اقتصاد فیزیک: تئوری بازی و تئوری اطلاعات برای مکانیک کوانتومی

Rationality is the universal invariant among human behavior, universe physical laws and ordered and complex biological systems. Econophysics isboth the use of physical concepts in Finance and Economics, and the use of Information Economics in Physics. In special, we will show that it is possible to obtain the Quantum Mechanics principles using Information and Game Theory.

At the moment, Information Theory is studied or utilized by multiple perspectives (Economics, Game Theory, Physics, Mathematics, and Computer Science). Our goal in this paper is to present the different focuses about information and to select the main ones, to apply them in Economics of Information and Game Theory. Economics and Game Theory1 are interested in the use of information and order state, in order to maximize the utility functions of the rational2 and intelligent3 players, which are part of an interest conflict. The players gather and process information. The information can be perfect4 and complete5 see Refs. [1], [2], [3] and [4], (Sorin, 1992). On the other hand, Mathematics, Physics and Computer Science are all interested in information representation, entropy (disorder measurement), optimality of physical laws and in the living beings’ internal order see Refs. [5], [6] and [7]. Finally, information is stored, transmitted and processed by physical means. Thus, the concept of information and computation can be formulated not only in the context of Economics, Game Theory and Mathematics, but also in the context of physical theory. Therefore, the study of information ultimately requires experimentation and some multidisciplinary approaches such as the introduction of the Optimality Concept [8]. The Optimality Concept is the essence of the economic and natural sciences [9] and [10]. Economics introduces the optimality concept (maximum utility and minimum risk) as equivalent of rationality and Physics understands action minimum principle, and maximum entropy (maximum information) as the explanation of nature laws [11] and [36]. If the two sciences have a common backbone, then they should allow certain analogies and to share other elements such us: equilibrium conditions, evolution, uncertainty measurement and the entropy concept. In this paper, the contributions of Physics (Quantum Information Theory)6 and Mathematics (Classical Information Theory)7 are used in Game Theory and Economics being able to explain mixed strategy Nash's equilibrium using Shannon's entropy [12], [13], [14] and [15]. According to [16, p. 11] “quantities of the formView the MathML sourceH=-∑pilogpiplay a central role in information theory as measures of information, choice and uncertainty. The form of H will be recognized as that entropy as defined in certain formulations of statistical mechanics wherepipiis the probability of a system being in cell i of its phase space,…,…” In Quantum Information Theory, the correlated equilibria in two-player games means that the associated probabilities of each-player strategies are functions of a correlation matrix. Entanglement, according to the Austrian physicist Erwin Schrödinger, which is the essence of Quantum Mechanics, has been known for long time now to be the source of a number of paradoxical and counterintuitive phenomena. Of those, the most remarkable one is the usually called non-locality which is at the heart of the Einstein–Podolsky–Rosen paradox (ERP) see Ref. [17, p. 12]. Einstein et al. [18] which consider a quantum system consisting of two particles separated long distance. “ERP suggests that measurement on particle 1 cannot have any actual influence on particle 2 (locality condition); thus the property of particle 2 must be independent of the measurement performed on particle 1.” The experiments verified that two particles in the ERP case are always part of one quantum system and thus measurement on one particle changes the possible predictions that can be made for the whole system and therefore for the other particle [5]. It is evident that physical and mathematical theories have had a lot of utility for the economic sciences, but it is necessary to highlight that Information Theory and Economics also contribute to the explanation of Quantum Mechanics laws. Will the strict incorporation of Classic and Quantum Information Theory elements allow the development of Economics and Game Theory? The definitive answer is yes. Economics has carried out its own developments around information theory; especially it has demonstrated both that the asymmetry of information causes errors in the definition of a optimal negotiation and that the assumption of perfect markets is untenable in the presence of asymmetric information see Refs. [19] and [20]. The asymmetry of the information according to the formalism of Game Theory can have two causes: incomplete information and imperfect information. As we will see in the development of this paper, Information Economics does not even incorporate in a formal way neither elements of Classical Information Theory nor Quantum Information concepts. The creators of Information Theory are Shannon and von Newmann see Ref. [15, Chapter 11]. Shannon the creator of Classical Information Theory introduces the entropy as the heart of your theory, endowing it of a probabilistic characteristics. On the other hand, von Newmann also creator of Game Theory, uses the probabilistic elements take into account by Shannon but defines a new mathematical formulation of entropy using the density matrix of Quantum Mechanics. Both entropy formulations developed by Shannon and von Newmann, respectively, permit us to model pure states (strategies) and mixed states (mixed strategies). In Eisert et al. [21] and [35], they not only give a physical model of quantum strategies but also express the idea of identifying moves using quantum operations and quantum properties. This approach appears to be fruitful in at least two ways. On one hand, several recently proposed quantum information application theories can already be conceived as competitive situations, where several factors which have opposing motives interact. These parts may apply quantum operations using a bipartite quantum system. On the other hand, generalizing decision theory in the domain of quantum probabilities seems interesting, as the roots of game theory are partly rooted in probability theory. In this context, it is of interest to investigate what solutions are attainable if superpositions of strategies are allowed [22], [23] and [24]. As we have seen before, from a historical perspective we can affirm that Game Theory and Information Theory advances are related to Quantum Mechanics especially regarding nanotechnology. Therefore, it is necessary to use Quantum Mechanics for five reasons: • The origin of quantum information and its potential applications: encryption, quantum nets and correction of errors has wakened up great interest especially in the scientific community physicists, mathematicians and economists. • Quantum Game Theory is the first proposal of unifying Game Theory and Quantum Mechanics with the objective of finding synergies between both. • In this paper we present an immediate result, product of using these synergies (possibility theorem). Possibility theorem allows us to introduce the concept of rationality in time series. • A perfect analogy exists between correlated equilibria that fall inside the domain of Game Theory, and entanglement that falls inside the domain of Quantum Information see Refs. [25] and [15]. • The reason of being of this paper is to demonstrate theoretically and practically that Information Theory and Game Theory permit us to obtain Quantum Mechanics Principles. This paper is organized as follows. Section 1 is a revision of the existent bibliography. In Section 2, we show the main theorems of quantum games. Section 3 is the core of this paper; here we present the Quantum Mechanics Principles as a consequence of maximum entropy and minimum action principle. In Section 4 we can see the conclusions of this research.

