The quantum mechanics of high-order kinematic values

Accepted Manuscript The quantum mechanics of high-order kinematic values E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva

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S0003-4916(18)30318-X https://doi.org/10.1016/j.aop.2018.12.001 YAPHY 67810

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Annals of Physics

Received date : 26 October 2018 Accepted date : 1 December 2018 Please cite this article as: E.E. Perepelkin, B.I. Sadovnikov and N.G. Inozemtseva, The quantum mechanics of high-order kinematic values, Annals of Physics (2018), https://doi.org/10.1016/j.aop.2018.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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THE QUANTUM MECHANICS OF HIGH-ORDER KINEMATIC VALUES E.E. Perepelkina, B.I. Sadovnikova, N.G. Inozemtsevab a

Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia E. Perepelkin e-mail: [email protected], B. Sadovnikov e-mail: [email protected] b Dubna State University, Moscow region, Moscow,141980 Russia N. Inozemtseva e-mail: [email protected]

Abstract In this paper, within the framework of a unified mathematical model, the new formulation of quantum mechanics — quantum mechanics of higher order kinematic values — is proposed. In contrast to the well-known formulations of quantum mechanics in the new formulation, the wave function has not only a coordinate or momentum representation, but also representations through acceleration and accelerations of higher orders. The representations of the wave function in terms of higher order kinematic values make it possible to consider the dissipative systems problem and the variable entropy systems one within a single mathematical apparatus. The new formulation is not built phenomenologically, but from first principles and in a particular case contains the formulation of the de Broglie–Bohm «wave-pilot» quantum mechanics. The new formulation is based on the Vlasov equations chain. This approach allows us to obtain a natural connection between classical and quantum systems. Examples of such systems are considered in this paper. Key words: quantum mechanics, high order mechanics, Wigner function, Vlasov equations,

rigorous result, PSI- algebra.

Introduction The quantum mechanics theory is known to base on the introduction of the wave function

  r , t  [1] and to use the function f  r , t     r , t  [2-4] as the density probability one. The 2

function f  r , t  defines the probability of a random variable R to take values r . For comprehensive consideration of systems described by random variables of the coordinate ( R ) and velocity ( V ), it is required to know the function f  r , v , t  [5, 6]. The consideration of the function f  r , v , t  requires introduction of a new equation so that the function should satisfy it. The construction of the functions f  r , v , t  from the function f  r , t  was performed by E.P. Wigner and H. Weyl [7-9]. Thus, the Wigner function satisfies the Moyal equation [10]. In quantum mechanics and during construction of the Wigner function [7-9], the Fourier transform is used as a «tool» for transition to the momentum representation f  r , t   f  p, t  [11-13]. Here we face a number of problems.

First, for purposes of the probability theory, the distribution density function f  r , v , t  construction of the two random variables R and V from the distribution density function f  r , t  of the random variable R requires solid argumentation. In the general case, such a construction is wrong at least because the Fourier transform is an automorphism with a period 4 [14, 15], that is 4       0    . Thus, a limitation appears at constructing the probability





density function f r , v , v , v , v , t and higher. In addition, the distribution function f  r , t  of the 1

random variable R does not contain complete information on another random variable V . In the general case, the random variables R and V may be independent. Secondly, a question arises: «What equations should the functions f  r , v , t  ,









f r , v , v , t , f r , v , v , v , t , … satisfy?» For the Wigner function, the Moyal equation was constructed in an «artificial» way (not from the first principles). Thirdly, when considering the problem of a quantum harmonic oscillator the question of the positivity of the function f  r , v , t  arises. Only the ground state of the oscillator provides positive probability density. In the paper [16,17] phase space  r , p, t  representation of the time-dependent Schrödinger equation was studied. By the generalized Husimi transform [18] these phase space representation directly from the coordinate one was obtained. In the papers [22-27] the quantum mechanics in  r , p, t  representation in Weyl-Wigner-Moyal-Groenewold formalism [19-21] was described. In 1850 were considered the high order kinematic physical systems by M.V. Ostrogradsky. It is proposed that physical systems contain as variable terms the finite number of  n

the set characteristics r , v , v , v ,... v that is coordinate and high order accelerations. In the scientific literature this approach is known as Ostrogradsky mechanics [29, 30] or high order mechanics. The are several cases when the high order mechanics was used. For example it is possible to show general relativity [34-39], the quantum field theory, the gauge theories [31-33] and some string models [40, 41] use the high order mechanics. In the paper [42-45] A.A. Vlasov proposed the statistical distribution functions theory. In this theory the variables r , v , v , v ,... for probability distribution density functions was considered as independent each other’s one. In this case the distribution function depends on not only coordinate r and velocity v but also on acceleration and high order accelerations: f  r , t  ,





f  r , v , t  , f r , v , v , t ,... and r , v , v , v ,... are appeared as infinite number variable set. In the paper, we present a new approach to the construction of quantum mechanics and statistical physics described in kinematic variables of higher orders. The approach is based on the infinite coupled chain of Vlasov equations [42, 43, 46] for the probability density functions f  r , t  , f  r , v , t  , f r , v , v , t , f r , v , v , v , t ,…. Such an approach has a number of strengths.



 



First, A.A. Vlasov obtained the infinite chain of equations from the first principles and it determines the infinite set of equations for the probability density functions. Secondly, the chain of Vlasov equations agrees with the mathematical apparatus of the probability theory. Thirdly, the chain of Vlasov equations explains the reason of negativity and positivity of a probability density function. Fourthly, in particular cases it gives the same results as the «classical» apparatus of quantum mechanics. Fifth, the Vlasov equation chain gives the opportunity to construct an infinite chain of equations of quantum mechanics for the wave functions   r , t  ,   r , v , t  ,  r , v , v , t ,









 r , v , v , v , t ,…, for which   f . Thus, it solves the question of positivity of the 2

probability density functions f    0 . Sixthly, the given approach allows one to extend the Lagrangian and Hamiltonian formalisms in a generalized phase space [46, 47]. The basis of this approach was laid by M.V. Ostrogradsky [28]. 2

2

Seventh, a new  -algebra is being constructed which makes it possible to write generalized equations of the electromagnetic field taking into account higher kinematic characteristics. In the particular case, generalized equations become the known Maxwell’s equations. The paper has the following structure. In §1, based on the chain of Vlasov equations, a chain of quantum mechanics equations is constructed for the wave functions   r , t  ,   r , v , t  ,









 r , v , v , t ,  r , v , v , v , t ,…, for which   f . The construction is performed with the use 2

of the approach we described in [48]. The equation for the wave function   r , t  coincides with





the Schrödinger equation. The equations for the wave functions   r , v , t  ,  r , v , v , t ,





 r , v , v , v , t ,… are new ones [16-18]. In §2, we consider the extension of the Lagrangian and Hamiltonian formalisms to the case of higher kinematic characteristics. The expression for the generalized potential energy U  r , t  , U  r , v , t  , U r , v , v , t , U r , v , v , v , t ,… is obtained. The energy is contained in new

















wave equations for the functions   r , t  ,   r , v , t  ,  r , v , v , t ,  r , v , v , v , t ,… The concept of the quantum potential Q  r , t  is expanded, which is used in the de Broglie-Bohm





«wave-pilot» theory [49-52]. The result is the obtained relations for Q  r , v , t  , Q r , v , v , t ,





Q r , v , v , v , t ,… and consideration of their relation to the classical potentials. The expression for the generalized Lagrangian and Hamiltonian is obtained as well as for the Legendre transformation linking them. The Lagrangian in the general case has a complex representation [53]. The complex form of the action Z is obtained and the map   eZ is considered. In §3 the generalized equations of motion are obtained for the change of the mean kinematic variables v , v , v ,…. In form, the generalized equations resemble the classical motion equations in the electromagnetic field. There is an analogue of «the electric and magnetic» field in the equations. In the particular case, the generalized motion equations become the classical motion equations. The higher-order kinematic variables taking into consideration modify the fields and its equations. The hydrodynamic representation of generalized motion equations is considered. The expression for the generalized quantum pressure is obtained. In §4 the equations for analogues of «electromagnetic» fields are constructed. The obtained equations can be divided into 4 groups. In the particular case each group transforms into one of equations from the Maxwell’s equations system. In §5 the example of the quantum harmonic oscillator is considered. The probability density functions f n  r , t  , f n  r , v , t  , f n r , v , v , t , f n r , v , v , v , t ,… are obtained, where n - is









the state number. The function f n  r , t  coincides with the known solution of the Schrödinger equation for the harmonic oscillator. The function f n  r , v , t  coincides with the Wigner function









for the harmonic oscillator. The functions f n r , v , v , t , f n r , v , v , v , t ,… are the new functions. As an example, the solution of the wave equation for the function   r , v , t  is given. It is shown that the wave equation for the function   r , v , t  has only one solution satisfying the ground state. The expression for the potential U  r , v , t  and the motion equation are obtained. The example of calculating the mean values of velocity, accelerations and accelerations of higher orders and their standard deviations is given. In conclusion, the main results of the work are given. 3

