Hidden Optimal Principle in Quantum Mechanics and Quantum Chemistry

Quantum Mechanic axioms are the result of the microscopic cooperative equilibrium among symmetric p layers (quantum object and human subject). On the other hand, we introduce Nash’s equilibrium in Hilbert space, which has two characteristics: it is a fixed point and it maximizes a utility function. Moreover, evolution and selfadjoint operators have interesting properties which allow us to study steady state in Evolutionary Game Theory. Also, we present cooperative games in complex systems language.The concept of cooperation is important in game theory but is somewhat subtle. The term cooperate means "to act together, with a common purpose". Incentive compatibility is equivalent to synergy principle, which appears naturally (to add or to multiply utilit ies, ec[34]). Finally, wein the relevant results not only proved the main theorems of Quantum Mechanics, Quantum Chemistry and its applications to salt-water but also we resolved some main questions such us: Do hydrogen-bonded networks, in which tunneling plays an important role, exist?, How cooperation and entropy affect water cluster equilibria?. Can salt (ClNa) p lay an Nobel role in photo-catalysis?. The answers are: tunnel effect it is possible in salt water but not in water, because ClNa, visib le light or electricity incentive to produce catalysis or photo-catalysis. Also, If hydrogen-bonded networks and water cluster size increases complexity of an global equilibrium, then tunnel effect appears as an local equilibrium.


Introduction
We will use a systemic perspective which is authorized by research methodology of comp lex systems [2], [3], in order to present a vision that integrates quantum computing concepts [7] and game theory elements to study cooperative games. The angular stone that unifies these two approaches is classic informat ion theory. However, there is a narrow and formal bridge to carry out these tasks denominated quantum probability. More easily, we can understand quantum probability as the Hilbert space of random variables with finite second mo ment, see [13], [16].
First, approaches related to cooperation appear. After that we carry out a rev ision o f existent literature, such us communicat ion, correlation, entanglement, dependence and co mmon ob ject ives, see [12]. Cooperat ive game theo ry enriched by Nash´s product focus approaches common objective top ic (resolution of co mmon prob lems) using bargaining roads wh ich allo w us to d iscover and imp lement equilibria. Concepts such us correlation, entanglement and To sum up, Quantum Mechanic axio ms are the result of the microscopic cooperative equilibriu m among symmetrical players (quantum-object and human-subject). Also, properties of selfadjoint and evolution operators here proved, can be applied to two-player symmetric games. Moreover, this work is an applicat ion of econophysics in Quantum Mechanics, see [8], [9]. This paper is organized as follows. The first section is a revision of the existent literature about cooperation, entanglement and the Hilbert space of random variab les. In the second section, we show the main theorems of co mplex games and its relat ionship with entanglement and ERP paradox.
In the third section we can see Nash's equilib riu m in Hilbert spaces, evolution operator and steady state. The fourth section is the conclusion of this research.

Cooperation in Economics and Quantum Mechanics
We can denote any strategic-form game Гas Γ = �N, �S j � jϵN , �u j � jϵN �, where N is the set of players, S j is the set of strategies for player j; and u j : S → ℝ is the utility payoff for player j. Here S = ∏ S j jϵN denotes the set of all possible comb inations or profiles of strategies that may be chosen by the various players.
In general, a randomized strategy is any probability distribution over a set of strategies. We may denote such a randomized strategy in general by σ = (σ(s) sϵS  The concept of cooperation is important in game theory but is somewhat subtle. The term cooperate means "to act together, with a co mmon purpose".
The common purpose can be explicit, when players add or mu ltip ly energy or utility (synergy principle). Co mp lex game theorem shows us that if we cooperate (to add or to mu ltip ly utility), then the results are correlated strategies and actions in order to maximize �∑ u j n j=1 . Definition 1.Complex-Game. We may define a cooperative transformation to be any mapping Ψ, such that if Гis a simp le-game in strategic-form, then Ψ(Г) is another complex-game in strategic-form.
Entanglement concept and dependence are related. Entanglement is a general physical concept; nevertheless, sometimes it is used in correspondence with mathematical correlation. On the other hand, dependence is a mathematical concept which is more general than correlation because if two stochastic variables ξand ηare uncorrelated (ρ(ξ,η)= 0), it does not follow that they are independent, see [16].
Definition 2.Dependent Strategy. A dependent (correlated) strategy for N players is any classic probability distribution Proof. Let U jϵN be random variables with joint distribution function F U 1 …U n , and φ ( u ) = ∑ u j N j=1 be a Borel function. If we put Z = φ(u) we see at once that (3) using the last equations we can write the density function of Z.
Let U jϵN be random variables with joint distribution functionsF U 1 …U n , and φ ( u ) = ∏ u j n j=1 be a Borel function. The synergy principle is taken into account because in the integral ∏ u j n j=1 ≤ z. If we put = φ(u) ; we see at once that: using the last equations we can write the density function of Z.

