Quantum-Theoretic Schrödinger Equation in Abstract Hilbert Space

If quantum theory could ‘see’ the intricate structure of the invariant set, it would ‘know’ whether a particular putative measurement orientation θ was counterfactual or not. It must be noted that counterfactuals in quantum mechanics appear in discussions of (a) non-locality, (b) pre- and post-selected systems, and (c) interaction-free measurement; Quantum interrogation. Only the first two issues are related to counterfactuals as they are considered in the general philosophical literature:

If it were that A, then it would be that B.

The truth value of a counterfactual is decided by the analysis of similarities between the actual and possible counterfactual worlds. The difference between a counterfactual (or counterfactual conditional) and a simple conditional: If A, then B, is that in the actual world A is not true and we need some “miracle” in the counterfactual world to make it true. In the analysis of counterfactuals out of the scope of physics, this miracle is crucial for deciding whether B is true. In physics, however, miracles are not involved. Typically:

A : A measurement M is performed
B : The outcome of M has property P

Physical theory does not deal with the questions of which measurement and whether a particular measurement is performed? Physics yields conditionals: “If Ai, then Bi“. The reason why in some cases these conditionals are considered to be counterfactual is that several conditionals with incompatible premises Ai are considered with regard to a single system. The most celebrated example is the Einstein–Podolsky–Rosen (EPR problem) argument in which incompatible measurements of the position or, instead, the momentum of a particle are considered. Stapp has applied a formal calculus of coun- terfactuals to various EPR-type proofs and in spite of extensive criticism, continues to claim that the nonlocality of quantum mechanics can be proved without the assumption “reality”.

However, since, by hypothesis, quantum theory is blind to the intricate structure of the invariant set I, it is unable to discriminate between factual and counterfactual measurement preparations and therefore admits them all as theoretically valid. Hence the quantum-theoretic notion of state is defined on a quantum sub-system in preparation for any measurement that could conceivably be performed on it, irrespective of whether this measurement turns out to be real or counterfactual. This raises a fundamental question. If we interpret the quantum-theoretic notion of state in terms of a sample space defined by a h ̄ neighbourhood on the invariant set, how are we to interpret the quantum-theoretic notion of state associated with counterfactual world states of unreality, not on the invariant set, where no corresponding sample space exists?

Hence, when p ∈ I (the real axis for the Gaussian integers), then α|A⟩ + β|B⟩ can be interpreted as a probability defined by some underlying sample space. However, when p ∉ I (the rest of the complex plane for the Gaussian integers), we define a probability-like state α|A⟩ + β|B⟩ from the algebraic properties of probability, i.e. in terms of the algebraic rules of vector spaces. Under such circumstances, α|A⟩+β|B⟩ can no more be associated with any underlying sample space. This ‘continuation off the invariant set’ does not contradict Hardy’s definition of state, since if p ∉ I, then its points are not elements of physical reality, and hence cannot be subject to actual measurement. A classical dynamical system is one defined by a set of deterministic differential equations. As such, there is no requirement in classical physics for states to lie on an invariant set, even if the differential equations support such a set.  As a result, for a classical system, every point in phase space is a point of ‘physical reality’, and the counterfactual states discussed above are as much states of ‘physical reality’ as are the real world states. Hence, the world of classical physics is perfectly non-contextual, and is not consistent with the invariant set postulate.

The following interpretation of the two-dimensional Hilbert Space spanned by the vectors |A⟩ and |B⟩, emerges from the Invariant Set Postulate. At any time t there corresponds a point in the Hilbert Space, where α|A⟩ + β|B⟩ can be interpreted straightforwardly as a frequentist probability based on an underlying sample space of trajectory √h ̄ neighbourhood on the invariant set. However, since the invariant set and hence its underlying deterministic dynamics are themselves non-computable, it is algorithmically undecidable as to whether any given point in the Hilbert Space can be associated with such a sample space or not; as such, each point of the Hilbert Space is as likely to support an underlying sample space as any other. For points in the Hilbert Space which have no correspondence with a sample space on the invariant set, α|A⟩ + β|B⟩ must be considered an abstract mathematical quantity defined purely in terms of the algebraic rules governing a vector space.

Consistent with the rather straightforward probabilistic interpretation of the quantum-theoretic notion of state on the invariant set, it is reasonable to suppose that, on the invariant set, the Schrödinger equation is itself a Liouville equation for conservation of probability in regions where dynamical evolution is Hamiltonian. Since quantum theory is blind to the intricate structure of the invariant set, the quantum-theoretic Schrödinger equation must be formulated in abstract Hilbert Space form, i.e. in terms of unitary evolution, using algebraic properties of probability without reference to an underlying sample space.

One algebraic property inherited from the Schrödinger equation’s interpretation as a Liouville equation on the invariant set is linearity: as an equation for conservation of probability, the Liouville equation, is always linear, even when the underlying dynamics X ̇ = f(X) are strongly nonlinear. This suggests that attempts to add nonlinear terms (deterministic or stochastic) to the Schrödinger equation, e.g. during measurement, are misguided.

Honey-Trap Catalysis or Why Chemistry Mechanizes Complexity? Note Quote.