§1 Chain of quantum mechanics equations Let us consider an infinite coupled chain of Vlasov equations [42, 43, 54] for the



probability density function f n r , v , v ..., t



 n Sn  Qn , n  ,

(1.1)

where det

n 

 det    n 1    n d n det    r ,  r  ...   r ,  n2    r ,  n1     n u , n   , r  r dt t    t





r  r det  n u  n , t  ...    n  r n , t 







 



 v    v    ...      n 1   v n , t  

 

 

det

  r            r    det det  r   v   ,  n   ...  , n     ...    ...  , (1.2)     n 2          n1    n2   v   v   r   

 

Sn n , t  ln f n n , t ,

det

Qn n , t  div n1

 n 1

 , t  ,

v

n

r

Probability density functions satisfy the relations det

f0  t   N t    ... 



 f  r , t  d r    f  r , v , t  d rd v  ...  3



3

1

 



,



3 3 3  ... f r , v , v ,..., t d rd vd v...

  

3

2

(1.3)

where normalization factor N  t  is a number of the particles. The mean values are determined according to the relations [42, 43] f1  r , t  v  r , t  

 f  r , v , t  vd v, 3



(1.4)

2

f2  r , v , t  v  r , v , t  

 f  r , v , v , t  vd v, 3

3



… The chain of equations (1.1) was written initially by A.A. Vlasov in the form

f1  div r  f1 v   0, t f 2   v ,  r f 2   divv  f 2 v   0, t ...

4

(1.5)

f n   v ,  r f n   v , v f n  ...  div n2 f n v t





 n 1

v

 0,

...

In [48], from the first equation in the chain (1.5), (1.1) we obtained the Schrödinger 2 equation for the wave function 1  r , t  , f1  1 . Let us construct the following quantum





mechanics equations for the wave functions  2  r , v , t  ,  3 r , v , v , t , …., corresponding to the





distribution functions f 2  r , v , t  , f3 r , v , v , t ,… We use the procedure described in [48] for constructing.

 

Since the probability density function f n  n , t is a positive function by definition, it can

 

be represented as a squared absolute value of some complex function  n  n , t , that is:

 

 

f n n , t   n n , t

2

  n  n  0.

(1.6)  n 1

By the Helmholtz decomposition, let us represent the vector field

v

in the form of

superposition of the fields:  n 1

v

 , t     n

n

 n2 

v

 

 

 n  n , t   n An  n , t , divn2 An  0.

(1.7)

v

 n 1

 

where  n ,  n are some real constants,  n  n , t is the scalar potential of the field

 

An  n , t is a vortex component of the field  n 1

v

v

and

 n 1

v

. Let us rewrite the expression (1.7)

  n  n2  n   n An  i 2 n  n2  n   n An  v

v

    i n n2  0  i n    n An  i n  n2  ln n  i n    n An . v v  n 

(1.8)

 

Let us write the complex  n  n , t function in the exponential form:

 

 

 n n , t   n n , t e

 



in n ,t



,

where n  n , t is a phase. Taking into account the representation (2.9), the function the form: 5

(1.9)

n is of n

det   n i 2  ,t   e n n  Arg  n   2n  2 k   n . n  n 

Thus, the potential  n . of the field

v

 n 1

v

in (1.10) is defined as an argument of the complex

n . Considering the notation (1.10), we rewrite the expression (1.8) and obtain: n

function  n 1

(1.10)

 n      i Arg  n     n An  i n  n2 Ln  n    n An   ln v v  n    n   n    n2   n   n2   n     n An .  i n  n2  Ln   n   Ln   n     n An  i n  v  v  n v n   

 , t   i  n

n

 n2 

(1.11)

Considering (1.6) and (1.11), we rewrite the equation (1.1)/(1.5)

    n 2  i  n 1   v , pˆ 1   n  2 v , pˆ 2  n  ...  n1  v , pˆ n1   n   n t n n n  





(1.12)

2

2   n 1  n An  n  n  pˆ n  An   n   n  U n n . 2 n  n  2 n  n 2 

where det

pˆ n  

i

n

 n1 , pˆ n  r

i

n

n1 , pˆ n2  pˆ n2   r

1

 n2

  n1 . r





The (1.12) equation is the required one for the wave function  n r , v , v ,..., t . We define the constant values  n , n ,  n as follows det

n  

det

n

2m

,

n 

1

n  

,

n

e , m

(1.13)

det

where

1



is the reduced Planck constant. In this case, the Heisenberg uncertainty principle

for the phase space is fulfilled  r , v 

 r  v  1 

1

2m

.

(1.14)

The values of the rest constants n , n  1 we also associate with the generalized uncertainty principle [46, 55] for the rest projections of the generalized phase space [46, 47]

6

 v v  2 

 v  v  3 

2

2m 3

2m

(1.15)

,

,

… In the particular case, at n  1 , the equation (1.12) becomes the known Schrödinger equation for the wave function 1  r , t  [10]

i 1  11 1 t

2

2   1 1  1 A1 A1  1  1  U11 ,  pˆ 1  211  21 1 2 

(1.16)

In the case of n  2 we obtain a new equation for the wave function  2  r , v , t  2

2   2  i  2 1  2 A2 ˆ   2  2  p2  A2   2   2  1  v , pˆ 1   2  U 2  2 , (1.17)  2 t 2 2  2  2 2  2 2 2 

According to (1.17), the expression for the potential U n is of the form





U n n , t  

n nn



f n n , t



(1.18)

,

where n



      n  2   i  1   v , pˆ 1   2 v , pˆ 2  ...  n 1  v , pˆ n 1    n  n  pˆ n2  n An , pˆ n  .  n t  n n n   n n   









The expression (1.18) determines the generalized Hamilton-Jacobi equation, which is considered in the next section. §2 Lagrangian and Hamiltonian formalism Proceeding form (1.18), we obtain the expression for the potential U n . Taking into consideration (1.12), the operator n can be written as follows

n



  i  i i i   n  2   v , r   v , v  ...   v , n3   n v   n t  n n n  n





 i n    An , n2   . (2.1)   n2  v v  n  

Let us perform intermediate transformations





U n n , t  

2  1 n 1  Q n  n   n2n    n An ,   n2n   v  n t n v n  

  n 2  1    v ,  rn   v ,  vn  ...   v ,   n3n   , v n   





where 7

(2.2)

det

Qn 

2   n  v   n   1  n     Sn    Sn  , n  n 2 n  v 2 v  n2

n2

(2.3)

n 2

where Q n is a generalized quantum potential. In the particular case at n  1 , the generalized quantum potential Q n equals the «classical quantum potential» [11,12 (49-52)] Q1  Q 

2  r 1 1  r 1  . 1 1 2m 1

(2.4)

In the case of n  1 , the expression (2.2) is of the known form [10-12 (48-52)]

U1  r , t   

 f 1  2  1  1  r 1   r1    1 A1 ,  r1  1  t f1  





.  

(2.5)

At n  2 , the potential is of the form

U2 r , v, t   

  f 1  2  2    2  v 2  v2    2 A2 , v2   v ,  r2   .   2  t f2    





(2.6)

The expression (2.2) can be rewritten in terms of the in terms of the operator  nn 

d nn , dt

indeed





U n n , t  

    n2   n 1  d nn  v     n2n  n  n v  n  dt  

2

  .  

(2.7)

From the expression (2.7) we can obtain the representation for the generalized Lagrangian Ln  n , t , indeed

 

1 d nn 1 1    n dt 2 n  n 2

 n 1

vp

2

det

 U n  Qn   Ln .