The Hil bert Space of Random Variables with Fini te Second Moment
1. An impo rtant role among Banachspaces L p ,p ≥ 1, is played by the space L 2 = L 2 (Ω, ℱ, P) , the space of (equivalence classes of) random variables with finite second mo ments, see [16], pag 260-280.
3. Let M = {η 1 , … , η n }be an orthonormal system and any random variab le in L 2 . Consequently the best (in the mean-square sense) estimator forξ in terms of η 1 , … , η n is ξ = ∑ (ξ, η j )η j n j=i (14) Hilbert space methods are extensively used in probability theory to study properties that depend only on the first two mo ments of random variables ("L 2 -Theory"  � … � � α j n | j n 1 > m n j n =1 � Definition 3. We say that the compound system is entangled if it is not decomposable.

Correlated Game and ERP Paradox
In Quantum Information Theory, the correlated equilib ria in t wo-player games mean that the associated probabilities of each-player strategies are dependent. Entanglement, according to the Austrian physicist Erwin Shrödinger, which is the essence of Quantum Mechanics, has been known for a long time now to be the source of a number of parado xical and counterintuitive phenomena. Of those, the most remarkable one is the so-called non-locality, wh ich is at the heart of the Einstein-Podolsky-Rosen paradox (ERP) see [12]pag 12. Einstein, Podolsky and Rosen consider a quantum system consisting of two particles separated by a great distance.
"EPR 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".
Nevertheless, the experiments verified that t wo particles in the EPR 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]. Moreover, the essence of the EPR a rgument is as follo ws. EPR was interested in what termed "elements of reality". Their belief was that any such element of reality must be represented in any complete physical theory. The goal of the argument was to show that Quantum Mechanics is not a complete physical theory, by identifying elements of reality that were not included in Quantum Mechanics. The way they attempted to do this was by introducing what they claimed was a sufficient condition for a physical property to be an element of reality, namely, that it be possible to predict with certainty the value that property will have, immed iately before measurement, see [15].
To illustrate, applying "revelation principle" for strategic-form games, the EPR parado x is solved automatically, because this principle demonstrates that all correlated game (entanglement) can be rep laced by a communicat ion game, see [14].
"...any equilibriu m o f any co mmunication game can be generated fro m a strategic-form game Г by adding a system for preplay commun ication mustbe equivalent to a correlated equilibriu m..." Thus, according to [1], "a correlated equilibriu m is any correlated strategy for the players in Г that could be self-enforcing ly implemented with the help of a med iator who can make nonbinding confidential reco mmendations to each player." Finally, nature is optimal and does not need to have replied or equivalent properties simultaneously such as (entanglement /\ co mmunicat ion).Therefore, we are in the presence of an exclusive-or (entanglement \/ communicat io n).
If in a theoretical or experimental way we can demonstrate that a system (n -player game) is entanglement, then ifis not necessary to speak of co mmunicat ion. In short, game theory helps to resolve EPR parado x.

