^{1}

^{*}

^{2}

^{3}

In this paper, we present the quantum mechanics as a reactive system. To show this , “holography” is introduced as a reactive system. Illumination of the object and reference beam create sources (hologram) that are activated by the same reference beam, which is reacted in a way to produce the image. Controlled chemical system react s to the external or internal change generate d new sources (chemical product) that diffuse in the body environment , and establish es a new equilibrium. The droplets bouncing on a vertical vibrating fluid bath that simulate s the main quantum phenomena is a reactive system between droplets and vibrating fluid. It is shown that quantum mechanics is a probabilistic reactive system between quantum potential and pseudo kinetic energy in which an integral is the Fisher information. Information and probability are the key point s in quantum mechanics. Quantum mechanics can be built by only the probability normalization properties and associate d information without assuming any other hypothesis. With joint probability and Fisher information by Euler Lagrange equation , we can f i nd the quantum potential and the continuity equation. With only probability approach, it is possible to give a meaning to the Schrodinger equation without any thermo-dynamical model of quantum mechanics. The reactive system can be denoted as a morphogenetic system where the form is generated by some designed rules or properties.

In this paper, we introduce the meaning of the reactive system (morphogenesis) with the field and sources that react in a controlled way to the field in a way to generate needed constraint in the system sources field. First example of the reaction system is the “holography” [

Given this wave model

∂ 2 ρ ∂ x 2 + ∂ 2 ρ ∂ y 2 − 1 c 2 ∂ 2 ρ ∂ t 2 = 0 (1)

The solution of it is a function of the space time variables, which is an initial value of the density ρ (

The homogeneous Equation (1) can substituted with a non-homogeneous equation

∂ 2 ρ ∂ x 2 + ∂ 2 ρ ∂ y 2 − 1 c 2 ∂ 2 ρ ∂ t 2 = S ( x , y , t ) (2)

where S is an ordinary source that changes in the space and time. The source S superpose its ρ density fields on the homogeneous solution of the wave Equation (1). The Equation (2) is a non-reactive system because the source S is completely independent from the density ρ (

All the previous equation describes the non-reactive sources.

Now we make a transformation of the density by the transformation U. So we have

∂ 2 U ρ ∂ x 2 + ∂ 2 U ρ ∂ y 2 − 1 c 2 ∂ 2 U ρ ∂ t 2 = U S ( x , y , t ) + ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) U ρ − U ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) ρ = U S ( x , y , t ) + [ ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) , U ] ρ = U S + [ A , U ] ρ (3)

Now this becomes a reactive system which is generated by the transformation U of the density given by Equation (2). The reactive sources are defined in this way

S R = [ A , U ] ρ (4)

The Equation (3) is the “prototype of the morphogenetic system (reactive system)” which organizes itself by the detection of the local field in (2). With the information of the local field, it generates new waves actively, which superpose with the previous field, to create the designed transformation U.

Now, if the reactive sources are equal to zero, then in this case

( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) U ρ − U ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) ρ = 0

We have

( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) U ρ = U ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 − 1 c 2 ∂ 2 ∂ t 2 ) ρ = U S

For which when the source S change by U the density ρ change in the same way. Without the reactive sources the density can change only in the same way of the sources. Only reactive sources can give to the field of waves any type of constrain or form.

When the source depend on the field the commutator S R = [ A , U ] ρ is different from zero. We remark that to generate the reactive source is necessary a non-local connection among field and sources as in the holographic process. All the sources must be active in that same time (synchronic condition).

Example of reactive sources process

The first

“Holographic process” is again a morphogenetic process for which a coherent light interfere with the object light. The superposition of the two optical field is memorized in a sensitive paper (hologram). So we print the hologram of the

light in the paper. This hologram [

We know that diffusion process destroy any structure in order to obtain an homogeneous density with time. “Turing” discovers that when we include in the diffusion process a chemical system with a network of chemical reactions, we generate in particular cases structures far from homogeneous states. When we want generate structures or forms we can design special chemical reactions as active sources of molecule. The chemical reaction with diffusion generate different wanted density of molecule in the body. These sources are created by chemical reactions so that they are dependent from the initial density of all the molecule in the system (building of the sources) after the generate molecules

gets diffused in all the system to generate a pattern or a form in all the system. The wanted structure or conceptual object is the engine that control the sources that in the diffusion process generate the image of the conceptual object. The Turing process is a morphogenetic system (reactive system) between the chemical reaction system (sources) and the field of the diffused molecule. In a mathematical way, we have the diffusion process by sources independent from the initial density [