(2.8)

In this case the phase  n has the meaning of a generalized action. Note that the Lagrangian L n does not contain the information on the vortex field An . A similar situation occurs also in the classical case. We obtain the generalized Hamilton-Jacobi equation. Let us rewrite the expression for the potential (2.2).

8

 n 1

1 n 1 1    n t 2 n  n 2

v

1

 n An

2 n  n

2

2

 U n  Qn 

2



   n 2   det  v ,     v ,     ...      v ,  n  n3  n    H n , n r n n v n v 2 n  n   



1

(2.9)



 

where we call the function H n  n , t

the generalized Hamiltonian function. Note that the

Hamiltonian H n , unlike the Lagrangian L n , contains the information on the vortex field An . A similar situation occurs also in the classical case. We call the obtained equation (2.9) the generalized Hamilton-Jacobi equation. In the particular case at n  1 , the equation (2.9) becomes a classical Hamilton-Jacobi equation [10-13 (48-53)]



1 m  v t 2

2

 e1  H1 ,

(2.10)

where e1 is a classical potential det

e1  U1  Q1 

2 e2 A1 . 2m

(2.11)

At n  2 we obtain «new Hamilton-Jacobi equation»



2

2 m  v t 2

2

 e 2  H 2 ,

(2.12)

where det

e 2  U 2  Q 2 

2 m  2 A2  2

2

 v ,  r 2  .

(2.13)

Using (2.8) and (2.9), we obtain the generalized Legendre transformation in the form

Ln  H n 

1

n

 n u , n n  .

(2.14)

In the particular case at n  1 , the transformation (2.14) becomes a classical Legendre transformation for the Lagrangian and the Hamiltonian





L1  H1  m v , v p .

(2.15)

At n  2 , from (2.14) we obtain the transformation which relates to L 2 and H 2

L2  H 2 

2

 v ,  r 2   m 



v , vp .

(2.16)

By analogy with [53], let us construct a complex action taking into account kinematic characteristics of the higher value. We use the notation introduced in §2 9

  i n  Ln  n   Ln  n  Ln  n ,  n  Sn  Ln f n  Ln   n  n   Ln  n  Ln  n .

(2.17) (2.18)

We define the complex function M n as

 

 

det

 

M n n , t  Sn n , t  i n n , t ,

(2.19)

where Sn  Re M n , n  Im M n . Taking into account the definitions (2.17) and (2.18) we obtain det

Zn 

Mn  Ln  n ,  n  eZn . 2

(2.20)

Taking into account the relation (1.1) and (2.8), we calculate  n Z n det dZ 1 1 d n Sn i d n  n 1 i n Zn  n M n  n n     Qn  Ln   n . 2 dt 2 dt 2 dt 2 n

(2.21)

By analogy with [53], we call the function  n the complex Lagrangian. The imaginary side of Im  n corresponds to the generalized Lagrangian L n , which is related to the generalized action  n . The real part Re  n corresponds to the sources of the flow of probability (1.2) Qn   n  n2  n . Thus, the variable Z n corresponds to the complex action. v

Let’s obtain an expression for the generalized complex Hamiltonian Legendre transformation connecting the functions  n and



 n 1

1 4 n  n

n

n

and the

.

2

 QnM  

vp

n

,

(2.22)

where

i

det n

 Qn 

det

2 n

Qn , QnM  

2  n  1  M   M    n   n . 2 n  v 2 v  n2

n2

Taking into account (2.21), (2.8), and (2.22), we obtain



i

n

n  

 n 1

1

vp

4 n  n

2

e

n

,

det

e

n

 Un 

n

,

or 2  1  i dn M n i  n  n   n2  M n  n   n2  M n  U n , 2 n dt n 2 n 2 v 2 n v

or 10

(2.23)

i

n

n 

2 det  n  M  M     Z n     Z n   U n  Qn  U n  e n . v n  v  n 2

n 2

Let’s obtain an expression for the generalized complex Hamiltonian

n Zn 

Z n   n u , n Z n    n   n u , n Z n    n ,   t  n  n   n u , n Z n  , 

(2.24)

Z n is the generalized complex Hamilton-Jacobi equation. Expression (2.24) t defines the Legendre transformation for the functions  n and n . Substituting (2.23) into (2.24) we obtain det

where

n



n

 n  1 1  S 1    n i   Qn   n u , n Sn    i n H n , 2  t t  2 2

or n

  n u , n Z n 



 1  in   4 n  n

 n 1

vp

2

e

n

    n u , n Z n   i  n enM  .  

The map (2.20) in the domain





  Sn n , t  ,





0   n n , t  4



(2.25) (2.26)



will be univalent. The function  n  n , t corresponds an action [53]. The quantum system cases may be correspond the domain (2.25)-(2.26). If  n  4 then the map (2.20) will be multivalent one (classical mechanics case) [53].

Fig. 1 The Mapping (2.20)

11

§3 Generalized motion equation Based on the expression (2.2), it is possible to obtain motion equations. Let us calculate the gradient of the potential U n . From (1.7) it follows that

1 1  n2n   n2  n   v 2 v 2 n

 n 1

v



n An . 2 n

(3.1)

Taking into account (3.1) and (2.2), the expression  n2U n is as follows: v

 n 1

  n2U n  v 2 n  n t 1

n  1  An    2 n  n t 4 n  n v

v

n2

 n 1

v

2



2    n  2  2 1  n   An   Qn      v ,  rn    v ,  vn   ...   v ,   n   . v v 4 n  n v n v    n2

n2

The addend  n2

 n 1

n2

(3.2)

n3

2

v

in the expression (3.2) can be represented in the form

v

2

 n 1

1  n1 2 r

v

  n 1   n 1   n 1    v ,  n1  v   n  v ,  n  Bn  , r    

(3.3)

where k 

det

Bn  rot  k 1 An  rot  k 2 An , r

v

we obtain a generalized motion equation



dn 2 n  n dt 1

 n 1

v



2 n   n2 An    U n     Q n     An  v v 2 n  n t 4 n  n v n2

n2

n2

    n  2    r v ,  n An   v v ,  n An  ...    n3  v ,  n An    n  n3  n   v  v 2 n  n    n  2 n  1        n   1    2   n 1    Bn    v ,  n  Bn   .  v , Bn   v , Bn   ...   v ,  2 n  n      

1









(3.4)

The equation (3.4) can be rewritten in a compact form, considering the formalism of the  algebra (see Appendix). Let us introduce the following notations

12

      1   1 Bn   1  n   v , An  cn   2   2    2     Bn      n   v , An  cn       n    ...  , n B   ... .    ...  n 1     n 1      n  2   Bn  n        v , An   n  n   n 1 cn n   n B    n       n  n      2  2   1 n    n n U Q   n An   cn   n n   4   n n n    

 

where

k 

 

(3.5)

cn are some functions having the properties:  n  2   cn  1 cn  v , v ,..., v , t  ,   n  2     2 cn   2 cn  r , v ,..., v , t  ,   ...  n  4  n  2    n 1 cn   n 1 cn  r , v ,..., v , v , t  ,   n  3      n cn   n  cn  r , v ,..., v , t  .   1

As a result, we obtain



dn dt

 n 1

  n

v

dn dt

 n 1

v

 An   n  n  n     n  n u , n B  ,   t





  n En   n u , n B  ,

where det

En  



 An   n  n   .  t

The motion equations (3.4) can also be written in other form

 

 n 1

dn 2 n  n dt 1 1 2 n  n

v

  n2U n    n2Qn  v

 n3 n  n  v

v

 n   n 1  n    v , Bn   2 n  n  

1  2   n 1   n dn An ,    n2  n An   v ,   n2   n An   v 2 n  n  2 v  2 n  n dt   1

or

13

(3.6)

dn dt

 n 1

v

   n 1    n  En   v ,  n  Bn   ,     

(3.7)

where 2 2 n  n  1  U n  Qn   n An  , n  2    n 1  d  En   n An   n2  n   n  n  n3  n   v ,  n2  An . v v dt n v  

 n

n 

The representations (3.6) and (3.7) of motion equations are equivalents and differ only in the form they are written. In the case n  1 , the equations (3.6), (3.7) are of the classical form [10, 48]





d v   1 E1   v , 1 B1  , dt where

(3.8)

