who has not hear or read about Richard Feyman’s famous quote about quantum mechanics: « I think I can safely say that nobody understands quantum mechanics » [3]? Was this a simple joke from a facetious man, or a deep truth about the way Nature is working? Nobody knows, but there are clues in scientific literature that it was indeed a joke. A first clue is given by a series of very interesting papers published in the forties by a couple a French mathematicians [4-7]. The main conclusion that could be drawn from these papers is that quantum mechanics could be defined in one single sentence made of very few words: « There is no state variable ». Any quantum formalism is in fact deeprooted in this simple principle. In order to understand how quantum rules emerge from this simple sentence, one must first consider that if all physical quantities are in theory simultaneously measureable, then a state variable should exist. What is called a state variable is a variable from which all others variables may be derived. This is the standpoint of
classical mechanics. In turn, if there are quantities that are not simultaneously measureable, then there can be no state variable. This is the standpoint of quantum mechanics. The absence of state variable gives an essential uncertainty and means that there exist by right two quantities that are not simultaneously observable or measureable. Such uncertainty implies that some predictions are prone to fundamental errors and should thus be expressed in terms of probabilities. Thus, considering for a physical quantity A the whole set of certain predictions (X1, X2,…, Xn) one should by principle express any prediction X as a linear combination X = c1·X1 + c2·X2 +…+ cn·Xn (spectral decomposition principle). Then, the probability of observing the element Xi is an arbitrary function f(ci) of the associated coefficient ci. If one further requires that the unknown function f should be the same for any spectral decomposition, it is possible to show that it should be such that f(x·y) = f(x)·f(y), where x and y are complex numbers, that is to say numbers having both a magnitude r and a phase φ: z = r·exp(i·φ). The further requirement that f is a continuous function then leads to the only acceptable solution: f(x) = |x|k, with k > 0 [6]. Finally, using a generalized version of Pythagoras’ theorem, it was possible to show that only two values of k are possible: k = 1 (existence of a state variable, Boolean logic) and k = 2 (no state variable, non-Boolean logic) [8]. For this last case, the probability of observing a given element is thus given by the square of the corresponding complex coefficient in the linear superposition, one of the base postulates of quantum theory [8]. The next step is to introduce the basic two quantities that cannot be simultaneously measured. As we are seeking a quantum theory for high-density macroscopic systems, such as living bodies, the number of available quanta N emerges as a crucial variable. Allowing some fluctuations ∆N in the number of quanta, it may be shown that the conjugate variable is necessarily the quantum phase φ with a fundamental uncertainty relationship [9]:
∆N·∆φ ≥ ½ (1)
Figure 2: Emergence of collective many-body coherent behavior in the cases of swarms of birds or fishes as a consequence of the fundamental uncertainty relationship of quantum field theory involving the total number of quanta N and the common phase angle φ of the swarm.
The crucial point here is that only pure numbers having no units are involved. This means that such an uncertainty relationship is scale invariant, that is to say, that quantum mechanics is the right way of thinking at any scale, atomic or macroscopic. The implication is that there is no fundamental distinction between observer and observed system, between quantum and classical system. Here, the classical way of thinking is obtained when the total number of quanta can be determined with certainty, i.e. when ∆N = 0. The consequence is that the phase φ is then a random variable that changes in a completely arbitrary way from one quantum to another one (incoherence). But, if the total number of quanta is not known with certainty, then the quantum phase may take a well-defined value (∆φ → 0) when ∆N → +∞. This situation describes a quantum regime that is routinely observed at a macroscopic scale in ferromagnetism, ferroelectricity, superconductivity and superfluidity phenomena for instance. Quantum means here appearance of a collective many-body coherent behavior that cannot be explained by considering only pairs of interacting quanta. It is this kind of quantum coherence typical of high-density situations that is pertinent in liquid water [9] and of course in a living body [10]. Most importantly, it cannot be observed in diluted cases (∆N → 0). The quite interesting point is that a “quantum” is not necessarily an elementary particle, but it can be an ion, a molecule, a protein, cell or even a bird or a fish. That is to say a