Nash's Equilibrium in Hilbert Space
The modern description of Quantum Mechanics is profoundly based on linear mappings. Here, we represent the features of linear mappings which are most essential for Quantum Mechanics. Because we main ly concentrate on fin ite level quantum systems, the vector spaces that are treated hereafter will be assumed to have a fin ite dimension, unless exp licitly stated otherwise, see [7].
Let us begin with some terminology: a linear mapping H→H is called an operator. The set of operators on H is denoted L(H): For an operator T; we define the norm of the operator by ‖ T ‖ = sup ‖x‖=1 ‖ Tx ‖ = λ max (18) A nonzero vector x∈ H is an eigenvector of T belonging to eigenvalue λ∈ℂ if T x = λ x. Remark 1. With a fixed basis{e 1 , . . . , e n }of H; any operator T can be represented as n × n matrix over the field of complex nu mbers. It is not difficult to see that the matrix representing the adjoint operator T* is the t ransposed complex conjugate of the matrix representing T.
Theorem 2. The matrix of eigenvectors X is unitary X* = X -1 which permits us to write the spectral theorem A = x n (j) � represents a colu mn vector and is a n × n -matrix. Also it is easy to see that AX = ΛX where Using matrix properties and vector orthogonality < x (i) |x (j) >= 0 when i ≠ j and < x (j) |x (j) >= 1 for all j.
Thus, we conclude that X * = X −1 . Here the symbol (* ) means hermit ian transpose. Finally, using AX = ΛX , it is possible to write the spectral theorem.
We should revise the following postulates with the purpose of finding a relat ionship between Nash's equilibriu m and the eigenvalues of Quantum Mechanics operators.
First Postulate:At a fixed point t 0 , the state of a physical system is defined by specifying a ket | ( ) > belonging to the Hilbert space (state space) H.
Second Postulate:Every measurable physical quantity A is described by an operator A acting in the state space H.
Third Postulate:The only possible result of the measurement of a physical quantity A is one of the eigenvalues of the corresponding observable A. A measurement of A always gives a real value, since A is by definit ion Hermit ian (selfad joint operator).
A measurement process in Quantum Mechanics is the best example of strategic interaction between "human-subject and quantum-object". Therefore, we have a min imu m of t wo symmetric players when we carry out a measurement p rocess of a physical variab le.
Theorem 3.The norm o f a selfadjo int operator. The maximu m eigenvalue λ max = max ‖x‖=1 < | | > represents the maximu m expected value of the selfad joint operator A and its eigenvector |x max > is a fixed point A λ max |x max >= |x max > Proof.The problem to maximize has the next Lagrangian: Nash's equilibrium in Hilbert space. In mixed strategy Nash's equilibriu m, the eigenvector which maximizes < > H =< | | > also maximizes expected utility, where max ‖x‖=1 E ( u ) = max < > H =< p max |A|p max > and Proof. Let A be the matrix of utilit ies in a bimatrix game where expected utility is given by E ( u ) =< > H =< | A | p > (29) taking into account the properties of the eigenvector matrix X; and of the orthogonal eigenvector ‖ x ‖ = 1 and equations (20), (21), (22). Moreover, in the case of a min imu m, the problem can be resolved in a similar way. Finally, obtaining Nash's equilibriu m not only requires fixed point theorem but also ma ximization of expected utility.

Cooperati ve Games in a Complex System Language
We present cooperative games in the language of Co mplex Systems. Incentive compatib ility is equivalent to synergy principle, which appears naturally (to add or to mu ltip ly utilities); on the other hand, coordination and communicat ion have the same imp lications of correlat ion, see [14] page 256. Moreover, correlated strategies have a similar concept in Quantum Co mputing named entanglement. Correlation doesn't guarantee cooperation although this is equivalent to communication.
First, we need to replace the Nash Bargaining solution in two-person symmetric games. We define a bimatrix bargaining problem Г o consist of a pair (F; u) where F is a closed convex subset of ℝ 2 ; u = (u; u) is a vector in ℝ 2 and the set is nonempty and bounded. Here F represents the set of feasible payoff allocations of the feasible set, and v represents the disagreement payoff allocation of the disagreement point. The utility is represented by Second, there is a unique solution Φ(.,.) that satisfies the axio ms of Nash´s bargaining solution. This solution function satisfies, for every two-person bargaining problem ( , ) ; ( , ) , ≥ ( − ) (34) Quantum elements such us photons, electrons, subject-object have symmetrical physical propert ies and its two-player cooperative solution is the same for each one.
Finally, the special case of cooperation between symmetric part icles or "human-subject andquantum-object" is when v = 0, u = ⟨ p | A | p ⟩ , | p 〉 = � | x 1 | 2 … | x n | 2 � and the Hermitian operator A can have complex or real values aij. Consequently, we can write Φ(F; 0) in Hilbert

Evo lution and Density Operators in Two-Player Symmetric Games
Let us begin to write the expected utility E(u) = u of the operator (symmetric matrix) A that represents utilit ies. Here, we use Dirac´s notation (braket ), where | . 〉 represents a column vector or ket and 〈 . | is a row vector or bra, and The evolution operator has integrated properties of symmetric games in strategic form and replicator dynamics in matrix form.
Simp lifying equation (46) Thus, if steady state is gotten when

3.2.Tunneling an Explicit Phenomenon of Opti mal Cooperati on
Fro m viewpoint of classical mechanics, an electron cannot overcome a potential barrier higher than its energy. However, according to quantum mechanics, electrons are not defined by a precise position, but by a cloud of probability. This means that in some systems this probability cloud extends to the other side of a potential barrier. Therefore, the electron can cross the barrier, and for examp le, to generate electric current. This current is called the tunnel current and it is the control parameter that allows us to describe the topography of any surface. [5], [16]. A body with energy less than what is required to overcome or pass through a potential barrier can do it. There is a probability greater than zero, that a body goes through a potential barrier even if less energy is needed.
There is a probability g reater than zero, to find an object or phenomena associated due to its presence outside the potential barrier of energy higher than the same.

