Von Neumann's first axiom for Q.M is that the state of a quantum system is completely described by a vector in a complex Hilbert space.
However, if there is 'entanglement', then the axiom is false.
Graciella Chichilnisky ignores this.
Quantum Theory uses infinite dimensional complex Hilbert Spaces which are important to represent certain notions such as wave particle duality. Here, we focus instead on topological issues, and for those, in view of Bohr’s and von Neumann’s axioms, it suffices to focus on finite dimensional real spaces. The reason is that “observables” in quantum theory are by definition self-adjoint operators, this is von Neumann’s first axiom of quantum theory (see Gudder [5]) and it explains the focus on (a) finite dimensions and (b) real spaces.
Sadly, because of 'entanglement', the axiom fails.
This is because quantum theory’s operators are self-adjoint (observables) and an operator is self-adjoint if and only if it is unitarily equivalent to a real valued multiplication operator (see Gudder [5]).
But entanglement is at the level of the quantum state not the operator which, in any case, isn't 'real'. It is a mathematical artefact or 'intension'. Quantum events, however, are real. An 'intensional fallacy' is committed when an intension of an epistemic type is given a real extension. One might say 'Idealism is the correct ontology. Reality is merely the shadow of what is mental or epistemic'. But that is a philosophical, not a scientific, position.
As in Arrow's Theorem, which is a cascading intensional fallacy, there can be a useless and stupid sort of QMT which is based on the intensional fallacy briefly outlined above.
Chichilnisky's claim is as follows
Based on the axioms of quantum theory we identify a class of topological singularities that separates classic from quantum probability,
They are already separate because the latter features 'superposition', the latter does not. That's why the latter's 'amplitude' is a complex number.
and explains many quantum theoryís puzzles and phenomena in simple mathematical terms so they are no longer quantum pardoxes.
As opposed to intensional fallacies. But quantum paradoxes are real and very useful. Logical fallacies are useless or mischievous.
The singularities
there are classical (e.g. when two particles collide and the system is undefined) as well as quantum singularities (e.g. black holes).
provide new experimental insights and predictions that are presented in this article and establish surprising new connections between the physical and social sciences.
People who commit intensional fallacies in Social Sciences can do the same in Physics. But they are ignored by actual Physicists.
The key is the topology of spaces of quantum eventsTopological quantum field theories exist and quantum events are considered as fluctuations or interactions within them. But these fields aren't real. They are mathematical constructs.
and of the frameworks postulated by these axioms. These are quite different from their counterparts in classic probability
they are similar to phase transitions except, in the quantum field, the transition is in information.
and explain mathematically the interference between quantum experiments and the existence of several frameworks or violation of unicity that characterizes quantum physics.
Because of superposition. But that is a fact about the world.
They also explain entanglement, the Heisenberg uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam-Berry phases.
Pancharatnam is in classical optics. There is a similar result in molecular physics. The problem with a theory which explains phenomena in both the classical and the quantum realm is that it is either part of a Grand Unified Theory or it is nonsense. If the former is the case, there would already be a crucial experiment to confirm or reject it. If the latter is the case, nobody will bother with it.
Somewhat surprisingly we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences: which is Arrow ís social choice paradox:
We don't that. It is possible that everybody will prefer to prefer some particular SWF if they knew about it. Preferences are epistemic and do not correspond to any set and can't give rise to a relation algebra. They violate Liebniz's laws. Thus nothing mathematical or logical can be said about them. One might say there is 'superposition' in Social Choice Theory. If we had enough information, we all might choose the same thing. Consider what happened between 1938 and 1939 in England. There was a 'phase transition' such that 'appeasers' (peace at any cost!) turned into belligerents (we must win this war at any cost!). What had changed was information about the nature of the Nazi regime and its plan for Europe as a whole. Suddenly, Hitler (the little Corporal) was seen not as a little man with a genuine grievance but rather as the incarnation of the genocidal 'Hun' determined to crush Christendom.
the foundations of social choice and of quantum theory are therefore mathematically equivalent.
No. The nature of quantum information is different from the nature of information for Social Choice. The former arises from a 'game against nature'. The latter is wholly strategic and features coordination and discoordination games of a volitional type.
