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Europhysics Letters 32 (8),pp.621-626 (1995) 10 December 1995

A Linear Solution of the Four-Dimensionality Problem

by Vladimir Trifonov trifonov@e-math.ams.org

Sternberg Astronomical Institute, Moscow State University - Moscow 119899, Russia*;
(*) Present address: AMS, P.O. Box 7621, Cumberland, RI 02864, USA.
(received 28 August 1995; accepted in final form 31 October 1995).
PACS. 02.10-v - Logic, set theory, and algebra. PACS. 02.10By - Logic and foundations. PACS. 02.10Ws - Category theory and homological algebra.
EDP Sciences (Les Editions de Physique).

Introduction. - Topos theory (see ref. [1-6]) offers an independent (of the set theory) approach to the foundations of mathematics. Topoi are categories with "set-like", objects, "function-like" arrows and "Boolean-like" logic algebras. Handling "sets", and "functions", in a topos may differ from that in classical mathematics (i.e. the topos Set of sets): there are non-classical versions of mathematics, each with its non-Boolean version of logic. One possible view on topoi is this: abstract worlds, universes for mathematical discourse, "inhabitants" (researchers) of which may use non-Boolean logics in their reasoning. From this viewpoint the main business of classical physics is to construct models of the objective (absolute) universe with a given "bivalent Boolean" model of the researcher, and choose the most adequate one. In a sense, our task is inverse: with a given model of the absolute universe, to construct models of the researcher, and find out how the researcher’s proper universe changes if his logic is changed. Thus, not the universe itself, but rather its "differential" is what interests us here. We build a functional scheme describing the researcher’s interactions (actions and observations) with the environment.

Action principle (principle I). - The major intuition-based attribute of actions (elementary influences of the researcher upon the world) is that they can be associatively composed (i.e. performed in sequence), the compositions also being actions, and there is an identity action (changing nothing). The set of the researcher’s actions (or effectors), together with an associative composition, is his motor space. Example: in quantum theory the observer’s actions are represented by operators on a linear space and constitute, together with an associative composition, a semi-group with an identity (monoid).

Superposition principle (principle II). - The major intuition-based property of observations (mental and visual pictures of fragments of reality and appearance) is their ability to be superposed, with some real (later we shall generalize the situation for an arbitrary field F ) "weight" factors, assigned by the observer to each item. Intuitively, they measure the "participatory degree" of observations in a particular observational situation. In formal language, there are two algebraic operations on the set of observations: summation and multiplication (by reals). The set of the researcher’s observations (or reflexors), together with the two operations, is his sensory space. Example: in relativity the tangent space T_a at an event a can be interpreted as the set of observations (mental and visual images of events) of an observer residing at a: he considers nearby events as superpositions of some observations taken with some real "weight" factors (decomposition of an event in a basis). Since T_a is a real linear space, there are, indeed, two operations on it - summation and multiplication by reals.

Quantum principle (principle III). - "No elementary phenomenon is a phenomenon until it is an observed phenomenon" (any observation of the researcher is one of his actions). This simply means that obtaining constructive information about reality changes its appearance. The quantum principle couples observations with actions into new entities, called by constructivists states of knowledge: any rational researcher performs an action in accordance to, and interprets an observation on the basis of, his particular state of knowledge. It is said that rational knowledge consists of two fundamental (sensory and motor) components.

Paradigm. - We shall call the set of a researcher’s states of knowledge his paradigm. Observations, then, induce superposition of states of knowledge, with weight factors (extensive development or accumulation of knowledge), and actions induce associative composition (intensive development or elevation of knowledge). Thus we have three operations on the researcher’s paradigm, which endows it with an algebraic structure. The linear case of this structure is, of course, a real linear associative algebra A with an identity. The sensory space S_A then is the additive linear space of the algebra, and the motor space M_A is, one would say, its multiplicative monoid M. However, it is quite difficult to interpret 0 (the zero of the algebra) as an action. The identity 1 of the algebra is the identity action, but what is 0? We would rather take M\0 as the motor space, but in the former a composition of two actions is not always an action (i.e. M\0 is not always a monoid), which violates the intuitive notion of action and, moreover, will not let us define the logic of the researcher. To make a compromise, we assume that M_A = M\0 if the latter is a monoid, otherwise M_A = M. In other words, the motor space is the monoid generated by the set of non-zero elements of M.