quantum is anything that can be considered at a given scale as a whole unit, not separable into smaller parts. For instance, in homeopathy, the whole living body in a state of healthiness is considered as a quantum. To make the argument clear, let’s talk about birds, considering not a single bird but a swarm of birds (figure 2). Have you noticed that when it is possible to count exactly the number of birds in the swarm (dilute case), then the movement of each bird appears as erratic and unpredictable? But, above a certain number of birds per unit volume, it becomes impossible to count them individually (∆N > 0) and as a consequence, a collective behavior emerges above a certain critical threshold. The whole swarm is now moving with a wonderful collective and coherent smooth behavior instead of being erratic and random. The same applies of course in shoals of fishes. These examples show that the scale invariance of relation (1) associated to the existence of creation and annihilation operators in quantum field theories [9] is a tangible reality at all scales, provided that one faces a high density situation. In fact, it can be shown that any coherent quantum system would display as a whole a classical behavior, with a predictable trajectory computable according to the least action principle [11]. In other words, classical mechanics is just a high-density approximation of quantum mechanics when phase coherence emerges. But we also know from thermodynamics that any macro-state M whether in equilibrium or not, has a non mechanical entropy S(M) = kB·log W(M), where W(M) is the phase space (position, momenta) volume compatible with a set macro-variables allowing to define precisely the macro-state. As shown by Edwin Thompson Jaynes, the second law stating that S(initial) ≤ S(final) directly follows from Liouville’s theorem, expressing the necessary condition for a change from a macro-state Mi to another macro-state Mf be reproducible by any observer able to control only the set of macrovariables defining the macro-state M [12]. In order to set a quantitative link between entropy and quantum mechanics, let’s consider a set of N particles of mass m enclosed in a cube of edge L in thermal contact with a thermal bath fixing the number of microstates (positions q and associated momenta p) that are accessible to this ensemble. Owing to Brownian motion, any particle has equal probability to be found at any location in the volume V = L3 leading to
an uncertainty in position ∆q = L along x-, y- or z-directions. Consequently the number of position states should be Wq ∝ VN = (∆q)3N. On the other hand, kinetic gas theory gives us the probability density of observing a given speed v at temperature T with a width at half-height measuring the uncertainty ∆v in the knowledge of molecular speeds. Consequently, the number of dynamical states corresponding to a spreading ∆p = m·∆v, should be Wp ∝ (∆p)3N. The total number of states in the phase space should then be W ∝ Wq·Wp ∝ (∆q·∆p)3N, with an universal proportionality constant that should have the dimension of a mechanical action in order to insure that W be a dimensionless number. Identifying this proportionality constant with Planck’s constant ħ, and taking into account that S ≥ 0, it comes:
S = kB·Ln (∆q·∆p/ħ)3N ≥ 0 ⇒ 3N·kB·Ln(∆q·∆p/ħ) ≥ 0 ⇒ ∆q·∆p/ħ ≥ 1 ⇔ ∆q·∆p ≥ ħ (2)
In contrast with (1) that applies at any scale, this second kind of uncertainty relationship imposes that atoms should have a characteristic size close to 0,1 nm. Moreover, it does not depend on N, the total number of identical particles that may be as high as the Avogadro constant. Instead, owing to the very small value of Planck’s constant, it gives us the illusion of a quantum world that exists at a very small scale that cannot be directly experimented with our macroscopic senses. This comes from the fact that, historically speaking, (2) was derived before (1). In fact, one should realize that (2) is just a special case of (1) in the limit of infinite dilution (N → 1 ⇒ ∆N → 0 and ∆φ → +∞). This could be easily shown by realizing that as ħ is a quantum of action, it allows to view any kind of energy E as a pulsation ω = ∆φ/∆t, according to the Planck-Einstein relationship: E = ħ·ω. Now, a set of N undistinguishable quanta is expected to have a total energy E = N·ħ·ω, meaning that any uncertainty ∆N in the number of quanta ∆N would translate into a corresponding uncertainty in energy ∆E = ∆N·ħ·ω. Consequently, the fundamental uncertainty relationship ∆N·∆φ ≥ ½ may be reworded as (∆E/ħ·ω)·(ω·∆t) ≥ ½, leading to:
∆E·∆t ≥ ħ/2 (3)
