Several days ago, Rolfe was kind enough to provide me with links to several papers which purport to explain how high-dilution homeopathic remedies work in terms of quantum mechanics.
Although I've been busy this week, tonight I finally got to the first of the bunch: Weak Quantum Theory: Complementarity and Entanglement in Physics and Beyond by Atmanspacher, Romer, and Walach. This paper is cited by most or all of the others, so I decided to look at it first even though it does not discuss homeopathic remedies. This is a bit long, and I might be the only person interested in it, but what the hell.
The Authors
I don't know much about the authors or their qualifications, so I won't say too much. Atmanspacher mentions PEAR on his website. Walach is the author of another truly surreal paper which I may discuss later.
1. Introduction
In the introduction to the paper, the authors discuss their motivations for developing Weak Quantum Theory. Basically, they are attempting to formulate a theory of measurements of systems which is not restricted to physics, but which retains certain features of quantum mechanics, specifically, entanglement and complementarity. They claim that the theory might be applicable to psychology, philosophy and "psychophysical situations" whatever that means. Perhaps the latter refers to the paranormal, but it is not explicitly stated.
2. Complementarity and Entanglement: some examples
The authors are fascinated with the quantum mechanical phenomena of complementarity and entanglement. Unfortunately, they conflate these two somewhat, which turns out to be a problem later.
Complementarity is one way of describing incompatible measurements. The best illustration is that old standby, the Heisenberg uncertainty principle: in quantum mechanics, you can't precisely know both where a particle is and what its momentum is simultaneously. Measuring one disturbs the other. Actually, this can be illustrated with a strictly classical example: a wave with a definite frequency (which corresponds to momentum in quantum mechanics) is of infinite extent, whereas a spatially-localized pulse requires adding up waves of a bunch of different frequencies, so that "the" frequency is not precisely defined.
Entanglement, on the other hand, refers to the phenomenon of correlations between what appears to be seperate parts of a system. The classic example is spin correlation. When two particles are produced as a pair, and fly off in different directions, measuring the spin of one particle will determine the spin of the other particle. This is because of conservation of angular momentum, in this example: the total spin after the particles are produced must equal the total spin before. Entanglement is only mysterious because the actual value isn't determined until the measurement is actually performed, but the two particles will be consistent even if there isn't time for a signal to get from one particle to the other. This is what is meant by non-locality in quantum mechanics.
When you put entanglement together with complementarity (or more precisely, incompatible observables) you get interesting correlations between two different measurements on the pair, but that doesn't mean entanglement and complementarity are the same thing.
In any case, enough of that digression; back to WQT.
The authors begin by explaining, to some degree, the familiar complementary relationship between position and momentum. They do allude to entanglement, and briefly switch to the example of photon spin, but do not really explain the difference between the two properly. They also mention energy-time complementarity.
One particularly embarressing error is referring to "spin-1/2 systems, i.e. spin measurements on photons." Photons are spin-1 particles; electrons (for example) are spin-1/2.
The example of frequency-time uncertainty (which I mentioned above) is also introduced. This is the classical wave example I described above. Mathematically speaking, this is very much like position-momentum uncertainty in simple 1D, linear systems, since position and momentum are actually related by Fourier transforms as well. Like position vs. momentum, going from the time domain to the frequency domain resembles (in fact, is) a change of basis in a vector space. When (loosely) applied to quantum mechanical particles, you can multiply frequency by Planck's constant to get energy, and this gives you energy-time uncertainty. The authors go on to say that this doesn't imply anything like quantum mechanical entanglement, but of course it doesn't: incompatible observables and entanglement are different things.
Before going on to more bizarre examples, the paper mentions some information theory related to chaotic systems; the authors use this as a more significant example, later, so I'll come back to it.
Finally, the section segues to "examples" that have nothing to do with physics at all, or, for that matter, mathematics. References are made to "conscious and unconscious processes," Jungian psychological states, philosophical propositions, and a host of other odd notions. Here this section totally falls apart. Nowhere do the authors explain what complementarity (in the sense of incompatible measurement outcomes) or entanglement mean in the context of these "systems", nor why we should expect these systems to have such features even if they were meaningful ideas. The authors do not even motivate the inclusion of these subjects, let alone develop them. They are simply dropped in.