Was browsing through Yuri Tarnopolsky’s Pattern Chemistry and its affect on/from humanities. Tarnopolsky’s states “chemistry” + “humanities” connectivity ideas thusly:

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Practically all comments to the folk tales in my collection contained references to a book by the Russian ethnographer Vladimir Propp, who systematized Russian folk tales as ‘molecules‘ consisting of the same ‘atoms‘ of plot arranged in different ways, and even wrote their formulas. His book was published in the 30’s, when Claude Levi-Strauss, the founder of what became known as structuralism, was studying another kind of “molecules:” the structures of kinship in tribes of Brazil. Remarkably, this time a promise of a generalized and unifying vision of the world was coming from a source in humanities. What later happened to structuralism, however, is a different story, but the opportunity to build a bridge between sciences and humanities was missed. The competitive and pugnacious humanities could be a rough terrain.

I believed that chemistry carried a universal message about changes in systems that could be described in terms of elements and bonds between them. Chemistry was a particular branch of a much more general science about breaking and establishing bonds. It was not just about molecules: a small minority of hothead human ‘molecules’ drove a society toward change. A nation could be hot or cold. A child playing with Lego and a poet looking for a word to combine with others were in the company of a chemist synthesizing a drug.

Further on, Tarnopolsky, following his chemistry then thermodynamics leads, then found the pattern theory work of Swedish chemist Ulf Grenander, which he describes as follows:

In 1979 I heard about a mathematician who tried to list everything in the world. I easily found in a bookstore the first volume of Pattern Theory (1976) by Ulf Grenander, translated into Russian. As soon as I had opened the book, I saw that it was exactly what I was looking for and what I called ‘meta-chemistry’, i.e., something more general than chemistry, which included chemistry as an application, together with many other applications. I can never forget the physical sensation of a great intellectual power that gushed into my face from the pages of that book.

Although the mathematics in the book was well above my level, Grenander’s basic idea was clear. He described the world in terms of structures built of abstract ‘atoms’ possessing bonds to be selectively linked with each other. Body movements, society, pattern of a fabric, chemical compounds, and scientific hypothesis—everything could be described in the atomistic way that had always been considered indigenous for chemistry. Grenander called his ‘atoms of everything’ generators, which tells something to those who are familiar with group theory, but for the rest of us could be a good little metaphor for generating complexity from simplicity. Generators had affinities to each other and could form bonds of various strength. Atomism is a millennia old idea. In the next striking step so much appealing to a chemist, Ulf Grenander outlined the foundation of a universal physical chemistry able to approach not only fixed structures but also “reactions” they could undergo.

The two major means of control in chemistry and organic life: thermodynamic control (shift of equilibrium) and kinetic control (selective change of speed). People might not be aware that the same mechanisms are employed in social and political control, as well as in large historical events out of control, for example, the great global migration of people and jobs in our time or just the one-way flow of people across the US-Mexican border!!! Thus, with an awful degree of simplification, the intensification of a hunt for illegal immigrants looks like thermodynamic control by a honey trap, while the punishment for illegal employers is typical negative catalysis, although both may lead to a less stable and more stressed state. In both cases, new equilibrium will be established, different equilibria housed upon different sets of conditions.

dna_broken_wide

Should I treat people as molecules, unless I am from the Andromeda Galaxy. Complex-systems never come to global equilibrium, although local equilibrium can exist for some time. They can be in the state of homeostasis, which, again, is not the same as steady state in physics and chemistry. Homeostasis is the global complement of the classical local Darwinism of mutation and selection.

Taking other examples, the immigration discrimination in favor of educated or wealthy professionals is also a catalysis of affirmative action type. It speeds up the drive to equilibrium. Attractive salary for rare specialists is an equilibrium shift (honey trap) because it does not discriminate between competitors. Ideally, neither does exploitation of foreign labor. Bureaucracy is a global thermodynamic freeze that can be selectively overcome by 100% catalytic connections and bribes. Severe punishment for bribe is thermodynamic control. The use of undercover agents looks like a local catalyst: you can wait for the crook to make a mistake or you can speed it up. Tax incentive or burden is a shift of equilibrium. Preferred (or discouraging) treatment of competitors is catalysis (or inhibition).

There is no catalysis without selectivity and no selectivity without competition. Equilibrium, however, is not selective: it applies globally to the fluid enough system. Organic life, society, and economy operate by both equilibrium shift and catalysis. More examples: by manipulating the interest rate, the RBI employs thermodynamic control; by tax cuts for efficient use of energy, the government employs kinetic control, until saturation comes. Thermodynamic and kinetic factors are necessary for understanding Complex-systems, although only professionals can talk about them reasonably, but they are not sufficient. History is not chemistry because organic life and human society develop by design patterns, so to speak, or archetypal abstract devices, which do not follow from any physical laws. They all, together with René Thom morphologies, have roots not in thermodynamics but in topology. Anything that cannot be presented in terms of points, lines, and interactions between the points is far from chemistry. Topology is blind to metrics, but if Pattern Theory were not metrical, it would be just a version of graph theory.