One crucial point that is often overlooked in quantum mechanics literature is that the uncertainty time t appearing in (3) is definitively not the time t used in Lorentz’s transformations or the time t ruling the time-dependent Schrödinger equation. As evidenced by its definition, ∆t = ∆φ/ω, it corresponds to an “internal” time intimately associated with the rotation of a phase vector embedded with the quantum object. It thus corresponds to an intrinsic lifetime, or if one prefers to the “proper” time of the theory of relativity. Similarly, the energy E appearing in (3) is not the eigenvalue of a hamiltonian operator, but rather a thermodynamic internal energy associated of a set of undistinguishable quanta. Consequently, there is no need to invoke the existence of a time operator conjugated to the hamiltonian operator to justify (3). Relation (3) is just a reformulation of (1) using total energy and lifetime instead of the number of quanta and quantum phase. The non-commuting operators responsible for (3) are then just the number operator and the phase operator of quantum field theories, as for (1). Now, in the case of a free single particle (N = 1) of mass m and speed v, one have E = p2/2m or ∆E = p·∆p/m, where p = m·v is the associated momentum. Writing v = p/m = ∆q/∆t, then leads to ∆t = m·∆q/p and the relationship ∆E·∆t ≥ ħ/2 becomes, (p·∆p/m)·(m·∆q/p) = ∆p·∆q ≥ ħ/2. This clearly shows that (2) is in fact deep-rooted in (1) or (3) and should be considered an infinite dilution approximation of quantum field
theories. As with energy, one may take into account the fact that ħ is a quantum of action, allowing to associate to any momentum value p a characteristic wave-vector k = p/ħ, according to the de Broglie-Schrödinger relationship. This means that both Schrödinger’s or Dirac’s equations should also be considered as an infinite dilution approximation of quantum field theories applying to a single quantum object (N = 1) isolated from other similar objects. As (1) and (3) applies to high-density situations, it becomes mandatory not to neglect virtual excitations coming from the vacuum between quantum objects that are responsible for Lamb’s shifts of energy levels in atoms [13] or for the occurrence of attractive forces between conductive plates (Casimir’s effect) [14]. In order to understand this importance of the vacuum, let’s go back to the quantum explanation of atomic stability. Basically, based on (2), it transpires that it is possible to furnish a high momentum ∆p to the electron as soon as ∆q becomes too small. This is this quantum effect that prevents electrons to collapse onto the nuclei, as predicted by classical phyics. Let’s now show that moving to (3) allow giving a deeper insight of what is really happening when the electron becomes too close from a nucleus. Accordingly, the relativistic nature of quantum electrodynamics allows the creation of matter and/or radiation out of a vacuum as well as their annihilation in the same vacuum, creating a de facto equivalence between the concepts of energy and vacuum. Specifically, the more we reduce the scale of the spatio-temporal observation, the more it becomes possible to have a mass M or an energy E = M·c2. The electron and its antiparticle the positron are the least massive particles in nature, but still one must have energy of at least 1.02 MeV to create an electron-positron pair. This becomes possible as soon as the size of this vacuum is of the order of 0.1 pm, since ħ·c ≈ 197 MeV·fm. Consequently; any electron that approaches the atomic nucleus close to 1 fm would encounter a virtual electron-positron pair. The electron will never reach the nucleus, because when it encounters the virtual positron it will disintegrate, releasing the partner electron, which in turn becomes observable, with excess kinetic energy that is amply sufficient to escape the Coulomb attraction. All this takes place at a time scale of the order of the attosecond, leaving any potential observer with the illusion of a permanent electron escaping “miraculously” the Coulomb attraction of the nucleus. This mechanism works fine owing to the impossibility of distinguishing two electrons. Consequently, at a quantum field level, on may argue that electrons around any nucleus are in fact permanently created and destroyed by the atomic vacuum. On average, any nucleus holding Z protons will always be surrounded by Z electrons, but these electrons are definitively not permanent particles, but quite ephemeral entities that gives the illusion of indestructible corpuscles owing to their undistinguishable character linked to our inability to differentiate between an electron popping up from the vacuum and another disappearing quite simultaneously in the same vacuum. The proof that such a strange picture of atomic stability is right comes from the existence of the Lamb’s shift, the smoking gun for this active participation of the vacuum to the bonding energy between electrons and nuclei. Accordingly, it may be shown that by “eating” their vacuum, atoms increases in size by a tiny universal amount δL that depends only on Sommerfeld’s fine structure constant α and the Compton wavelength λC computed from the rest mass me of the electron [15]:
ℎ ·
⟹ = · √2(

) ·
1 2