3. Algebraic quantum theory in a nutshell
The authors (actually, I expect it was just one of them) then go on to summarize a formal theory of quantum mechanics in terms of algebras of linear operators. This is a bit more formal than the physicist's usual Dirac notation, but if you're familiar with the subject, it is pretty easy to pick up the notation and perform the mental construction of a Hilbert space representation.
4. Weak quantum theory
This ought to be the second most interesting part of the paper.
In this section, the authors define the axioms of a generalized theory of measurements of a system. Instead of defining measurement outcomes as numbers and the spectrum as the set of possible outcomes, they begin with the spectrum as an arbitrary set, and define measurement outcomes, not necessarily numbers, as elements of the spectrum. This is an interesting and unfortunately understated point, and the section would have been much clearer had they been more explicit about it. Once that is understood, the rest of the "set theory" makes sense; there don't appear to be any big surprises.
The authors identify several features of WQT to differentiate it from quantum mechanics.
First, they bring up that it does not contain Planck's constant. This is specious for two reasons. First, their theory is not numerical, it is a set theory, so of course it does not contain numbers. Second, if you go through section 3 (real quantum mechanics) you won't find Planck's constant there, either. This is because Planck's constant is not actually fundamental to the theory - it's a matter of how the operators are defined. In fact, Planck's constant is a matter of units. Physicists occasionally use "natural units" where <del>h</del> and c are both set to 1 with no dimensions, and measure energy and mass in units of 1/length.
Second, the authors point out that observables don't add along with several implications. This pretty much boils down to not having numerical measurements as well, and life would have been simpler had they simply said so at the outset. But this introduces a very serious problem: how are we to interpret the action of an operator on a state variable? The authors don't allow for a probability interpretation without additional axioms (in fact, they explicitly state that some systems don't lend themselves to probabilities, and disturbingly refer to such airy notions as interpreting art or feeling an emotion), so its not clear what the theory would actually mean if you were able to define what your states and operators were. There is, unfortunately, no answer to this problem. The paper never presents any hint of how results in WQT are to be interpreted.
Third, say the authors, there are no Bell's inequalities. Again, this is the difference between sets and numbers.
Finally, the authors show how to add axioms to recover conventional, numerical quantum mechanics.
5. Complementarity and entanglement in weak quantum theory: two applications
The authors finish the paper with two examples. This should be the best part of the paper, because we finally get to see how this new set theory is applied to a real, nonphysical example. Sadly, this section is a complete loss.
The first example, as mentioned above, has to do with information in chaotic systems. This was apparently developed from an earlier paper by one of the authors (Atmanspacher). Although the section is supposed to be about applications of WQT, the new theory is never mentioned or applied here; what math is shown is just more examples of non-commuting operators. This example gives no insight into WQT.
The second example is even worse: countertransference in psychotherapy! The authors do not even make a pretense of showing math here. It is just talk. The authors draw the analogy (metaphor?) of an entangled system consisting of the patient's and the therapist's states of mind. I'm not even sure what this is supposed to suggest: are the authors really implying that some non-material process is responsible for transferring the patient's state of mind to the therapist's? As in telepathy? Perhaps they merely intend to use WQT as a metaphor or model for subliminal, nonverbal cues the therapist picks up on, but if so, this seems awfully obscure. It certainly doesn't give any insight into how to apply WQT or what its good for.
Again, there is no insight gained as to how to apply or interpret WQT given in this example. The paper concludes without showing a single example of constructing observable operators and evaluating whether observables are compatible or not. There are no examples in the paper that don't involve convential linear algebra.
Finally...
To summarize, WQT is a cute little set theory, but it appears to have no utility. Certainly in this paper, there are no examples of applying it, no attempt to explain how to construct operators or interpret outcomes, nothing that shows what the theory is actually good for.
Most importantly, the authors utterly fail to demonstrate that WQT is applicable to any problem they are interested, and for that matter, that the problems they are interested in even require anything resembling WQT. WQT is a toy, and the authors did not even bother to have any fun with it.