∂ 2 ρ j ∂ x 2 + ∂ 2 ρ j ∂ y 2 − 1 D ∂ ρ j ∂ t = S j

Suppose, for given the chemical reactions system

A → k 1 B B → k 2 C B + 2 C → k 3 3 C C → k 4 D (5)

The muster equation for this chemical system is

{ ∂ ρ A ∂ t = − k 1 ρ A ∂ ρ B ∂ t = k 1 ρ A − k 2 ρ B − k 3 ρ B ρ C 2 ∂ ρ C ∂ t = k 2 ρ B + k 3 ρ B ρ C 2 − k 1 ρ C ∂ ρ D ∂ t = k 4 ρ C 5(a)

So the diffusion equation can be written as

∂ 2 ρ A ∂ x 2 + ∂ 2 ρ A ∂ y 2 − 1 D ∂ ρ A ∂ t = 0 ∂ 2 ρ B ∂ x 2 + ∂ 2 ρ B ∂ y 2 − 1 D ∂ ρ B ∂ t = 0 ∂ 2 ρ C ∂ x 2 + ∂ 2 ρ C ∂ y 2 − 1 D ∂ ρ C ∂ t = 0 ∂ 2 ρ D ∂ x 2 + ∂ 2 ρ D ∂ y 2 − 1 D ∂ ρ D ∂ t = 0 5(b)

Now, we have the reactive system of equations, by applying 5(a) in 5(b) as

∂ 2 ρ A ∂ x 2 + ∂ 2 ρ A ∂ y 2 − 1 D ( − k 1 ρ A ) = 0 ∂ 2 ρ B ∂ x 2 + ∂ 2 ρ B ∂ y 2 − 1 D ( k 1 ρ A − k 2 ρ B − k 3 ρ B ρ C 2 ) = 0 ∂ 2 ρ C ∂ x 2 + ∂ 2 ρ C ∂ y 2 − 1 D ( k 2 ρ B + k 3 ρ B ρ C 2 − k 1 ρ C ) = 0 ∂ 2 ρ D ∂ x 2 + ∂ 2 ρ D ∂ y 2 − 1 D ( k 4 ρ C ) = 0 (6)

So we have two systems, the first one tells us the “behavior of the metabolite A, B, C, D when we change the time for the same position. The second one is a “reactive diffusion process” for which we can compute the metabolites for the same time as parameter with different positions. For the previous chemical system, we have that “A” is eliminated from the system so we have a negative source for “A”. For “B”, we have a positive source that is originated from A, and also a negative source that eliminate B and generate C and 3C. For “C” this is generated from internal system of B and in the same time is eliminated to produce “D” which is included in the system. In conclusion, A is destroyed, D is generated by B and C and are oscillated elements that one time they are generated and in a second time they are destroyed. Because we have a diffusion process the oscillation process alimented by A and D that is eliminated, move in the system and generate waves. For the oscillation reaction we have the space time behavior (

For the simulation of the quantum mechanics by droplets bouncing on a vertical vibrating fluid bath as we can see in this image (

The simple equation of the droplets bouncing on the vertical vibrating fluid is [

m d 2 x d t 2 + a d x d t = ρ sin ( w d x d t ) (7)

where

S R = ρ sin ( w d x d t ) is a reactive source.

In the polar coordinates we have

m d 2 x d t 2 + a d x d t = ρ ( i sin ( w d x d t ) + ρ cos ( w d x d t ) )

We remark that

m d 2 x d t 2 + a d x d t = 0 is the model for the “dumping process”.