 1 A1   r 1 1 ,   B1  rot r A1 , t 2 e2 e 1 1  U1  Q1  A1 . 2 2m E1  

At n  2 , the equation (3.7) is of the form





d2 v   2 E2   v ,  2 B2  , dt

(3.9)

where

E2    2

2 

A2  2  v    2  2  r  2   v ,  r  A2 , t 2 2 2 2  2  1  U 2  Q2   2 A2  . 2  2 

The equation (3.6)





d2 v   2 E2  v , 1 B2    v ,  2 B2  , dt

(3.10)

where E2   1

2 

A2  v  2  2   r 1  2 , t

2    v , A2  . 2 2

Hydrodynamic representation In paper [54] we obtained a chain of relations for the derivatives and average values of kinematic characteristics of the for

14

f0 v

f1 v f2

v

d0 d r  r 0 f0 , dt dt d1  f1 v    v  v  v  v ,  r f 2  d 3v, dt  

 f0

 f2



 v  v

d2 v   v v dt 

(3.11)



,  v f 3 d 3v ,

… The relations (3.11) can be rewritten in terms of the hydrodynamic stress-elasticity tensor [42, 43] 1

 f  r , v , t   v 

P 

 2



P 

2

v

 f  r , v , v , t   v  3



 v

 v

v

 d v,

 v





3

 v

 d v, 3

(3.12)

.... Let us calculate the derivative

 1 P x



  2 P v



 1 P x 

 

v 

 xf  d v.

 v



 v

 v d v,

 v



 v

 vf  d v,

v

 v

  v 

v

  v 

v





 v

  v 



 3

 P

  2 P  3 P , ,... we obtain , v  v 





2

3

f3

3

4

(3.13)

3

...

Substituting (3.13) into (3.11), we obtain a set of motion equations  d1 1  P v    v dt f1 x  1

,

 2 d2 1  P v    v dt f 2 v 

(3.14)

,

... The motion equations (3.14) are related to the obtained generalized motion equations v , v ,… in the hydrodynamic representation (3.6), (3.7). The mean values of correspond to external interactions. The derivatives

 1 P

representation are associated with «forces of pressure».

15

x 

,

  2 P v 

,… in the hydrodynamic

When considering quantum mechanics, the generalized quantum potential Q n leads to the generalized quantum pressure

 n  q

P of the form P   n2 f n

 2 Sn ,   n x    n x

 2 

x  v ,... . Indeed, calculating the ―force‖ of

 n  q

where

 n 

det

 n 1

x   x   , that is

1 

x  x ,

the generalized quantum pressure 



x  v ,

(3.15)

3 

 n  q 1  P , we obtain f n   n x

 n  q 1  P   2 n2  n   f n   n x  x

 1 2 fn   f  n x  n x  n

 Q   2 n  n  n  n .   x 

(3.16)

The addend (3.16) is included into the generalized motion equations (3.6), (3.7). According to the hydrodynamic representation (3.14), the addend (3.16) is associated with the «force» of the generalized quantum pressure (3.15). §4 Equations of the «electromagnetic» field Based on the generalized motion equations (3.6), (3.7), we can obtain equations foe the fields E , B , D , H . From (3.3) it follows that

divr

1

Bn  0, divv

 2

Bn  0,...,

(4.1)

or in the form of  -vectors (see Appendix)

 An      n B   n , n A , n A   ...  , 0    det



n



0    n , n B   ...  , 0   





, n B  0.

(4.2)

(4.3)

From the equations (4.1) and (4.3) it follows that the divergence and  -divergence of the fields B equals zero. At n  1 , the equations (4.1) and (4.3) become a single equation

divr

1

B1  0.

(4.4)

The equation (4.4) is included into the Maxwell’s equation system. Let us calculate the rotation of the expression (3.6), rot  k  En for k  0,..., n we obtain r

16



 1 Bn  rot r En  rot r   v  2   ...    n2  n    , v t  



  2 Bn 1 3 n  rot v En  rot v   r      v     ...    n2     , v t  

(4.5)

...    Bn 1 2 n 1  rot  n2 En  rot v   r      v     ...    n3     . v v t   n



At n  1 , the equations (4.5) become a single equation



 1 B1  r , t   rot r E1  r , t  . t

(4.6)

The equation (4.6) is also included in the Maxwell’s equation system. Before considering the system of equations (4.5) for the cases n  1 , let us define the following relations, making relations between the fields En and En 1 , An and An 1 as well as between the potentials  n   and  n 1

.  n  2   n 1   n 1   En  r , v ,.., v , t    En 1  r , v ,.., v , t  d 3 v ,     

(4.7)

 n  2   n 1   n 1   An  r , v ,.., v , t    An1  r , v ,.., v , t  d 3 v ,     

(4.8)

k 

 n  2





 n  r , v ,.., v , t   







k 



 n 1



 n 1

 n1  r , v ,.., v , t  d 3 v , k  1,..., n. 



(4.9)

The correctness of the applied conditions (4.7)-(4.9) should be seen from the further constructions. For all k  1,..., n  1 the condition (4.9) is fulfilled automatically. In the case k  n , the condition (4.9) is equivalent to the expression U n  Qn 

1

 n An   n cn  2

4 n  n

  n1   n1   n 1 n  n 1   n  cn 1  d 3 v .  v , An 1    2 n  n      n 1 

If the vortex component An of the field is absent, the expression (4.10) is as follows U n  Qn   n  cn 

  n 1  3  n 1 n  n   c v .  n 1 n 1  d 2 n  n     n 1 

Note that it follows from the condition (4.8) the field property

k 

Bn  rot  k 2 An v

17

(4.10)

k 



 n  2   n 1   n 1   Bn  r , v ,.., v , t   rot  k 1 An   rot  k 1 An 1  r , v ,.., v , t  d 3 v  r r     





k 

 n 1   n 1  Bn 1  r , v ,.., v , t  d 3 v .  

(4.11)

So, let us consider the case n  2 . We obtain from (4.5) two new equations

 1 B2  r , v , t   rot r E2  r , v , t   rot r v  2  2  r , v , t  , t 2    B2  r , v , t  1   rot v E2  r , v , t   rot v  r    2  r , v , t  . t 

(4.12)

Let us integrate the equations (4.12) over the velocity space using of the relations (4.7), (4.11). For the first equation from (4.12), taking into consideration (A.19), we obtain



 1 B2 d 3v  rot r  E2 d 3v   rot v r  2  2 d 3v,  t    



 1 B1  r , t   rot r E1  r , t  . t

(4.13)

When obtaining (4.13), it was supposed that the field r  2  2 vanishes rapidly «enough» at v   , that is

 rot



v

 r  2   2 d 3v 

 d



v

,  r  2  2   0.

The equation (4.13) completely coincides with the equation (4.6). Integration of the second equation from (3.6) gives a trivial result «0=0». Let us calculate the divergence from En (3.6) and take into account (1.7), we obtain

 div r An  div r En   r 1  n  divr   v  2  n  ...    n2  n   n  , v t     divv An  divv En   v  2  n  divv   r 1  n   v 3  n  ...    n2  n   n  , (4.14) v t   ...   div n2 An  0  div n2 En    n2  n   n  div n2   r 1  n   v  2  n  ...    n3  n 1  n  . v v v v  v t  

In the case n  1 , the equations (4.14) become a single equation

divr E1   r 1 1 ,

18

(4.15)

2 e2 A1 . When considering the model of «strongly self-consistent 2 2m field» [48], the right side of the equation (4.15) corresponds the density f1  r , t  .

where e 1 1  U1  Q1 

At n  2 , the system (4.14) consists of two equations  1 2 div r A2  div r E2   r    2  div r  v    2 , t   divv A2  divv E2   v  2  2  divv  r 1  2 . t 

(4.16)

Let us integrate the equations (4.16) over the velocity space using the relations (4.7-9) and the property

divr v    r , v     r , v     v , r    divv r  . For the first equation from (4.16) considering (1.7), we obtain 

 divr  A2 d 3v  divr  E2 d 3v   r  1  2 d 3v   divv r  2  2 d 3v, t    



 divr A1  divr E1   r 1 1 , t divr E1   r 1 1.