Now, the scale invariance of (1) insures that this mechanism is not limited to the interaction between electrons and protons in atoms. It also explains the van der Waals attraction between molecules and more importantly, the hydrogen bonding between water molecules (see next section). The quantum vacuum is also probably responsible for the amazing catalytic power of enzymes in biology that are able to perform at room temperature and ambient pressure with high yields chemical reactions that are quite difficult to make in a beaker. As ħ·c ≈ 197 eV·nm, it comes that a vacuum of 1 nm is capable of generating X-ray photons having energy close to 200 eV, which no known chemical bond is capable of withstanding. Thus an enzyme accurately deploying nanometric cavities manages to achieve, at room temperature and pressure, chemical reactions impossible at a macroscopic scale. The secret is simply in the control of natural fluctuations of the quantum vacuum. Obviously, as far as van der Waals interactions are concerned, there is no need to invoke in that case creation-annihilation of electron/positron pairs responsible for covalent chemical bonding. Here, it is creation/annihination of massless photons from the vacuum, a much less energy-demanding situation, that are responsible for van der Waals sticking between molecules. Accordingly, owing to (3), any molecule is permanently surrounded by a virtual field of photons popping in and out from the vacuum around it. When two molecules become close enough, their respective photonic virtual fields overlap breaking the spherical symmetry of the isolated molecules. Owing to this symmetry breaking, virtual photons having a wavelength larger than intermolecular distance are excluded from the space between the two molecules, forcing them to stick. Consequently, at a quantum field level, van der Waals attraction should be considered as a kind of depletion interaction involving virtual photons instead of matter particles. This beautiful universal sticking mechanism welding together matter and vacuum is usually kept hidden by invoking chemical bonding for the atomic scale and atomic polarizabilities or Hamaker’s constants for the molecular scale. The main trouble in doing such a masquerade is to replace a basically many-body quantum effect involving photons having any energy between 0 and several hundreds of eV, by a two body interaction involving a few eV for chemical bonds and a few meV for van der Waals bonding. This quite drastic approximation works fine only for diluted situations and completely break down as soon as the density becomes higher than a critical threshold, highly dependent on the number of electrons and nuclei involved in the interaction. In such cases, one is obliged to invoke so-called “specific effects” to explain experimental facts. These specific effects are in fact just the price to pay for separating matter from its surrounding vacuum.
Water revisited One of the first substances that benefits from a full quantum treatment including the vacuum is liquid water. Details have been published elsewhere [9,16] and only conclusions will be reported here. The main result is that hydrogen bonding between water molecules should be considered as coherent van der Waals bonding. Coherent means here that water molecules and vacuum becomes welded together through an internal electromagnetic field borrowed from the vacuum and trapped in so-called “coherence domains”. The role of this internal electromagnetic field is to perform virtual excitations of water molecules towards a Rydberg level localized on oxygen atoms lying about 0.5 eV
below their ionization threshold (ca. 12.6 eV). As always in quantum field approaches, all the possible paths leading to this localized level should be taken into account. This means direct excitation from the ground state as well as in indirect transitions through other discrete levels (summation) as well as any energy level located in the continuum (integral). Consequently, water properties are ruled by oscillator strengths involving the whole photonic excitation spectrum (0-200 eV) and not by a single HOMO-LUMO excitation or a single ionization potential. As these electromagnetic excitations comes from the internal vacuum of water molecules and not from outside, they are necessarily virtual excitations, i.e. such that ∆E·∆t < ħ/2, with photons acquiring a negative mass in order to remain trapped between water molecules. The main observable consequence of this coherent internal but invisible mixing of matter, radiation and vacuum is the opening of a non-classical coherence gap that obliges water molecules to stick together in a much efficient way than by a mere van der Waals attraction involving only a symmetry breaking. In this case, there is both symmetry breaking and Bose-Einstein condensation of the virtual photons that are constantly exchanged between water molecules (figure 3). This emergence of quantum phase coherence between matter and radiation is usually masqueraded as “hydrogen bonding” in order to reduce the basically many-body problem to a two-body specific interaction.
