"Quaestionem de mensione expellas furca, tamen usque recurret." Horace&IV, Epist., I, 10, 24 ("You may chase the measurement problem away with a pitchfork, yet in the end it will come back.")
2. A theoretical model: resonance, basophils and succussion
This is an attempt to clarify, to myself as well as to the reader, some ideas on the theoretical issues which are at stake in the debate about the memory of water. The antibody's amplitude in any sample being extremely thin, one may wonder how such a small perturbation of the null-state may trigger the basophiles' response. In my opinion this question arises from the attempt to apply a classical paradigm to a quantum phenomenon. I will propose here an alternative picture.
Among all the possible space-time paths of the aqueous solution there is a subset, a very small one, that leads to a state where at least one antibody molecule will be measured in the selected sample. This is the same as saying that there is a very small probability that an antibody molecule may actually be found in a selected sample at the end of Benveniste's experiment. For this subset the results of Benveniste's experiment pose no problem.
The subset is very small. This is the problem, which in Nature's Editorial Note is formulated as the solution being "diluted to such an extent that there is a negligible chance of there being a single molecule in any sample". Does this mean that the probability of the basophils reacting to the antibody's presence is equally small? No, as I will try to explain.
The system is governed by a Schrödinger equation whose Hamiltonian contains an interaction term between basophils and antibody, which we may assume to be null if the molecule's amplitude is below a certain threshold. The interaction term induces a transition probability between the unperturbed states (state-1) of the basophils and the state where they will degranulate (state-2). We can imagine that the basophils relie on sensors, i.e. appropriate molecules, to detect the presence of the antibody. This corresponds to the current orthodox explanation based on the ligand-receptor docking model. The presence of the antibody amplitude will be detected by the sensors, which will switch on an activated state corresponding to the presence of the antibody. The amplitude of the activated state will be proportional to the amplitude of the antibody in the sample, i.e. very small. The point now is that the amplitude of the excited states of the basophils' sensors may be boosted by an appropriate resonance mechanism, triggered by the Hamiltonian's interaction term. This means that once the excited amplitude is triggered by the antibody's amplitude, it may be be pumped up by an appropriate resonant device. We may say that the presence of the small amplitude "dopes" the system, in a similar way as the presence of impurities dopes certain materials in solid state physics, radically modifying their properties. The process of "amplitude amplification" is a hot topic in quantum computing, based on the on previous work by Shor, Grover and others. Actually my current considerations were originally inspired first by Shor's, then by Grover's algorithm. According to my conjecture, the mechanism on which the basophils relie to detect the antibody's presence would be the analogous to the device underlying Grover`s algorithm for databank search. The mechanism required here is quite simple. It is well-known that any two-state system with non-null tunneling coefficients between the two states will undergo a periodic shift from its state-1 to its state-2, irrespective of its initial condition. A textbook-instance of such a process is given by the ammonia molecule. In this setting, degranulating basophils represent state-2 of the system, where the non-null tunneling coefficient is induced by the presence of a small but non-null antibody amplitude. I provide here a technical appendix, where a sketch a simple model of a device amplifying the impact of the antibody amplitude so as enable the basophils to detect it.
In this framework the need to shake the solution (or succussion, as it is called in homeopathy) is clear, since the turbulent mixing induces spreading and filamentation of the antibody's wave-packet through interaction with other molecules' wave-packets. Without shaking the wave-packet may remain confined in a small region of the solution and fail to become entangled with the basophils' sensors. From an heuristic point of view it is convenient to visualize the antibody's wave-packet as a continuum of weighted copies of the antibody's molecule, dispersing and interacting with the water molecules' wave-packets. The antibody's amplitude diffusion takes place locally along the same lines as the spread of the classical probability distribution of a body subjected to the Brownian motion of smaller particles. The amplitude is then spread around by turbulent mixing. The basic assumption here is that the antibody molecule will undergo scattering induced by prolonged interaction with water molecules. It is worth noting that macroscopic delocalization of relatively massive objects, such as rubidium atoms, has been experimentally verified (Dürr&al, Nature, 395, 33. 7/2/2004: Recent experiments exhibit the wave nature of biological molecules such as porphyrins).
Since we are not able to write down the Hamiltonian for the aqueous solution, the above "explanation" is highly speculative, based on the assumption that the sensors pick up the small antibody amplitude, giving rise to an interaction term in the system`s Hamiltonian. The role of basophiles however is precisely to detect the presence of antibodies and react accordingly. The idea that they may relie on a quantum mechanical device to implement their existential function seems fairly natural. Turbulent mixing may create a uniform amplitude distribution, facilitating amplitude concentration by a quantum-Fourier-transform-like detection device.
In this setting it is worth recalling how one of the founders of modern physics described the quantum: "... this wave is not the reality. This wave is a probability -- this wave is a tendency. [...] this idea of the wave field being a tendency was something just in the middle between reality and non-reality. So you invented the old Aristotelian term of possibility or potential. At least it was something in the middle between the actual fact and the non-fact. And it was the tendency that a fact should happen. That was the striking thing about it, you know, this new invention of a possibility which was a reality in some way but not a real reality -- a half reality." [Heisenberg to Kuhn]
An aspect of the model described here is the idea that turbulent mixing spreads the antibody's amplitude. If this hypothesis is correct it should equally apply to any molecule or indeed to any object, under the proviso that the time needed to spread the amplitude may get unmanagably longer for a bigger object. The model can be applied and tested in situations where the antibody's wave-packet acts on its environment without being localized. In other words the antibody acts only as a cathalizator. This suggests that the antybody's amplitude may conceivably be detected using Nuclear Magnetic Resonance (NMR) techniques, analyzing the changes induced on sample molecules by their environment. If the solution is modified by the presence of the antibody's non-null amplitude, its effect may be detected through NMR analysis of the sample molecules.
A quantum model for homeopathy has been proposed by H.Walach.