(4.17)

The equation (4.17) completely coincides with the equation (4.15). Integrating the second equation over the velocity space gives a trivial result «0=0». The conditions (5.9-10) on the potential  for the case n  2 is as follows 1

U1  Q1 

1  r , t  

1





 2  r , v , t  d 3v ,

2    1  1 A1  1 c1  1   2  2  v , A2  1c2  d 3v, 2 21 1      2 





(4.18)

Using the notations from Appendix, we rewrite the Vlasov equation (1.1)/(1.5) as follows

f n   n , n u f n   0. t

(4.19)

The function f n is a scalar function, which can be represented in the form of the divergence of a field n D , that is det

fn  or

f n  divr

1



n



, n D ,

(1.20)



Dn  divv

 2

Dn  ...  div n2 v

19

 n

Dn .

Note that the representation (1.20) agrees with the conditions (1.3). Substituting (4.20) into the equation (4.19), we obtain  nD  nu  n , t 



n

 , n D   0.  



(4.21)

Taking into account the property (A.17) from the expression (4.21), we can obtain the equation nD  n J   n , n H  , n  t det

where n J  n u



n

2

,

(4.22)



, n D , and n H is a field. At n  1 the equation (4.22) becomes formally one 

of Maxwell’s equations

 1 D1  r , t  1  J1  r , t   rot r t where

1

J1  f1 v , f1  divr

1

H1  r , t  ,

(4.23)

D1 . In the case n  2 we obtain two equations

  1 D2     vf 2   rot r  t        2 D2   v f 2   rot r    t  or

1

1

 2

H2  , 1  2 H 2  rot v   H 2  H 2  rot v

 1 D2  r , v , t   vf 2  r , v , t   rot r 1 H 2  r , v , t   rot v  2 H 2  r , v , t  , t  2  D2  r , v , t  2 1  v  r , v , t  f 2  r , v , t   rot r   H 2  r , v , t   rot v   H 2  r , v , t  . t

(4.24)

As before (4.7-9), we impose analog conditions on the fields D and H  n  2   n 1   n 1   Dn  r , v ,.., v , t    Dn 1  r , v ,.., v , t  d 3 v ,     

(4.25)

 n  2   n 1   n 1   H n  r , v ,.., v , t    H n 1  r , v ,.., v , t  d 3 v .     

(4.26)

Let us integrate the first equation from (4.24) over the velocity space taking into account (1.4), (4.25-26), we obtain  1 D2  r , v , t d 3v   vf 2  r , v , t d 3v  rot r  1 H 2  r , v , t  d 3v   rot v t     

20

 2

H 2  r , v , t d 3v,

 1 D1  r , t  1  J1  rot r t

1

H1  r , t  ,

(4.27)

The obtained equation (4.27) completely coincides with the Maxwell equation (4.23), obtained at n  1 . Integrating the second equation in (4.24) over the velocity space gives a new equation   2 D2 d 3v   v f 2  r , v , t d 3v  rot r   2 H 2 d 3v   rot v  t       

  2 D1  r , t   f1 v t

 rot r

 2

1

H 2 d 3v ,

H1  r , t  ,

(4.28)

The equation (4.28) significantly differs from the Maxwell equation (4.23), as it contains new «current» density f1 v or rather the current density of an external force (3.11) or (3.14). According to the physical content, the equation (4.28) corresponds to the motion equation in the hydrodynamic representation (3.14). Thus, a system of the generalized field equations (4.3), (4.5), (4.14), (4.22) is obtained. In the particular case, at n  1 or when integrating over the corresponding kinematic spaces d 3vd 3v... , the obtained system becomes the known system of Maxwell’s equations. §5 Example of the quantum harmonic oscillator in the generalized phase space Let us consider a model of the one-dimensional harmonic oscillator. The oscillation equation is of the form x  12 x  0,

(5.1)

where 1 is the oscillation frequency. In follows from the equation (5.1) that the following relations are true

v  12 x, v  12v, v  12v, v  12v,....,

(5.2)

From (5.1) according to (1.2) and (1.4), we obtain

v  0,

v  12 x,

v  12v,

v  12v,

v  12v,....,

(5.3)

Q1  divr v  0, Q2  divv v  0, Q3  divv v  0, Q4  divv v  0,.... The chain of Vlasov equations (1.1)/(1.5) for the harmonic oscillator in the onedimensional stationary case is as follows

f1  0, x f f v 2  v 2  0, x v v

21

(5.4)

f3 f f  v 3  v 3  0, x v v f 4 f 4 f 4 f 4 v v v  v  0, x v v v ... v

The characteristic equations for (5.4) are of the form

dx dx dv dx dv dv dx dv dv dv  0,  ,   ,    ,... v v v v v v v v v v

(5.5)

From (5.5) we obtain the following first integrals

1  x 2 , 2  v 2  12 x2 , 3  v2  12v2 , 3  v  12 x, 4  v 2  12v 2 , 4  v  12v,  4  1  v  12 x   4 arcsin

(5.6)

v

4

,

… As a result, the solution of the equation chain (5.4) can be represented as follows

f1,n  x   F1,n 1  x   ,

f 2,n  x, v   F2,n 2  x, v   ,

f3,n  x, v, v   F3,n 3  v, v  ,3  x, v   ,

(5.6)

f 4,n  x, v, v, v   F4,n 4  v, v  ,4  v, v  ,  4  x, v, v, v   , ... where F is a functions, which can be defined from the initial-boundary conditions. The form of the functions F1,n and F2,n for the harmonic oscillator is known. The function F1,n is obtained from the solution of the Schrödinger equation for the harmonic oscillator

F1,n  ψn  x  , 2

1  En  1  n   , n  0,1, 2,... 2  1

 m1  4  ψn  x     e 2n n !    1

m1 x 2 2

 m1  H n  x  ,  

(5.7)

where H n is the Hermite polynomials [15, 56]. The function F2,n is the Wigner function for the harmonic oscillator. The form of the functions Fk ,n , k  2 can be constructed from the characteristic equations as follows

22

F1,n  c1e



F2,n  c2e F3,n 3 ,3   c3e

1

 1 H n2  2  2 r

2 r2





2 2 v2

3 2 v2

F4,n  4 ,4 ,  4   c4e

  , 

(5.8)

  Ln  22  ,  v 

(5.9)

  Ln  32   3  ,  v 

(5.10)



4 2 v2

  Ln  42   4    4  ,  v 

(5.11)

... where Ln is polynomials Laguerre [56, 57], and  is the Dirac delta function. As a result, the solution of the Vlasov equation chain (5.4) for the harmonic oscillator is of the form

f1,n  x   c1e f 2,n  x, v   c2e f3,n  x, v, v   c3e f 4,n  x, v, v, v   c4e





v 2 12 v 2

v 2 12 v 2 2 v2



2 v2



x2 2 r2

v 2 12 x 2

 x H n2   2 r

  , 

(5.12)

 v 2  12 x 2  Ln  , 2  v 

(5.13)

 v 2  12v 2  2 Ln     v  1 x  , 2  v  

(5.14)

2 v2

 v 2  12 v 2  2 Ln     v  1 v   2  v  

 v    1  v  12 x    v  12 v  arcsin 2  v  12 v 2  …

 ,  

(5.15)

Remark As it is seen from the solutions (5.12)-(5.15), starting from the third function f3,n , f 4,n ,... and further all the solutions are obtained from the previous solutions by multiplying by the Dirac delta function of the following characteristic. Such a solution structure is because only the first two solutions f1,n and f 2,n are «independent». For the harmonic oscillator all the kinematic characteristic of higher orders v, v, v ,... according to (5.2), are expressed only in terms of coordinate and velocity, indeed v  12 x, v  12v, v  14 x, v  14v,....

(5.16)

The coefficients ck in (5.12)-(5.15) can be defined from the normalization condition of

the probability density function (1.3). Indeed, for the function f3,n  x, v, v  we obtain

23

f3,n  x, v, v  





f 4,n  x, v, v, v dv 





c4



12 12 v2  v2 2 v2

1

e

c4

, 1 



 12 12 v 2  v 2      v  12 x  , Ln     v2  

It follows that

c3 

1

v . v

(5.17)

By analogy, for the function f 2,n  x, v 

f 2,n  x, v  





f3,n  x, v, v dv  c3e





12 v 2 12 x 2



2 v2



 12  v 2  12 x 2   , Ln     v2  

from this we obtain

c2  c3 

c4

1

, 1 

v v  . v v

(5.18)

As a result, the function f 2,n  x, v  is of the form

f 2,n  x, v   c2e



v

2

12 x 2



2 v2

 v 2  12 x 2  Ln   2  v 

(5.19)

Since the function f 2,n  x, v  is the Wigner function Wn  x, p  

 1

n



e2  x , p  Ln  4  x, p   ,   x, p  

1  p 2 m12 x 2     , (5.20) 1  2m 2 

or det

Wn  x, v   mWn  x, mv 

 1 

n

m



m

e

v 2 12 x 2

1

 v 2  12 x 2  Ln  2m , 1  

then comparing Wn  x, v  (5.20) and f 2,n  x, v  (5.19), we obtain

 v2 

1 2m

, c2  c3 

 1

n

m



, c4 

n  1 m1 .