Figure 3: The physics behind the formation of water coherence domains. Absorption of virtual photons popping out continuously from the quantum vacuum within an assembly of N electrons belonging to water molecules transforms their shape from a trigonal structure to a tetrahedral one with a size increase in total volume ruled by the universal constant δL function of the Compton wavelength of the electron λC and Sommerfeld’s fine structure constant α. Depending on the coupling constant G between matter and zero-point electromagnetic fields through an excited state located below the ionization threshold of the water molecule, a coherence gap could be opened welding together water molecules and photons into a coherence domain. This quantum Bose-Einstein condensation is named hydrogen bonding by chemists and biologists.
This masquerading explains why it is so difficult to define clearly what is the real nature of hydrogen bonding with a lot of confusion lurking around its putative covalent versus electrostatic nature [17]. In fact, this is a completely useless debate, as one tries to capture through a Schrödinger equation or through a Coulomb’s interaction that rigorously applies to a infinitely diluted single molecule a phenomenon welding together several millions of undistinguishable molecules.
A coherence domain is thus made of a large amount of similar densely packed water molecules that display a coherent collective behavior as a densely packed swarm of birds in the sky behave as a whole autonomous inseparable entity. The size of these coherent domains is fixed by the energy gap existing between the ground state and the focused excited state. For water, this gap is about 12 eV, corresponding to a photon wavelength of about 100 nm. It follows from this full quantum treatment, that liquid water should be considered as a nanostructured medium and not as a homogeneous random liquid. Unfortunately, such a theoretical prediction seems to be in complete disagreement with NMR [18,19] or neutron diffusion [20,21] measurements that rather points to a homogeneous structure down to the molecular scale in the form of a flickering network of hydrogen bonds. In fact, some other spectroscopic measurements such as IR [22,23], Raman [24], X-ray absorption [25] or diffusion [26-28] (but see [29] for a refutation) really need a two-state model for liquid water, supporting the coherence domain picturing. This lack of agreement among scientists working in the field of water is one of the main reasons why the coherence domains hypothesis is still considered with suspicion. Quite recently, new quantum modeling has clarified the situation by demonstrating that coherence domains cannot form in uniform 3D systems such as pure liquid water owing to the fact that the minimum number of electrons needed for stability in 3D is much greater than the number of atoms N, so stability cannot be achieved [30]. However, the model does admit the appearance of coherent structures on the surface of existing mesoscopic volumes. This means that coherent water domains may occur in bulk liquid water provided that nanobubbles or other colloids are present. It follows that quantum effects coming from vacuum fluctuations leading to coherent domains formation should really needed to be taken seriously into account before attempting modeling liquid water properties. Neglecting the influence of the vacuum and the coherence domains structure in the presence of nanobubbles should then be responsible for the contradictory and controversial results reported in past literature where the exact status of dissolved gases were not studied. It is worth stressing that concepts such as atoms, ions, molecules and bonds are conventional ways of thinking that replaces the abstract matter/vacuum coupling through exchanged virtual photons by more tangible things having a characteristic size or length (atomic or ionic radii as well as bond lengths). A characteristic length λ measures in fact the spatial range where quantum phase coherence exists among nuclei and electrons. To this characteristic length is automatically associated a characteristic frequency f = ∆E/h, ruled by the depth of the coherence gap ∆E created when matter, vacuum and photons remain locked together oscillating in phase at this characteristic frequency f. It is worth noting that such a frequency is systematically red-shifted relative to the frequencies f’ = c/λ of the virtual excitations that are responsible for emergence of coherence. In water for instance, it is virtual excitations in the UV domain (∆E ≈ 12 