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The Quantum Field Theory (QFT) Dual Paradigm in Fundamental Physics and the Semantic Information Content and Measure in Cognitive Sciences

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Representation and Reality in Humans, Other Living Organisms and Intelligent Machines

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 28))

Abstract

In this paper we explore the possibility of giving a justification of the “semantic information” content and measure, in the framework of the recent coalgebraic approach to quantum systems and quantum computation, extended to QFT systems. In QFT, indeed, any quantum system has to be considered as an “open” system, because it is always interacting with the background fluctuations of the quantum vacuum. Namely, the Hamiltonian in QFT always includes the quantum system and its inseparable thermal bath, formally “entangled” like an algebra with its coalgebra, according to the principle of the “doubling” of the degrees of freedom (DDF) between them. This is the core of the representation theory of cognitive neuroscience based on QFT. Moreover, in QFT, the probabilities of the quantum states follow a Wigner distribution, based on the notion and measure of quasiprobability, where regions integrated under given expectation values do not represent mutually exclusive states. This means that a computing agent, either natural or artificial, in QFT, against the quantum Turing machine paradigm, is able to change dynamically the representation space of its computations. This depends on the possibility of interpreting QFT system computations within the framework of category theory logic and its principle of duality between opposed categories, such as the algebra and coalgebra categories of QFT. This allows us to justify and not only to suppose, like in the “theory of strong semantic information” of L. Floridi, the definition of modal “local truth” and the notion of semantic information as a measure of it, despite both measures being defined on quasiprobability distributions.

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Notes

  1. 1.

    For the notion of “finitary” computation, as distinguished from “infinitistic” (second-order computation) and “finististic” (Turing-like computation), see [11]. This notion depends on the category theory (CT) interpretation of logic and computation [12], as far as based on Aczel’s non-well founded (NWF) set theory [13], justifying a coalgebraic semantics in quantum computing [14], as far as based on the CT principle of the dual equivalence between a Boolean initial algebra and a final coalgebra [15, 16]. The key notion of the doubling of the degrees of freedom between a q-deformed Hopf algebra and a q-deformed Hopf coalgebra, as representing each quantum system in quantum field theory, perfectly satisfies such a logic, as we see below.

  2. 2.

    It is useful to recall here that the canonical variables (e.g. position and momentum) of a quantum particle do not commute among themselves, like in classical mechanics, because of Heisenberg’s uncertainty principle. The fundamental discovery of D. Hilbert consists in demonstrating that each canonical variable of a quantum particle commutes with the Fourier transform of the other (such a relationship constitutes a CCR), allowing a geometrical representation of all the states of a quantum system in terms of a commuting variety, i.e. the relative “Hilbert space”.

  3. 3.

    We recall that typical example of function composition is a recursive, iterated function: x n+1 = f (x n ).

  4. 4.

    We recall here that by an “ultrafilter” we mean the maximal partially ordered set defined on the power set of a given set ordered by inclusion, and excluding the empty set.

  5. 5.

    Two corollaries of the Lövenheim–Skolem theorem, demonstrated by Skolem himself in 1925, are significant for our aims, i.e. (1) that only complete theories are categorical, and (2) that the cardinality of an algebraic set depends intrinsically by the algebra defined on it. Think, for instance of the principle of induction by recursion for Boolean algebras, allowing a Boolean algebra to construct the sets on which its semantics is justified, blocking however Boolean computability on finite sets. It is evident that Zermelo’s strategy of migrating to second-order set-theoretic semantics grants categoricity to mathematics on an infinitistic basis.

  6. 6.

    Recall that set self-inclusion is not allowed for standard sets because of Cantor’s theorem. This impossibility is the root of all semantic antinomies in standard set theory, from which the necessity of a second-order set-theoretic semantics ultimately derives.

  7. 7.

    This depends on the trivial observation that a coalgebra C = 〈C,γ : C → ΩC 〉, where γ is a transition function characterizing C, over an endofunctor Ω: C → C can be seen also as an algebra in the opposite category C op, i.e. Coalg(Ω) = (Alg(Ω op)op [16, p. 417].

  8. 8.