(5.21)

Taking into account the uncertainty relation for the harmonic oscillator  v r 

2m

, we

obtain

1 

v v v   . r v v 24

(5.22)

Thus, the solutions (5.12)-(5.15) of the Vlasov chain are of the form x2

 2  x 1 1 f1,n  x   n e 2 r H n2  2 n ! 2 r  2 r

 1

f 2,n  x, v   f3,n  x, v, v   f 4,n  x, v, v, v  

 1

n

2 v r

 1

n



e

2 v r



e

v2 x2  2 v2 2 r2

v2 v2  2 v2 2 v2

  , 

(5.23)

  v2 x2   Ln  2  2  2   ,   2 v 2 r  

(5.24)

  v2 v2   Ln  2  2  2     v  12 x  ,   2 v 2 v  

(5.25)

v2 v2  2 v2 2 v2

  v2 v2   Ln  2  2  2     v  12v   2 v  r   2 v 2 v     v  12 v v ,    v  12 x  arcsin 2 2 2   1 v   v 1   … n



e

(5.26)

The functions (5.23) and (5.24) are known functions. The functions (5.25), (5.26),.. are new solutions and have never been obtained before. The mean values of the kinematic variables (1.4) are of the form

1



f 2,n  x, v  vdv  0,

v 

 f1,n  x  

v 

1 2  f3,n  x, v, v  vdv  1 x, f 2,n  x, v  

v 

1 f 4,n  x, v, v, v  vdv  12v,  f3,n  x, v, v  

v  0,



v  12 x,



v

 0,

v

 12v,

v

 0,

(5.27)

v

 0,

... The obtained expressions (5.27) correspond to (5.3). Let us calculate standard deviations  2 of the kinematic variables. Note that in the general case, the variables introduced earlier  r2 ,  v2 ,  v2 ,  v2 ,... do not have to coincide with the standard deviations  2 of the corresponding random variables X ,V ,V ,V ,... . As an example, let us calculate V2 ,0 .    

V2 ,n 





f 4,n  x, v, v, v  v 

    v 1 14    2   e

2

n

2 v  r



 

2 v



12 x 2 2 v2

 dxdvdvdv  2

v

  v2  2 x2   Ln  2  2  1 2  v 2 dxdv,   2 v 2 v  

it follows that

25

V2 ,0   v2 ,

(5.28)

Remark Note that in deriving the Vlasov equation chain [42, 43, 46] (1.1)/(1.5) the condition of the positivity of the probability density function is not imposed anywhere. Therefore, the presence of negative quantities for the Wigner function (5.24), for example, or for subsequent functions (5.25), (5.26) is consistent. The question arises if there are positive probability density functions in the phase space  x, v  and further in the generalized phase space  x, v, v, v,... . The answer to this question can be obtained on the base of the chain of Vlasov equations (1.1). Let us suppose there is a solution f1,n  x  (5.23), obtained from the Schrödinger equation for the quantum harmonic oscillator. The function f 2,n  x  is related to the functions f1,n  x  by the relation (1.3)

f1,n  x  





f 2,n  x, v  dv.

(5.29)



The solution of the equation chain (1.1) is obtained by the method of characteristics. Along the characteristics (5.6), the probability density remains constant. For the function f 2,n  x  according to (5.6), circles (ellipses) are the characteristics v 2  12 x 2  const.

(5.30)

If the function f1,n  x  has zeros, that is xk , k  1,..., n : f1,n  xk   0 , then it follows from (5.29) that 



f 2,n  xk , v  dv  0.

(5.31)



Let the probability density function satisfies the condition f 2,n  x, v   0 , then it follows from (5.31) that f 2,n  xk , v   0 almost on the whole interval v   ,   . Consequently, on the whole circle (5.30) passing through the point  xk , 0  , the function f 2,n also equals zero (see Fig.2). Fig. 2 shows the position of the first zero x1 of the function f1,n . All circles (5.30) of a larger radius intersect the vertical line x  x1 , for example, in points P and P . Consequently, on all such circles the value of the function f 2,n equals zero. That is the function f 2,n differs from zero only inside a circle of the radius x1 . Thus, of all functions f 2,n , only the functions for which f1,n does not have zeros are positive in the whole domain. Only one function in (5.23) does not have zeros – it is f1,0 . Therefore of all functions in (5.24), only f 2,0 is positive in the whole domain. Analog statements are true for the rest functions in the chain.

26

Fig. 2 Values of the function Positive solutions can be obtained from the solution of the wave equations (1.12). As an example, we consider the equation (1.17) for the wave function  2  r , v , t  . For the onedimensional harmonic oscillator, the equation (1.17) is as follows

i  2  2  2 i    2  v  2  U 2 2 . 2  2 t  2 v  2 x

(5.32)

Let us represent the solution in the form

 2  r , v , t   ψ  r , v  ei2E 2t ,

(5.33)

where E 2 is a constant value. Substituting (5.33) into (5.32), we obtain

 2ψvv - ivψ x  2 U 2 - E 2  ψ  0.

(5.34)

It follows from (5.24) and from Fig.2 that the solution (5.34) can be obtained on the form

ψ  ce



v2 4 v2



12 x 2  i xv 4 v2

,

(5.35)

where  is some constant variable. Substituting (5.35) into (5.34), we obtain v2

2  vx  v 2   2 2 x 2   2 U 2 -E 2   22  i 12  2 2  2  0. 4 4 v 2 v 2 v

(5.36)

The expression (5.36) is complex, therefore it is equivalent to two conditions

v2

2    v 2   2 2 x 2   2 U 2 -E 2   22  0, 4 4 v 2 v

  2 2  0. 2 1

It follows from the second condition (5.37) that 27

(5.37)

12 . 2 2



(5.38)

From the first condition (5.37) taking into consideration (5.38), we obtain

v2

2 12 2 14 2   v  x   2U 2   2E 2  22  0. 4 4 v 2 2 4 2 2 v

(5.39)

The variable E 2 by the definition (5.33) is constant, therefore we suppose

2

E2  

2 2 v2

v



2 2 v



2

2 2



2

2

2

,

(5.40)

where for the harmonic oscillator the conditions (1.13), (1.15) are taken into consideration

 v  v   2 

2

2m

, 2 

v . v

(5.41)

As a result, from (5.39) with consideration of (5.40) for the potential U 2 we obtain the expression  2   2 x2  U 2  x, v   m12 1  2 2  v 2  1  . 2   21 

(5.42)

The probability density function f 2,0 is of the form f 2,0  x, v   c e 2



v2 2 v2



12 x 2 2 v2

c e



2

v2 x2  2 v2 2 r2

,

(5.43)

which responds to (5.23) f 2,0  x, v  . The expression for the phase of the wave function  2 is of the form (5.33), (5.35)

2  

m12 2

xv 

E2

t

2

m12 2

xv 

2 2

t.

(5.44)

It follows from (5.44) and (1.4) that the mean accelerations of the form

v  2 2

2 2 m12 x 2 2  2  12 x. v 2m v m 2

(5.45)

The expression (5.45) coincides with the Vlasov approximation and with (5.3), (5.27) v 

1 U1 1  m12 x 2   12 x. m x m x 2 28

(5.46)

The expression (5.42) for the potential U 2 can be obtained also by the general formula (2.2), (2.3), (2.6) 2    v S2,0  m214 x 2  m12v 2  1  1     U 2  x, v      2E 2   2  v S2,0  , 2 2    2  2 2 2       

where S2,0  ln f 2,0

S2,0  x, v   

v 2 12 x 2   const. 2 v2 2 v2

(5.47)

As a result,    2   2 x2  U 2  x, v   m12 v 2 1  2 2   1  . 2    21 

(5.48)

The expression (5.48) completely coincides with the expression (5.42), which was to be proved. Let us write the motion equations (see §3). For the representation (3.9), (3.10) ( n  2 ), we obtain

v  12 xex , 2 2  2  12 x 2    2 2 2 (5.49)  E 2  m1  v     c2 , 2 2  2    2  m 2 E  1  2  2  2  1 c2   2  1 xv  2 t   1 c2 , 2 2  2 2  2 2  m 2m12  2 m12  2v E2  v  2  2   r 1  2  ex 2  1 v  v   2 vex  1 ex , 2  2 2 2 2  2 1  2 B2   , B2   .