Tunnel Effect in Hydrogen B onds
Consider us the interaction of two water mo lecules, wh ich form part of a larger co mplex, such as salt water clusters, where the NaCl is completely dissociated. If two water mo lecules form a hydrogen bond, then the hydrogen atom may or may not link to one of the oxygen atoms. [17], [18].
In our study we have families of clusters formed as follows: n-Na, n-Cl, n-Na-Cl, n-Cl-Na, n-H2O, with n, m, s element of the natural nu mbers. n(water mo lecules)-m(sodium ato ms)-s(chlorine ato ms). Hydrogen and oxygen atoms of two water molecu les are a distance below the threshold. On the other hand, a hydrogen appears uncoupled between two oxygen atoms, that is not lin ked specifically to any of the two oxygens and your extreme oscillatory motion penetrates the molecular o rbitals of the two oxygen atoms, then we have a potential or feasible tunnel effect.
When hydrogen atom penetrates the orbital of one of the oxygen atoms, evidently out of the potential barrier of the other oxygen atom and vice versa. This implies that is constantly out of one of the potential barriers or put in another way: there is a probability greater than zero, that an atom leaves a potential barrier having a kinetic energy less than the height of the potential barrier.
Theorem.Tunnel effect is a phenomenon that emerges fro m external incitement such us catalysts presence and when the interior cooperation is maximu m.
Proof.-Let it be a co mplex system of n players (quantum objects) i = 1, ..., n, with a total energy E, wh ich is the sum of each of energies Ei. Given a set of thresholds ui, which correspond one to one to each player i, where Ei<ui. An infinitesimal change in the total system energy dE / dt, can cause a considerable variation in a specific energy of one players i, ((ΔEi)/(Δt)), wh ich exceeds the respective threshold that player ui/Δt.