eV, λ ≈ 100 nm, f’ ≈ 3 PHz) that gives rises to hydrogen bonding involving mid-IR virtual photonic exchanges (∆E ≈ 200 meV, f ≈ 39 THz, λ ≈ 10 µm) between water molecules. Accordingly, let’s us consider an electron belonging to a given water molecule borrowing a virtual photon of energy ∆E from the vacuum. Following (3), the maximum lifetime ∆t of this virtual photon is given by ∆t ≈ ħ/∆E. During this time, the virtual photon is able to propagate at maximum distance ∆x ≈ c/∆t, carrying a momentum ∆p ≈ ħ/∆x towards another electron belonging to a neighboring water molecule. Consequently, the force existing between the emitter and the absorber is given by: f = ∆p/∆t ≈ ħ·c/(∆x)2. Now introducing the dimensionless Sommerfeld’s fine-structure
constant α, measuring the coupling of the virtual photon with other vacuum virtual particles that may be encountered during the distance ∆x travelled in the vacuum (figure 3) leads to a force f = α·ħ·c/(∆x)2. This is just the classical Coulomb’s force between two electrons belonging to two water molecules that are close enough to emit and absorb a virtual photon during the time window framed by the quantum-mechanical uncertainty relationships. Averaging this kind of electromagnetic interactions over all electrons and all water molecules in a coherence domain explains the occurrence of a specific interaction named “hydrogen bonding”. This of course explains why water, despite its quite low molecular weight, is a liquid and not a gas, as well as why density increases upon heating the liquid from T ≈ 230K up to T = 4°C [31]. The key point behind this so-called “density anomaly” is the active participation of the physical vacuum through many-body electronic interactions that cannot be reduced to mere pairwise interactions between fictive objects named “water molecules”. The physical quantum reality lying behind liquid water is thus like an inhomogeneous highly correlated electron gas stabilized by atomic nuclei over distances two or three order of magnitude larger than the characteristic size of the molecular object. This long-range collective behavior is a characteristic signature of the emergence of quantum coherence among electronic clouds.
Figure 4: Morphogenic water is a generic term covering all the cases where water molecules adopt a 4-monolayers 2D structure in order to hydrate any kind of organic as well as inorganic matter. Coupling with virtual photons having a wavelength of about 100 nm allowing reaching an excited state located 10 eV above the ground-state energy of water molecules defines a mosaic of coherence domains welding together about 300 000 water molecules and able to hold about 1 Go·cm-2 of information.
Understanding high dilution effects We are now in position to discuss what may happen when a concentrated solution is diluted far below the Avogadro’s threshold. In fact, quantum-mechanically speaking there is absolutely no dilution, because the density at every step remains that of a liquid. The only thing that happens upon diluting a solute is that one replaces a quantum coherence ruled by the solute to another quantum coherence ruled by the water. Another way of saying that is to consider that when the solute concentration is high, the concentration of water coherence domains is low. Upon dilution, the concentration of the solute decreases, meaning that the concentration of water coherence domains increases. The higher the dilution of the solute, the higher the number of water coherence domains. As shown in figure 4, for an excitation energy ħω ≈
10 eV, a coherence domain gathers about 300 000 water molecules, welding them together as a single quantum object sharing a definite common quantum phase with an internal fluctuating infrared electromagnetic field. It is this quantum object that could be manipulated during dilution. Depending on the chemical nature of the solute, a given characteristic partition should be established between the solute, the coherence domains and the other water molecules that remain in a incoherent state. It is worth noting that water molecules in this incoherent state share necessarily the same spatial extension (about 100 nm) as water molecules in a coherent state. The difference is that is the incoherent state; attraction between water molecules involves the van der Waals interaction that is much weaker than the hydrogen bonding. Another consequence is that the density of incoherent water is little bit higher than that of coherent water as in this last case, water molecules are a little bigger owing to their coupling with the