 2

2 

2 2  2

 Q2  U 2    2c2 

As a result, considering (5.49), the motion equation (5.16) is as follows

d2 v   2 E2 , dt

(5.50)

which is a valid identical equation as

    2 2 12   v  v  x  1 v   2 E2  1 v,  t  x  v   which was to be proved. The Hamilton-Jacobi equation (2.16) with (5.44), (5.49) gives a valid identical equation 29

2 2 m  v  e 2  H 2 , t 2 4 2 m1 x  m14 x 2 22   2 2  H2 , 2 2 2 2



2

(5.51)

where

e 2  U 2  Q2 

2

v

2  x

2

  2 x2   m12  v 2  1   m12v 2  2 2  

2

2

2

2



m14 x 2 . 2

For the quantum pressure (3.15) and the quantum pressure force (3.16), we obtain  2  q 

P

  f

2 2 2,0

 2 S2,0 v 2

v2

  v2  e 2 2 v r



x2 2 r2

,

1  Q2 1    P    22v. m v f 2,0 v 2



2 v

q

The generalized Lagrangian (2.8) with (5.44), (5.49) is of the form 



1 d 22 1 1  vp  2 dt 2 2  2 2

2

 U 2  Q2   L2 ,

12 x 2     m 4 2  2 2 22  v  v     x   m  v      L2 , 2 1 1  2 x v  2 2 2   t 

which gives a valid identical equation

2

2

2

 m12v 2  m14 x 2 

2

2

2

 m12v 2  m14 x 2   L2 ,

(5.52)

The Legendre transformation (2.16) for H 2 and L 2 taking (5.51), (5.52), (5.49), (5.44) into account also gives a valid identical equation

L2  H 2 



2

2

 v ,  r 2   m 

2  m14 x 2 , 2 2 x m12v2  m14 x2  m12v2  m14 x2 .

2

 m12v 2  m14 x 2 

2



v , vp .

2



2

v

At n  1 the potentials Q1,0  x  and U1  x  are as follows m12 x 2 , 2 2      2  4 x2  1   S1,0    1   12  1 4   Q1,0  x   1  S1,0 21  2  21   v 2 v  U1  x  

30

(5.53)

1

1

2



m12 x 2 , 2

where S1,0  x   

2 S1,0 12 x 2 12 x  S1,0 12  const ,   ,   . 2 v2 x  v2 x 2  v2



Having the energy of the ground state denoted E1,0 

1 1

2

, for the quantum potential we obtain

the expression m12 x 2 . 2

Q1,0  x   E1,0 

(5.54)

Taking into account (5.49), the condition (4.10) is as follows

  1   2 2  m12 E  xv  2 t   1 c2  dv,     211    2  2 2  

U1  Q1,0  1 c1 

where

1

c2 and

1

(5.55)

c1 are some functions (3.5) to be defined. The expression (5.55) becomes a

valid identical equation if we suppose

1

1

c2 and

c2 

1

c1 as follows

2 2 E 2

2

1

t,

c1  E1,0 .

2

According to (4.7) and (5.49), the field E1 is of the form

E1  r , t  





E2  r , v , t  dv  ex



12   vdv   .  2 

(5.56)

As a result, the motion equation (3.8) is of the form

d v   1 E1   . dt

(5.57)

According to (5.3), (5.27), the equation (5.57) is a valid identical equation as v   .

v   . For the quantum pressure (3.15) and the quantum pressure force (3.16), we obtain (ground state) 1  q 

P

  f

2 1 1,0

 2 S1,0 x

2

2 r

1  Q1,0 1    P    12 x. m x f1,0 x 1



x2

  v2  e 2 , 2 r

31

q

The generalized Hamilton-Jacobi equation (2.14)/(2.15), the expressions for the generalized Lagrangian (2.8) and the generalized Hamiltonian (2.9)/(2.10) in the case n  1 coincide with the corresponding classical representations.

Conclusion The approach considered in this paper is internally coherent and in the particular case at n  1 becomes in the classical apparatus of the quantum mechanics and statistical physics. The results of the paper can be used in constructing models that take into account higher orders of kinematic characteristics.

Acknowledgements This work was supported by the RFBR No. 18-29-10014. Appendix (   algebra ) Let us construct  -algebra. Let 2

2

be a set of natural numbers of the form

 nk  : nk 1  2nk , n1  1, k 

,

(A.1)

that is 2

 1, 2, 4,8,16,32,... .

Definition 1. Let us consider a finite ordered set n 

  v        r   v  v     , where   ,  ,  ,...,   Let us call the set of values

2

of the scalar variables

  n1 

 r     v      , n   ...      nv1         

1 n

 

 in the general case are hypercomplex numbers.

det



r

det

,   2 n

v

  n1  det  v   

 ,...,   n n

,

(A.2)

 - scalar of the order n or  -set (PSI-set) of the order n and rank 0. Let us denote the set of 0 0 elements n  as n   , that is n   n   . Let is denote the addition operation as «+» and the multiplication « » for the elements

 0 . n  , n  n 

32

Definition 2. The element n   n  0 is called the sum n   n of the elements n  , n  n  0

  r      r     r     r    v   v   v  v                  .         n n n  ...   ...   ...    n1      nv1      nv1      nv1  v                         

(A.3)

To define the multiplication operation « », let us introduce the notion of the correspondence matrix. Definition 3. Let n  , n  n  0 . The matrix n M    of the size n  n of the form

n





n

   r    r    M      v  r  n ...     n1    v   r    

r 

v

 

  n1   v     

...



  n1   v     

  v v   ...  v      ... ... ...    n1    n1    n1    v   v   v    v          ...         n1   v     

we call the correspondence matrix of the elements n 

.

M T     n M   is the correspondence matrix for n

Definition 4. The convolution operation of the matrix

nk

n

nk 

2

nk

 , where n  , n  n  0 , then

n

.

M

 nk 1  2 nk 2    2  2 ... 2  2 2  2 ...    2  2 ... 2  2 2  2 ... nk M    rs     2  2 ... 2  2 2  2 ...    2  2 ... 2  2 2  2 ...  nk  2 nk 1  2  2 is a matrix of the size 2  2 ,

is the operation of transforming the matrix

2 2 2 2 2 2 2 2

M in k  1, k 

elements according to the following rule 33

(A.4)

n

Corollary 1. If n M    is the correspondence matrix for n  n





      ,       

steps in a column

(A.5)

* nk

M of

for n1  1 det

1

M  rs    11   1*M ,

for n2  2

 11 2 M  rs     21

12  step 1  11  22  det * ~    2M, 22   21  12 

for n3  4

   4M     

11 12 21 22 31 32 41 42

13 14   11  22  33  44   23 24   21  12  43  34  det    *M , ~  4        33 34  31 32 13 24   43 44   41  12  23  14 

for n4  8

 11   22  33   44  55  66  77  88     21  12   43  34  65  56  87  78   31   42  13   24  75  86  57   68     41  32   23  14  85  76  67  58  det *  M. 8M   51  62  73  84  15   26  37   48  8    61  52  83  74   25  16   47  38     71  82  53  64  35   46  17   28                  72 63 54 45 36 27 18   81 By analogy, the matrix convolution

* nk

M , nk 

2

is defined. The correspondence matrix

nk

M of

the size nk  nk factors in k  1 steps onto matrices of the size nk 1  nk 1 , nk 2  nk 2 ,..., 2  2 . Remark Let us pay attention to the structure of the matrix convolution n*M . The first element in a column of the convolution n*M is the trace of the correspondence matrix n M , that is Sp n M . As is known, a matrix trace is the invariant in the transition from one coordinate system to another, that is Sp n M  inv . The second element consists of the sum of symmetric elements ij   ji of the matrix n

M , which indices  i, j  pairwise differ by 1, that is 1, 2  ,  3, 4  ,  5,6  ,  7,8 ,... .

