Lionel Milgrom's latest paper, “A New Geometrical Description of Entanglement and the Curative Homeopathic Process” [1], as introduced by Alex Hankey (“Self-Consistent Theories of Health and Healing” [2]) quotes Hahnemann saying that

“The unprejudiced observer is well aware of the futility of transcendental speculations which can receive no confirmation from experience.”

Milgrom's futile transcendental speculations have been going on for six years. This latest paper is light on equations but heavy on pictures and mysticism and further from science (and indeed reality) than ever. But it's still possible to find some things which are meaningful enough to be wrong.First we find complaints and special pleading to be allowed to overturn evidence-based medicine (EBM) and the double-blind randomized-controlled trial (DBRCT):

“EBM and the DBRCT, like much of biomedical science, are rooted in the reductionist philosophy of logical positivism combined with local realism. The latter states that: (a), the universe is real and it exists whether we observe it or not; (b), legitimate conclusions and predictions can be drawn from consistent experimental outcomes and observations; and (c), no signal can travel faster than light [3,4,5]. In questioning (a) and (c) above, quantum theory transcends local realism [4] and the reductionism of biomedicine [5]. Attempts at explaining homeopathy's efficacy have made use of concepts generalized from the discourses of semiotics [6,7] and quantum theory [8,9,10].”

EBM just means that someone's checked that it actually works - if it could be demonstrated that homeopathy worked at actually curing diseases then it would be part of EBM. In fact anything in CAM would quickly become part of EBM if it worked. What we actually get is subjective reports of improvements in self-limiting or cyclic conditions, while journals publish flawed, biased articles on effects at the fringes of statistical significance [11,12,13,14,15].

There's nothing magical about DBRCTs either, they are just the most rigorous way of trying to sort out if there is actually any weak effect there. If homeopathy really worked as well as its proponents seem to suggest then the results should be blatantly obvious and there would be no need to dig so hard to find them. Let's pedantically consider each letter.

- T for Trial
- You have to test something to make sure that you aren't just remembering the positive anecdotes and forgetting the negative ones.
- C for Control
- You compare your treatment group with a group receiving no treatment, to make sure that it's really the treatment having an effect. It's usual to give the control group a placebo.
- R for Randomized
- To make sure that the patients in the treatment and control group are similar, so that similar disease progressions would be expected in each group if the treatment were ineffective. Otherwise you could deliberately or subconciously put the healthier people into the treatment group and then of course they likely to be healthier at the end of the trial. It's good to have large groups.
- B for Blind
- The patient shouldn't know whether they are getting the treatment or the control because this could bias their self-reported symptoms and also their expectations.
- D for Double
- The doctor shouldn't know whether a patient is in the treatment or control group either, or else he or she can deliberately or subconciously influence the patient.

A DBRCT is just the best way of minimizing all the possible biasing factors in the case that the effect of the treatment is less than blatantly obvious. So it's not surprising that good quality DBRCTs tend to come out negative for homeopathy while less well-controlled trials show positive effects - that shows exactly that the positive effects of homeopathy are nothing to do with the remedies themselves [16,17,18,19].

And then, on with the entanglement, as if I haven't already explained how the Greenberger-Horne-Zeilinger [20] system actually works, or why most of what he says about entanglement isn't correct.

“Entanglement is said to occur in a quantum system when its seemingly separate parts are so holistically matched or correlated, measurement of one part of the system

instantaneously(i.e., not limited by the speed of light and without classical signal transmission) provides information about all its other parts,regardless of their separation in space and time, or their size[21].”

Italics are his. I've tried to explain entanglement in other posts, and I've tried to clarify that “size” doesn't mean “number of interacting particles” since even maintaining *seven* nuclear spins in a state of coherent quantum phase is quite hard [22]; macroscopic coherent states do not persist for very long at all [23,24]. Superconductors and superfluids work because of ways in which the particles in question are prevented from interacting [25,26]. Apparently,

“the Memory of Water [27] also relies on macro-entangled coherence, albeit between large numbers of water molecules [28].”

which isn't true at all, and not just because there is no such thing [29]. The Memory of Water is supposed to be a physical effect whereby the structure of a sample water depends on what used to be in solution in it: it's nothing to do with coherence of the quantum phase. (Del Giudice *et al.* [28] seem to be talking about coherence of dipoles in an electric field, not coherence of the quantum phase.) Milgrom then goes on to explain:

‘Nonlocal correlation is not the only prerequisite for entanglement. A quantum system's processes must also be describable in terms of a “non-commuting algebra of complementary observables.” [4]’

All this means is that it matters what order you do certain pairs of measurements in, since eigenstates of one operator are not eigenstates of another. I just found the quote marks interesting, as if he's pasted that in without knowing how to explain it. To be fair, I can't be bothered to explain it either. But this complementarity means, according to Milgrom, that

“To fully explain quantum phenomena, therefore, it is necessary to have two different but complementary concepts. The answer one obtains performing two different sets of observations depends entirely on the order in which they are performed; yet

bothare necessary in order to acquire a complete picture of the system.”