Although I've been busy this week, tonight I finally got to the first of the bunch: Weak Quantum Theory: Complementarity and Entanglement in Physics and Beyond by Atmanspacher, Romer, and Walach. This paper is cited by most or all of the others, so I decided to look at it first even though it does not discuss homeopathic remedies. This is a bit long, and I might be the only person interested in it, but what the hell.
The Authors
I don't know much about the authors or their qualifications, so I won't say too much. Atmanspacher mentions PEAR on his website. Walach is the author of another truly surreal paper which I may discuss later.
1. Introduction
In the introduction to the paper, the authors discuss their motivations for developing Weak Quantum Theory. Basically, they are attempting to formulate a theory of measurements of systems which is not restricted to physics, but which retains certain features of quantum mechanics, specifically, entanglement and complementarity. They claim that the theory might be applicable to psychology, philosophy and "psychophysical situations" whatever that means. Perhaps the latter refers to the paranormal, but it is not explicitly stated.
2. Complementarity and Entanglement: some examples
The authors are fascinated with the quantum mechanical phenomena of complementarity and entanglement. Unfortunately, they conflate these two somewhat, which turns out to be a problem later.
Complementarity is one way of describing incompatible measurements. The best illustration is that old standby, the Heisenberg uncertainty principle: in quantum mechanics, you can't precisely know both where a particle is and what its momentum is simultaneously. Measuring one disturbs the other. Actually, this can be illustrated with a strictly classical example: a wave with a definite frequency (which corresponds to momentum in quantum mechanics) is of infinite extent, whereas a spatially-localized pulse requires adding up waves of a bunch of different frequencies, so that "the" frequency is not precisely defined.
Entanglement, on the other hand, refers to the phenomenon of correlations between what appears to be seperate parts of a system. The classic example is spin correlation. When two particles are produced as a pair, and fly off in different directions, measuring the spin of one particle will determine the spin of the other particle. This is because of conservation of angular momentum, in this example: the total spin after the particles are produced must equal the total spin before. Entanglement is only mysterious because the actual value isn't determined until the measurement is actually performed, but the two particles will be consistent even if there isn't time for a signal to get from one particle to the other. This is what is meant by non-locality in quantum mechanics.
When you put entanglement together with complementarity (or more precisely, incompatible observables) you get interesting correlations between two different measurements on the pair, but that doesn't mean entanglement and complementarity are the same thing.
In any case, enough of that digression; back to WQT.
The authors begin by explaining, to some degree, the familiar complementary relationship between position and momentum. They do allude to entanglement, and briefly switch to the example of photon spin, but do not really explain the difference between the two properly. They also mention energy-time complementarity.
One particularly embarressing error is referring to "spin-1/2 systems, i.e. spin measurements on photons." Photons are spin-1 particles; electrons (for example) are spin-1/2.
The example of frequency-time uncertainty (which I mentioned above) is also introduced. This is the classical wave example I described above. Mathematically speaking, this is very much like position-momentum uncertainty in simple 1D, linear systems, since position and momentum are actually related by Fourier transforms as well. Like position vs. momentum, going from the time domain to the frequency domain resembles (in fact, is) a change of basis in a vector space. When (loosely) applied to quantum mechanical particles, you can multiply frequency by Planck's constant to get energy, and this gives you energy-time uncertainty. The authors go on to say that this doesn't imply anything like quantum mechanical entanglement, but of course it doesn't: incompatible observables and entanglement are different things.
Before going on to more bizarre examples, the paper mentions some information theory related to chaotic systems; the authors use this as a more significant example, later, so I'll come back to it.
Finally, the section segues to "examples" that have nothing to do with physics at all, or, for that matter, mathematics. References are made to "conscious and unconscious processes," Jungian psychological states, philosophical propositions, and a host of other odd notions. Here this section totally falls apart. Nowhere do the authors explain what complementarity (in the sense of incompatible measurement outcomes) or entanglement mean in the context of these "systems", nor why we should expect these systems to have such features even if they were meaningful ideas. The authors do not even motivate the inclusion of these subjects, let alone develop them. They are simply dropped in.