We identify necessary and sufficient conditions on how to restrict experiments to avoid these singularities and recover unicity,
I can identify such a condition. Don't spend any money on doing experiments. Give me the money and I will order Pizzas and beer. We can then say 'The Great Spaghetti Monster has uniquely determined everything. All hail the Great Spaghetti Monster!'
avoiding possible interference between experiments and also quantum paradoxes; the same topological restriction is shown to provide a resolution to the social choice impossibility theorem of Chichilnisky (1980)
Social Choice features no sets and thus no topology. Saying the Great Spaghetti Monster is the sole efficient cause of everything in the Universe resolves all the problems which make our tiny brains hurt.
Events are physical phenomena that either occur or don't occur. They are central to any probabilistic theory.
Probability density functions or cumulative distribution functions do without them.
In classic probability all experiments are part of one large experiment and events are described within one sample space with a single basis of coordinates or framework : this is the unicity assumption of classic physics (Griffiths, 2003 [10]).
This is a useful enough simplification but isn't true. The fact is, in General Relativity, there is no global inertial frame. But since it is easier to work in inertial frames, we just go ahead and do say anyway. Moreover, it is possible that there are experiments outside our 'light cone' which can never be part on 'one large experiment' within our light cone.
In quantum probability, instead, quantum events are projections maps on a Hilbert space.
a complex Hilbert space. But we are welcome to do the same thing with classical events. Suppose two things are indiscernibly identical from one point of view but may not be from some other point of view. Then we can use a Hermitian.
Quantum theory considers all possible experiments on a physical system and breaks tradition by explicitly accepting that there may be no universal experiment and no single framework to describe all observed events
The same thing can be done in molecular physics or classical optics or when considering phase transitions which are multiply realizable.
. The multiplicity of frameworks in quantum theory violates unicity: there may be no unique basis of coordinates to describe the results of all possible experiments on a physical system.
The same may be true in chemistry. Superposition is like multiple realizability.
When two frameworks or coordinate systems fail to be orthogonal to each other they give rise to so called interaction or interference among experiments that is at the heart of quantum theory and distinguishes it from classic physics
But something similar can arise in Crystallography which is why it evolved into Quantum Crystallography. But there are plenty of other macroscopic physical systems which are non-deterministic or chaotic. What distinguishes QMT is the importance of information. Non-locality does not mean the exchange of 'real' information but there appears to be 'phase transition' in how information is shared between things.
We show that the topological structure of the spaces of quantum events
i.e. the invariants postulated by a topological quantum field theory
- which are also the propositions of quantum logic
they may be or they may not be. You could have a category theoretical approach which is not topological though it features an 'internal logic' and has ambient toposes. No doubt, this can be translated into a topology for some specific purpose. My point is that the Birkhoff / von Neumann approach isn't the only option on the menu.
- explains why experiments interfere,
An explanation is an interpretation. We have many such.
why we typically have no common frameworks, and why quantum logic is more complex and richer than the binary logic of classic physics
Yet it is a lot less complex than 'natural deduction' though, the latter can use an "orthomodular lattice" structure to capture the non-distributive nature of quantum measurements. The problem here is that an arbitrary lattice of this sort which captures a particular Von Neumann algebra may not have many of the desirable properties of lattices of projections. But this is just another way of saying logic isn't magic. It does not enable you to discover things about reality without ever getting out of your armchair.
Chichilnisky's fundamental error in this paper is to think unicity is a feature of classical systems. It isn't. Consider the weather. That's a chaotic system which may or may not be deterministic. It has no unique solution. The fact is QMT postulates unicity at the macroscopic scale. There is no such postulate for Classical theory. What we are speaking off is merely an ad hoc 'working assumption'.
Why Unicity fails:
there are too many degrees of freedom or path dependence is too complex to calculate
impossibility theorems for selecting frameworks
There is no way to show that a non-deterministic way of selecting frameworks with particular properties is impossible. This is because an algorithm can't do what a non-algorithmic process or lawless choice sequence can do. Such impossibility theorems are only possible because of an obvious intensional fallacy- like Arrow's.
The next step is to define restricted domains of experiments within which one can recover unicity, and show why the recovery cannot be obtained in general.
Why restrict what experimenters are doing more particularly if they are smarter than you and can help bring into existence cool new tech- e.g. Quantum Computers?