A “complete picture of the system” is not actually possible in these terms. It is impossible to have a system in two complementary states at the same time. A “complete picture” in terms of macroscopic variables (such as position and momentum) therefore does not exist. We just have the idea that there's a wavefunction which exists but is not directly observable, on which we can operate in various ways in order to obtain observable results.

Having misunderstood and misrepresented quantum theory, Milgrom now goes on to do the same for weak quantum theory (WQT) [30]. Leick has already pointed out [31] that

‘Milgrom writes “Complementarity and indeterminacy are epistemological in origin not ontological”, [5] which is a serious misquote of the original paper, where it says that “[...] there is no way to argue that complementarity and indeterminacy in weak quantum theory are of ontic rather than epistemic nature.[...] one would expect them to be of rather innocent epistemic origin

in many cases.” [30] The difference between the two versions cannot be emphasized enough, as quantum effects such as entanglement are due to the ontic nature (ie not simply to our incomplete knowledge) of complementarity and indeterminacy!’

But what does it mean that WQT “relaxes several of its nanoscopically limiting axioms, including dependence on Planck's constant.”? Planck's constant *h* is what connects quantum theory with reality - it turns out that light comes in photons and the energy of each photon is proportional to the frequency of the light, with the constant of proportionality being *h*. This is how Planck was able to solve the problem of black-body radiation. If “complementarity and entanglement are not restricted by a constant like Planck's constant” then what do we have in its place, to connect WQT with reality? The simple answer is that there is no connection to reality so it's not even a sensible question. The more involved answer is that “WQT has no interpretation in terms of probabilities” which amounts to more or less the same thing. How can Milgrom then write that “the product 〈Ψ_{PPR}|Ψ_{PPR}〉=|Ψ_{PPR}|^{2} presumably represents the probability of cure”? (If Ψ_{PPR} is properly normalized then 〈Ψ_{PPR}|Ψ_{PPR}〉=1 and it says nothing about the “probability of cure” or anything - to find that he’d have to define an “cure” operator and calculate its expectation value.) By the way, it's often more convenient to work with ℏ = *h*/2π so you'll see that in some equations later on.

I have already wondered what use WQT would be in answering objective questions like “does homeopathy work?” if it doesn't seem to have any interpretation in terms of observables. Medical effects *are* quantifiable. Anyway, Milgrom then goes on to introduce Walach's use of semiotics [7] and there's a box-out which contains the unintentionally ironic Hahnemann quote. Semiotics is more linguistics than science, it's got no place here. The way we interpret signs and produce meaning has got nothing to do with the molecular biology of how actual pharmaceuticals work.

The rest of the quote in the box-out explains that the observer

“can take note of nothing in every individual disease, except the changes in the health of the body and of the mind (morbid phenomena, accidents, symptoms) which can be perceived externally by means of the senses... All these signs represent the disease in its whole extent, that is, together they form the true and only conceivable portrait of the disease.”

The first part of that may have been true a couple of hundred years ago but it isn't true now. The second part was never true: we now know about germs, viruses, genes, DNA and molecular biology. Symptoms are part of the body's reaction to an underlying pathology. They are not the pathology itself. The same pathology can present in different ways in different people, and many symptoms are shared between different diseases.

A few kets finally turn up now, as Milgrom once again formulates his patient, practitioner and remedy wavefunctions. He then decides to attach one of Walach's semiotic sign-object-meaning triangles (each corner of which seems to represent an operator or possibly the expectation value of it) to each of the three corners of the patient-practitioner-remedy triangle. It's meaningless, but where it becomes actually wrong is in the invocation of complex numbers and a strange sort of quantum origami. Already in part C of his Fig. 2 the bra-ket notation seems to have broken down - and how he manages to fold the “corners of the large triangle to create a pyramid with a hexagonal base” is beyond me, since a pyramid with a hexagonal base needs six sides and a triangle only has three corners. This folding appears to have turned the states into their complex conjugates, but then Milgrom reflects the whole thing so that it's upside down and then unfolds it and it turns out to be twisted through 60°. How is that supposed to happen? It's nonsense mathematically (not to mention scientifically) and I don't even think it makes geometrical sense. Which directions are real and which are imaginary doesn't seem to be made clear for fairly obvious reasons - taking the complex conjugate means mirroring in the Real line but each of the three corners is flipped over a different line in the 2-d plane, and then the whole “pyramid” is mirrored in the whole 2-d plane which apparently represents the “homeopathic operator, Π*r*”. This is all I suppose taking place in the

‘“therapeutic state space” [32] (an analogue of the complex mathematical Hilbert space more familiar from orthodox quantum theory) [4].’