Results
First, according to our interpretation, the second law of thermodynamics is fulfilled, as it exp licit ly states that "given entropy, system energy is minimal since given energy, system entropy is maximu m." We also know that in a state of equilibriu m, the values taken by the characteristic parameters of a closed thermodynamic system are such that maximize the value of a certain magnitude called entropy. In this case, we are analy zing two related systems, which change entropy and energy value, this evolution cannot be explained by the second law of thermodynamics but can be explained by Theory of Quantum Games.
Second, following Game Theory and the Theory of Quantum Games, we say the optimal evolution of a co mplex system min imizes the entropy. Figure 1, Figure 2 and Figure  3. Cooperation and Minimum Entropy. We can observe, the left side cluster, where there are one sodium atom, one Chloride atom, and twelve water molecules, which are in equilibrium. That is, the balance of forces is zero and the energy of the system is minimal. The intermediate cluster consists of a sodium atom, a Chlorine atom and ten water molecules, also it is in equilibrium. The cluster on the right is the union of the previous clusters. Meanwhile also the right cluster is in equilibrium. The question is what happens to the value of entropy and energy when you bind left cluster on intermediate cluster. Just note that the resulting entropy of joining two clusters is less than the sum of individual entropies. We can verify that 303,718 Cal / Mol-Kelvin < 369.267 Cal / Mol-Kelvin. On the side of the energy we observe to increases: 457.768 Kcal > 451.420 Kcal. Entropy system is decreasing and energy is increasing. What Happens with second law of Thermodynamics with Third, on the other hand, the cooperative equilibriu m, maximize an objective function called utility, which is exactly equivalent to the energy of a thermodynamic system. In this particular examp le, we see that in both cases, entropy is minimized but not maximized energy, there is cooperation in the o xygen atoms that are the same type, whereas when there are ato ms of different type converge to an equilibriu m but not necessarily in a cooperative manner. Figure 2, Figure  3.
Fourth, using the analogy in question, we demonstrate experimentally and with the help of Quantum Chemistry that clusters of mo lecules that interact cooperatively, min imize entropy and maximize energy, which is the fundamental assumption of the Quantum Theory of Cooperative Games, which is the cornerstone of our d iscussion of tunneling in salt water and its future use. Figure 4. That is, the balance of forces is zero and the energy of the system is minimal. Right side cluster is composed by ten water molecules. Meanwhile also the right cluster is in equilibrium. It is easy to see that entropy resulting from joining two identical clusters is less than the sum of individual entropies. We  Fifth, it is evident that pure water clustering tunnel effects not occur due to the non-presence of a NaCl-type catalyst. Figure 2.
Finally, theory of cooperative games makes clear that despite obtaining an invariant represented by entropy minimizat ion, it is required energy maximization. Figure  4and Figure 4, while the second spectrum corresponds to cluster four. We note in the graph, the spectral line 2825.32 (1/cm) corresponding to a frequency of tunneling, which disappears in the cluster number four Figure 6. Using a equipment to measure O2 concentration (AGS-688 y EGA-688 de BRAINBEE), we can verify presence of hydrogen because, when we do photo splitting salt water, appears oxygen and hydrogen simultaneously. In this figure we can see that O2 concentration increase with time, in reference with O2 base concentration equal to 20,64 % and salt concentration 30% wt The presence of a positive catalyst lo wers the energy of the system, allowing it to develop a reaction with less energy and faster. Increasing effectiveness and reaction output power.
Theoretical Results.
One player (molecu le, quantum element, water) wh ich verifies Nash´s equilibria is intelligent and optimize one utility function. That is the case of salt water.
Each physical variable is represented by anHermit ian operator whose norm allows us to obtain mixed strategy Nash's equilibriu m in Hilbert space. Figure 7. Differences of energy and entropy in the presence of a catalyst, as the case of sodium. The inner energy of the system without catalyst is lower, while the entropy is higher, which is logical, since the mere presence of the catalyst increases entropy Figure 8. Variation of internal energy by presence of catalyst. The graphs left, right and middle explain that the internal energy is less when there is no catalyst (left graph). In the case of entropy, we know that it is minimal when there is no catalyst. To sum up, in a system it is verified that the entropy is minimal and the energy is maximum when there is no catalyst By virtue of the theorems demonstrated in this paper, properties of Hermitian operators can only be used in symmetric games, which can be represented by complex matrix.
Mixed strategy Nash's equilibriu m in a bimatrix symmetric game represents a cooperative solution when exist correlation exclusive-or co mmunicat ion.
The idea of isolated physical systems has its explanation, the simp lificat ion of variab les and relat ionships among the parts. On the contrary, co mp lex systems analyze entirety, synergy and interactions as the cause of a common objective denominated cooperation.
The measurement process in Quantum Mechanics and Photocatalysis in Quantum Chemistry is the best examp le of strategic interaction between "quantum-subject (catalysts) and quantum-object (water) ." Therefore, we have a minimu m of two players when we carry out a measurement of a physical variable such us energy, frequency, symmetry, and so on. Lemma named Nash's equilibriu m in Hilbert space explains that In mixed strategy Nash's equilibriu m, the eigenvector which maximizes <A> H = <x|A|x> also maximizes expected utility, where: maxE(u ) = max<A> N =< |A| >.To use an analogy between utility and energy is exp lained in Jiménez, E.H (2003a, 2003b).
Experi mental Results.
The presence of a positive catalyst lo wers the energy of the system, allowing it to develop a reaction with less energy and faster. Increasing effectiveness and reaction output power. Figure 8.
In Figure 4, we can verify that standard deviation of spectrum has a relationship with system entropy.
Simp le Quantum Catalysis does not verifies Nash´s equilibria (min imu m entropy and maximu m energy).

Conclusions
• Each physical variab le is represented by anHermitian operator whose norm allows us to obtain mixed strategy Nash's equilibriu m in Hilbert space.
• By virtue of the theorems demonstrated in this paper, properties of Hermitian operators can only be used in symmetric games, which can be represented by complex matrix.
• Mixed strategy Nash's equilib riu m in a bimatrix symmetric game represents a cooperative solution.
• The idea of isolated physical systems has its explanation, the simp lificat ion of variab les and relat ionships among the parts. On the contrary, co mp lex systems analyze entirety, synergy and interactions as the cause of a common objective denominated cooperation.
• The measurement process in quantum mechanics is the best examp le of strategic interaction between "human-subje ct and quantum-object." Therefore, we have a minimu m of two p layers when we carry out a measurement of a physical variable such us energy, speed, mo mentu m.
• In physics, the existence of isolated systems with different unique objectives and disconnected to each other is untenable. This paper shows the necessity of introducing other elements like cooperation and optimality as foundations of the laws of quantum mechanics. We proved that Nash's equilibriu m is the hidden optimal principle of quantum mechanics.
• The apparent ERP parado x is completely resolved by using revelation principle for strategic-form games, because it demonstrates that a correlated game is co mp letely equivalent to a communicat ion game.
Consequently, it is not necessary to speak of communicat ion and worse even of information speed if a game is already correlated.
• Econophysics is both to use rational approaches of economics in the foundations of physics (in special, quantum mechanics) and to transpose physics formalism of quantum theory, statistical mechanics, electrodynamics and others in economics.