quantum vacuum. This supramolecular organization of water molecules into coherence domains has several important consequences. First, as there coherence domains cannot be tridimensional, they cannot be observed in pure liquid water. In order to self-organize, another substance that is not water should be present. For liquid water, best candidates are nanobubbles of gas and/or nanoparticles. This obviously explains the importance of succussion in homeopathic preparations. Without vigorous agitation of the liquid, incorporation of atmospheric gases as nanobubbles would be not very efficient. Similarly, dispersion of nanometric colloidal particles arising either from the preparation (lactose) or from the walls of the container would not be possible. The fact that nano-objects are systematically detected at any homeopathic dilution fully confirms this viewpoint. As coherence domains are welding together water molecules through virtual photons, it follows that water should be highly sensitive to electromagnetic fields in certain frequency ranges (mainly ultraviolet and far-infrared). Moreover, as electrons form coherent trapped plasma, coherence domains may behave as little magnets, rendering water also highly sensitive to magnetic fields and sensible to radio waves. This readily explains the well-documented effects of magnetic fields and radio wave upon crystallization, fouling, biofilm growth, etc… By the way, this could also explain the large increases of chronic degenerative pathologies that could be induced by electromagnetic smog increasing without limits nowadays. It follows also that homeopathic preparations should be done in a place where any source of electromagnetic field has been carefully controlled. Now, let’s consider how one could imprint in water an information pattern coming from a solute. A first possibility would be a mosaic structure where coherence domains alternate with incoherent domains of roughly the same size. Figure xx gives an schematic representation of this hypothesis where one could affect the number ‘1’ to a coherence domain and the number ‘0’ to an incoherent domain. This transforms water into an array of bits that could memorize some information. In order to evaluate the amount of information that could be stored on this morphogenic water, one may assume a structure of 4 layers of water molecules, leading to a layer thickness b ≈ 1 nm (figure 4). It then transpires that on square centime of morphogenic water is able to hold about 1 Go of information. In order to understand the meaning of this number, let’s assume that a cell surrounded by its lipid bilayer has an area of about 500 µm2. As a membrane has two sides, each square centimeter is able to hold 2 Go of information, leading to a storage capacity of 500×10-8×2×109 octets ≈ 10 000 octets = 10 ko per cell. As human beings, our body contains about 3.7·1013 cells, meaning that the whole body has a
storage capacity of 3.7×1013×104 ≈ 4×1017 octets = 400 Po (1 Po = 106 Go = 1015 octets). Taking into account that five minutes of video H.264 (MP4 format) correspond to 4 Go of data, the human body is able to memorize through its membrane 5·108 minutes of video H.264. As one year is about 525 600 minutes, a human body may be though as a hard disk storing 951 years of video H.264. Alternatively, one may consider that at least 109 bits per seconds coming from our environment are reaching our body. With our 400 Po of storage capacity in membranes, this translates into 3.2·109 seconds ≈ 100 years of human life as 1 year ≈ 3.1·107 sec. Consequently, a human body has the capacity to memorize its interaction with the environment during his whole biological life. Ranking the three main organs able to display neuronal activity may also be of some interest (figure 5). Ranking through the number of neurons leads to the order: brain (≈ 1011 neurons), gut (≈ 108 neurons) and heart (≈ 40 000 neurons). Let’s now rank these three organs according to membrane storage capacity. For the brain, with 146 000 neurons per square meter, one gets about 1 m2 for 1011 neurons, i.e. a storage capacity of 2×109×104 = 2×1013 octets = 20 To. For the heart made of 3 billions of cells assuming an area of 500 µm2 per cell, we get a storage capacity of 3×109×104 = 3×1013 octets = 30 To. Finally, the intestinal area being about 200 m2 = 2·106 cm2, the gut is the great winner with its 2×109×2×106 = 4×1015 octets = 4 Po of storage capacity.
Figure 5: Comparative information storage capacities in lipidic membranes for organs displaying neuronal activity, bacteria and mycelia.
