“Antropomorphic” mathematics and the crisis of science

Created by humans, our science is anthropomorphic, but not universal. Even in the basics of mathematical logic and axioms of set theory, it emanates from the everyday experience of people. This science meets fundamental challenges when trying to describe, for example, quantum systems,

One of the main achievements of mathematics of XX century are Godel’s incompleteness theorems which state that any formal system in mathematical logic, capable of expressing elementary arithmetic, can not be both consistent and complete. Namely, there are statements expressible in its language that are unprovable. Godel’s theorems developed an axiomatic theory of natural numbers of R. Dedekind and G. Peano. Published in 1931, they showed the failure of Hilbert's program to formalize mathematics. At present, Godel’s incompleteness theorems provide the main principle of methodology of modern science.

Indeed, contemporary theoretical physics forces us to conclude that any complicated physical system is not described by a unique theoretical model, but one needs several models, each of them has its own area of application and describes only a part or a certain aspect of a physical system. Moreover, these models at the intersection of their application areas fail to be consistent in principal.

In particular, until recently, theoreticians followed famous Dirac’s thesis: "A physical law should have mathematical beauty", written by him on the wall of D.D. Ivanenko’s office in Moscow State University. However, almost none of existent realistic theories satisfy this thesis. For example, the unified Standard Model of electroweak interaction is far from to be mathematically elegant. At present, only classical field theory admits the comprehensive mathematical formulation in terms of fibre bundles. Fundamental problems remain in classical mechanics: for instance, there is no intrinsic definition of inertial reference frames. In quantum mechanics, we have different non-consistent quantization techniques, e.g., algebraic quantization (the GNS construction) and canonical quantization.

However, the main "headache" of contemporary theoretical physics is quantum field theory. Some its parts (algebraic quantum theory, perturbative quantum theory, quantum electrodynamics) themselves look rather satisfactory. However, an integrated mathematical formulation of quantum field theory fails to exist yet. Moreover, there are doubts whether such a formulation within the existent mathematics is possible at all.

This mathematic is based on the mathematical logic which formalizes the logic of human thinking. It results from evolution of mental processes of a human mind, and it is the logic of statements in a language of words. This logic is not universal, it is "anthropomorphic". For example, an intelligent ocean in "Solaris" of Stanislaw Lem exemplifies a different logic, not the logic of statements.

In addition to the mathematical logic, the foundation of contemporary mathematics also contains the axiomatic set theory. In the initial period of its development at the fall XIX century (e.g., by G. Cantor), set theory was based on the intuitive notion of a set. However, soon it turned out that the uncertainty of this notion led to contradictions. The most famous of them are antinomies of Russell (1902) and Cantor (1899). Unfolded around antinomies debate has stimulated the development of axiomatic set theory, although its axioms are based on intuitive ideas, too. First axioms of set theory were suggested by Zermelo in 1908. At the present, there are several axiomatic systems of set theory, which are divided into four groups. Let us mention the Zermelo - Fraenkel system and the Von Neumann – Bernays – Godel one. The latter mainly is used in mathematical physics since it is a base of theory of categories. In the framework of this axiomatics, in addition to sets, another basic concept of the class is introduced in order not to consider too "big" sets that leads to contradictions. For example, all of the sets form a class, but not a set. Classes, unlike sets, can not be elements of classes and sets. With all the variety of axiomatic systems of set theory, all of them include some basic concepts and axioms, e.g., the notions of that a set consists of elements, the subset, the complement of a subset, the empty set, and axioms of the existence of the union and intersection of sets. All of these concepts came from the everyday experience of people dealing with classical macroscopic objects. However, they are not so evident, for example, in a quantum world. In particular, a quantum system may not consist of elements, or not admit a subsystem, or a subsystem has no a complement, etc.

Thus, our mathematics based on the logic of statements and set theory fails to be adequate in order to study the inanimate nature, where there are no “statements” and "words”. Therefore, our science fails to be universal, and it is both limited in its subject and incomplete in the image.

About twenty years ago, the idea was put forward to develop a new "quantum" logic and new "quantum" mathematics. However, the problem is not in that a new system of axioms must be offered, but in the fact that such a system could lead to “rich” mathematical theory. It is not possible yet. At the same time, the existent mathematics is meaningful because it simply follows an observable reality. Figuratively speaking, it solves a problem which has a solution a posteriori, and this solution needs to be recorded only. Developing one or another "quantum" mathematics, we do not know whether the problem has a solution in principle. Unfortunately, we can not put ourselves in the place of quarks and, therefore, we do not understand something important in a quantum world.

References:

Sardanashvily's blog post  Archive

What is a reference frame in field theory and mechanics

Non-relativistic mechanics as like as classical field theory is formulated in terms of fibre bundles.

In classical field theory on a fibre bundle Y->X, a reference frame is defined as an atlas of this fibre bundle, i.e. a system of its local trivializations.

If it is a gauge theory, Y->X is a fibre bundle with a structure Lie group G, and a reference frame equivalently defined as a system of local sections of an associated principal bundle P->X.

In particular, in gravitation theory on a world manifold X, a reference frame is a system of local frame fields, i.e. local sections of a fibre bundle P->X of linear frames in the tangent bundle TX of X.  This also is the case of relativistic mechanics. Therefore, one can treat the components of a tangent reference frame as relativistic velocities of some observers.

There are some reasons to assume that a world manifold X is parallelizable, i.e. its tangent bundle is trivial, and there exists a global section of a frame bundle P->X, i.e. a global reference frame. In this case, by virtue of the well-known theorem, there exists a flat connection K on a world manifold X which is trivial (K=0) with respect to this reference frame, and vice versa. Consequently, a reference frame on a parallelizable world manifold can be defined as a flat connection. The corresponding covariant differential provides relative velocities with respect to this reference frame.