Chichilnisky's theorem is trivial-
A necessary and sufficient restriction on the experiments of a quantum system with n degrees of freedom
Sadly, smart peeps keep discovering new quantum degrees of freedom. What is cool is that they can exploit this to develop new functional materials or circumvent performance bottlenecks in existing materials.
to satisfy unicity, is that the corresponding space of frameworks F n is topologically trivial or contractible
Field Expansion and Group Contraction fallacies, as I have explained elsewhere, vitiate Arrow's theorem. Essentially, by arbitrarily stipulating that a space is contractible, you are saying there is a 'fixed point'. But such a beastie means there is 'naturality' or even a punctum Archimedes. It is like saying 'there is an atomic proposition' or 'there is an absolute proof' or 'there is a 'natural' proof that P is not equal to NP.' You may as well say 'The Great Spaghetti Monster has ordained all that is! All hail the Great Spaghetti Monster!'
I have previously pointed out that if you think Arrow's theorem isn't nonsense, you must also believe Godel's proof of God.
Chichilnisky summarizes her own contribution to this stupidity-
. In 1980 social choice theory was redefined as follows: one seeks to define a map that assigns a social preference to any two or more individual preferences.
Sadly, preferences are epistemic (based on existing knowledge) and 'impredicative' ( e.g. 'I'll have what she's having!' as in 'Whey Harry met Sally'. The intensional fallacy arises when a fixed, well-defined, extension is supplied to an epistemic or impredicative 'intension'. True, for any practical purpose, we may arbitrarily do so in an ad hoc manner- e.g. looking at available data on 'Revealed Preference' or Opinion Polls or 'Focus Groups' etc. But this is an ideographic procedure. Nothing nomothetic can be said about it.
Reasonable conditions are that map must be continuous and symmetric, depending on individuals preferences but not on the order of the individuals, and that respect unanimity so that if both individuals have the same preferences, the social preference is the same.
This is wholly unreasonable. We may all want Mr. Garrison to become POTUS and fuck all the immigrants to death. But we are relieved that there are Constitutional Checks and Balances preventing this horrific outcome. We don't want only our preferences to be taken account of. We would rather outsource decisions to experts- even foreign experts though we ourselves are xenophobic. We definitely don't want 'continuous mappings'- e.g. deciding in 1939 to use only one quarter of our army to fight Hitler because in 1938 we didn't want more than ten percent of it to be used in any such way. Symmetric mappings preserve values under permutation of arguments. This means ignoring the fact that we are at war with Germany so as to increase welfare payments rather than spend that money on the Armed Forces.
Continuity means that it is possible to approximate the social preference by taking sufficiently accurate measurements of the individual preference
This can't be done because of impredicativity. Smart peeps face a 'Keynesian Beauty contest' problem- they can't opt for something the hoi polloi won't understand. Stupid people do want to outsource decisions to the smart but are worried smart peeps will place a disproportionate burden on them. The solution is to get a leader- e.g. Churchill- who will listen to smart peeps but who has a soft spot for the ordinary people.
Theorem 31 The existence of common preferences is equivalent to
saying that those concerned are indiscernibly identical in a particular respect. Suppose all cats like 'Whiskas', then any cat will be happy to be fed that brand of pet food. This may be true if the guys who run 'Whiskas' are super-smart and know everything there is to be known about cats.
the existence of common frameworks, or unicity.
Because we already know everything about quantum entities or events.
Proof. The equivalence can be seen formally by considering the necessary and sufficient conditions
there are no such conditions.
for the existence of a selection of a single framework in restricted domains of experiments.
this is a purely arbitrary matter.
It is equally arbitrary to say that anything is impossible for either Social Choice or QMT. As the knowledge base changes, both change. Both represent 'intensions' whose 'extensions' defy the law of identity. No logical or mathematical result can be proved about them.
Consider the following
Example 32 The conjunction fallacy Tversky & Kahneman, [16]1983 defined an important and common probability judgment error, called the conjunction fallacy, that is based on the lack of common frameworks.
Nope. It was based on guys who weren't probability theorists jumping to an unwarranted conclusion.
It is the famous Linda problem. Judges are provided a brief story of a woman named Linda who used to be a philosophy student at a liberal university and was active in the anti-nuclear movement. The judges are asked to rank the likelihood of the following events: that Linda is now (a) active in the feminist movement, (b) a bank teller, (c) active in the feminist movement and a bank teller, (d) active in the feminist movement and not a bank teller, and (e) active in the feminist movement and a bank teller. The conjunction fallacy occurs when option (c) is judged to be more likely that option (b) (even though the latter contains the former).