It is worth noting that the ability to memorize information in the morphogenic shell of water around any lipidic membranes is not limited to human beings. Any living cell has the capacity of doing that. In particular, it is known that a human being holds within his gut up to 4×1014 bacteria (figure 5), offering a storage capacity of 4×1014×104 = 4×1018 octets = 4 Eo. But the biggest memory displayed by a single organism is found in mycelia, some of them aged of 1 500 years being able to reach a mass of 100 tons corresponding to about 1021 cells translating into a memory capacity of 10 yotta-octets, i.e. one billion Po. These considerations should help to realize that living species are able to store huge amount of information within the morphogenic water wrapping their lipidic membranes. Through our morphogenic water, we are constantly eating and storing not only matter but also a whole range of immaterial information. Through the constant circulation and exchange of information within our body, any part of it
becomes entangled with other ones, giving rise to a quantum or an immaterial dynamic force using Hahnemann vocabulary. If ability to memorize information is crucial for any living entity, the ability to process and transfer it is no more crucial. The most obvious way for processing and transmitting information is to rely on electromagnetic fields that are able to travel close to the speed of light. As shown in figure 3, at the very heart of a water coherence domain, there is an electromagnetic field associated to any electronic transition between the ground state of water molecules towards an excited state having of pulsation ω such that ∆E = ħ·ω. This means that each coherence domain carry a monochromatic electromagnetic wave whose frequency f = ω/2π is ruled by the energy of the excited state involved in the electronic transition. Changing this excitation energy, means changing the associated frequency. Now, it is a well-known result of Fourier analysis that any kind of complex signals may be viewed as a sum of monochromatic waves. This means that any kind of electromagnetic signal, even a highly complex one could be generated if enough coherence domains are available, each of them encoding it own frequency slightly different form the others.
Figure 6: Energetic spectrum of a water molecule according to quantum mechanics. Each energy level displayed here could be used as the excited state assumed in figure 3. It follows that a collection of water coherence domains is able to store any kind of electromagnetic signal, whatever its complexity.
As shown in figure 6, a single water molecule is characterized by millions of excited energy levels depending on the exact electro-vibro-rotation state of the molecule. As the higher the dilution, the higher the number of coherence domains there should be no surprise that the higher the ability to encode the electromagnetic spectrum of a whole DNA, a complex drug and even a complete organ, plant or tree. It follows from this rigorous analysis that any homeopathic remedy may hold through its constitutive 2D coherence domains an immaterial electromagnetic signature that grows in complexity as the dilution increases. In this sense, going well beyond the Avogadro’s limit should not be seen as a problem and pretending that homeopathic preparations contains nothing is just the consequence of a short-seeing attitude focusing only on matter within an non-quantum frame of thinking. Taking into account the obvious fact that our universe is made of matter AND radiations ruled by quantum-mechanicals rules at all scales allows homeopathy to earn its scientific credentials.
Conclusion Within the conceptual frame of thought presented here, action of ultra-low doses should be considered as a real therapeutic alternative that do not “violate” any scientific knowledge but instead perform a specific action at the level of highest biological sensitivities and complexities. More specifically there should be no surprise by finding that homeopathic remedies could turn some important genes on or off, initiating a cascade of gene actions to correct the gene expression that has gone wrong and produced the disorder or disease [32,33]. It follows that the immune system, inflammation mechanisms and leukocytes are among the targets of homeopathic effects. Efforts trying to ban homeopathy from a rational medicinal practice [34] are thus completely pointless [35]. Worse, trying to block funding for homeopathic research on the ground that homeopathy = placebo, should be considered as a criminal attitude. In homeopathy, the self-healing capacity of the body is exploited at its full capacity, with a full avoidance of secondary effects, a problem that plagues allopathic practice. 
As currently, the majority of diseases are multifactorial processes, homeopathy with its multifaceted approach linked to the possibility of using ultra-high dilutions, should be considered as a viable answer to complex diseases, such as diabetes, schizophrenia, cancer, atherosclerosis, involving hundreds of genomic variants that interact with one another and with environmental factors. More generally, considering that everything is entangled, a quantum-mechanical thought is mandatory when considering living biological entities. This is the challenge facing any physician or biologist wishing to really heal people from their diseases.
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