    The fundamental property of \(\mathcal{V}\) is that it is the counterpart of the power set functor ℘ in the category of the topological spaces (i.e. for continuous functions) such as the Stone space category, Stone. This functor maps a set S to its power set ℘(S) and a function f : S → S’ to the image map ℘f given by (℘f) (X) ≔ f [X] (= { f(x) | x ∈ X}). Applied to Kripke’s relational semantics in modal logic, this means that Kripke’s frames and models are nothing but “coalgebras in disguise”. Indeed, a frame is a set of “possible worlds” (subsets, s) of a given “universe” (set, S) and a binary “accessibility” relation R between worlds, R ⊆ S × S. A Kripke’s model is thus a frame with an evaluation function defined on it. Now R can be represented by the function R[•]: S → ℘(S), mapping a point s to the collection R[s] of its successors. In this way, frames in modal logic correspond to coalgebras over the covariant power set functor ℘. For such a reconstruction see [16, p. 391].

  9. 9.

    However, see the fundamental remarks about the limits of decidability and computability in this first-order modal logic semantic approach in [76], in which it is said, just in the conclusion, that one of the most promising research programs in this field is related to the coalgebraic approach to modal logic semantics.

  10. 10.

    This depends on the fact that contravariance in QM algebraic representation theory can have only an indirect justification, as Abramsky elegantly explained in his just quoted paper. QM algebraic formalism is, indeed, intrinsically based on von Neumann’s covariant algebra, so that only Hopf algebras’ self-duality are “naturally” (in the algebraic sense of the allowed functorial transforms) justified in it [62, 80, 81].

  11. 11.

    In this regard, the famous Aristotelian statement synthesizing his “intentional” approach to epistemology—“not the stone is in the mind, but the form of the stone”—has an operational counterpart in the homomorphism algebra coalgebra of QFT neuro dynamics.

  12. 12.

    This result has been recently formally obtained [85]. For an intuitive explanation of this result, see below the two Appendices, Sects. 7.1–7.2.

  13. 13.

    In parenthesis, in the machine \({\mathbb{M}}\) the general coalgebraic principle of the observational (or behavioral) equivalence among states holds in the following way. Indeed, for every two coalgebras (systems) \({\mathbb{S}}_{1} ,{\mathbb{S}}_{2} \in {\mathbf{Coalg}}\left( {C \times {\mathcal{I}}} \right),\;\left( {!c_{{{\mathbb{S}}_{1} }} = !c_{{{\mathbb{S}}_{2} }} } \right) \Rightarrow \left( {!x_{{{\mathbb{S}}_{1} }} = !x_{{{\mathbb{S}}_{2} }} } \right).\) All scholars agree that this has an immediate meaning for quantum systems logic and mathematics, as a further justification for a coalgebraic interpretation of quantum systems.

  14. 14.

    A condition elegantly satisfied in the QFT formalism by the fractal structure of the systems phase space and, therefore, by the chaotic character of the macroscopic trajectories (phase transitions) defined on it, generally [88], and specifically in dissipative brain dynamics [32].

  15. 15.

    Remember that the transitive rule in the NWF set theory does not hold only for the inclusion operation, i.e. for the superset/subset ordering relation.

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    Gianfranco Basti

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Appendix A

Appendix A

1.1 A.1 Induction and Coinduction as Principles of Set Definition and Proof for Boolean Lattices

The collection of clopen subsets of a Stone space, to which a Boolean algebra is isomorphic, according to the Stone theorem is effectively an ultrafilter U (or the maximal filter F) on the power set, ℘(S), of the set S. Namely, it is the maximal partially ordered set (maximal poset) within ℘(S) ordered by inclusion, i.e. (℘(S), ⊆), with the exclusion of the empty set. Any filter F is dual to an ideal I, simply obtained in set (order) theory by inverting all the relations in F, that is, x ≤ y with y ≤ x, and by substituting intersections with unions. From this derives that each ultrafilter U is dual to a greatest ideal that, in Boolean algebra, is also a prime ideal, because of the so-called prime ideal theorem, effectively a corollary of the Stone theorem, demonstrated by himself. All this, applied to the Stone theorem, means that the collection of partially ordered clopen subsets of the Stone space, to which a Boolean algebra is isomorphic, corresponds to a Boolean logic complete lattice L for a monadic first-order predicate logic. From this, the definition of induction and coinduction as dual principles of set definition and proof is immediate, as soon as we recall that the fixed point of a computation F is given by the equality x = F(x) [68, p. 46]:

Definition 2

(sets inductively/coinductively defined by F). For a complete Boolean lattice L whose points are sets, and for an endofunction F, the sets

$$\begin{array}{*{20}l} {F_{\text{ind}} : = \bigcap {\left\{ {x|F(x) \le x} \right\}} } \hfill \\ {F_{\text{coind}} : = \bigcup {\left\{ {x|x \le F(x)} \right\}} } \hfill \\ \end{array}$$

are, respectively, the sets inductively defined by a recursive F, and coinductively defined by a co-recursive F. They correspond, respectively, to the meet of the pre-fixed point and the join of the post-fixed points in the lattice L, i.e. the least and greatest fixed points, if F is monotone, as required from the definition of the category Pos (see above, Sect. 5.1).