In the nicest possible way, how many readers of J. Alt. Complement. Med. are familiar with Hilbert spaces? He seems to think that in an equation such as 〈Ψ_{PPR}|Π*r*|Ψ_{PPR}〉=⟨Δ *S* *x*⟩ that it's the operator which is making the complex conjugate 〈Ψ_{PPR}| out of |Ψ_{PPR}〉, which just isn't the way it works at all. (Anyway, if you fold over the corners of an equilateral triangle so that you are left with a regular hexagon, the triangles will meet in the middle when they are flat against the hexagon - the pyramid they define has zero height. And each wavefunction exist in its own Hilbert space so I don't know what it's supposed to mean to put them all in each others’ spaces.) It's hardly worth looking at his Fig. 3 where he does it all again only with tetrahedra. The lack of any explicit conceptual difference between Figs. 2 and 3 demonstrates how arbitrary and meaningless it is, since he can apparently produce two completely different pictures to represent what is supposed to be the same things, and this makes it useless trying to work out on which level to take it seriously - there isn't a level on which it makes any sense. (There are probably lots more versions of this quantum homeopathic origamy nonsense coming soon to “peer reviewed” journals with low editorial standards near you.)

Meanwhile, there's a second box-out on the Kochen-Specker theorem [33]. This is a theorem which says that it's not possible to find a direct correspondence between quantum mechanical observables and classical quantities. The first half of the box seems to be ok, up until the part where he he claims that

‘signs and symptoms of disease are considered observable manifestations of an “invisible” disturbed vital force, Vf.’

This is apparently because Auyang [4] said

“Eigenvalues are analogous to symptoms of a disease, which are disturbances of the body that show up and indicate something that does not show up. Just as a cold persists though its symptoms are suppressed, so a quantum system's wave function has a definite amplitude, even though it has no eigenvalue...”

and Milgrom has taken this analogy far too seriously. Common cold viruses are not invisible. (I'm not sure what “no eigenvalue” means in this context either: is it that the eigenvalue is zero or that the state is not an eigenstate? Measurement is supposed to collapse a mixed state into an eigenstate.) There's a mention of self-adjoint operators, which are those operators which operate on states to give physical observables. (There are, for example, ladder operators which operate on states to give new states.) It's not exactly true that “they consist only of real numbers” because for example the momentum operator for the *x* direction is −*i* ℏ ∂/∂*x* - rather, it means that the operator is a Hermitian matrix which is equal to its own conjugate transpose and it has real eigenvalues (but see also the spectral theorem).

The Kochen-Specker theorem [33] knackers hidden variable theories, in which the quantum mechanical correlations leading to entanglement are explained by theorizing that the system somehow already “knows” which state it's going to turn out to be in when you measure, even if this information is not available from the wavefunction. It turns out that you can't have definite values of all the hidden variables corresponding to quantum mechanical observables all the time which are independent of the way in which they might be measured. This is because for classical quantities it shouldn't matter in what order you measure certain properties, but for certain complementary pairs of quantum mechanical observables it does indeed matter in what order you measure. This is actually only a problem if the Hilbert space has three or more dimensions [34], and Milgrom decides that since his homeopathy Π*r* mirror is a 2d plane, so the “therapeutic state space” is this 2d plane on which the Kochen-Specker theorem need not apply. In fact he's drawn his mirror as a 2d plane embedded in a 3d space, and if he wants a pyramid which goes upside-down then he needs at least a 3d Hilbert space to do it in. It's clear that in the real world there are wavefunctions which really do “exist” in Hilbert spaces with three or more dimensions out of which observable quantities can be extracted with the appropriate measurement operators: the theorem just says that these very observables were not some how “in there” before we did the extraction. What comes out actually depends on the interaction between the measuring operation and the wavefunction, so the intrinsic properties of the wavefunction (and I maintain that it *does* have them) are not those which correspond exactly to things we are intuitively familiar with, such as position or momentum. So I don't think that the Kochen-Specker theorem is particularly relevant to what Milgrom is trying to do, and he wouldn't be able to get around it anyway because he's working in 3d not 2d. (What he's drawn isn't a Hilbert space anyway: states exist as rays in a Hilbert space, not polygons.)