There is no fallacy here. People think Lindia is stupid because she studied nonsense in a 'safe space'. She is an obese cat-lady who has a low IQ job- e.g. Bank Teller or DMV clerk. 'Active in the feminist movement' just means she likes moaning about her period and the great injustice perpetrated on Wimmin by Neo-Liberalism in that they have to sit down to pee. In other words, in responding to this question, people are expressing their bigoted views derived, no doubt, from TV shows or bar-room chit-chat. Incidentally, any woman who does not want to have sex with me is ipso facto a Lesbian.
Chichilnisky doesn't get this. She
uses a geometric approach to quantum theory taken from Busemeyer and Bruza
Authors of ' Quantum Models of Cognition and Decision Making' which is Deepak Chopra level silly.
They argue that 'underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models.'. First we represent two answers to the feminism question by two different frameworks or basis of coordinates for euclidean space .
Nothing wrong with Feminism. Make things better for women and men too greatly benefit. I may be a misogynist (coz Beyonce didn't come for my Birthday party even though I sent her a 'best friends for ever and ever' bracelet. Mummy promised to speak to Beyonce's parents because it broke her heart to see me cry my little eyes out. Anyway, that's the reason my sixtieth birthday party was such a wash-out.
Each framework is given by two orthogonal rays that span a two dimensional space. The answer yes to feminism is represented in Figure 1 by the ray labeled F and the answer no to the feminism question is represented by an orthogonal ray labeled F.
The lady went to a liberal university and must have studied Feminist philosophers. It would have been odd if she wasn't a Feminist unless, obviously, she had converted to Islam or married JD Vance or something of that sort.
Our answer to the Bank Teller question has to do with our belief that Philosophy Degrees are not prized by employers. But 'Bank Teller' is a proxy for 'ill-paid, monotonous, work'.
and it means that being a feminist and not being a bank teller are close in this belief space.
Nonsense! We think of Feminists as independent people. This lady has a job and does it conscientiously. Shame she studied Philosophy rather than Accountancy.
What nobody who answers these questions is doing is making probability calculations. No mathematical axioms of any type are being used. True, had I been asked these questions and had access to my smart phone, I might have made 'frequentist' calculations based on asking questions to a Generative AI.
In this second framework, and according to the axioms of quantum theory, the square amplitude equals the probability of saying yes to the bank teller question starting from the initial state and this equals j< B j S >j 2= 0:0245 in the Figure.
Nonsense! If we interviewed the respondents we would find they said 'yes' to the Bank teller question because they thought Philosophy Majors find it difficult to get high paid jobs.
Now consider the sequence of answers in which the person says yes to the feminism question and then says yes to the bank teller question in that order.
Actually, if the order of question had been reversed, we might say no to the 'Feminism' question. We imagine Bank Tellers as returning home to their cat and eating chocolates while reading Romance Novels. Because we have already committed to the 'Feminism' answer, we modify our views. The lady may indulge in Romance Novels as a 'guilty pleasure', but feels it incumbent on herself to keep up some semblance of Feminist commitment.
The order that questions are processed is critical in quantum theory, and here we are assuming that the more likely event is evaluated first.
I think it is likely that a Philosophy Major from a liberal college would be Feminist, anti-Racist, etc.
Chichilnisky indulges in some mathsy masturbation to conclude that a simple geometric model reproduces the basic facts of the conjunction fallacy. If this were the case, then Quantum probability would accord higher probability to an event which, on a frequentist basis, is less probable. Such is not the case. What is true is that once an observation has been made then there is something like a Monty Hall problem. But this isn't the case when speaking of Linda. There, the 'conjunction fallacy' arises for a purely psychological reason which has to do with the way 'confabulation' fleshes out a hypothetical person to make them more 'real' and interesting to us. Thus, if you are telling me about a lady you think I might hit it off with, my imagination starts adding details out of my own imagination. I think of Linda as fat- like me- and liking chocolates and cats and RomComs. She might prove sympathetic if I tell her about how Beyonce cruelly refused to attend my birthday party because she was afraid of what the 'mean girls' would say. Still, if I meet Linda and she is slim and athletic and enjoys hearing me tear holes in Amia Srinivasan, I will be yet more delighted. True, Linda will brusquely tell me that she doesn't date fat losers, but still, the one occasion when we met at Starbucks will remain golden in my memory. On the other hand, I will definitely report her to the FBI because she revealed she was using her job as a Bank Teller to sabotage the SWIFT system and thus bring down Neo-Liberalism, Patriarchy and the manifest injustice that women have to sit down to pee. Hopefully, I will get a reward of some sort.