Time. - We employ the constructivistic concept of time: a fundamental attribute of thought process, the basis to distinguish one entity from another. No statement on time being a physical property of the universe is made. Constructivists describe time as a partial order on the set of states of knowledge. So do we, slicing the paradigm with a 1-form on its sensory space, which partially orders states of knowledge by the naturally ordered set R of reals. Example: The proper time of an inertial observer in special relativity is a 1-form t, such that for any event a with components aⁿ (n = 0,1,2,3) in his rest frame, t(a)=a⁰

Metric. - A researcher’s natural ability to "feel" angles and estimate distances between observations is represented by a metric on his sensory space S_A . We do not force metrics into the scheme because a natural metric is defined automatically, once the proper time of the researcher is known, as follows. Each real algebra is completely defined with the structure constant tensor C(s; a, b) on its additive linear space S_A. Tensor C is a multilinear function of two vector arguments a, b and one 1-form argument s. Choosing some 1-form t on S_A (i.e. a time) makes the tensor C(t; a, b) depend only on the vector arguments. If C(t; a, b) is symmetric in a and b, it is, of course, a (proper- or pseudo-)Euclidean metric on S_A.

Absolute universe. - The actual reality principle (principle IV) affirms that the absolute universe exists and consists of interacting systems. Each system is represented by its states. Given all states of a system, it is defined completely. Some different systems may have the same states (common states). A system X is a subsystem of a system Y if all states of X are common to X and Y . If two systems are subsystems of each other, it is natural to consider them equal. Given two systems X and Y , we can consider a system Z (the union of and Y) whose states are all states of and all states of Y. For two systems X and Y with common states there is a system Z (the intersection of X and Y ) whose states are their common states. A system X that can have only states that Y cannot, is the complement of Y . The behaviour of the ontological pair "system, state" resembles that of "set, element" in "naïve" set theory, although conceptually they are very different. Two systems interact if states of one system depend on states of the other one, which is described as a function in set theoretical terms. Thus, with systems as sets (of their states) and interactions as functions, the category Set of sets serves as a first-order model of the absolute-universe axioms.

Proper universe. - The researcher’s actions change states of a system X : a state x before an action a becomes a state y after that, so any action a induces a function X->X, and we have the influence of the researcher with the motor space M on a system X as a realization of the monoid M in the set X, i.e. a function J, assigning to each a from M a function f_a: X->X such that a) f_a°f_b= f_(a×b) , ° is the operation of composition of functions, × is the multiplication in M; b) f₁ is the identity function id, i e. id(x) =x for all x from X, 1 is the identity of M. A pair (X, J), where J is a realization of a monoid M in a set X, is called an M-system. The collection {M} of all M-systems describes all the possible influence of a researcher with the motor space M on the absolute universe. {M} is a topos in which arrows (X, J)->(Y, K) are functions f: X->Y preserving realizations: f°(f_a)=(h_a)°f, f_a: X->X, h_a: Y->Y, for any a from M. The principle of active comprehension (principle V: the logic of a researcher is developed in his interaction with the environment) defines the proper universe as the topos {M} and assigns to the researcher its logic and mathematics.

Summary. - We see that only intuition-based concepts, logically prior to physics, are used here. The technique engaged is extremely simple in the sense that it is just several steps from the set and category axioms. To compare, the notion of smooth affine manifold (a starting point for the working physicist) is far more complicated. Of course, some of the notions seem uncertain - it is often inevitable, and now, to improve the situation, we give a strict form of the above outline.