The reactive source gives the energy to the droplet to move under the action of the vibrating fluid. So we have a couple between the vibrating and the droplet bouncing. The energy move from the droplet to the fluid and the other way around. For the transformation U we have

m d 2 U x d t 2 + a d U x d t = m d d t ( d U x d t ) + a U d x d t + a x d U d t = m d d t ( U d x d t + x d U d t ) + a ( U d x d t + x d U d t ) = m ( U d 2 x d t 2 + 2 d U d t d x d t + x d 2 U d t 2 ) + a ( U d x d t + x d U d t ) = U ( m d 2 x d t 2 + a d x d t ) + 2 m d U d t d x d t + m x d 2 U d t 2 + a x d U d t = 2 m d U d t ( d x d t + a x ) + m d 2 U d t 2 = S R (8)

Given the dumping solution x, we can found the solution for U. So we can compute the transformation by which we can know the transformation of x. We can also design the transformation U and with “x” we can compute the reactive sources to generate the transformation U. This is the process by which we can compute the sources to obtain the transformation of x. Many experiments shows the analogy between quantum mechanics and the reactive previous phenomena.

The probability to have joined set of data and states is given by the expression

p ( v ) = p ( s 1 , ⋯ , s n , q 1 , ⋯ , q m ) = p ( s j , q k ) (9)

where we have two type of variables. The variables q_{k} are the internal or macro variable to the particle, and s_{j} are external variables that generate noise to the states of the particle as in the Brownian movement. Any particle in quantum mechanics is in correlation with all the other particles in the universe that externally change the state of any particle. In the real world we must consider the state of all the universe, but because this is impossible, we perceive the other part of universe as noise. In agreement with the previous chapter we have the same separation of sources and variables (states). States and sources are joined by the probability in the multidimensional space. We know that for all possible states q_{k} we have the fundamental properties

∫ p ( s j , q k ) d q 1 d q 2 ⋯ d q n = 1

So we have

∂ ∂ q k ∫ p ( s j , q k ) d q j = ∫ ∂ p ( s j , q k ) ∂ q k d q j = ∫ ∂ p ( s j , q k ) ∂ q k 1 p ( s j , q k ) p ( s j , q k ) d q j = ∫ ∂ log p ∂ q k p d q j = 0

With another derivatives we have

∂ ∂ q h ∫ ∂ log p ∂ q k p d q j = ∫ ∂ ∂ q h ( ∂ log p ∂ q k p ) d q j = ∫ ( ∂ 2 log p ∂ q h ∂ q k p + ∂ log p ∂ q k ∂ p ∂ q h ) d q j = ∫ ( ∂ 2 log p ∂ q h ∂ q k p + ∂ log p ∂ q k ∂ p ∂ q h 1 p p ) d q j = ∫ ( ∂ 2 log p ∂ q h ∂ q k p + ∂ log p ∂ q k ∂ log p ∂ q h p ) d q j = ∫ ∂ 2 log p ∂ q h ∂ q k p d q j + ∫ ∂ log p ∂ q k ∂ log p ∂ q h p d q j = 0

where

I = ∫ ∂ log p ∂ q k ∂ log p ∂ q h p d q j = E ( ∂ log p ( s , q ) ∂ q k ∂ log p ( s , q ) ∂ q h ) (10)

is the required Fisher information.

For instance, given the like-hood distributions F ( x , σ ) = e − x 2 σ 2 with σ = 1 we have as

It is easy to show that the Fisher information in the first case, i.e.

From the Fisher information we have

S ( s 1 , s 2 , ⋯ , s n ) = ∫ ρ ∂ log ρ ∂ q i ∂ log ρ ∂ q j d q k (11)

where we define ∂ log ρ ∂ q i as like the osmotic velocity that we see in Nelson [

principle of the like kinetic energy we have the average value of the like kinetic energy as action S function of the external sources s_{j}. The minimum variation of the action function of the external variables is given by the expression

δ S ( s 1 , s 2 , ⋯ , s n ) = δ ∫ ρ 1 2 ∂ log ρ ∂ q i ∂ log ρ ∂ q j d q k (12)

Derivation of Euler-Lagrangian interpretation of Schrodinger Equation

The Euler Lagrange equation the minimum variation of the Fisher entropy (12) is given by the expression

δ S = ∂ ( ρ 1 2 ∂ log ρ ∂ x i ∂ log ρ ∂ x j ) ∂ ρ − ∂ ∂ x μ ∂ ( ρ 1 2 ∂ log ρ ∂ x i ∂ log ρ ∂ x j ) ∂ ∂ ρ ∂ x η = 1 2 ( ∂ log ρ ∂ x i ∂ log ρ ∂ x j − 1 ρ ∂ ( ∂ ρ ∂ x i ∂ ρ ∂ x j ) ∂ ∂ ρ ∂ x μ ) = 1 2 ( 1 ρ 2 ∂ ρ ∂ x i ∂ ρ ∂ x j − 1 ρ ∂ ∂ x μ ∂ ( ∂ ρ ∂ x i ∂ ρ ∂ x j ) ∂ ∂ ρ ∂ x η )