Lagrangian and Hamiltonian non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles Q->R over the time axis R. Such fibre bundle is always trivial. Therefore, a reference frame in non-relativistic mechanics can be defined both as a trivialization of a fibre bundle Q->R and a connection K on this fibre bundle. Absolute velocities are represented by elements v of the jet bundle JQ of Q, and their covariant differential v-K are relative velocities with respect to a reference frame K.

This description of a reference frame as a connection enables us to formulate non-relativistic mechanics with respect to any reference frame and arbitrary reference frame transformations.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical field Theory (WS, 2009)

G.Giachetta, L.Mangiarotti, G.Sardanashvily,  Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

Sardanashvily's blog post Archive

 

Covariant (polysymplectic) Hamiltonian field theory (from my Scientific Biography)

Classical field theory is formulated as Lagrangian theory. All fundamental field equations are Euler – Lagrange equations derived from some Lagrangian. At the same time, classical autonomous (conservative) mechanics admits both Lagrangian and Hamiltonian formulations, which, however, are not equivalent. A Hamiltonian formulation of mechanics is based on symplectic geometry which a phase space is provided with. Naturally, a long time ago there arose a question about a Hamiltonian formulation of field theory. However, a straightforward application of symplectic Hamiltonian formalism to field theory, when canonical momenta are correspondent to derivatives of field functions only with respect to time, leads to an infinite-dimensional phase space, where  canonical variables are functions in any given instant. The Hamilton equations on such a phase space are not familiar differential equations and, in no way, comparable to the Euler - Lagrange equations of field theory. Such a symplectic Hamiltonian construction is utilized exclusively in quantum field theory to obtain the commutation relations of quantum field operators.

At the same time, a finite-dimensional phase space can be obtained if one considers canonical momenta correspondent to derivatives of field functions relative to all space-time coordinates. Such an approach is called the covariant Hamiltonian field theory. Its different variants are considered. These are polysymplectic, multisymplectic, k-symplectic Hamiltonian theories in accordance with a choice of a phase space and entered structure on it, generalizing symplectic geometry. In 1990, this question attracted attention of my student and collaborator Oleg Zakharov, who, in 1992, published an article in Journal of Mathematical Physics. However, he met a problem of constructing Hamilton equations of fields similar to the Euler – Lagrange ones. I built these equations, and then close interested in this topic.

We restricted our consideration to first order field theory, and developed polysymplectic Hamiltonian formalism on fibre bundles which, in the case of fibre bundles over the temporal axis X=R, led to non-autonomous Hamiltonian mechanics with the usual canonical variables. We constructed a globally defined polysymplectic form on a phase space, developed polysymplectic Hamiltonian formalism and, given a Lagrangian, built the associated Hamiltonians. The main results were presented in [65,66] in 1992 and, in 1993, we published already quite detailed theory [67]. The main problem was that Lagrangian and Hamiltonian formalisms on fibre bundles are not equivalent, unless only a Lagrangian is hyperregular, i.e., when the Legendre map of a configuration space to a phase space is a diffeomorphism. In a general case, one and the same Lagrangian is associated to different Hamiltonians, or no one. The comprehensive relationship between Lagrangian and polysymplectic Hamiltonian formalisms can be given in the case of the so-called semiregular and almost regular Lagrangians. The basic theorems are presented in papers [67,69] and books [10,11], and the final theory was published in the book [12] in 1997 and in the  article [88] in 1999.

Polysymplectic Hamiltonian formalism was considered in application to the basic field models, all of which are almost regular. We studied a possibility of quantization of fields in covariant canonical variables [70,114]. However, a question remains still open because additional gauge symmetries, arising in field models in covariant canonical variables, are not studied till now.

Reference:

G.Sardanashvily, My Scientific Biography 

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Why a classical system admits different non-equivalent quantization

A classical mechanical system admits equivalent description in different variables whose transformation law need not be linear.

In particular, a Hamiltonian classical system is equivalently described by variables related by arbitrary canonical transformations. 

If we have a completely integrable Hamiltonian system, its descriptions in original variables and the action-angle ones also are equivalent, though the transformation law between these variables is neither linear nor canonical in general.

In contrast with classical variables, quantum operators are linear operators in Hilbert spaces of quantum states and, therefore, they admit only linear transformations.  For instance, let a classical system be described in an equivalent way with respect to different variables (q,p) and (q’,p’) which possess some non-linear transformation law q’=Q(q,p), p’=P(q,p). Let (q, p) and (q’,p’) be quantization of these variables by operators in Hilbert spaces E and E’, respectively. Then the quantum systems characterized by quantum operators (q, p) and (q’,p’)  fail to be equivalent because there is no Hilbert space morphism E->E’ which transform (q, p)->(q’,p’).

In particular, there is no quantum partner of classical canonical transformations ubless they are linear.

Quantization of a completely integrable Hamiltonian system with respect to original variables and the action-angle ones is not equivalent and leads to different energy spectrums. For instance, this is the case of a Kepler system, whose familiar Schrodinger quantization provides the well-known energy spectrum of a hydrogen atom, but its quantization with respect to action-angle variables leads to a different energy spectrum.

Thus, a classical system can admit non-equivalent quantization. A problem is that nobody generally knows what its quantization is true.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005)

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables, Phys. Lett. A 301 (2002) 53-57; arXiv: quant-ph/0112083