On to Fig. 3 anyway. As I mentioned, for some reason this time he folds up the big triangle into a tetrahedron. Does this represent a mathematical transformation of some kind? (No.) There are no brakets arounds the Ψs this time, perhaps that's the difference. The practitioner has a wavefunction Ψ_{Pr} and therefore a triangle, but then apparently “sits at the center of tetrahedron” too. There's clearly no special reason for this apart from Milgrom wanted it that way and thereby made it up (and in the text it's “the patient notionally at the tetrahedral epicenters”.). And then of course the practitioner also has an operator Π*r* which is supposed to be a mirror which somehow also twists the tetrahedron in a way which doesn't make a huge amount of sense (and I don't think this is a self-adjoint operator if it flips between these two states). Then there's another box-out regarding chirality and there's nothing wrong with it, apart from that it's almost totally irrelevant, only serving to remind us that Milgrom used to be a chemist.

The final step is to combine the original tetrahedron and the twisted one into the shape called the stella octangula which Hankey got so excited about. (But he also folds up the big triangle into a small flat triangle which apparently introduces a 60° twist. I don't think he runs with this; he was just getting carried away. I don't know why the Ψs have now moved to the corners where previously we had operators.) The twisting is supposed to be the practitioner showing the cure to the patient or some nonsense like that. It's not a real-space twist: it doesn't matter which way the patient is “looking” or “going”. States evolve through Hilbert space according to the time-dependent Schrödinger equation:

HΨ = iℏ |
| (1) |

where the left side has the Hamiltonian operating on Ψ (which classically involves the kinetic and potential energies, where the former involves taking derivatives with respect to space - stationary states are energy eigenstates) and the right side involves taking the derivative with respect to time. (This equation is completely deterministic, by the way.) How should we describe pointing “the patient in the direction of cure” now exactly?

The problem I always have with Milgrom is that I try to read it as if it were science. I assume that there's sense and meaning in there but the concepts are difficult and require work to get to. The problem is that there's no sense or meaning, and I end up doing a lot of work trying to get the right level into focus when there *is* no right level. It's meaningless. I don't think it's even correct geometrically. It's nearly finished though so that's good.

We only have to deal with the stella octangula's role in quantum teleportation first [35,36,37,38]. I'm not interested in the stuff about the Platonic solids or the “classical four elements”, or the Merkabah. (Read Finding Moonshine if you want a more sensible discussion about symmetry and that.) It's the link back to quantum mechanics which is more troubling, since some might see that and think Milgrom's on to something. Let me assure you he isn't. The picture which Aravind [38] draws is a representation of operations described by Bennett *et al.* [36] when dealing with a entangled state of *two* spin-1/2 particles - it gives a way of understanding which combinations of spin states are more entangled than others, or something. The corners of a tetrahedron *A*, *B*, *C* and *D* represent four Bell states while the centre *E* represents a totally unpolarized state. Aravind explains:

“The twirl operation can also be visualized readily on the Horodecki diagram. The effect of a twirl on an arbitrary Bell diagonal mixture is to project it orthogonally onto the line

AEcontaining the Werner states. For a non-separable state in theA-sector of the tetrahedron, this reduction is achieved without any loss in entanglement but for states in theB,CandDsectors there is a complete loss of entanglement. The proper way to reduce the latter states is to either subject them to a modified twirl [36] that projects them onto Werner-like states in their own sectors or else to transfer them into theA-sector (by a suitable unilateral rotation) and then apply the standard twirl.”

There's an octagon embedded in the tetrahedron, formed by the intersection of the tetrahedron and its inverse, within which lie all the separable states. How does this compare to Migrom's picture? Milgrom built up his intersecting tetrahedra from at least three “particles” so he would need a different shape (probably in more than three dimensions) to represent all their states; the centre, representing complete unpolarization and being the most unentangled state in Aravind's picture, is the patient (probably) in Milgrom's picture, but the patient is also a face; in Aravind's picture the vertices of the tetrahedron represent maximally-entangled Bell states, while Milgrom seems to have expectation values or operators or something. So it's clear that just because he has contrived to arrive at the same shape doesn't mean that he's somehow doing something connected to what these guys are doing. (It may not be a total coincidence either that Sandu Popescu [39] is acknowledged by Aravind [38] and cited by Milgrom [40] in his reply to Leick [31].)

To conclude, then: in order to avoid facing the fact that quantum mechanics is simply not relevant to the system of a homeopath and a patient [41], Milgrom concludes that the “state functions representing each of the Px, Pr, Rx, and the PPR entangled state are not related to quantifiable physical observables”, admitting how useless it all is for actually working anything out; but when he states that “it is clear that the nature of the therapeutic process requires its initial separation and ‘isolation’ from the usual external environment, as a necessary prerequisite for the coherence of entanglement to occur, and cure to begin,” he admits something I think we already knew: that it is necessary to be out of touch with reality to be a homeopath.

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This document was translated from L^{A}T_{E}X byH^{E}V^{E}A.