Definition 1. - Let F be a partially ordered field. 1) An F-xenomorph is the category A[F] of linear algebras over F. 2) Paradigms of an F-xenomorph are A[F]-objects, his actions (or effectors) are A[F]-arrows. 3) For a paradigm A: a) states of knowledge are elements of the algebra A; b) a metric is the algebra-A structure constant tensor C together with a 1-form t on the additive linear space S_A of the algebra A, if C(t; a, b) is symmetric in a and b; t is the proper (or psychological) time; c) the sensory space is S_A together with all the metrics of the paradigm A, elements of S_A are observations (or reflexors); d) the motor space is the multiplicative subgroupoid M_A of the algebra A, generated by the set of non-zero elements of A; e) the dimensionality of the paradigm A is that of the algebra A. 4) A paradigm A is (ir)rational if M_A is (not) a monoid. 5) If A is a rational paradigm and the topos {M_A} is (not) Boolean, the paradigm A is (non-)classical. 6) A classical paradigm of maximal finite dimensionality, if it exists, is a classic paradigm.

Notes. - 1) We generalized the notion of action. Actions in the old sense (i.e. elements of the monoid M_A or internal actions or normal research) are, of course, A[F]-arrows term->A, where term is the terminal object of A[F], so the definition is correct. External actions (or extraordinary research) A->B may cause a paradigm change. 2) Irrational paradigms do not possess any definite logic (if the groupoid M_A is not a monoid we cannot construct the topos {M_A} and therefore define its logic). 3) The field F is what distinguishes one xenomorph from another, and sometimes we shall call F the (type of) psychology of the xenomorph, without assigning, of course, the standard meaning to the term. Here fields are taken together with partial orders on them, so two different orders on the same field deliver two psychologically different xenomorphs. 4) A paradigm may have several metrics or it may have none. 5) The absolute universe is a topos of realizations of a single-element monoid, therefore it is the proper universe of an "absolutely objective" paradigm whose motor space contains the identity action only. Informally, any absolutely objective researcher is absolutely inert. We now apply this scheme to the "current human parameters" (the psychology is R, logic is Boolean). The conclusion we shall obtain is that S_A is Minkowski space.

Theorem. - R-xenomorph has a unique classic paradigm E, of dimensionality 4, with a unique metric, of signature 2.

Proof. - If the logic of the topos {M_E} is Boolean then M_E is a group (see ref. [6], p. 121). Therefore E is associative, with an identity and without divisors of zero, and M_E = M\0 (because 0 has no inverse element), and we have isomorphisms E<->H (see ref. [7]) (H is the quaternion algebra) and M_E<->SU(2)xR⁺)=H\0 (R⁺ is the multiplicative group of positive reals, H\0 is the multiplicative group of non-zero quaternions). Thus the classic paradigm exists, it is unique and it is four-dimensional. For a basis e_n in S_E let t_m be the components of a 1-form t in the dual basis eⁿ (the indices run from 0 to 3). Then components G_pq of the metric G (summation on n is assumed) are G_pq = C(t;e_p, e_q) = C(t_neⁿ; e_p, e_q) =t_nC(eⁿ; e_p, e_q)=t_nCⁿ_pq, where Cⁿ_pq are the components of C. They are easily found in the basis of the unit quaternions 1, i, j, k, and we have G_nm:

t0  t1  t2   t3
t1 -t0 -t3   t2
t2  t3 -t0  -t1
t3 -t2  t1  -t0

G must be symmetric. Non-trivial symmetry demands t₁= -t₁, t₂= -t₂, t₃= -t₃ which yields t₁=t₂=t₃=0. Thus E has a unique (up to scalar factor) metric of signature 2, generated by a unique (up to scalar factor) psychological time (t₀, 0, 0, 0), which concludes the proof.