3. Algebraic quantum theory in a nutshell
The authors (actually, I expect it was just one of them) then go on to summarize a formal theory of quantum mechanics in terms of algebras of linear operators. This is a bit more formal than the physicist's usual Dirac notation, but if you're familiar with the subject, it is pretty easy to pick up the notation and perform the mental construction of a Hilbert space representation.
4. Weak quantum theory
This ought to be the second most interesting part of the paper.
In this section, the authors define the axioms of a generalized theory of measurements of a system. Instead of defining measurement outcomes as numbers and the spectrum as the set of possible outcomes, they begin with the spectrum as an arbitrary set, and define measurement outcomes, not necessarily numbers, as elements of the spectrum. This is an interesting and unfortunately understated point, and the section would have been much clearer had they been more explicit about it. Once that is understood, the rest of the "set theory" makes sense; there don't appear to be any big surprises.
The authors identify several features of WQT to differentiate it from quantum mechanics.
First, they bring up that it does not contain Planck's constant. This is specious for two reasons. First, their theory is not numerical, it is a set theory, so of course it does not contain numbers. Second, if you go through section 3 (real quantum mechanics) you won't find Planck's constant there, either. This is because Planck's constant is not actually fundamental to the theory - it's a matter of how the operators are defined. In fact, Planck's constant is a matter of units. Physicists occasionally use "natural units" where <del>h</del> and c are both set to 1 with no dimensions, and measure energy and mass in units of 1/length.
Second, the authors point out that observables don't add along with several implications. This pretty much boils down to not having numerical measurements as well, and life would have been simpler had they simply said so at the outset. But this introduces a very serious problem: how are we to interpret the action of an operator on a state variable? The authors don't allow for a probability interpretation without additional axioms (in fact, they explicitly state that some systems don't lend themselves to probabilities, and disturbingly refer to such airy notions as interpreting art or feeling an emotion), so its not clear what the theory would actually mean if you were able to define what your states and operators were. There is, unfortunately, no answer to this problem. The paper never presents any hint of how results in WQT are to be interpreted.
Third, say the authors, there are no Bell's inequalities. Again, this is the difference between sets and numbers.
Finally, the authors show how to add axioms to recover conventional, numerical quantum mechanics.
5. Complementarity and entanglement in weak quantum theory: two applications
The authors finish the paper with two examples. This should be the best part of the paper, because we finally get to see how this new set theory is applied to a real, nonphysical example. Sadly, this section is a complete loss.
The first example, as mentioned above, has to do with information in chaotic systems. This was apparently developed from an earlier paper by one of the authors (Atmanspacher). Although the section is supposed to be about applications of WQT, the new theory is never mentioned or applied here; what math is shown is just more examples of non-commuting operators. This example gives no insight into WQT.
The second example is even worse: countertransference in psychotherapy! The authors do not even make a pretense of showing math here. It is just talk. The authors draw the analogy (metaphor?) of an entangled system consisting of the patient's and the therapist's states of mind. I'm not even sure what this is supposed to suggest: are the authors really implying that some non-material process is responsible for transferring the patient's state of mind to the therapist's? As in telepathy? Perhaps they merely intend to use WQT as a metaphor or model for subliminal, nonverbal cues the therapist picks up on, but if so, this seems awfully obscure. It certainly doesn't give any insight into how to apply WQT or what its good for.
Again, there is no insight gained as to how to apply or interpret WQT given in this example. The paper concludes without showing a single example of constructing observable operators and evaluating whether observables are compatible or not. There are no examples in the paper that don't involve convential linear algebra.
Finally...
To summarize, WQT is a cute little set theory, but it appears to have no utility. Certainly in this paper, there are no examples of applying it, no attempt to explain how to construct operators or interpret outcomes, nothing that shows what the theory is actually good for.
Most importantly, the authors utterly fail to demonstrate that WQT is applicable to any problem they are interested, and for that matter, that the problems they are interested in even require anything resembling WQT. WQT is a toy, and the authors did not even bother to have any fun with it.