= 1 2 1 ρ 2 ∂ ρ ∂ x i ∂ ρ ∂ x j − 1 ρ ∂ 2 ρ ∂ x i ∂ x j = 2 m ( h 2π ) 2 Q = k Q

here, Q is the quantum potential and h is the Plank constant.

In fact we have

1 2 1 p 2 ( ∂ p ∂ q ) 2 − 1 p ∂ 2 p ∂ q 2 = − 1 p 1 2 ( − 1 4 p 3 2 ∂ p ∂ q + 1 2 p 1 2 ∂ 2 p ∂ q 2 ) = − 1 p 1 2 ∂ ∂ q ( 1 2 ρ 1 2 ∂ ρ ∂ x ) = − 1 p 1 2 ∂ ∂ q ( ∂ p 1 2 ∂ q ) = − 1 p 1 2 ( ∂ 2 p 1 2 ∂ q 2 ) = − Δ p 1 2 p 1 2 = − Δ R R = 2 m Q ( h 2π ) 2

For a set of states we have

Q = 1 k ( Δ p p − 1 2 ( ∇ p p ) 2 ) = 2 1 k ( ∇ ( ∇ p ) p − 1 2 ( ∇ p p ) 2 ) = 1 k ∇ ( 1 k ∇ log ( p ) ) − 1 2 ( 1 k ∇ log ( p ) ) 2 = 1 k ∇ u − 1 2 u 2

i.e. Q = 1 k ∇ u − 1 2 u 2 12(i)

where u is the pseudo osmotic velocity [

∂ S ∂ t + 1 2 m p i p j + V + ( h 2π ) 2 2 m ( 1 ρ 2 ∂ ρ ∂ x i ∂ ρ ∂ x j − 2 ρ ∂ 2 ρ ∂ x i ∂ x j ) = ∂ S ∂ t + 1 2 m p i p j + V + Q = 0

Now for

R 2 = ρ

and the Plank constant is equal to 1 we have

Q = − 1 2 m ∇ 2 R R

So for ∇ S = m v = p we have

∂ S ∂ t + 1 2 m p i p j + V + Q = ∂ S ∂ t + | ∇ S | 2 2 m + V + ( − 1 2 m ∇ 2 R R ) = 0 12(ii)

where Q is the Bohm quantum potential that is a consequence for the extreme condition of Fisher information (minimum or maximum condition for the Fisher information).

Now we write the continuity equation for the probability ρ = R 2 in this way

∂ ρ ∂ t + ∇ ( ρ v ) = 0

That can write in this way

∂ R 2 ∂ t + ∇ ( R 2 v ) = 2 R ∂ R ∂ t + ∇ ( R 2 v ) = 2 R ∂ R ∂ t + 2 R ∇ R v + R 2 ∇ v = 0

For R ≠ 0 we divide the previous expression for 2R so we have

2 R ∂ R ∂ t + 2 R ∇ R v + R 2 ∇ v = ∂ R ∂ t + ∇ R v + 1 2 R ∇ v = 0 ∂ R ∂ t + ∇ R v + 1 2 R ∇ v = ∂ R ∂ t + 1 2 m ( 2 ∇ R m v + R ∇ m v ) = 0

Because ∇ S = m v = p we have

∂ R ∂ t + 1 2 m ( 2 ∇ R ∇ S + 1 2 R ∇ ∇ S ) = ∂ R ∂ t + 1 2 m ( 2 ∇ R ∇ S + R ∇ 2 S ) = 0

Now we combine the continuity equation of the probability with the

∂ S ∂ t + ( | ∇ S | 2 2 m − 1 2 m ∇ 2 R R + V ) + i ( ∂ R ∂ t + 1 2 m ( R ∇ 2 S + 2 ∇ R ∇ S ) ) = 0

where the real part is consequence of the Fisher information and the imaginary part is due to the continuous equation for the probability.