Definition 3

(induction and coinduction proof principles). In the hypothesis of Definition 2, we have:

$$\begin{array}{*{20}l} {if\;F(x) \le x\;then\;F_{\text{ind}} \le x\,} & { ( {\text{induction}}\,{\text{as}}\,{\text{a}}\,{\text{method}}\,{\text{of}}\,{\text{proof)}}} \\ {if\;x \le F(x)\;then\;x \le F_{\text{coind}} } & {({\text{coinduction}}\,{\text{as}}\,{\text{a}}\,{\text{method}}\,{\text{of}}\,{\text{proof)}}} \\ \end{array}$$

These two definitions are the basis for the duality between an initial algebra and its final coalgebra, as a new paradigm of computability, i.e. Abramsky’s finitary one, and henceforth for the duality between the universal algebra and the universal coalgebra [15].

1.2 A.2 Extension of the Coinduction Method to the Definition of a Complete Boolean Lattice of Monadic Predicates

The fundamental result of the above-quoted Goldblatt–Thomason theorem and van Benthem theorem is that a set-tree of NWF sets—effectively a set represented as an oriented graph where nodes are sets, and edges are inclusion relations with subsets governed by Euclidean rule—corresponds to the structure of a Kripke frame of his relational semantics, characterized by a set of “worlds” and by a two-place accessibility relation R between worlds, e.g. the second graph from left below corresponds to the graph of the number 3, with u = 3, v = 2, w = 1. Therefore for understanding intuitively the extension of the coinduction method to the domains of monadic predicates of a Boolean lattice, let us start from (1) the “Euclidean rule (ER)” ‹∀u,v,w ((uRv ∧ uRw)→vRw)› (see the second from left graph below), driving all the NWF set inclusions and that is associated by van Benthem’s correspondence theorem to the modal axiom E (or 5): ‹◊α→⎕◊α›, of the modal propositional calculus, and (2) from the “seriality rule (SR)” ‹∀u∃v (uRv)› (an example of this axiom is given by the fourth or fifth graph below)—that has an immediate physical sense, because it corresponds to whichever energy conservation principle in physics, e.g. the I Principle of Thermodynamics—and that is associated to the modal axiom D: ‹⎕α → ◊α›. The straightforward first-order calculus, by which it is possible to formally justify the definition/justification by coinduction (tree unfolding) of an equivalence class as the domain of a given monadic predicate, through the application of the two above rules to whichever triple of objects ‹u, v, w›, is the following:

For ER, ‹∀u,v,w ((uRv ∧ uRw) → vRw)›; hence, for seriality, ‹∀u,v (uRv → vRv)›; finally: ‹∀u,v,w [((uRv ∧ uRw) → (vRw ∧ wRv ∧ vRv ∧ wRw)) ↔ ((v≡w) ⊂ u)]›, I.e. (v≡w) constitutes an equivalence subclass of u, say Y, because a “generated” transitiveFootnote 15-symmetric-reflexive relation holds among its elements, which are therefore also “descendants” of their common “ascendant”, u. More intuitively, using Kripke’s relational semantics graphs for modal logics, where ‹u, v, w› are also “possible worlds” (models) of a given universe W, and where R is the two-place “accessibility relation” between worlds, the above calculus reads:

The final graph constitutes a Kripke-like representation of the KD45 modal system, also defined in literature as “secondary S5”, since the equivalence relationship among all the possible worlds characterizing S5 here holds only for a subset of them, in our example, the subset of worlds {w, v}.

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Basti, G. (2017). The Quantum Field Theory (QFT) Dual Paradigm in Fundamental Physics and the Semantic Information Content and Measure in Cognitive Sciences. In: Dodig-Crnkovic, G., Giovagnoli, R. (eds) Representation and Reality in Humans, Other Living Organisms and Intelligent Machines. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-43784-2_9

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