The third element also consists of the sum ij   ji , and the indices  i, j  differ by 2, that is 1,3 ,  2, 4  ,  5,7  ,  6,8 ,... . In the fourth line, the difference of the indices equals 3, etc. In the last line, the difference between the indices is maximal and equals n  1 . 34

Definition 5. Let n  , n  n  0 , n 

2

. By the product n   0

 -scalar n we mean the  -scalar n   n 

 or n  n of the  -scalar n  on the

n

which is equal to the convolution n*M    of

the correspondence matrix n M    of the  -scalars n  and n , that is

n



det

det

det

  n  n  n*M  n .

(A.6)

n

Lemma 1. Let n  , n  n  0 and n M    are the correspondence matrix of their product, then * n

M     n*M T    .

Lemma 2 For  -scalars the multiplication is a commutative operation, that is n  n  n n  . Remark Note, that the  - set space dimension coincides with the space dimension Cayley’s algebra, according to the Cayley-Dickson procedure. In the case of real and complex numbers, the operation of addition and multiplication are commutative and associative. The  - set can consist not only of scalar variables (A.2), but also of vectors. Definition 6. The  -set (A.7) will be called the  -vector of the order n  set) of the order n and rank 1

  r u   v   u   , n u  ...      nv1     u   

det

u

1 n

r 

u,

2 n

det

u

v

  n1  det  v   

u ,..., u  n n

2

or  -set (PSI-

(1.7)

u.

The set of  -vectors (A.7) we denote as n   , that is n u  n   . 1

1

For  - vectors, the addition «+» and multiplication « » operations can be defined. Definition 7. We will call the  -vector

n

d  n 1 the sum

1

n

u, n s  n 

35

n

u  n s of the  - vectors

  r u    r  s    r u   r  s   v   v   v  v  u   s   u   s       . d  u  s    n n n  ...   ...   ...    n1      nv1      nv1      nv1   v    u     s    u    s       

(A.8)

The multiplication operation « » allows two variants: scalar multiplication and vector one. Definition 8. Let n u , n s n  . The matrix of the size n  n ( n  1

 n u, n s 

   r u ,  r  s     v r  u, s  n M u  s     ...      n1     v   r   u, s     









...





  u ,   s 

...

...

...

r 

v

u, s

v

   

v

  n1   v     

 u ,  v  s  ...  

2

) n M u  s 

   nv1    nv1      u ,   s          n1    v        v u ,   s     .    ...    n1    n1    v   v      u ,   s      

is called the correspondence matrix of the scalar product of the  - vectors n u and matrix elements (A.9) are the scalar products of the vectors

 , .

(A.9)

n

s . The

The correspondence matrix n M  u  s  of the vector product of the  - vector n u by the

 - vector n s is of the form

 n u, n s 

    r u ,  r  s        v r    u , s  n M u  s     ...      n1     v   r   u, s     

 r 

v

  u, s 

...

  v u ,  v  s   

...

...

...

    

  n1  v   

 u ,  v  s  ...  

   nv1    nv1      u ,   s          n1    v      v    u,  s     .    ...    n1    n1    v   v      u ,   s      

The elements of the matrix (A.10) are the vector products of the vectors

36

 , .

(A.10)

Definition 9. We will call the convolution of the correspondence matrix n*M  u  s  the scalar

 n u , n s  . We will call the convolution of the correspondence matrix M  u  s  the vector product of the  - vectors  n u , n s  .

product of the  - vectors * n

Remark The scalar product of complex ( )  -vectors is commutative. For hypercomplex ( , , ,... )  -vectors (quaternions, octanions, sedenions, etc.), commutativity is violated in the general case. The vector product is not commutative in the general case. For the complex ( )  -vectors, the vector product is anti-commutative, that is u , s  n n     n s , n u  and n M  u  s    n M T  u  s    n M  s  u  . Lemma 3. For n   n  0 and n u  n 1 at n  n

2

it is true that

 n u  n u n .

The elements of the  -set are matrices or tensors, in the general case. Definition 10. We call the  -set (1.11) the  -tensor of the order n    r  rq1......rql 1 k    v  rq1......rql 1 k l  nTk   ...    n1    v  q1 ...ql   r1 ...rk 

2

    n1   det det  v   r  q1 ...ql 2 l  v  q1 ...ql n l  , 1T l det   r1 ...rk , nTk   r1 ...rk ,..., nTk    rq11......rkql .  n k    

and the rank  k  l 

(A.11)

The elements of the  -set (A.11) are k-times covariant and 1-times contrvariant tensors. We k l k l denote the set of the  -tensors (A.11) as n   , that is nTkl  n   . Remark The  -tensor (A.11) of the zero rank corresponds to the  -scalar (A.2). The  -tensor (A.11) of the first rank corresponds to the  -vector (A.7), etc. The addition and multiplication operations for the  -tensors of the second rank are performed with account of the rules of matrix addition and multiplication. Operations with the  -tensors of the third rank and higher are performed on the base of the tensor calculus rules. Differential  -sets. Differential operators can be used as  -vectors. Without limiting the generality, let us consider the case n  4

 r     v  . 4   v     v  37

(A.12)

1.  -gradient. The  -gradient is a  -vector.

 r     v  ,   4  v     v 

 r  r    v v   v v   v v     r       v   r v   v  r    v v    v v     ,  ,  4  4     v 4    v v r  v   r   v   v   v      v  v  v v r      r   v   v   v   Sp 4 M  r  r    v  v   v v    v  v   .

(A.13)

2.  -divergence. The  -divergence is a  -scalar.

 r  u   v   u 4 u   v  ,  u  v   u

 div r  r u  divv  v u  div v  v u  div v  v u     div r  v u  divv  r u  div v  v u  div v  v u  .  4 , 4 u    v v r  v   div r u  divv u  divv u  div v u   v v v r    div r u  divv u  divv u  div v u  Sp 4 M  divr  r u  divv  vu  divv  vu  divv  v u.

(A.14)

3.  -rotor. The  -rotor is a  -vector.

 rot r  r u  rot v  v u  rot v  v u  rot v  v u     rot r  v u  rot v  r u  rot v  v u  rot v  v u  .  4 , 4 u     v  v  r  v   rot r u  rot v u  rot v u  rot v u   v v v r    rot r u  rot v u  rot v u  rot v u  Sp 4 M  rot r  r u  rot v  vu  rot v  vu  rot v v u.

(A.15)

Remark Previously in monograph [46, 47] we obtained the expressions (A.13-15) for the generalized phase space for consideration of the Vlasov equation chain. It is the expressions (A.13-15) that, due to their «invariance», makes sense when constructing kinetic equations such as the Vlasov equation chain. In [46, 47] the representations (A.13-15) had the notations  , div , rot  respectively. In the given paper, for brevity, we also use the mentioned notations. 4. The properties of the  -rotor, divergence and gradient. The following expressions are true a. The  -rotor from the  -gradient equals to the zero  -vector, that is

0   n , n  n     ...  . 0    b. the  -divergence from the  -rotor equals to the zero  -scalar, that is 38

(A.16)

0   n ,  n , n u    ...  . 0   

(A.17)

Let us verify the given identical equations for the case n  2 . For the expression (A.17), we obtain

  r u  u   v  , 2  u  



2

 rot r

r 

v

u , r  v  rot u  rot u r  v   divr rot v  v u div r rot v  r u   .  div rot  r u div rot  v u  v r  v r 

 2 , 2 u   

,  2 , 2 u 

u  rot v

Taking into consideration the properties of the scalar triple product divr rot v a   r , v , a    v , r  , a     v , r , a    divv rot r a,

(A.18)

we obtain



2

 divr rot v u  v    div rot u  r   v r

,  2 , 2 u 

 divv rot r u  r   .  div r rot v u  v  

As a result,



2

0 ,  2 , 2 u     . 0

By analogy, for the expression (A.16), we obtain

 r  r    v  v        ,  2 2   r     v   r  v  v   rot    rot v  r  v     0   2 , 2  2     r v ,   rot   r    rot   r     0  r v  v r 



where it is taken into account that

rot r v   r , v    r , v     v , r     v , r     rot v r . Thus, we can define the  -algebra

n

 k l  , ,

. The

n

(A.19)

 k l  is the set of the  -

algebra. The operations «+» and « » are the binary operation of the  -algebra. For each set of n, k , l one can consider the existence of an identity element, an inverse element, a commutativity and associativity properties.

39

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