Relativity. - Defining (changes of) viewpoints of the xenomorph as (isometric changes of) bases e_n, we easily obtain a "sensory-motor" version of special relativity. In fact, though the 1-form t plays the role of psychological time (no such thing in special relativity), once t generates a metric, the latter in turn generates time in its standard sense. If we ignore the motor structure of the paradigm, four-dimensionality and Lorentz metric become a mystery, which is the case in standard physics. Let us call the paradigm E the Einsteinian paradigm. It is easily checked that besides the four-dimensional classic paradigm, there are two (and only two) non-trivial classical paradigms (of dimensionalities 1 and 2), both subalgebras of E (see ref. [7]). Informally, physics of the Einsteinian paradigm is a "superposition" of three versions of Boolean physics of different dimensionalities, which may account for the existence of three particle generations.

Virtual reality. - Since the "objective" absolute universe Set is a Boolean topos, we can separate a Boolean part - the most "objective", in a sense, in any rational paradigm A. Obviously, it is the set expA of invertible elements of the algebra A: they constitute a group. For any finite-dimensional rational paradigm of an R-xenomorph (decoherent paradigm), its set of invertible elements is a Lie group, so it has a natural topology. The sensory space S_A is the tangent space of expA at the identity. If A has a metric G, then there is a natural metric field g (left translations of G) on expA, i.e. expA possesses a natural (pseudo- or proper-)Riemannian structure, so we take it as spacetime of "objective" (or physical) events of the paradigm A. Thus any spacetime metric is induced by a psychological time, which means, in particular, that the notion of distance is undefined in the absence of an observer. Although always Hausdorff, expA may have non-trivial global topology.

Definition 2. - For a decoherent paradigm A: 1) If G is a metric of A, a physical metric is a tensor field g on the Lie group expA of invertible elements of the algebra A, such that at any point u from expA, g(u)=G(u×a, u×b), where a,b belong to A. 2 a) Spacetime is expA together with all the physical metrics of A; b) elements a from A, such that a belongs to (A\)expA are (non-)physical events; c) topology of spacetime is that of the Lie group. 3) A cosmology is a pair (expA, g), where g is a physical metric of A.

Notes. - 1) (Non-)physical observations and actions are defined automatically. Compositions of physical actions are always physical actions, while compositions of non-physical actions may result in physical actions (A\expA is not always closed under the composition). 2) In Lie theory terms, expA is an exponent-image of the algebra A. Then the psychological time t generates its exponent-image T (cosmological time). In other words, psychological time is a logarithm-image of cosmological time. 3) The virtual-reality principle (principle VI) assigns to every decoherent paradigm its spacetime.

Einsteinian paradigm E. - In this case spacetime expE has a "cylindric" product topology (S³)xR (S³ is three-sphere), and, for the metric g, expE looks like a four-dimensional "funnel" opening up into the future. Space-like hypersurfaces (of constant cosmological time T) are, of course, three-spheres. In fact, what we have here is an ideal big bang (the universe is expanding, it was arbitrarily small in the past, but no singularity). The scheme gives an "open" universe with a compact space-like hypersurface (the closest classical analogue is the model with positive cosmological constant > _crit). Since expE receives a locally compact (non-Abelian) Lie group structure, the corresponding Haar measure should be used in integration over spacetime. The Einsteinian paradigm is rather "objective": it contains a single non-physical reflexor 0, and no non-physical effectors.

E-mathematics. - The theorem offers the mathematics of the topos {SU(2)xR⁺}={expE} as the real-world math. Boolean though, it has an unpleasant property: the axiom of choice fails in {expE} (see ref.[6], p. 300). This means, for example, that it is impossible to prove countable additivity of the Lebesgue measure, without which the strict form of modern analysis is, of course, impossible, which, in turn, makes the basic technique of quantum theory invalid. Fortunately, we have a substitution - the axiom of determinateness that in many cases works better; for example, it does not create the difficulties associated with the algebra of cardinals (see ref. [8]). E-versions of the renorm procedures may eliminate the unreasonable ineffectiveness of classical mathematics in local-field schemes. If A is a non-trivial Grassmann algebra, the paradigm A is a Grassmannian (or supersymmetric) paradigm of an R-xenomorph. Since A has divisors of zero, M_A cannot be a group.Therefore, the logic of a Grassmannian paradigm is always non-Boolean, and mathematics is always non-classical.

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