1. The Universe is structured in optimal laws. Random processes are those that maximize the mean information and are strongly related to symmetries, therefore, to conservational laws. Random processes have to do with optimal processes to manage information, but they do not have anything to do with the contents of it, this is the anthropic principle: Laws and physical constants are designed to produce life and conscience see Ref. [34]. 2. Every physical process satisfies that the action is a locally minimum. It is the most important physical magnitude, after information, because through it is possible to obtain the energy, angular momentum, momentum, charge, etc. Every physical theory must satisfy the necessary (but not sufficient) condition dS=0dS=0, because it is an objective principle. 3. We have demonstrated that the concepts of information and the principle of minimum action dS=0dS=0 leads us to develop the concepts of Quantum Mechanics and to explain the spontaneous decay transitions. Also, we have understood that all physical magnitudes are quadratic mean values of random fluctuations. 4. Information connects every thing in nature, each phenomenon is an expression of totality. Moreover, in quantum gravity, this fact is explained by Wheeler–DeWitt equation. 5. Mass appears were certain symmetries are broken, or equivalently, when the mean information decreases. A very small decrease in information produces enormous amounts of energy. Also, we have shown that there exist a relation between the amount of information and the energy of a system (Eq. (187)). Information, mass and energy are conservative quantities which are able to transform one into another. 6. A interdisciplinary scope it is possible, only, if we suppose that General System Theory is valid because games and information can be seen as particular cases of complex systems. Quantum Mechanics is obtained in the form of a generalized complex system where entities, properties and relations conserve his primitive form and add some other ones related with nature of the physical system.