Now for the equation ψ = R e i S , from the previous equation for S we have the Schrodinger equation. [

i ∂ ψ ∂ t = ( − 1 2 m ∇ 2 + V ) ψ with h 2π = 1

In conclusion we can make a reverse process used by Schrodinger we can generate the Schrodinger equation by the information space and the continuity equation of the probability. In this way the Hilbert mechanism can be explained only by probability normalization constraints.

Given the terms in (10)

F 1 = ∂ 2 log p ∂ q h ∂ q k , F 2 = ∂ log p ∂ q k ∂ log p ∂ q h (13)

We compare the classical reaction process between the kinetic energy of a particle with the potential energy with the quantum potential and the pseudo kinetic term in a way to show that also the quantum mechanics is a reactive system like the particle in the field.

The terms (13) can be write in this way

F 1 = ∂ 2 log p ∂ q h ∂ q k = ∂ ∂ q h ∂ log p ∂ q k = ∂ ∂ q h ( 1 p ∂ p ∂ q k ) = − 1 p 2 ∂ p ∂ q h ∂ p ∂ q k + 1 p ∂ 2 p ∂ q h ∂ q k F 2 = ∂ log p ∂ q k ∂ log p ∂ q h = 1 p 2 ∂ p ∂ q k ∂ p ∂ q h

The sum of the two functions is

F 1 + F 2 = F = − 1 p 2 ∂ p ∂ q h ∂ p ∂ q k + 1 p ∂ 2 p ∂ q h ∂ q k + 1 p 2 ∂ p ∂ q k ∂ p ∂ q h = 1 p ∂ 2 p ∂ q h ∂ q k F = 1 p ∂ 2 p ∂ q h ∂ q k − 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k + 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k

where we have

k Q = 1 p ∂ 2 p ∂ q h ∂ q k − 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k

And pseudo kinetic energy is

K = 1 2 ∂ log p ∂ q h ∂ log p ∂ q k

So we have

k Q + K = 1 p ∂ 2 p ∂ q h ∂ q k

The term ∂ 2 p ∂ q h ∂ q k is the Hessian of the probability. We remember that the

eigenvalues of the hessian give us the stable condition of the quantum system. If the eigenvalue is positive we have the stable condition when the eigenvalue is negative the state is unstable. Only the stable condition for the energy is possible in quantum system. Given

G = K − k Q = K + k Q − 2 k Q = 1 p ∂ 2 p ∂ q h ∂ q k − 2 k Q

When we substitute the values we have

G = ( 1 p ∂ 2 p ∂ q h ∂ q k − 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k ) + ( 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k ) − 2 ( 1 p ∂ 2 p ∂ q h ∂ q k − 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k ) = 1 p ∂ 2 p ∂ q h ∂ q k − 2 ( 1 p ∂ 2 p ∂ q h ∂ q k − 1 2 1 p 2 ∂ p ∂ q h ∂ p ∂ q k ) = − 1 p ∂ 2 p ∂ q h ∂ q k + 1 p 2 ∂ p ∂ q h ∂ p ∂ q k = − ∂ 2 log p ∂ q h ∂ q k

In conclusion we have the reactive system

G = L = K − k Q = − ∂ 2 log p ∂ q h ∂ q k

That is comparable with the (12)

G = K − k Q = − ∂ 2 log p ∂ q h ∂ q k and S = ∫ ( K − k Q ) ρ d q k = − ∫ ∂ 2 log p ∂ q h ∂ q k ρ d q k = ∫ ∂ log p ∂ q k ∂ log p ∂ q h ρ d q k = ∫ L d q k

So the classical action potential that is the difference between kinetic energy and potential energy can be found also in quantum mechanics by the like kinetic energy and quantum potential.

In this paper, we present reactive systems (morphogenetic system) in different contexts. The first is the wave reactive system, the second is the Holographic system, the third is the diffusion and chemical morphogenetic system and the last one is the walking droplets [

We thank the editor and reviewers for their valuable suggestions for the improvement of this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Resconi, G., Patro, S.K. and Amrit, S.S. (2019) Quantum Mechanics as a Reactive Probabilistic System. Journal of Applied Mathematics and Physics, 7, 686-701. https://doi.org/10.4236/jamp.2019.73048