Busemeyer and Bruza state that, given that the two questions are treated as incompatible, we must also be violating unicity.
In Social situations there is always superposition- i.e. no unicity. Indiscernibly identical things aren't in fact identical. A Feminist Bank Teller may not want to bring down the SWIFT system. Indeed, she may vote for Trump. Alas, what is unlikely is that she will want to have sex with me.
Indeed, they say, we are assuming that the person is unable to form a single description (i.e. a single sample space) containing all the possible conjunctions
Indeed. The truth is many of our interactions, under Yoneda lemma, are ontologically dysphoric- they are not at home in the world.
: What they do not explain is why this is assumed. This article shows that, for topological reasons that are akin to those of the social choice paradox, this assumption is unavoidable.
The reverse is the case. The assumption is silly. In the Social World we only have intensions with unknowable, perhaps ontologically dysphoric, extensions.
In other words, it is unavoidable that the person will be unable to form a single description for some basis of coordinates, or frameworks.
Any yet the thing can be done easily enough even without the invocation of the Great Spaghetti Monster.
The results presented here explain the violation of unicity. This implies that necessarily in some cases, the person would have never thought about conjunctions - for example those involving feminism and bank tellers - sufficiently to assign probabilities to all these conjunctions.
Nonsense! People play betting games of this type in pubs all the time. I recall a pretty girl from Newcastle approaching me at a bar in Gloucester Road near the Coach Station. She wanted to know if I was an Accountant. I admitted as much. She then said 'are you Gay and do you live with a hair-dresser'. I denied it. The girl was upset. She had lost her bet. Still I bought drinks for her and her fat friend. Had I thrown in a packet of chips, they would have felt obliged to have sex with me.
Incidentally, the reason they thought my boy-friend was a hairdresser was because I cut my hair myself using an electric hair clipper. This meant that portions of my scalp were bare. The girls assumed this was some avant garde hair style used by submissive homosexuals to signal their willingness to participate in highly degrading sex acts of a type, the two Geordie lasses firmly believed, were widespread in the West End.
If we did assume unicity in this example, then we could not explain the conjunction fallacy because the joint probabilities can be defined under unicity, and they will always be less than (or equal to) the marginal probabilities.
Unless, there is an observer effect. In the example I gave above, it is possible that when the Geordie lasses suggested I was a homosexual who enjoyed being sodomized and then shat upon, that I suddenly realized this is exactly what I wanted to be.
Therefore as stated by Busemeyer and Bruza, to explain the experimental result requires the violation of unicity.
It is possible that we have a Society in which, if the public assigns a higher probability to Linda being a Feminist Bank teller than to her being a Bank Teller, then, immediately this becomes common knowledge, any non-Feminist Bank Teller immediately quits her job. If 'Expectations create Reality' you have impredicativity and thus no unicity.
The results of this article go further: they explain why the violation of unicity is a necessary logical implication when considering all possible experiments of a given physical system - as is the goal of quantum theory.
This is not an explanation. It could be an interpretation of QMT which would be falsified once a GUT is found. But, as presented here, it is merely nonsense.
And they illustrate why violation of unicity is, at its core, the same as the paradox of social choice.
In other words, if there is a sole efficient cause or a 'slingshot' proposition such that all true statements are equivalent to it, then, for sure, there is unicity and determinism. But we can't be sure both arise in any given field of enquiry because there is as yet unsuspected 'naturality' or a way to 'carve reality along its joints'.
Chichilnisky ends her paper with the following enigmatic remark=
It can be shown that the topological problem posed by Jules Verne (viz that if you around the world, you gain a day) is the same as in the Pancharatnam Berry phases.
No. Phineas Fogg, travelling East, loses about 16 minutes a day. This adds up to the 24 hours he 'gains' on returning. There is no singularity or 'phase transition' because day and date are locally determined. You have to adjust your watch and calendar as you travel. This is not an adiabatic process acquired over the course of a cycle, which is why a real life, nineteenth century, Phineas Fogg would not have made a mistake attributed to Magellan's sailors who had journeyed over vast spaces where the locals had no calendars or clocks.
It is well accepted that Barry phases arises from the existence of a singularity,
i.e. a place where a function is undefined. But it can be defined well enough for practical purposes.
which is the same origin that is postulated here for the basic properties of quantum theory that are described above, a topic that to be discussed in further writings.
Yet crazier nonsense. I've said it before and I will say it again, Arrow's theorem is stoooopid. Obsessing over it will drive you nuts.
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